Shape decomposition algorithms for laser capture microdissection

Leonie Selbach; Tobias Kowalski; Klaus Gerwert; Maike Buchin; Axel Mosig

doi:10.1186/s13015-021-00193-6

Shape decomposition algorithms for laser capture microdissection

Selbach, Leonie; Kowalski, Tobias; Gerwert, Klaus; Buchin, Maike; Mosig, Axel 2021-07-08 00:00:00 Background: In the context of biomarker discovery and molecular characterization of diseases, laser capture microdissection is a highly effective approach to extract disease-specific regions from complex, heterogeneous tissue samples. For the extraction to be successful, these regions have to satisfy certain constraints in size and shape and thus have to be decomposed into feasible fragments. Results: We model this problem of constrained shape decomposition as the computation of optimal feasible decompositions of simple polygons. We use a skeleton-based approach and present an algorithmic framework that allows the implementation of various feasibility criteria as well as optimization goals. Motivated by our application, we consider different constraints and examine the resulting fragmentations. We evaluate our algorithm on lung tissue samples in comparison to a heuristic decomposition approach. Our method achieved a success rate of over 95% in the microdissection and tissue yield was increased by 10–30%. Conclusion: We present a novel approach for constrained shape decomposition by demonstrating its advantages for the application in the microdissection of tissue samples. In comparison to the previous decomposition approach, the proposed method considerably increases the amount of successfully dissected tissue. Keywords: Laser capture microdissection, Shape decomposition, Skeletonization Introduction processed with LCM provide more accurate molecular Laser capture microdissection (LCM) [1] is a highly effec - markers of diseases [4, 5]. With LCM being used more tive approach to extract specific cell populations from and more commonly in clinical studies, there is a need to complex, heterogeneous tissue samples. In the dissection, automate all procedures involved in sample processing. a laser cuts around the boundary of a selected region and a subsequent laser pulse catapults the fragment into a Practical application collecting device. LCM has been used extensively in the Our contribution is motivated by an application intro- context of biomarker discovery [2] as well as the molec- duced in [2] in which a region of interest (ROI) to be ular characterization of diseases [3]. Since LCM sepa- dissected from the tissue sample is identified using label- rates homogeneous and disease-specific regions from free hyperspectral infrared microscopy. In this approach, their heterogeneous and unspecific surrounding tissue an infrared microscopic image of the sample yields infra- regions, the characterizations obtained from genomic, red pixel spectra at a spatial resolution of about 5 μm. A transcriptomic or proteomic characterizations of samples previously trained random forest classifier assigns each pixel spectrum to one tissue component such as healthy or diseased, with the diseased class being further subdi- *Correspondence: leonie.selbach@rub.de vided into inflamed tissue as well as several subtypes of Department of Computer Science, Faculty of Mathematics, Ruhr thoracal tumors. The general sample preparation task in University Bochum, Bochum, Germany the context of LCM is to dissect all tumor regions (or all Full list of author information is available at the end of the article © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. The Creative Commons Public Domain Dedication waiver (http:// creat iveco mmons. org/ publi cdoma in/ zero/1. 0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data. Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 2 of 17 regions identified as one specific tumor subtype) from a samples consist of complex shapes of varying sizes, they sample. The current standard approach for the dissection oftentimes do not satisfy these constraints and therefore with LCM is to draw shapes manually. This is severely cannot be extracted from the tissue sample without any limited, not only because it takes an inacceptable amount previous processing. This either increases the amount of time for large numbers of samples, but also because of necessary user-interaction or negatively affects the it is required that the human operator will subjectively sample quality and thus compromises the advantages of decompose complex regions into smaller fragments. In LCM-based sample preparation. this paper, we propose a novel automated decomposi- Given a binary mask of a microscopic slide with the tion approach. While our current contribution deals with ROIs as the foreground, the image is preprocessed for the specific context of label-free infrared microscopy, our LCM in such a way that the ROIs are reduced to a num- approach equally applies more broadly to LCM in the ber of connected components without holes. By inter- context of other microscopic modalities, most notably preting each of these connected components as a simple H&E stained (hematoxylin and eosin stained) images [4] polygon, we can model the given problem of constrained for which recent digital pathology approaches facilitate shape decomposition as the computation of optimal fea- reliable computational identification of disease specific sible decompositions of polygons (see Fig. 1). The con - regions [6, 7]. straints can be modeled as certain feasibility criteria and optimization goals. Our decomposition method utilizes Problem statement and solution a skeleton of the shape and follows a dynamic approach. In this paper, we address one central problem of pro- Specifically, we restrict our cuts to certain line segments cessing samples with LCM. That is, not all dissected based on the skeleton. This not only results in simple cuts fragments can be successfully collected due to various but also in a flexible framework that allows to integrate possible circumstances. Besides technical reasons as for various criteria. example an incorrect focus of the laser, the main cause With this paper, we present a novel approach for the is assumed to be the size and morphology of the frag- automated decomposition of tissue samples with lim- ment. The fragments must not exceed certain limits of ited user-interaction. Unlike previous decomposition minimal or maximal size and should be of approximately methods used in the context of LCM, we placed a focus round shape. As the regions of interest (ROIs) in tissue on the morphological properties of the fragments. In Fig. 1 Polygon decomposition in a histopathological tissue sample. Top: Regions of interest are selected from a histopathological tissue sample (H&E-stained image of a subsequent sample on the top left) in which different tissue types have been identified using the method in [2]. Bottom: After a preprocessing, each connected component is given as a simple polygon without holes, which is then decomposed using the proposed skeleton-based approach S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 3 of 17 the experimental evaluation on lung tissue samples of morphology. We propose an algorithm for constrained patients with non-small-cell lung carcinoma, the pro- polygon decomposition using a skeleton-based approach. posed approach achieved a higher success rate and the amount of successfully collected tissue was increased by Skeletonization 10-30%. This paper is an extended version of the prelimi - Our approach is based on the medial axis or skeleton of nary work presented in [8]. the shape. The medial axis is defined as the set of points The paper is organized as follows: In "Related work" that have more than one closest point on the bound- section, we discuss related work on decomposition algo- ary of the shape. The medial axis was introduced for rithms. In "Method" section, we introduce our algorith- the description of biological shapes [20, 21] but is now mic framework and discuss different possible feasibility widely used in other applications such as object recog- criteria as well as optimization goals. In "Experimental nition, medical image analysis and shape decomposition results" section, we present experimental results and (see [10] for a survey). An important property is that the demonstrate the advantages of our method in compari- medial axis represents the object and its geometrical and son to a heuristic decomposition approach. We conclude topological characteristics while having a lower dimen- with a summary of our results and future improvements sion [22, 23]. in "Conclusion" section. Formally, the medial axis of a shape D is defined as the set of centers of maximal disks in D. A closed disk B ⊂ D is maximal in D if every other disk that contains B is not Related work contained in D. A point s is called skeleton point if it is Polygon decomposition is an important tool in compu- the center of a maximal disk B(s) (see Fig. 2). For a skel- tational geometry, as many algorithms work more effi - eton point s, we call the points where B(s) touches the ciently on certain polygon classes, for example convex boundary the contact points—every skeleton point has at polygons [9]. Moreover, polygon decomposition is fre- least two contact points. A skeleton S is given as a graph quently used in applications such as pattern recognition consisting of connected arcs S , which are called skeleton or image processing [9]. Object recognition, biomedi- branches and meet at branching points. Given a simple cal image analysis and shape decomposition are typical polygon without holes the skeleton is an acyclic graph. areas of application that utilize skeletons [10]. Skeletons There are various methods for the computation of the are oftentimes used to analyze the morphology of a given medial axis in practice [10]. In general, the medial axis is shape and work especially well on elongated structures, very sensitive to noise in the boundary of object. This is such as vessels [11], pollen tubes [12] or neuron images a problem that often occurs in digital images and leads [13, 14]. There are several shape decomposition meth - to spurious skeleton branches. Procedures that remove ods based on the skeleton or some other medial repre- these uninformative branches are known as pruning sentation of a shape. However, most of these methods methods. Pruning can be applied after skeletonization are designed for object recognition and thus focus on [24–26] or is included in the computation of the skeleton decomposing a shape into “natural” or “meaningful” [27–30]. For our application, we utilize the skeletoniza- parts [15–17]. In some approaches, even decompositions tion and pruning method of Bai et al. [29], which was with overlapping parts are allowed [18, 19]. None of the previously used for other bioimaging applications [12, established decomposition methods facilitate a straight- forward introduction of adjustable size and shape con- straints as needed for our application. We utilize the skeleton for two main reasons: it is well-established to represent shape morphology and has proved useful for shape decomposition. As cancerous tis- sue regions often present themselves as highly complex and ramified shapes, we apply the skeleton to obtain a morphological representation, based on which we com- pute a decomposition that includes the morphological features. Method To improve the success rate of LCM, a shape decomposi- Fig. 2 Medial axis of a simple shape. This medial axis consists of five branches connected by two branching points. The skeleton point tion method is needed that computes feasible fragments, s is the center of a maximal disk B(s) and has three contact points i.e. fragments that fulfill certain constraints in size and {c , c , c } on the boundary of the shape 1 2 3 Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 4 of 17 14]. This algorithm produces a discrete and pruned skel - eton, which consists of a finite number of skeleton pixels as our skeleton points. This is favorable for our practical application as we have a discrete input and a discrete out- put is expected. Furthermore, the computed skeleton has the property that every branching point has a degree of exactly three. Skeleton‑based polygon decomposition We consider the following problem: Given a simple poly- gon P, compute an optimal feasible decomposition of P. A decomposition is feasible if every subpolygon is feasi- ble, in the sense that it fulfills certain conditions on for instance its size and shape. We present an algorithmic framework that allows the integration of various criteria for both feasibility and optimization, which are discussed later. As for now, we only consider criteria that are locally evaluable. In our skeleton-based approach, we only allow cuts that are line segments between a skeleton point and its corresponding contact points. Thus, the complexity of our algorithm mainly depends on the number of skel- eton points rather than the number of boundary points of the polygon. Every subpolygon in our decomposition is generated by two or more skeleton points. We pre- sent two decomposition algorithms: One in which we restrict the subpolygons to be generated by exactly two skeleton points and a general method. In the first case, each subpolygon belonging to a skeleton branch can be Fig. 3 Domain decomposition lemma. The domain is decomposed decomposed on its own and in the second case the whole based on the contact points of skeleton point p. The partial skeletons share only p as a common point. All contact points of any other polygon is decomposed at once. skeleton point q are contained in exactly one of the connected components Decomposition based on linear skeletons First, we consider the restriction that the subpolygons are generated by exactly two skeleton points. In this case, Moreover, we have the corresponding skeleton points have to be on the same S(D ) ∩ S(D ) = p ∀ i �= j. i j skeleton branch S . In our computed skeleton, a branch- ing point belongs to exactly three branches and thus has Corollary 2 Let p ∈ S(D) and A , A , . . . , A be as 1 2 k three contact points. Each combination of two out of above. For each skeleton point q =p exists an i such that the three possible cut line segments corresponds to one all contact points of q are contained in A . of these branches. Due to the Domain Decomposition Lemma (see Fig. 3, proof in [22]) and the following corol- lary, we can decompose each skeleton branch on its own. Let S be a skeleton branch with a linear skeleton of size n and let P be the polygon belonging to this branch. By k k Theorem 1 (Domain Decomposition Lemma) Given a P (i, j) , we denote a subpolygon that is generated by two domain D with skeleton S(D), let p ∈ S(D) be some skele- skeleton points i and j on S (see Fig. 4). Thus, we have ton point and let B(p) be the corresponding maximal disk. P (1, n ) = P . First, we consider the decision problem, k k k Suppose A , A , . . . , A are the connected components of 1 2 k which can be solved by using dynamic programming. For D \ B(p) . Define D = A ∪ B(p) for all i. Then: i i each skeleton point i from n to 1, we determine X(i). X(i) k is True if there exists a feasible decomposition of the S(D) = S(D ). polygon P (i, n ) . This is the case if either i k k i=1 S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 5 of 17 we can combine those to obtain a decomposition of the entire polygon. This leads to the following result. Theorem 4 Given a simple polygon P with skeleton S consisting of n points, one can compute a feasible decom- position of P based on the skeleton branches of S in time O(n F ) , with F being a factor depending on the feasibility criteria. Fig. 4 Subpolygon induced by skeleton points. The polygon P belongs to the skeleton branch S consisting of the points 1 to n . k k Note that there might not exist a feasible decomposi- Two skeleton points i and j together with line segments to their tion of the entire polygon or for certain subpolygons. By corresponding contact points induce a subpolygon P (i, j) using this method, we are able to obtain partial decompo- sitions. Thus, this approach can be favorable in practice. General decomposition In the general setting, subpolygons are allowed to be gen- erated by more than two skeleton points. In this paper, Fig. 5 Decomposition based on linear skeletons. The polygon we will briefly explain the idea of our method (see [31] for P (i, n ) has a feasible decomposition if either polygon itself is feasible k k a more detailed description and the corresponding for- (left) or there exists a point j such that P (i, j) is feasible and P (j, n ) k k k has a feasible decomposition (right) mulas). Recall that our skeleton is an acyclic graph con- sisting of a finite number of vertices, i.e. skeleton points. The skeleton computed for our application (method of Bai et al. [29]) has the property that the maximal degree a) P (i, n ) is feasible or k k of a skeleton point is three. We represent the skeleton as b) there exists j > i such that P (i, j) is feasible and a rooted tree by selecting one branching point as the root P (j, n ) has a feasible decomposition. k k (see Fig. 6). Since branching points belong to three differ - ent branches, these nodes are duplicated in the skeleton This is illustrated in Fig. 5. By choosing optimal points j tree such that each node corresponds to the cut edges on during the computation, we can include different opti - the respective branch. Our method and its runtime are mization goals. If X(1) is True, the entire polygon has based on two main observations. a feasible decomposition, which can be computed via backtracking. Observation 5 The maximal number of skeleton points that can generate a subpolygon is equal to the number of Lemma 3 Given a subpolygon P with a linear skel- endpoints in the skeleton, i.e. the number of leaves in the eton S consisting of n points, one can compute a feasible k k skeleton tree. decomposition of P based on S in time O(n F ) , with F k k k being a factor depending on the feasibility criteria. Observation 6 Every subpolygon can be represented as the union of subpolygons generated by just two skeleton Proof We initialize X(n ) = True . For every skeleton points. point i, for i = n − 1 down to 1, we compute X(i) such that X(i) equals True if there exists a feasible decompo- sition of P (i, n ) . To compute X(i), we consider O(n ) k k k Let i be a node in the skeleton tree and T the subtree other values X(j) for i < j ≤ n and check in time O(F ) rooted in i. By P(i), we denote the subpolygon ending if the polygon P (i, j) is feasible. The correctness follows in the skeleton point i. This polygon corresponds to the inductively. subtree T in the given tree representation (see Fig. 7). For each node i (bottom-up), we compute if there exists The factor F is determined by the runtime it takes a feasible decomposition of the polygon P(i). Such a to decide whether a subpolygon is feasible. This fac - decomposition exists if either tor depends on for instance the number of points in the skeleton or in the boundary of the polygon. We discuss a) P(i) is feasible or examples in the following "Feasibility constraints and b) There exists a feasible polygon P ending in i and fea- optimization" section. After computing decompositions sible decompositions of the connected components for each subpolygon corresponding to a skeleton branch, of P(i) \ P . Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 6 of 17 Fig. 6 Tree representation of the skeleton graph. Representing the skeleton graph as a tree rooted at the point r. Every node in the tree represents a possible cut in the polygon. Therefore the branching points are duplicated to provide the cuts on the respective branches Fig. 7 A subpolygon and its corresponding subtree. The subpolygon P(i) ending in the skeleton point i is represented by the subtree T Thus, we have to consider all different combinations approach does not depend on the initial choice of the of skeleton points that together with i can form such root node. a polygon P . In a top-down manner, we consider the different combinations of nodes [i , i , . . . , i ] such that Theorem 7 Given a simple polygon P with skeleton S 1 2 l ′ ′ i ∈ T and T ∩ T =∅ for all j =j . The polygon P consisting of n points with degree at most three, one can j i i i ′ corresponds to the subtree rooted in i with i , i , . . . , i compute a feasible decomposition of P based on S in 1 2 l as the leaves, depicted in blue in Fig. 8. Note that we O(n F ) time, with k being the number of leaves in the skel- can compute P as a union of subpolygons iteratively. eton tree and F as above. We check if P is feasible and if we have feasible decom- positions for each P(i ) , meaning every subtree T (gray j i in Fig. 8). Because of Observation 5, we know that l ≤ k , Feasibility constraints and optimization for k being the number of leaves in the skeleton tree. The proposed polygon decomposition method is a versa - We have a feasible decomposition of the whole polygon tile framework that can be adjusted for different feasibil - if there exists one of the polygon P(r). This computation ity constraints and optimization goals. With regard to the dominates the runtime with the maximum number of application in LCM, we considered criteria based on size combinations to consider being in O(n ) . Note that this and shape. As stated before, it is assumed that the main S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 7 of 17 Fig. 8 Possible combination of skeleton points considered in the decomposition of a subpolygon. In the decomposition of a subpolygon P(i), different combinations of skeleton points in the subtree T are considered. The resulting subpolygons can be represented as subtrees (blue) in the tree representation spanning between nodes of these skeleton points cause for unsuccessful dissections lies in an incorrect size fragment into a collecting device. As the laser burns part or morphology of the considered fragments. In LCM, a of the boundary, the fragment has to have a certain mini- laser separates a tissue fragment from its surrounding mal size to ensure that enough material is supplied to be sample leaving a small connecting bridge as the impact analyzed. On the other hand, the size cannot be too large point of a following laser pulse, which catapults the or otherwise the force of the laser pulse does not suffice Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 8 of 17 for the transfer process. Furthermore, observations show optimization goal is finding a minimal decomposition that the dissection often fails due to an irregular shape of by minimizing the number of fragments (MinNum). We the fragment. Specifically, elongated shapes or fragments define X(i) as the number of subpolygons in an optimal with narrow regions (bottlenecks) seem to be problem- feasible decomposition of P(i, n), set X(n) = 0 and com- atic. The tissue can tear at these bottlenecks and is only pute X(i) = min X(j) + 1 . By minimizing the length of j∈I transferred partly or not at all because the laser pulse is cut edges (MinCut), shorter cuts are preferred and thus concentrated on only a small part of the boundary. preferably placed at bottlenecks in the polygon. Since In the following, we describe the implementation of every skeleton point i is the center of a maximal disk, we different constraints that we considered based on our can obtain the cut length by the corresponding radius r(i). application. For simplicity, we limit the description to We can define X(i) either as the length of the longest cut the decomposition algorithm for linear polygons, but or as the sum of cut lengths in an optimal decomposition all mentioned constraints can be applied to the general of P(i, n) and compute X(i) = max{min X(j), r(i)} or j∈I decomposition method as well. X(i) = min X(j) + r(i) . The runtime for both MinNum j∈I and MinCut is the same as for the decision problem. Fur- Feasibility constraints thermore, we considered maximizing the fatness (Max- For the size constraint, we restricted the area of the Fat) as an optimization goal. A decomposition is optimal subpolygons. Given two bounds l and u, a polygon P is if the smallest aspect ratio is maximized. We define x(i) feasible if l ≤ A(P) ≤ u , for A(P) being the area of the as the value of the smallest aspect ratio and compute polygon. One could also apply this constraint on the X(i) = max {min{X(j), AR(P(i, j))}} . Applying fatness j∈I number of boundary points instead of the area. We as a feasibility constraint or an optimization goal results implemented different shape constraints. On the one in the same runtime because both approaches require the hand, we considered approximate convexity. Then, a calculation of the aspect ratios of all subpolygons. polygon P is feasible if every inner angle lies between two given bounds. As this criterion does not prevent elon- Comparison of criteria gated shapes, we considered fatness instead. Fatness can Our algorithm facilitates the use of a wide range of feasi- be used as a roundness measurement and is defined by bility criteria and optimization goals, which can be com- the aspect ratio AR(P) of a polygon, which is the ratio bined with each other. Note that for certain combinations between its width and its diameter [32–34]. For a sim- other (faster) methods might exist. One example is find - ple polygon, the diameter is defined as the diameter of ing the minimal (MinNum) decomposition in which the the minimum circumscribed circle and the width as the area of the subpolygons is bounded. For polygons with diameter of the maximum inscribed circle. A polygon P linear skeletons this can be modeled as finding the mini - is called α-fat if AR(P) ≥ α . For the fatness constraint, we mal segmentation of a weighted trajectory (in O(n log n) define a polygon as feasible if it is α-fat for some given time [35]). For general polygons, this problem can be parameter α ∈ (0, 1] . Higher values of α result in frag- modeled as computing the minimal (l, u)-partition of a ments that are more circular and less elongated in shape. weighted cactus graph (in O(n ) time [36, 37]). For area as well as approximate convexity, we can com- The selection of constraints used for the algorithm pute the required values incrementally if the values for all obviously affects the resulting decomposition. The num - subsequent subpolygons are given beforehand. Therefore, ber of subpolygons as well as the position of cuts varies we can check the feasibility in constant time. u Th s, a fea - noticeably. Depending on the underlying application, sible decomposition using these criteria can be computed one might choose suitable constraints. In the following, in time O(n + m) for n being the number of skeleton we present decompositions for different combinations of points and m the number of boundary vertices. If the criteria and assess their suitability for our specific appli - fatness criterion is used, one has to calculate the aspect cation. The typical results are exemplified using a ROI ratio of each polygon, which takes O(m logm) time and polygon of a lung tissue sample (see Fig. 9). therefore results in a runtime of O(n m log m). Panel A and B in Fig. 9 illustrate the effect of the size criterion. Having a larger upper bound obviously results Optimization goals in fewer subpolygons. The MinNum optimization goal The algorithm computes the value X(i) for each skeleton minimizes the number of subpolygons, but the solutions i. For the decision problem, we defined X(i) to be True are not necessarily unique and one optimal decomposi- if there exists a feasible decomposition of the polygon tion is chosen arbitrarily. This can be observed at the P(i, n). With a redefinition of X(i), we can implement a bottom-most skeleton branch in both these decompo- variety of optimization goals. For a point i, let I be the sitions as the cuts in A would be feasible with the con- set of points j such that P(i, j) are feasible. One possible straints from B as well. In panel C and D of Fig. 9, the S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 9 of 17 Fig. 9 Decompositions based on different criteria. Exemplary decompositions obtained by applying the algorithm based on linear skeletons with different feasibility constraints and optimization goals. A area in [50, 300] , M inNum. B area in [50, 500] , M inNum. C area in [50, 300] , fatness ≥ 0.4, MinNum. Darea in [50, 300] , fatness ≥ 0.5, MinNum. E area in [50, 300] , M inCut. F area in [50, 300] , MaxF at fatness constraint was applied in form of a lower bound different optimization goal will not influence the amount on the aspect ratio of subpolygons. In comparison to the of area in the decomposition, but the quantity and posi- decomposition depicted in A, this criterion avoids the tions of cut edges may change considerably. These tendency towards elongated fragments. However, tighter changes are expected to affect the amount of successfully bounds do not necessarily result in better outcomes as a dissected tissue fragments in the microdissection. When feasible decomposition might not exist at all. This case is looking at the decompositions of the top left skeleton illustrated in panel D, where the algorithm did not find branch in those three polygons, one notices that the ones a feasible decomposition for the polygon parts that are in A and E have the same number of fragments, but with depicted in gray. For our application, this would not be MinCut a cut with a lower length is chosen. Maximizing favorable as it reduces the amount of extracted tissue the fatness usually results in a higher number of subpol- material. ygons. As can be seen in panel F, the resulting subpoly- We applied the different optimization goals denoted gons are less elongated and more circular in shape. We by MinNum (panel A), MinCut (panel E) and MaxFat expect these to be the desired shapes for our application. (panel F) with the same feasibility constraint. Choosing a Hence, we used the area constraint in combination with Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 10 of 17 the MaxFat optimization in our experiments and the fol- Both decomposition methods were applied with the lowing comparison of decomposition methods for LCM. same area bounds, namely a minimal and maximal area 2 2 of 100 px (ca. 1800 μm ) and 2800 px (ca. 50000 μm ) Experimental results respectively. The main focus of the development of a Experimental setup novel decomposition method lies in the reduction of tis- For the evaluation of our algorithms, we conducted LCM sue loss. This is determined by the amount of area in the experiments on shapes obtained from infrared micro- decomposition itself as well as the amount of successfully scopic images of 10 thin sections of FFPE (formalin-fixed dissected fragments. First, we compare the methods and paraffin-embedded) lung tissue samples from patients the resulting decompositions on a computational level. with non-small-cell lung carcinoma. The pixel spectra of Then, we analyze their performance in the practical set - the images were classified into different tissue types using ting with LCM. a random forest classifier as described in [2]. All pixel positions belonging to the tumor class were chosen as the Computational results regions of interest (ROI). The binary mask of the ROI was We examine the results of the MaxFat and BiSect decom- preprocessed by a morphological opening followed by a position on three different levels: fragments (subpoly - morphological closing and subsequent hole filling. Each gons), components (ROI polygons) and samples. We connected component of the preprocessed binary mask examine the size of the decompositions, the area loss and is given as a simple polygon on which we applied two dif- the morphology of the fragments. ferent decomposition approaches. The number of input polygons for our experiments ranged from 14 to 109 per Decomposition size sample with a total amount of 441. The resulting decom - The sampling consisted of 441 components with an aver - positions serve as the input for LCM. Each fragment is age area of 4500 px (ca. 81300 μm ). The MaxFat decom - transmitted in form of a circular list of discrete boundary position over all ten samples contained 4143 fragments points. with an average of 9.36 fragments per component. The For the experiment, we used our algorithm based BiSect decomposition consisted of considerably less frag- on linear skeletons such that each skeleton branch is ments with an average of 2.36 fragments per component decomposed separately. This follows the practical con - and 1089 for the entire sampling. sideration that a polygon as a whole may not possess a BiSect achieves a smaller decomposition size because feasible decomposition, while some individual branches most components did not require many bisections for the do. The resulting skeletons consisted of roughly 80 to fragments to fulfill the given area constraints (see Fig. 10 1000 points, involving around five to ten branches for A2, B2). With MaxFat, every skeleton branch in decom- each polygon to be decomposed, see Figs. 9, 10 for typi- posed individually. Therefore, most decompositions con - cal examples. We applied a size constraint as lower and sist of at least as many fragments as there are branches in upper bounds on the area of the subpolygons and com- the skeleton (see Fig. 10 A1, B1). puted an optimal decomposition in which the fatness, i.e. the minimum aspect ratio of the subpolygons, is maxi- Area loss mized. We denote this approach by MaxFat. Figure 11 depicts the area loss on the level of individual We compare our approach to a heuristic decomposi- components. The mean area loss with MaxFat is slightly tion method, which was used to decompose tissue sam- lower than the one with BiSect (MaxFat M = 8.96%, ples for LCM in previous work. As this method follows a BiSect M = 10.81%). However, the distribution of Max- bisection approach, we denote it by BiSect. Unlike Max- Fat shows a greater variability in values, a larger stand- Fat, this method includes merely a size constraint and no ard deviation and some high-loss outliers (MaxFat SD shape criterion or optimization goal. A polygon is decom- = 11.26, BiSect SD = 6.11). For BiSect, the variability is posed by recursive bisection if its area exceeds an upper lower and there are less outliers. On the level of samples, size bound. If the area of a (sub)polygon is below a given one can see that in 8 of 10 cases the area loss with Max- lower bound, it is discarded. Every bisection is designed Fat is lower than the one with BiSect (see Table 1). The to leave a strip of tissue behind such that each subpoly- decompositions with MaxFat contained up to 10% more gon retains contact to the surrounding membrane of the area. The area loss averages around 10.77% for MaxFat microscopic slide in order to meet a technical require- and 16.35% for BiSect. ment of the specific LCM system used in this study for Both methods inherently involve area loss. With BiSect, the dissection to be possible. The MaxFat decomposition area loss occurs due to the strips left behind by every does not include these strips because all subpolygons bisection. Therefore, the amount increases proportional intersect with the boundary of the input polygon. to the size of the components and the necessary cuts S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 11 of 17 Fig. 10 Exemplary decompositions with MaxFat and BiSect. Decompositions of four exemplary components from the tissue samples. Left: decomposition with MaxFat. Right: decomposition with BiSect Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 12 of 17 Table 1 Comparison of area loss for samples as the combined loss over all components for MaxFat and BiSect Sample 1 2 3 4 5 6 7 8 9 10 MaxFat 8.58 21.43 5.22 23.85 3.42 11.71 3.40 13.56 10.45 6.07 BiSect 10.62 18.04 13.44 23.41 13.47 19.02 13.00 14.59 18.48 19.46 area loss. This contributes to the higher standard devia - tion and outliers that were observable on the level of individual components. The results for entire samples suggest that the samples are dominated by components that cause small or no area loss when decomposed with MaxFat. Since the resulting fragments for each compo- nent in one tissue sample are collectively gathered, the quality of the decomposition should be assessed on the level of samples. Regarding area loss during decompo- sition, MaxFat generally achieved better results. How- ever, the quality of the methods is ultimately determined by their performance in practice and their success with LCM. Therefore, practical evaluations are necessary. Here, we expect the morphology of the fragments to be Fig. 11 Comparison of area loss for components. Distribution of a critical factor. the area loss (in%) in the decompositions of individual components contained in the sampling (n = 441). Comparison of MaxFat (M = 8.96%, SD = 11.26) and BiSect (M = 10.81%, SD=6.11) Morphology We compared both decomposition methods based on the resulting fatness, i.e. aspect ratio, of the fragments. This (see Fig. 10). With MaxFat, area is lost for each skeleton value measures the circularity of the shape. On the level branch for which a feasible decomposition did not exist. of individual fragments, the aspect ratios in BiSect pre- This mainly occurs if the corresponding (sub)polygon is sent themselves in the pattern of a normal distribution either too slim or too wide. The first case is depicted in whereas the distribution for MaxFat is clearly left-skewed panel A1 of Fig. 10: Because the area of the gray polygons (see Fig. 12). The average aspect ratio of fragments over belonging to the bottom two branches was below the all samples is considerably higher with MaxFat (MaxFat given lower bound, a feasible decomposition did not exist M = 0.58, BiSect M = 0.39). We observe similar results and their area was lost. This can be attributed to short - when considering the average aspect ratio in the decom- comings of the underlying skeleton pruning method. positions of components (see Fig. 13). The values for Improving the pruning of the skeleton may avoid such MaxFat are larger and the variability is smaller (MaxFat short branches. The second case of too wide shapes is M = 0.58, BiSect M = 0.36). The standard deviation for exemplified in panel D1. If the upper area bound is rela - MaxFat is half as high as the one of BiSect (MaxFat SD tively small, the MaxFat decomposition of a wide shape = 0.05, BiSect SD = 0.1). For over 75% of components, leads to either thin-slicing or no feasible solution at all. the average fatness in the decompositions computed with This is due to our definition of the cut edges, which do MaxFat was higher than 0.5, whereas with BiSect nearly not allow internal decompositions. This also illustrates 75% have an average fatness lower than 0.4. that our approach is tailored towards more complex, BiSect applies only a size constraint and a compo- ramified shapes rather than fat objects. It is noteworthy nent is only decomposed if its area exceeds the given that the polygon depicted in panel D1/D2 of Fig. 10 cov- upper bound. Therefore, many components are not ers an area of around 43,000 px (ca. 7,77,000 μm ) and decomposed, but their shape is oftentimes elongated thus represents a huge outlier in our sampling. and ramified as can be seen in panel A2 of Fig. 10. These observations coincide with the presented results. Because this method follows a bisection approach, the For MaxFat, the first cause of area loss might occur fre - cut placement creates fragments with irregular shapes quently but merely contributes a small value to the over- and narrow bottlenecks (see Fig. 10 C2). The exception all loss. The second cause does not appear as often in the can be observed in large, round components as their samples because the average area of the components is decomposition resembles a grid pattern (see panel D2 fairly small, but obviously results in a large amount of of Fig. 10). In this case, the resulting fragments achieve S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 13 of 17 Fig. 12 Comparison of the fatness for fragments. Distribution of aspect ratios of individual fragments contained in the decompositions with MaxFat in A (n = 4151, M = 0.58, SD = 0.1) and BiSect in B (n = 1089, M = 0.39, SD = 0.11) MaxFat consistently obtains such fragments, we sus- pect this to be the main advantage of this decomposi- tion method in the practice. Hence, we expect a higher success rate in the practical application with LCM. Running times The computations were executed on a Windows PC (Intel Core i5-8600 CPU, 16 GB RAM). The proposed approach consists of a skeletonization and a subsequent decomposition with MaxFat. The average computation time for one input polygon was 0.63 s for the skeletoni- zation, 14.38 s for the decomposition with MaxFat and 0.43 s for the decomposition with BiSect. When consid- Fig. 13 Comparison of the average fatness for components. ering median values, we see that one for MaxFat (0.24 Distribution of average aspect ratios in the decompositions of all individual components contained in the sampling (n = 441). s) is lower than the one for BiSect (0.47 s). This suggests Comparison of MaxFat (M = 0.58, SD = 0.05) and BiSect (M = 0.36, that MaxFat can perform very fast on the majority of SD = 0.1) inputs but more slowly on others. In general, the run- ning time for BiSect fairly low, as many polygons are not decomposed at all. For MaxFat, on the other hand, a higher fatness. The results suggest that without the the time complexity depends on the number of bound- application of some shape criterion the BiSect decom- ary points as well as the number of skeleton points. position does not naturally result in fragments of large We looked at one sample in more detail. Sample 7 fatness. MaxFat, on the other hand, utilizes both size consisted of 63 polygons with different boundary (M and shape criteria and tries to maximize the fatness of = 274.63, Min = 131, Max = 1104) and skeleton (M a decomposition. This results in smaller fragments that = 152.55, Min = 81, Max = 507) sizes. Therefore, the are less elongated and rounder in shape (see Fig. 10). MaxFat decomposition showed a variation in runtimes The computational evaluation reveals that MaxFat con - (M = 1.77 s, Min = 0.21 s, Max = 29.57 s). The runt - sistently obtains higher fatness values. This strengthens imes for BiSect were consistent (M = 0.47 s, Min = our choice to include the fatness criterion in the opti- 0.47 s, Max = 0.5 s). In total, the decomposition with mization goal rather than the feasibility constraints. BiSect required 35.62 s and resulted in 103 fragments. Even without applying a strict bound on the fatness, The proposed approach was performed in ca. 3.52 min we were able to achieve high fatness values without the (1.69 min skeletonization and 1.83 min decomposition) risk of area loss due to the non-existence of a feasible and resulted in 487 fragments. Note that the runtime of decomposition. The success of a dissection using LCM MaxFat can be optimized by parallelizing the execution depends on the size and morphology of the tissue frag- of the algorithm not only on the different polygons in ment. We hypothesize that approximately round shapes one sample but also on the different skeleton branches. have a higher chance to be successfully collected. As Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 14 of 17 Practical results with LCM view) after the laser pulse. One might only conjecture the The practical evaluation of both shape decomposition reasons: As mentioned before, the size and shape of the approaches consisted of the dissection of all computed fragments, the focus of the laser pulse as well as its posi- fragments with LCM. Because one tissue sample cannot tion on the boundary of the fragment affect the transfer - be dissected twice, the experiment was performed on ring process and its success. The last category “too small” empty microscopic slides. Therefore, it was not possible contains fragments of such a small size that the laser did to compare the amount of successfully dissected tissue by not leave enough material to be collected. measuring for example the protein content. The evalua - Figure 14 depicts the distribution of assigned labels tion was restricted to visual assessment. based on the number of fragments in each category. In both decompositions, the majority of fragments was Classification of dissected fragments labeled as successful. The amount of successfully dis - The dissection of each fragment was observed and classi - sected fragments of MaxFat is consistently over 90% fied into the following categories. A fragment was labeled for all samples. The distribution for BiSect shows more “successful” if it disappeared from the field of view after variation between the samples. On average, 95.44% of the the laser pulse. In this case, we expect it to be success- fragments of MaxFat were successful, 4.34% were labeled fully transferred into the collecting device. Unsuccessful as fallen and merely 0.22% as torn. None of the fragments fragments were further divided into three categories. The were too small. BiSect averages around 80.98% successful, label “torn” describes fragments that tore during the dis- 14.99% fallen, 2.39% torn and 1.64% too small fragments. section. Because they were only partially transferred, the collected area is not measurable. A fragment was labeled Area loss and success rates “fallen” in the following two cases. The fragment fell Tables 2, 3 show the success rates of MaxFat and BiSect before the transferring process, which might be the case with regard to tissue yield, the results are also visualized if all connections to the surrounding membrane were in Fig. 15. We distinguish two different success rates. The already severed before the laser pulse. The other case cov - success rate of the microdissection (Table 2) represents ers fragments that fell back onto the slide (in the field of the ratio of the area of the fragments that was successfully Fig. 14 Distribution of labels assign during LCM. Comparison of the percentage of labels assigned to fragments during laser capture microdissection for MaxFat (A) and BiSect (B) Table 2 Comparison of microdissection success rates for MaxFat and BiSect sample 1 2 3 4 5 6 7 8 9 10 MaxFat 90.00 92.72 98.05 96.58 94.77 95.74 97.04 95.75 96.49 97.43 BiSect 80.39 76.61 73.02 81.79 81.79 79.19 81.47 81.09 85.10 95.80 The LCM success rate (in%) describes the amount of tissue area that was collected from the fragments by LCM S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 15 of 17 Table 3 Comparison of overall success rates for MaxFat and BiSect sample 1 2 3 4 5 6 7 8 9 10 MaxFat 82.28 72.85 92.93 73.54 91.53 84.53 93.74 82.77 86.41 91.52 BiSect 71.85 62.79 63.21 62.64 70.78 64.13 70.88 69.26 69.38 77.16 This success rate (in%) represents the overall tissue yield from as sample by combining the LCM success rate with the area loss during the decomposition of components, i.e. how much area of the original ROI is contained in the computed fragments at least 10% higher. In sample 3, the amount of lost tis- sue could potentially be decreased by 29.72% when using MaxFat rather than BiSect. On average, the tissue yield with the proposed decomposition approach is 17.55% higher. The practical evaluation confirms our conjecture that the proposed decomposition method performs better in practice than the heuristic bisection approach. The amount of successfully dissected fragments is consist- ently higher with the MaxFat approach. With BiSect, this rate varies more noticeably and the algorithm was not able to filter out fragments that did not fulfill the lower area constraint. Besides the quantity of successfully dis- sected fragments, the tissue area that was collected with MaxFat was larger as well. Together with the smaller area loss in our decomposition, which we observed in the computational assessment, the proposed method proved to minimize the tissue loss considerably. When used on Fig. 15 Comparison of success rates. Comparison of the amount of actual tissue samples, our decomposition method will tissue loss and the overall success rates of MaxFat and BiSect. With increase the tissue yield and thus the amount of protein 100% being the ROI area in the sample, the two lighter bar segments or DNA available for further analysis. correspond to the percentage of tissue loss in the decomposition and the microdissection, respectively. The darkest segment represents the percentage of the original area that was successfully collected during Conclusion the microdissection. This value represents the overall success rate In this paper, we presented a skeleton-based decompo- sition method for simple polygons as a novel approach to decompose disease-specific regions in tissue samples while aiming to optimize the amount of tissue obtained collected as computed by the LCM system. Over all ten by laser capture microdissection (LCM). The lack of samples, the values for MaxFat are higher than for BiSect. previous benchmark methods and results is somewhat The largest difference can be observed for sample 3 with remarkable. It indicates that previous studies utilizing a value of 25.03%. Overall, the success rate for the micro- LCM relied on manual decomposition of the regions to dissection averages at 95.46% for MaxFat and 80.99% for be dissected, which is clearly impractical in clinical study BiSect. Using these percentages, we calculated the overall settings involving dozens or hundreds of samples. As the success rate (Table 3) of both decomposition methods by first fully automated approach, we provide a conceptual combining the following factors: The ROI area contained contribution that may pave the way for making LCM in the samples (combined area of all components), the feasible in large clinical studies. Our approach will also amount of area lost due to the decomposition algorithm facilitate systematic assessment of optimal size and mor- and lastly the success rate of the microdissection. For phology criteria for LCM experiments, which would be example, the MaxFat decomposition resulted in 8.58% difficult if not impossible to conduct based on manual area loss for sample 1 (see Table 1). Thus, 91.42% of the original area was contained in the fragments for LCM. shape decomposition. The microdissection showed a success rate of 90%, which Size and morphology of the fragments are assumed to results in an overall success rate of 82.28%. This means be the key factors that influence the success of dissec - that 82.28% of the tissue contained in sample 1 could be tions using LCM. Our approach is designed to minimize collected using the MaxFat decomposition approach. For tissue loss by utilizing a size constraint and optimizing all ten samples, the overall success rate of MaxFat was the shapes towards fat or circular fragments. As we Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 16 of 17 Abbreviations demonstrated, this translates into practice when com- LCM: Laser capture microdissection; ROI: Region of interest; H[MYAMP: E-stain] paring our approach to a recursive bisection method Hematoxylin and eosin stain; FFPE: Formalin-fixed paraffin-embedded; M: that is currently used and only applies a size constraint. Arithmetic mean; SD: Standard deviation. Our approach is tailored towards complex morpho- Acknowledgements logical structures that are commonly found in can- We acknowledge support by the Open Access Publication Funds of the Ruhr- cerous tissue and are usually the most challenging to Universität Bochum. Moreover, we gratefully acknowledge Nina Goertzen for providing infrared microscopic imaging data for our study. dissect using LCM without major loss of tissue mate- rial. Not surprisingly, the algorithm does not perform Authors’ contributions as well when decomposing relatively large and fat AM proposed the project. LS, MB and AM developed the method. LS imple- mented the method. TK carried out the microdissection supervised by KG. LS shapes. However, such shapes do not occur frequently analyzed the experimental results. LS, MB and AM participated in writing the and the impact on the overall success seems to be mini- manuscript. All authors read and approved the final manuscript. mal. These shapes can be easily decomposed using Funding simple approaches, e.g. the bisection-based approach, Open Access funding enabled and organized by Projekt DEAL. This work was without major tissue loss. Thus, we plan to improve our supported by the Ministry for Culture and Science (MKW ) of North Rhine- method by distinguishing these shapes and compute Westphalia (Germany) through grant 111.08.03.05-133974 and the Center for Protein Diagnostics (PRODI). their decompositions separately. The implementation of our approach relies on a (dis - Availability of data and materials crete) skeletonization of the underlying polygons. Spe- Source code and all shapes from our validation samples as well as computa- tional results are available from https:// github. com/ Lahut ar/ Skele ton- based- cifically, we utilize the approach of Bai et al. [30], which Shape- Decom posit ion. git. uses a heuristic pruning approach. While other high- quality implementations of discrete skeletonization Declarations algorithms exist [38, 39], the approaches lack pruning strategies that are essential for our approach to pro- Consent for publication Not applicable. duce practically relevant results. It is likely that recent improvements for skeletonization and pruning will fur- Competing interests ther improve our results. For example, recent methods The authors declare that they have no competing interests. of Durix et al. [27, 28] promise to avoid short and other Author details spurious branches, which contributed to area loss in Department of Computer Science, Faculty of Mathematics, Ruhr University our decomposition. More broadly, concepts for robust Bochum, Bochum, Germany. Department of Biophysics, Faculty of Biology and Biotechnology, Ruhr University Bochum, Bochum, Germany. Bioinformat- skeletonizations have been proposed based on the ics Group, Faculty of Biology and Biotechnology, Ruhr University Bochum, -medial axis [25, 40], which are built on solid theoreti- Bochum, Germany. Center for Protein Diagnostics, Ruhr University Bochum, cal grounds and thus may provide useful concepts for Bochum, Germany. further improved shape decomposition approaches. Received: 3 February 2021 Accepted: 21 June 2021 Here, skeletonization is considered as an input to our decomposition method. While better and more robust skeletonization may further improve performance, a detailed investigation is beyond the scope of our pre- References sent study. 1. Emmert-Buck MR, Bonner RF, Smith PD, Chuaqui RF, Zhuang Z, Gold- stein SR, Weiss RA, Liotta LA. Laser capture microdissection. Science. The practical evaluation with LCM showed the 1996;274(5289):998–1001. advantage of using the proposed decomposition 2. Großerueschkamp F, Bracht T, Diehl HC, Kuepper C, Ahrens M, Kallenbach- method. Over the entire sampling, our decompositions Thieltges A, Mosig A, Eisenacher M, Marcus K, Behrens T, et al. Spatial and molecular resolution of diffuse malignant mesothelioma heterogeneity contained more successful fragments and we achieved by integrating label-free ftir imaging, laser capture microdissection and a success rate of 95% in the dissection. In combination proteomics. Sci Rep. 2017;7(1):1–12. with the lower area loss during the decomposition, we 3. Kondo Y, Kanai Y, Sakamoto M, Mizokami M, Ueda R, Hirohashi S. Genetic instability and aberrant DNA methylation in chronic hepatitis and were able to minimize the overall tissue loss for all sam- cirrhosis-a comprehensive study of loss of heterozygosity and microsatel- ples and increased the tissue yield by 10–30%. Over- lite instability at 39 loci and dna hypermethylation on 8 cpg islands in all, our work contributes to further optimization and microdissected specimens from patients with hepatocellular carcinoma. Hepatology. 2000;32(5):970–9. automation of LCM and thus promises to contribute to 4. Datta S, Malhotra L, Dickerson R, Chaffee S, Sen CK, Roy S. Laser capture the further maturing of the technology and enhancing microdissection: Big data from small samples. Histol Histopathol. its suitability for systematic use in larger scale clinical 2015;30(11):1255. 5. Ong C-AJ, Tan QX, Lim HJ, Shannon NB, Lim WK, Hendrikson J, Ng WH, Tan studies. JW, Koh KK, Wasudevan SD, et al. An optimised protocol harnessing laser capture microdissection for transcriptomic analysis on matched primary and metastatic colorectal tumours. Sci Rep. 2020;10(1):1–12. S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 17 of 17 6. Vu QD, Graham S, Kurc T, To MNN, Shaban M, Qaiser T, Koohbanani NA, 24. Shaked D, Bruckstein AM. Pruning medial axes. Comput Vision Image Khurram SA, Kalpathy-Cramer J, Zhao T, et al. Methods for segmenta- Understand. 1998;69(2):156–69. tion and classification of digital microscopy tissue images. Front Bioeng 25. Chazal F, Lieutier A. The “ -medial axis’’. Graph Models. 2005;67(4):304–31. Biotechnol. 2019;7:188. 26. Sud A, Foskey M, Manocha D. Homotopy-preserving medial axis sim- 7. Wang S, Rong R, Yang DM, Fujimoto J, Yan S, Cai L, Yang L, Luo D, Behrens plification. In: Proceedings of the 2005 ACM symposium on solid and C, Parra ER, et al. Computational staining of pathology images to study physical modeling. 2005, p 39–50 the tumor microenvironment in lung cancer. Cancer Res. 2020;7:548. 27. Durix B, Chambon S, Leonard K, Mari J.-L, Morin G. The propagated skel- 8. Selbach L, Kowalski T, Gerwert K, Buchin M, Mosig A. Shape Decomposi- eton: a robust detail-preserving approach. In: International conference tion Algorithms for Laser Capture Microdissection. In: 20th International on discrete geometry for computer imagery. Berlin: Springer; 2019, p workshop on algorithms in bioinformatics ( WABI 2020). Leibniz interna- 343–354 tional proceedings in informatics (LIPIcs), 2020, p 13:1–13:17. 28. Durix B, Morin G, Chambon S, Mari J-L, Leonard K. One-step compact 9. Keil JM. Polygon decomposition. Handbook of computational geometry. skeletonization. Eurographics. 2019;5:21–4. Berlin: Springer; 2000. p. 491–518. 29. Bai X, Latecki LJ, Liu W-Y. Skeleton pruning by contour partitioning 10. Saha PK, Borgefors G, di Baja GS. A survey on skeletonization algorithms with discrete curve evolution. IEEE transactions on pattern analysis and and their applications. Pattern Recogn Lett. 2016;76:3–12. machine intelligence. 2007;29:3. 11. Drechsler K, Laura CO. Hierarchical decomposition of vessel skeletons for 30. Bai X, Latecki L.J. Discrete skeleton evolution. In: International workshop graph creation and feature extraction. In: 2010 IEEE international confer- on energy minimization methods in computer vision and pattern recog- ence On Bioinformatics and biomedicine (BIBM). 2010, p 456–61. nition. Springer; 2007, p 362–374. 12. Wang C, Gui C-P, Liu H-K, Zhang D, Mosig A. An image skeletonization- 31. Buchin M, Mosig A, Selbach L. Skeleton-based decomposition of simple based tool for pollen tube morphology analysis and phenotyping F. J polygons. In: Abstracts of 35th European workshop on computational Integr Plant Biol. 2013;55(2):131–41. geometry. 2019. 13. Narro ML, Yang F, Kraft R, Wenk C, Efrat A, Restifo LL. Neuronmetrics: Soft- 32. Efrat A, Katz MJ, Nielsen F, Sharir M. Dynamic data structures for fat ware for semi-automated processing of cultured neuron images. Brain objects and their applications. Comput Geomet. 2000;15(4):215–27. Res. 2007;1138:57–75. 33. Damian M. Exact and approximation algorithms for computing optimal 14. Schmuck MR, Temme T, Dach K, de Boer D, Barenys M, Bendt F, Mosig A, fat decompositions. Comput Geomet. 2004;28(1):19–27. Fritsche E. Omnisphero: a high-content image analysis (HCA) approach 34. Sui W, Zhang D. Four methods for roundness evaluation. Phys Proc. for phenotypic developmental neurotoxicity (DNT ) screenings of orga- 2012;24:2159–64. noid neurosphere cultures in vitro. Archiv Toxicol. 2017;91:4. 35. Alewijnse S.P.A, Buchin K, Buchin M, Kölzsch A, Kruckenberg H, Westen- 15. Reniers D, Telea A. Skeleton-based hierarchical shape segmentation. In: berg M.A. A framework for trajectory segmentation by stable criteria. In: SMI’07. IEEE international conference on shape modeling and applica- Proceedings of the 22nd ACM SIGSPATIAL international conference on tions, 2007, p 179–188. advances in geographic information systems. SIGSPATIAL ’14. 2014, p 16. Leonard K, Morin G, Hahmann S, Carlier A. A 2d shape structure for 351–360 decomposition and part similarity. In: 2016 23rd international conference 36. Buchin M, Selbach L. Decomposition and partition algorithms for tissue on pattern recognition (ICPR). IEEE. 2016, p 3216–3221 dissection. In: Computational geometry: young researchers forum. 2020, 17. Papanelopoulos N, Avrithis Y, Kollias S. Revisiting the medial axis for p 24–27 planar shape decomposition. Comput Vision Image Understand. 37. Buchin M, Selbach L. Partitioning algorithms for weighted cactus graphs, 2019;179:66–78. 2020;.arXiv: 2001. 00204 v2 18. di Baja GS, Thiel E. (3, 4)-weighted skeleton decomposition for pattern 38. Cacciola F. A CGAL implementation of the straight skeleton of a simple representation and description. Pattern Recogn. 1994;27(8):1039–49. 2D polygon with holes. In: 2nd CGAL user workshop. 2004, p 1 19. Tănase M, Veltkamp RC. Polygon decomposition based on the straight 39. Fabri A, Giezeman G.-J, Kettner L, Schirra S, Schönherr S. On the design line skeleton. Geometry morphology, and computational imaging. Berlin: of CGAL a computational geometry algorithms library. Softw Pract Exp. Springer; 2003. p. 247–68. 2000;30(11):1167–202. 20. Blum H. A transformation for extracting new descriptors of shape. Models 40. Chaussard J, Couprie M, Talbot H. Robust skeletonization using the Percep Speech Visual Forms. 1967;1967:362–80. discrete -medial axis. Pattern Recogn Lett. 2011;32(9):1384–94. 21. Blum H. Biological shape and visual science (part I). J Theor Biol. 1973;38(2):205–87. Publisher’s Note 22. Choi H.I, Choi S.W, Moon H.P. Mathematical theory of medial axis trans- Springer Nature remains neutral with regard to jurisdictional claims in pub- form. Pac J Math. 1997;181(1):57–88. lished maps and institutional affiliations. 23. Wolter F-E. Cut locus and medial axis in global shape interrogation and representation. Sea Grant College Program: Massachusetts Institute of Technology; 1993. Re Read ady y to to submit y submit your our re researc search h ? Choose BMC and benefit fr ? Choose BMC and benefit from om: : fast, convenient online submission thorough peer review by experienced researchers in your ﬁeld rapid publication on acceptance support for research data, including large and complex data types • gold Open Access which fosters wider collaboration and increased citations maximum visibility for your research: over 100M website views per year At BMC, research is always in progress. Learn more biomedcentral.com/submissions http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algorithms for Molecular Biology Springer Journals http://www.deepdyve.com/lp/springer-journals/shape-decomposition-algorithms-for-laser-capture-microdissection-Vjt0mHVHnd

Loading next page...

References (40)

(2007)
Neuronmetrics: Software for semi-automated processing of cultured neuron images
Brain Res, 1138
(Durix B, Chambon S, Leonard K, Mari J.-L, Morin G. The propagated skeleton: a robust detail-preserving approach. In: International conference on discrete geometry for computer imagery. Berlin: Springer; 2019, p 343–354)
Durix B, Chambon S, Leonard K, Mari J.-L, Morin G. The propagated skeleton: a robust detail-preserving approach. In: International conference on discrete geometry for computer imagery. Berlin: Springer; 2019, p 343–354
Durix B, Chambon S, Leonard K, Mari J.-L, Morin G. The propagated skeleton: a robust detail-preserving approach. In: International conference on discrete geometry for computer imagery. Berlin: Springer; 2019, p 343–354, Durix B, Chambon S, Leonard K, Mari J.-L, Morin G. The propagated skeleton: a robust detail-preserving approach. In: International conference on discrete geometry for computer imagery. Berlin: Springer; 2019, p 343–354
(2000)
Genetic instability and aberrant DNA methylation in chronic hepatitis and cirrhosis-a comprehensive study of loss of heterozygosity and microsatellite instability at 39 loci and dna hypermethylation on 8 cpg islands in microdissected specimens from patients with hepatocellular carcinoma
Hepatology, 32
(2012)
Four methods for roundness evaluation
Phys Proc, 24
(2000)
Dynamic data structures for fat objects and their applications
Comput Geomet, 15
S Wang (2020)
548
Cancer Res, 7
(2019)
Methods for segmentation and classification of digital microscopy tissue images
Front Bioeng Biotechnol, 7
(1996)
Laser capture microdissection
Science, 274
(Cacciola F. A CGAL implementation of the straight skeleton of a simple 2D polygon with holes. In: 2nd CGAL user workshop. 2004, p 1)
Cacciola F. A CGAL implementation of the straight skeleton of a simple 2D polygon with holes. In: 2nd CGAL user workshop. 2004, p 1
Cacciola F. A CGAL implementation of the straight skeleton of a simple 2D polygon with holes. In: 2nd CGAL user workshop. 2004, p 1, Cacciola F. A CGAL implementation of the straight skeleton of a simple 2D polygon with holes. In: 2nd CGAL user workshop. 2004, p 1
(2016)
A survey on skeletonization algorithms and their applications
Pattern Recogn Lett, 76
JM Keil (2000)
10.1016/B978-044482537-7/50012-7
Polygon decomposition. Handbook of computational geometry
(1967)
A transformation for extracting new descriptors of shape
Models Percep Speech Visual Forms, 1967
(2011)
Robust skeletonization using the discrete \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ-medial axis
Pattern Recogn Lett, 32
(2013)
An image skeletonization-based tool for pollen tube morphology analysis and phenotyping F
J Integr Plant Biol, 55
(1994)
(3, 4)-weighted skeleton decomposition for pattern representation and description
Pattern Recogn, 27
(Buchin M, Mosig A, Selbach L. Skeleton-based decomposition of simple polygons. In: Abstracts of 35th European workshop on computational geometry. 2019.)
Buchin M, Mosig A, Selbach L. Skeleton-based decomposition of simple polygons. In: Abstracts of 35th European workshop on computational geometry. 2019.
Buchin M, Mosig A, Selbach L. Skeleton-based decomposition of simple polygons. In: Abstracts of 35th European workshop on computational geometry. 2019., Buchin M, Mosig A, Selbach L. Skeleton-based decomposition of simple polygons. In: Abstracts of 35th European workshop on computational geometry. 2019.
S Datta (2015)
1255
Histol Histopathol, 30
(Alewijnse S.P.A, Buchin K, Buchin M, Kölzsch A, Kruckenberg H, Westenberg M.A. A framework for trajectory segmentation by stable criteria. In: Proceedings of the 22nd ACM SIGSPATIAL international conference on advances in geographic information systems. SIGSPATIAL ’14. 2014, p 351–360)
Alewijnse S.P.A, Buchin K, Buchin M, Kölzsch A, Kruckenberg H, Westenberg M.A. A framework for trajectory segmentation by stable criteria. In: Proceedings of the 22nd ACM SIGSPATIAL international conference on advances in geographic information systems. SIGSPATIAL ’14. 2014, p 351–360
Alewijnse S.P.A, Buchin K, Buchin M, Kölzsch A, Kruckenberg H, Westenberg M.A. A framework for trajectory segmentation by stable criteria. In: Proceedings of the 22nd ACM SIGSPATIAL international conference on advances in geographic information systems. SIGSPATIAL ’14. 2014, p 351–360, Alewijnse S.P.A, Buchin K, Buchin M, Kölzsch A, Kruckenberg H, Westenberg M.A. A framework for trajectory segmentation by stable criteria. In: Proceedings of the 22nd ACM SIGSPATIAL international conference on advances in geographic information systems. SIGSPATIAL ’14. 2014, p 351–360
(Leonard K, Morin G, Hahmann S, Carlier A. A 2d shape structure for decomposition and part similarity. In: 2016 23rd international conference on pattern recognition (ICPR). IEEE. 2016, p 3216–3221)
Leonard K, Morin G, Hahmann S, Carlier A. A 2d shape structure for decomposition and part similarity. In: 2016 23rd international conference on pattern recognition (ICPR). IEEE. 2016, p 3216–3221
Leonard K, Morin G, Hahmann S, Carlier A. A 2d shape structure for decomposition and part similarity. In: 2016 23rd international conference on pattern recognition (ICPR). IEEE. 2016, p 3216–3221, Leonard K, Morin G, Hahmann S, Carlier A. A 2d shape structure for decomposition and part similarity. In: 2016 23rd international conference on pattern recognition (ICPR). IEEE. 2016, p 3216–3221
(2000)
On the design of CGAL a computational geometry algorithms library
Softw Pract Exp, 30
(Drechsler K, Laura CO. Hierarchical decomposition of vessel skeletons for graph creation and feature extraction. In: 2010 IEEE international conference On Bioinformatics and biomedicine (BIBM). 2010, p 456–61.)
Drechsler K, Laura CO. Hierarchical decomposition of vessel skeletons for graph creation and feature extraction. In: 2010 IEEE international conference On Bioinformatics and biomedicine (BIBM). 2010, p 456–61.
Drechsler K, Laura CO. Hierarchical decomposition of vessel skeletons for graph creation and feature extraction. In: 2010 IEEE international conference On Bioinformatics and biomedicine (BIBM). 2010, p 456–61., Drechsler K, Laura CO. Hierarchical decomposition of vessel skeletons for graph creation and feature extraction. In: 2010 IEEE international conference On Bioinformatics and biomedicine (BIBM). 2010, p 456–61.
(1998)
Pruning medial axes
Comput Vision Image Understand, 69
(2020)
An optimised protocol harnessing laser capture microdissection for transcriptomic analysis on matched primary and metastatic colorectal tumours
Sci Rep, 10
(2017)
Omnisphero: a high-content image analysis (HCA) approach for phenotypic developmental neurotoxicity (DNT) screenings of organoid neurosphere cultures in vitro
Archiv Toxicol, 91
(2005)
The “\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ-medial axis”
Graph Models, 67
(Buchin M, Selbach L. Partitioning algorithms for weighted cactus graphs, 2020;.arXiv:2001.00204v2)
Buchin M, Selbach L. Partitioning algorithms for weighted cactus graphs, 2020;.arXiv:2001.00204v2
Buchin M, Selbach L. Partitioning algorithms for weighted cactus graphs, 2020;.arXiv:2001.00204v2, Buchin M, Selbach L. Partitioning algorithms for weighted cactus graphs, 2020;.arXiv:2001.00204v2
(Bai X, Latecki LJ, Liu W-Y. Skeleton pruning by contour partitioning with discrete curve evolution. IEEE transactions on pattern analysis and machine intelligence. 2007;29:3.)
Bai X, Latecki LJ, Liu W-Y. Skeleton pruning by contour partitioning with discrete curve evolution. IEEE transactions on pattern analysis and machine intelligence. 2007;29:3.
Bai X, Latecki LJ, Liu W-Y. Skeleton pruning by contour partitioning with discrete curve evolution. IEEE transactions on pattern analysis and machine intelligence. 2007;29:3., Bai X, Latecki LJ, Liu W-Y. Skeleton pruning by contour partitioning with discrete curve evolution. IEEE transactions on pattern analysis and machine intelligence. 2007;29:3.
(2017)
Spatial and molecular resolution of diffuse malignant mesothelioma heterogeneity by integrating label-free ftir imaging, laser capture microdissection and proteomics
Sci Rep, 7
(2015)
Laser capture microdissection: Big data from small samples
Histol Histopathol, 30
(Selbach L, Kowalski T, Gerwert K, Buchin M, Mosig A. Shape Decomposition Algorithms for Laser Capture Microdissection. In: 20th International workshop on algorithms in bioinformatics (WABI 2020). Leibniz international proceedings in informatics (LIPIcs), 2020, p 13:1–13:17.)
Selbach L, Kowalski T, Gerwert K, Buchin M, Mosig A. Shape Decomposition Algorithms for Laser Capture Microdissection. In: 20th International workshop on algorithms in bioinformatics (WABI 2020). Leibniz international proceedings in informatics (LIPIcs), 2020, p 13:1–13:17.
Selbach L, Kowalski T, Gerwert K, Buchin M, Mosig A. Shape Decomposition Algorithms for Laser Capture Microdissection. In: 20th International workshop on algorithms in bioinformatics (WABI 2020). Leibniz international proceedings in informatics (LIPIcs), 2020, p 13:1–13:17., Selbach L, Kowalski T, Gerwert K, Buchin M, Mosig A. Shape Decomposition Algorithms for Laser Capture Microdissection. In: 20th International workshop on algorithms in bioinformatics (WABI 2020). Leibniz international proceedings in informatics (LIPIcs), 2020, p 13:1–13:17.
(2019)
Revisiting the medial axis for planar shape decomposition
Comput Vision Image Understand, 179
(1997)
Mathematical theory of medial axis transform
Pac J Math, 181
(1973)
Biological shape and visual science (part I)
J Theor Biol, 38
(2004)
Exact and approximation algorithms for computing optimal fat decompositions
Comput Geomet, 28
(2020)
Computational staining of pathology images to study the tumor microenvironment in lung cancer
Cancer Res, 7
(Reniers D, Telea A. Skeleton-based hierarchical shape segmentation. In: SMI’07. IEEE international conference on shape modeling and applications, 2007, p 179–188.)
Reniers D, Telea A. Skeleton-based hierarchical shape segmentation. In: SMI’07. IEEE international conference on shape modeling and applications, 2007, p 179–188.
Reniers D, Telea A. Skeleton-based hierarchical shape segmentation. In: SMI’07. IEEE international conference on shape modeling and applications, 2007, p 179–188., Reniers D, Telea A. Skeleton-based hierarchical shape segmentation. In: SMI’07. IEEE international conference on shape modeling and applications, 2007, p 179–188.
(2019)
One-step compact skeletonization
Eurographics, 5
(Buchin M, Selbach L. Decomposition and partition algorithms for tissue dissection. In: Computational geometry: young researchers forum. 2020, p 24–27)
Buchin M, Selbach L. Decomposition and partition algorithms for tissue dissection. In: Computational geometry: young researchers forum. 2020, p 24–27
Buchin M, Selbach L. Decomposition and partition algorithms for tissue dissection. In: Computational geometry: young researchers forum. 2020, p 24–27, Buchin M, Selbach L. Decomposition and partition algorithms for tissue dissection. In: Computational geometry: young researchers forum. 2020, p 24–27
(Sud A, Foskey M, Manocha D. Homotopy-preserving medial axis simplification. In: Proceedings of the 2005 ACM symposium on solid and physical modeling. 2005, p 39–50)
Sud A, Foskey M, Manocha D. Homotopy-preserving medial axis simplification. In: Proceedings of the 2005 ACM symposium on solid and physical modeling. 2005, p 39–50
Sud A, Foskey M, Manocha D. Homotopy-preserving medial axis simplification. In: Proceedings of the 2005 ACM symposium on solid and physical modeling. 2005, p 39–50, Sud A, Foskey M, Manocha D. Homotopy-preserving medial axis simplification. In: Proceedings of the 2005 ACM symposium on solid and physical modeling. 2005, p 39–50
(Bai X, Latecki L.J. Discrete skeleton evolution. In: International workshop on energy minimization methods in computer vision and pattern recognition. Springer; 2007, p 362–374.)
Bai X, Latecki L.J. Discrete skeleton evolution. In: International workshop on energy minimization methods in computer vision and pattern recognition. Springer; 2007, p 362–374.
Bai X, Latecki L.J. Discrete skeleton evolution. In: International workshop on energy minimization methods in computer vision and pattern recognition. Springer; 2007, p 362–374., Bai X, Latecki L.J. Discrete skeleton evolution. In: International workshop on energy minimization methods in computer vision and pattern recognition. Springer; 2007, p 362–374.

Publisher: Springer Journals
Copyright: Copyright © The Author(s) 2021
eISSN: 1748-7188
DOI: 10.1186/s13015-021-00193-6
Publisher site: See Article on Publisher Site

Abstract

Background: In the context of biomarker discovery and molecular characterization of diseases, laser capture microdissection is a highly effective approach to extract disease-specific regions from complex, heterogeneous tissue samples. For the extraction to be successful, these regions have to satisfy certain constraints in size and shape and thus have to be decomposed into feasible fragments. Results: We model this problem of constrained shape decomposition as the computation of optimal feasible decompositions of simple polygons. We use a skeleton-based approach and present an algorithmic framework that allows the implementation of various feasibility criteria as well as optimization goals. Motivated by our application, we consider different constraints and examine the resulting fragmentations. We evaluate our algorithm on lung tissue samples in comparison to a heuristic decomposition approach. Our method achieved a success rate of over 95% in the microdissection and tissue yield was increased by 10–30%. Conclusion: We present a novel approach for constrained shape decomposition by demonstrating its advantages for the application in the microdissection of tissue samples. In comparison to the previous decomposition approach, the proposed method considerably increases the amount of successfully dissected tissue. Keywords: Laser capture microdissection, Shape decomposition, Skeletonization Introduction processed with LCM provide more accurate molecular Laser capture microdissection (LCM) [1] is a highly effec - markers of diseases [4, 5]. With LCM being used more tive approach to extract specific cell populations from and more commonly in clinical studies, there is a need to complex, heterogeneous tissue samples. In the dissection, automate all procedures involved in sample processing. a laser cuts around the boundary of a selected region and a subsequent laser pulse catapults the fragment into a Practical application collecting device. LCM has been used extensively in the Our contribution is motivated by an application intro- context of biomarker discovery [2] as well as the molec- duced in [2] in which a region of interest (ROI) to be ular characterization of diseases [3]. Since LCM sepa- dissected from the tissue sample is identified using label- rates homogeneous and disease-specific regions from free hyperspectral infrared microscopy. In this approach, their heterogeneous and unspecific surrounding tissue an infrared microscopic image of the sample yields infra- regions, the characterizations obtained from genomic, red pixel spectra at a spatial resolution of about 5 μm. A transcriptomic or proteomic characterizations of samples previously trained random forest classifier assigns each pixel spectrum to one tissue component such as healthy or diseased, with the diseased class being further subdi- *Correspondence: leonie.selbach@rub.de vided into inflamed tissue as well as several subtypes of Department of Computer Science, Faculty of Mathematics, Ruhr thoracal tumors. The general sample preparation task in University Bochum, Bochum, Germany the context of LCM is to dissect all tumor regions (or all Full list of author information is available at the end of the article © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. The Creative Commons Public Domain Dedication waiver (http:// creat iveco mmons. org/ publi cdoma in/ zero/1. 0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data. Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 2 of 17 regions identified as one specific tumor subtype) from a samples consist of complex shapes of varying sizes, they sample. The current standard approach for the dissection oftentimes do not satisfy these constraints and therefore with LCM is to draw shapes manually. This is severely cannot be extracted from the tissue sample without any limited, not only because it takes an inacceptable amount previous processing. This either increases the amount of time for large numbers of samples, but also because of necessary user-interaction or negatively affects the it is required that the human operator will subjectively sample quality and thus compromises the advantages of decompose complex regions into smaller fragments. In LCM-based sample preparation. this paper, we propose a novel automated decomposi- Given a binary mask of a microscopic slide with the tion approach. While our current contribution deals with ROIs as the foreground, the image is preprocessed for the specific context of label-free infrared microscopy, our LCM in such a way that the ROIs are reduced to a num- approach equally applies more broadly to LCM in the ber of connected components without holes. By inter- context of other microscopic modalities, most notably preting each of these connected components as a simple H&E stained (hematoxylin and eosin stained) images [4] polygon, we can model the given problem of constrained for which recent digital pathology approaches facilitate shape decomposition as the computation of optimal fea- reliable computational identification of disease specific sible decompositions of polygons (see Fig. 1). The con - regions [6, 7]. straints can be modeled as certain feasibility criteria and optimization goals. Our decomposition method utilizes Problem statement and solution a skeleton of the shape and follows a dynamic approach. In this paper, we address one central problem of pro- Specifically, we restrict our cuts to certain line segments cessing samples with LCM. That is, not all dissected based on the skeleton. This not only results in simple cuts fragments can be successfully collected due to various but also in a flexible framework that allows to integrate possible circumstances. Besides technical reasons as for various criteria. example an incorrect focus of the laser, the main cause With this paper, we present a novel approach for the is assumed to be the size and morphology of the frag- automated decomposition of tissue samples with lim- ment. The fragments must not exceed certain limits of ited user-interaction. Unlike previous decomposition minimal or maximal size and should be of approximately methods used in the context of LCM, we placed a focus round shape. As the regions of interest (ROIs) in tissue on the morphological properties of the fragments. In Fig. 1 Polygon decomposition in a histopathological tissue sample. Top: Regions of interest are selected from a histopathological tissue sample (H&E-stained image of a subsequent sample on the top left) in which different tissue types have been identified using the method in [2]. Bottom: After a preprocessing, each connected component is given as a simple polygon without holes, which is then decomposed using the proposed skeleton-based approach S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 3 of 17 the experimental evaluation on lung tissue samples of morphology. We propose an algorithm for constrained patients with non-small-cell lung carcinoma, the pro- polygon decomposition using a skeleton-based approach. posed approach achieved a higher success rate and the amount of successfully collected tissue was increased by Skeletonization 10-30%. This paper is an extended version of the prelimi - Our approach is based on the medial axis or skeleton of nary work presented in [8]. the shape. The medial axis is defined as the set of points The paper is organized as follows: In "Related work" that have more than one closest point on the bound- section, we discuss related work on decomposition algo- ary of the shape. The medial axis was introduced for rithms. In "Method" section, we introduce our algorith- the description of biological shapes [20, 21] but is now mic framework and discuss different possible feasibility widely used in other applications such as object recog- criteria as well as optimization goals. In "Experimental nition, medical image analysis and shape decomposition results" section, we present experimental results and (see [10] for a survey). An important property is that the demonstrate the advantages of our method in compari- medial axis represents the object and its geometrical and son to a heuristic decomposition approach. We conclude topological characteristics while having a lower dimen- with a summary of our results and future improvements sion [22, 23]. in "Conclusion" section. Formally, the medial axis of a shape D is defined as the set of centers of maximal disks in D. A closed disk B ⊂ D is maximal in D if every other disk that contains B is not Related work contained in D. A point s is called skeleton point if it is Polygon decomposition is an important tool in compu- the center of a maximal disk B(s) (see Fig. 2). For a skel- tational geometry, as many algorithms work more effi - eton point s, we call the points where B(s) touches the ciently on certain polygon classes, for example convex boundary the contact points—every skeleton point has at polygons [9]. Moreover, polygon decomposition is fre- least two contact points. A skeleton S is given as a graph quently used in applications such as pattern recognition consisting of connected arcs S , which are called skeleton or image processing [9]. Object recognition, biomedi- branches and meet at branching points. Given a simple cal image analysis and shape decomposition are typical polygon without holes the skeleton is an acyclic graph. areas of application that utilize skeletons [10]. Skeletons There are various methods for the computation of the are oftentimes used to analyze the morphology of a given medial axis in practice [10]. In general, the medial axis is shape and work especially well on elongated structures, very sensitive to noise in the boundary of object. This is such as vessels [11], pollen tubes [12] or neuron images a problem that often occurs in digital images and leads [13, 14]. There are several shape decomposition meth - to spurious skeleton branches. Procedures that remove ods based on the skeleton or some other medial repre- these uninformative branches are known as pruning sentation of a shape. However, most of these methods methods. Pruning can be applied after skeletonization are designed for object recognition and thus focus on [24–26] or is included in the computation of the skeleton decomposing a shape into “natural” or “meaningful” [27–30]. For our application, we utilize the skeletoniza- parts [15–17]. In some approaches, even decompositions tion and pruning method of Bai et al. [29], which was with overlapping parts are allowed [18, 19]. None of the previously used for other bioimaging applications [12, established decomposition methods facilitate a straight- forward introduction of adjustable size and shape con- straints as needed for our application. We utilize the skeleton for two main reasons: it is well-established to represent shape morphology and has proved useful for shape decomposition. As cancerous tis- sue regions often present themselves as highly complex and ramified shapes, we apply the skeleton to obtain a morphological representation, based on which we com- pute a decomposition that includes the morphological features. Method To improve the success rate of LCM, a shape decomposi- Fig. 2 Medial axis of a simple shape. This medial axis consists of five branches connected by two branching points. The skeleton point tion method is needed that computes feasible fragments, s is the center of a maximal disk B(s) and has three contact points i.e. fragments that fulfill certain constraints in size and {c , c , c } on the boundary of the shape 1 2 3 Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 4 of 17 14]. This algorithm produces a discrete and pruned skel - eton, which consists of a finite number of skeleton pixels as our skeleton points. This is favorable for our practical application as we have a discrete input and a discrete out- put is expected. Furthermore, the computed skeleton has the property that every branching point has a degree of exactly three. Skeleton‑based polygon decomposition We consider the following problem: Given a simple poly- gon P, compute an optimal feasible decomposition of P. A decomposition is feasible if every subpolygon is feasi- ble, in the sense that it fulfills certain conditions on for instance its size and shape. We present an algorithmic framework that allows the integration of various criteria for both feasibility and optimization, which are discussed later. As for now, we only consider criteria that are locally evaluable. In our skeleton-based approach, we only allow cuts that are line segments between a skeleton point and its corresponding contact points. Thus, the complexity of our algorithm mainly depends on the number of skel- eton points rather than the number of boundary points of the polygon. Every subpolygon in our decomposition is generated by two or more skeleton points. We pre- sent two decomposition algorithms: One in which we restrict the subpolygons to be generated by exactly two skeleton points and a general method. In the first case, each subpolygon belonging to a skeleton branch can be Fig. 3 Domain decomposition lemma. The domain is decomposed decomposed on its own and in the second case the whole based on the contact points of skeleton point p. The partial skeletons share only p as a common point. All contact points of any other polygon is decomposed at once. skeleton point q are contained in exactly one of the connected components Decomposition based on linear skeletons First, we consider the restriction that the subpolygons are generated by exactly two skeleton points. In this case, Moreover, we have the corresponding skeleton points have to be on the same S(D ) ∩ S(D ) = p ∀ i �= j. i j skeleton branch S . In our computed skeleton, a branch- ing point belongs to exactly three branches and thus has Corollary 2 Let p ∈ S(D) and A , A , . . . , A be as 1 2 k three contact points. Each combination of two out of above. For each skeleton point q =p exists an i such that the three possible cut line segments corresponds to one all contact points of q are contained in A . of these branches. Due to the Domain Decomposition Lemma (see Fig. 3, proof in [22]) and the following corol- lary, we can decompose each skeleton branch on its own. Let S be a skeleton branch with a linear skeleton of size n and let P be the polygon belonging to this branch. By k k Theorem 1 (Domain Decomposition Lemma) Given a P (i, j) , we denote a subpolygon that is generated by two domain D with skeleton S(D), let p ∈ S(D) be some skele- skeleton points i and j on S (see Fig. 4). Thus, we have ton point and let B(p) be the corresponding maximal disk. P (1, n ) = P . First, we consider the decision problem, k k k Suppose A , A , . . . , A are the connected components of 1 2 k which can be solved by using dynamic programming. For D \ B(p) . Define D = A ∪ B(p) for all i. Then: i i each skeleton point i from n to 1, we determine X(i). X(i) k is True if there exists a feasible decomposition of the S(D) = S(D ). polygon P (i, n ) . This is the case if either i k k i=1 S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 5 of 17 we can combine those to obtain a decomposition of the entire polygon. This leads to the following result. Theorem 4 Given a simple polygon P with skeleton S consisting of n points, one can compute a feasible decom- position of P based on the skeleton branches of S in time O(n F ) , with F being a factor depending on the feasibility criteria. Fig. 4 Subpolygon induced by skeleton points. The polygon P belongs to the skeleton branch S consisting of the points 1 to n . k k Note that there might not exist a feasible decomposi- Two skeleton points i and j together with line segments to their tion of the entire polygon or for certain subpolygons. By corresponding contact points induce a subpolygon P (i, j) using this method, we are able to obtain partial decompo- sitions. Thus, this approach can be favorable in practice. General decomposition In the general setting, subpolygons are allowed to be gen- erated by more than two skeleton points. In this paper, Fig. 5 Decomposition based on linear skeletons. The polygon we will briefly explain the idea of our method (see [31] for P (i, n ) has a feasible decomposition if either polygon itself is feasible k k a more detailed description and the corresponding for- (left) or there exists a point j such that P (i, j) is feasible and P (j, n ) k k k has a feasible decomposition (right) mulas). Recall that our skeleton is an acyclic graph con- sisting of a finite number of vertices, i.e. skeleton points. The skeleton computed for our application (method of Bai et al. [29]) has the property that the maximal degree a) P (i, n ) is feasible or k k of a skeleton point is three. We represent the skeleton as b) there exists j > i such that P (i, j) is feasible and a rooted tree by selecting one branching point as the root P (j, n ) has a feasible decomposition. k k (see Fig. 6). Since branching points belong to three differ - ent branches, these nodes are duplicated in the skeleton This is illustrated in Fig. 5. By choosing optimal points j tree such that each node corresponds to the cut edges on during the computation, we can include different opti - the respective branch. Our method and its runtime are mization goals. If X(1) is True, the entire polygon has based on two main observations. a feasible decomposition, which can be computed via backtracking. Observation 5 The maximal number of skeleton points that can generate a subpolygon is equal to the number of Lemma 3 Given a subpolygon P with a linear skel- endpoints in the skeleton, i.e. the number of leaves in the eton S consisting of n points, one can compute a feasible k k skeleton tree. decomposition of P based on S in time O(n F ) , with F k k k being a factor depending on the feasibility criteria. Observation 6 Every subpolygon can be represented as the union of subpolygons generated by just two skeleton Proof We initialize X(n ) = True . For every skeleton points. point i, for i = n − 1 down to 1, we compute X(i) such that X(i) equals True if there exists a feasible decompo- sition of P (i, n ) . To compute X(i), we consider O(n ) k k k Let i be a node in the skeleton tree and T the subtree other values X(j) for i < j ≤ n and check in time O(F ) rooted in i. By P(i), we denote the subpolygon ending if the polygon P (i, j) is feasible. The correctness follows in the skeleton point i. This polygon corresponds to the inductively. subtree T in the given tree representation (see Fig. 7). For each node i (bottom-up), we compute if there exists The factor F is determined by the runtime it takes a feasible decomposition of the polygon P(i). Such a to decide whether a subpolygon is feasible. This fac - decomposition exists if either tor depends on for instance the number of points in the skeleton or in the boundary of the polygon. We discuss a) P(i) is feasible or examples in the following "Feasibility constraints and b) There exists a feasible polygon P ending in i and fea- optimization" section. After computing decompositions sible decompositions of the connected components for each subpolygon corresponding to a skeleton branch, of P(i) \ P . Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 6 of 17 Fig. 6 Tree representation of the skeleton graph. Representing the skeleton graph as a tree rooted at the point r. Every node in the tree represents a possible cut in the polygon. Therefore the branching points are duplicated to provide the cuts on the respective branches Fig. 7 A subpolygon and its corresponding subtree. The subpolygon P(i) ending in the skeleton point i is represented by the subtree T Thus, we have to consider all different combinations approach does not depend on the initial choice of the of skeleton points that together with i can form such root node. a polygon P . In a top-down manner, we consider the different combinations of nodes [i , i , . . . , i ] such that Theorem 7 Given a simple polygon P with skeleton S 1 2 l ′ ′ i ∈ T and T ∩ T =∅ for all j =j . The polygon P consisting of n points with degree at most three, one can j i i i ′ corresponds to the subtree rooted in i with i , i , . . . , i compute a feasible decomposition of P based on S in 1 2 l as the leaves, depicted in blue in Fig. 8. Note that we O(n F ) time, with k being the number of leaves in the skel- can compute P as a union of subpolygons iteratively. eton tree and F as above. We check if P is feasible and if we have feasible decom- positions for each P(i ) , meaning every subtree T (gray j i in Fig. 8). Because of Observation 5, we know that l ≤ k , Feasibility constraints and optimization for k being the number of leaves in the skeleton tree. The proposed polygon decomposition method is a versa - We have a feasible decomposition of the whole polygon tile framework that can be adjusted for different feasibil - if there exists one of the polygon P(r). This computation ity constraints and optimization goals. With regard to the dominates the runtime with the maximum number of application in LCM, we considered criteria based on size combinations to consider being in O(n ) . Note that this and shape. As stated before, it is assumed that the main S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 7 of 17 Fig. 8 Possible combination of skeleton points considered in the decomposition of a subpolygon. In the decomposition of a subpolygon P(i), different combinations of skeleton points in the subtree T are considered. The resulting subpolygons can be represented as subtrees (blue) in the tree representation spanning between nodes of these skeleton points cause for unsuccessful dissections lies in an incorrect size fragment into a collecting device. As the laser burns part or morphology of the considered fragments. In LCM, a of the boundary, the fragment has to have a certain mini- laser separates a tissue fragment from its surrounding mal size to ensure that enough material is supplied to be sample leaving a small connecting bridge as the impact analyzed. On the other hand, the size cannot be too large point of a following laser pulse, which catapults the or otherwise the force of the laser pulse does not suffice Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 8 of 17 for the transfer process. Furthermore, observations show optimization goal is finding a minimal decomposition that the dissection often fails due to an irregular shape of by minimizing the number of fragments (MinNum). We the fragment. Specifically, elongated shapes or fragments define X(i) as the number of subpolygons in an optimal with narrow regions (bottlenecks) seem to be problem- feasible decomposition of P(i, n), set X(n) = 0 and com- atic. The tissue can tear at these bottlenecks and is only pute X(i) = min X(j) + 1 . By minimizing the length of j∈I transferred partly or not at all because the laser pulse is cut edges (MinCut), shorter cuts are preferred and thus concentrated on only a small part of the boundary. preferably placed at bottlenecks in the polygon. Since In the following, we describe the implementation of every skeleton point i is the center of a maximal disk, we different constraints that we considered based on our can obtain the cut length by the corresponding radius r(i). application. For simplicity, we limit the description to We can define X(i) either as the length of the longest cut the decomposition algorithm for linear polygons, but or as the sum of cut lengths in an optimal decomposition all mentioned constraints can be applied to the general of P(i, n) and compute X(i) = max{min X(j), r(i)} or j∈I decomposition method as well. X(i) = min X(j) + r(i) . The runtime for both MinNum j∈I and MinCut is the same as for the decision problem. Fur- Feasibility constraints thermore, we considered maximizing the fatness (Max- For the size constraint, we restricted the area of the Fat) as an optimization goal. A decomposition is optimal subpolygons. Given two bounds l and u, a polygon P is if the smallest aspect ratio is maximized. We define x(i) feasible if l ≤ A(P) ≤ u , for A(P) being the area of the as the value of the smallest aspect ratio and compute polygon. One could also apply this constraint on the X(i) = max {min{X(j), AR(P(i, j))}} . Applying fatness j∈I number of boundary points instead of the area. We as a feasibility constraint or an optimization goal results implemented different shape constraints. On the one in the same runtime because both approaches require the hand, we considered approximate convexity. Then, a calculation of the aspect ratios of all subpolygons. polygon P is feasible if every inner angle lies between two given bounds. As this criterion does not prevent elon- Comparison of criteria gated shapes, we considered fatness instead. Fatness can Our algorithm facilitates the use of a wide range of feasi- be used as a roundness measurement and is defined by bility criteria and optimization goals, which can be com- the aspect ratio AR(P) of a polygon, which is the ratio bined with each other. Note that for certain combinations between its width and its diameter [32–34]. For a sim- other (faster) methods might exist. One example is find - ple polygon, the diameter is defined as the diameter of ing the minimal (MinNum) decomposition in which the the minimum circumscribed circle and the width as the area of the subpolygons is bounded. For polygons with diameter of the maximum inscribed circle. A polygon P linear skeletons this can be modeled as finding the mini - is called α-fat if AR(P) ≥ α . For the fatness constraint, we mal segmentation of a weighted trajectory (in O(n log n) define a polygon as feasible if it is α-fat for some given time [35]). For general polygons, this problem can be parameter α ∈ (0, 1] . Higher values of α result in frag- modeled as computing the minimal (l, u)-partition of a ments that are more circular and less elongated in shape. weighted cactus graph (in O(n ) time [36, 37]). For area as well as approximate convexity, we can com- The selection of constraints used for the algorithm pute the required values incrementally if the values for all obviously affects the resulting decomposition. The num - subsequent subpolygons are given beforehand. Therefore, ber of subpolygons as well as the position of cuts varies we can check the feasibility in constant time. u Th s, a fea - noticeably. Depending on the underlying application, sible decomposition using these criteria can be computed one might choose suitable constraints. In the following, in time O(n + m) for n being the number of skeleton we present decompositions for different combinations of points and m the number of boundary vertices. If the criteria and assess their suitability for our specific appli - fatness criterion is used, one has to calculate the aspect cation. The typical results are exemplified using a ROI ratio of each polygon, which takes O(m logm) time and polygon of a lung tissue sample (see Fig. 9). therefore results in a runtime of O(n m log m). Panel A and B in Fig. 9 illustrate the effect of the size criterion. Having a larger upper bound obviously results Optimization goals in fewer subpolygons. The MinNum optimization goal The algorithm computes the value X(i) for each skeleton minimizes the number of subpolygons, but the solutions i. For the decision problem, we defined X(i) to be True are not necessarily unique and one optimal decomposi- if there exists a feasible decomposition of the polygon tion is chosen arbitrarily. This can be observed at the P(i, n). With a redefinition of X(i), we can implement a bottom-most skeleton branch in both these decompo- variety of optimization goals. For a point i, let I be the sitions as the cuts in A would be feasible with the con- set of points j such that P(i, j) are feasible. One possible straints from B as well. In panel C and D of Fig. 9, the S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 9 of 17 Fig. 9 Decompositions based on different criteria. Exemplary decompositions obtained by applying the algorithm based on linear skeletons with different feasibility constraints and optimization goals. A area in [50, 300] , M inNum. B area in [50, 500] , M inNum. C area in [50, 300] , fatness ≥ 0.4, MinNum. Darea in [50, 300] , fatness ≥ 0.5, MinNum. E area in [50, 300] , M inCut. F area in [50, 300] , MaxF at fatness constraint was applied in form of a lower bound different optimization goal will not influence the amount on the aspect ratio of subpolygons. In comparison to the of area in the decomposition, but the quantity and posi- decomposition depicted in A, this criterion avoids the tions of cut edges may change considerably. These tendency towards elongated fragments. However, tighter changes are expected to affect the amount of successfully bounds do not necessarily result in better outcomes as a dissected tissue fragments in the microdissection. When feasible decomposition might not exist at all. This case is looking at the decompositions of the top left skeleton illustrated in panel D, where the algorithm did not find branch in those three polygons, one notices that the ones a feasible decomposition for the polygon parts that are in A and E have the same number of fragments, but with depicted in gray. For our application, this would not be MinCut a cut with a lower length is chosen. Maximizing favorable as it reduces the amount of extracted tissue the fatness usually results in a higher number of subpol- material. ygons. As can be seen in panel F, the resulting subpoly- We applied the different optimization goals denoted gons are less elongated and more circular in shape. We by MinNum (panel A), MinCut (panel E) and MaxFat expect these to be the desired shapes for our application. (panel F) with the same feasibility constraint. Choosing a Hence, we used the area constraint in combination with Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 10 of 17 the MaxFat optimization in our experiments and the fol- Both decomposition methods were applied with the lowing comparison of decomposition methods for LCM. same area bounds, namely a minimal and maximal area 2 2 of 100 px (ca. 1800 μm ) and 2800 px (ca. 50000 μm ) Experimental results respectively. The main focus of the development of a Experimental setup novel decomposition method lies in the reduction of tis- For the evaluation of our algorithms, we conducted LCM sue loss. This is determined by the amount of area in the experiments on shapes obtained from infrared micro- decomposition itself as well as the amount of successfully scopic images of 10 thin sections of FFPE (formalin-fixed dissected fragments. First, we compare the methods and paraffin-embedded) lung tissue samples from patients the resulting decompositions on a computational level. with non-small-cell lung carcinoma. The pixel spectra of Then, we analyze their performance in the practical set - the images were classified into different tissue types using ting with LCM. a random forest classifier as described in [2]. All pixel positions belonging to the tumor class were chosen as the Computational results regions of interest (ROI). The binary mask of the ROI was We examine the results of the MaxFat and BiSect decom- preprocessed by a morphological opening followed by a position on three different levels: fragments (subpoly - morphological closing and subsequent hole filling. Each gons), components (ROI polygons) and samples. We connected component of the preprocessed binary mask examine the size of the decompositions, the area loss and is given as a simple polygon on which we applied two dif- the morphology of the fragments. ferent decomposition approaches. The number of input polygons for our experiments ranged from 14 to 109 per Decomposition size sample with a total amount of 441. The resulting decom - The sampling consisted of 441 components with an aver - positions serve as the input for LCM. Each fragment is age area of 4500 px (ca. 81300 μm ). The MaxFat decom - transmitted in form of a circular list of discrete boundary position over all ten samples contained 4143 fragments points. with an average of 9.36 fragments per component. The For the experiment, we used our algorithm based BiSect decomposition consisted of considerably less frag- on linear skeletons such that each skeleton branch is ments with an average of 2.36 fragments per component decomposed separately. This follows the practical con - and 1089 for the entire sampling. sideration that a polygon as a whole may not possess a BiSect achieves a smaller decomposition size because feasible decomposition, while some individual branches most components did not require many bisections for the do. The resulting skeletons consisted of roughly 80 to fragments to fulfill the given area constraints (see Fig. 10 1000 points, involving around five to ten branches for A2, B2). With MaxFat, every skeleton branch in decom- each polygon to be decomposed, see Figs. 9, 10 for typi- posed individually. Therefore, most decompositions con - cal examples. We applied a size constraint as lower and sist of at least as many fragments as there are branches in upper bounds on the area of the subpolygons and com- the skeleton (see Fig. 10 A1, B1). puted an optimal decomposition in which the fatness, i.e. the minimum aspect ratio of the subpolygons, is maxi- Area loss mized. We denote this approach by MaxFat. Figure 11 depicts the area loss on the level of individual We compare our approach to a heuristic decomposi- components. The mean area loss with MaxFat is slightly tion method, which was used to decompose tissue sam- lower than the one with BiSect (MaxFat M = 8.96%, ples for LCM in previous work. As this method follows a BiSect M = 10.81%). However, the distribution of Max- bisection approach, we denote it by BiSect. Unlike Max- Fat shows a greater variability in values, a larger stand- Fat, this method includes merely a size constraint and no ard deviation and some high-loss outliers (MaxFat SD shape criterion or optimization goal. A polygon is decom- = 11.26, BiSect SD = 6.11). For BiSect, the variability is posed by recursive bisection if its area exceeds an upper lower and there are less outliers. On the level of samples, size bound. If the area of a (sub)polygon is below a given one can see that in 8 of 10 cases the area loss with Max- lower bound, it is discarded. Every bisection is designed Fat is lower than the one with BiSect (see Table 1). The to leave a strip of tissue behind such that each subpoly- decompositions with MaxFat contained up to 10% more gon retains contact to the surrounding membrane of the area. The area loss averages around 10.77% for MaxFat microscopic slide in order to meet a technical require- and 16.35% for BiSect. ment of the specific LCM system used in this study for Both methods inherently involve area loss. With BiSect, the dissection to be possible. The MaxFat decomposition area loss occurs due to the strips left behind by every does not include these strips because all subpolygons bisection. Therefore, the amount increases proportional intersect with the boundary of the input polygon. to the size of the components and the necessary cuts S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 11 of 17 Fig. 10 Exemplary decompositions with MaxFat and BiSect. Decompositions of four exemplary components from the tissue samples. Left: decomposition with MaxFat. Right: decomposition with BiSect Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 12 of 17 Table 1 Comparison of area loss for samples as the combined loss over all components for MaxFat and BiSect Sample 1 2 3 4 5 6 7 8 9 10 MaxFat 8.58 21.43 5.22 23.85 3.42 11.71 3.40 13.56 10.45 6.07 BiSect 10.62 18.04 13.44 23.41 13.47 19.02 13.00 14.59 18.48 19.46 area loss. This contributes to the higher standard devia - tion and outliers that were observable on the level of individual components. The results for entire samples suggest that the samples are dominated by components that cause small or no area loss when decomposed with MaxFat. Since the resulting fragments for each compo- nent in one tissue sample are collectively gathered, the quality of the decomposition should be assessed on the level of samples. Regarding area loss during decompo- sition, MaxFat generally achieved better results. How- ever, the quality of the methods is ultimately determined by their performance in practice and their success with LCM. Therefore, practical evaluations are necessary. Here, we expect the morphology of the fragments to be Fig. 11 Comparison of area loss for components. Distribution of a critical factor. the area loss (in%) in the decompositions of individual components contained in the sampling (n = 441). Comparison of MaxFat (M = 8.96%, SD = 11.26) and BiSect (M = 10.81%, SD=6.11) Morphology We compared both decomposition methods based on the resulting fatness, i.e. aspect ratio, of the fragments. This (see Fig. 10). With MaxFat, area is lost for each skeleton value measures the circularity of the shape. On the level branch for which a feasible decomposition did not exist. of individual fragments, the aspect ratios in BiSect pre- This mainly occurs if the corresponding (sub)polygon is sent themselves in the pattern of a normal distribution either too slim or too wide. The first case is depicted in whereas the distribution for MaxFat is clearly left-skewed panel A1 of Fig. 10: Because the area of the gray polygons (see Fig. 12). The average aspect ratio of fragments over belonging to the bottom two branches was below the all samples is considerably higher with MaxFat (MaxFat given lower bound, a feasible decomposition did not exist M = 0.58, BiSect M = 0.39). We observe similar results and their area was lost. This can be attributed to short - when considering the average aspect ratio in the decom- comings of the underlying skeleton pruning method. positions of components (see Fig. 13). The values for Improving the pruning of the skeleton may avoid such MaxFat are larger and the variability is smaller (MaxFat short branches. The second case of too wide shapes is M = 0.58, BiSect M = 0.36). The standard deviation for exemplified in panel D1. If the upper area bound is rela - MaxFat is half as high as the one of BiSect (MaxFat SD tively small, the MaxFat decomposition of a wide shape = 0.05, BiSect SD = 0.1). For over 75% of components, leads to either thin-slicing or no feasible solution at all. the average fatness in the decompositions computed with This is due to our definition of the cut edges, which do MaxFat was higher than 0.5, whereas with BiSect nearly not allow internal decompositions. This also illustrates 75% have an average fatness lower than 0.4. that our approach is tailored towards more complex, BiSect applies only a size constraint and a compo- ramified shapes rather than fat objects. It is noteworthy nent is only decomposed if its area exceeds the given that the polygon depicted in panel D1/D2 of Fig. 10 cov- upper bound. Therefore, many components are not ers an area of around 43,000 px (ca. 7,77,000 μm ) and decomposed, but their shape is oftentimes elongated thus represents a huge outlier in our sampling. and ramified as can be seen in panel A2 of Fig. 10. These observations coincide with the presented results. Because this method follows a bisection approach, the For MaxFat, the first cause of area loss might occur fre - cut placement creates fragments with irregular shapes quently but merely contributes a small value to the over- and narrow bottlenecks (see Fig. 10 C2). The exception all loss. The second cause does not appear as often in the can be observed in large, round components as their samples because the average area of the components is decomposition resembles a grid pattern (see panel D2 fairly small, but obviously results in a large amount of of Fig. 10). In this case, the resulting fragments achieve S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 13 of 17 Fig. 12 Comparison of the fatness for fragments. Distribution of aspect ratios of individual fragments contained in the decompositions with MaxFat in A (n = 4151, M = 0.58, SD = 0.1) and BiSect in B (n = 1089, M = 0.39, SD = 0.11) MaxFat consistently obtains such fragments, we sus- pect this to be the main advantage of this decomposi- tion method in the practice. Hence, we expect a higher success rate in the practical application with LCM. Running times The computations were executed on a Windows PC (Intel Core i5-8600 CPU, 16 GB RAM). The proposed approach consists of a skeletonization and a subsequent decomposition with MaxFat. The average computation time for one input polygon was 0.63 s for the skeletoni- zation, 14.38 s for the decomposition with MaxFat and 0.43 s for the decomposition with BiSect. When consid- Fig. 13 Comparison of the average fatness for components. ering median values, we see that one for MaxFat (0.24 Distribution of average aspect ratios in the decompositions of all individual components contained in the sampling (n = 441). s) is lower than the one for BiSect (0.47 s). This suggests Comparison of MaxFat (M = 0.58, SD = 0.05) and BiSect (M = 0.36, that MaxFat can perform very fast on the majority of SD = 0.1) inputs but more slowly on others. In general, the run- ning time for BiSect fairly low, as many polygons are not decomposed at all. For MaxFat, on the other hand, a higher fatness. The results suggest that without the the time complexity depends on the number of bound- application of some shape criterion the BiSect decom- ary points as well as the number of skeleton points. position does not naturally result in fragments of large We looked at one sample in more detail. Sample 7 fatness. MaxFat, on the other hand, utilizes both size consisted of 63 polygons with different boundary (M and shape criteria and tries to maximize the fatness of = 274.63, Min = 131, Max = 1104) and skeleton (M a decomposition. This results in smaller fragments that = 152.55, Min = 81, Max = 507) sizes. Therefore, the are less elongated and rounder in shape (see Fig. 10). MaxFat decomposition showed a variation in runtimes The computational evaluation reveals that MaxFat con - (M = 1.77 s, Min = 0.21 s, Max = 29.57 s). The runt - sistently obtains higher fatness values. This strengthens imes for BiSect were consistent (M = 0.47 s, Min = our choice to include the fatness criterion in the opti- 0.47 s, Max = 0.5 s). In total, the decomposition with mization goal rather than the feasibility constraints. BiSect required 35.62 s and resulted in 103 fragments. Even without applying a strict bound on the fatness, The proposed approach was performed in ca. 3.52 min we were able to achieve high fatness values without the (1.69 min skeletonization and 1.83 min decomposition) risk of area loss due to the non-existence of a feasible and resulted in 487 fragments. Note that the runtime of decomposition. The success of a dissection using LCM MaxFat can be optimized by parallelizing the execution depends on the size and morphology of the tissue frag- of the algorithm not only on the different polygons in ment. We hypothesize that approximately round shapes one sample but also on the different skeleton branches. have a higher chance to be successfully collected. As Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 14 of 17 Practical results with LCM view) after the laser pulse. One might only conjecture the The practical evaluation of both shape decomposition reasons: As mentioned before, the size and shape of the approaches consisted of the dissection of all computed fragments, the focus of the laser pulse as well as its posi- fragments with LCM. Because one tissue sample cannot tion on the boundary of the fragment affect the transfer - be dissected twice, the experiment was performed on ring process and its success. The last category “too small” empty microscopic slides. Therefore, it was not possible contains fragments of such a small size that the laser did to compare the amount of successfully dissected tissue by not leave enough material to be collected. measuring for example the protein content. The evalua - Figure 14 depicts the distribution of assigned labels tion was restricted to visual assessment. based on the number of fragments in each category. In both decompositions, the majority of fragments was Classification of dissected fragments labeled as successful. The amount of successfully dis - The dissection of each fragment was observed and classi - sected fragments of MaxFat is consistently over 90% fied into the following categories. A fragment was labeled for all samples. The distribution for BiSect shows more “successful” if it disappeared from the field of view after variation between the samples. On average, 95.44% of the the laser pulse. In this case, we expect it to be success- fragments of MaxFat were successful, 4.34% were labeled fully transferred into the collecting device. Unsuccessful as fallen and merely 0.22% as torn. None of the fragments fragments were further divided into three categories. The were too small. BiSect averages around 80.98% successful, label “torn” describes fragments that tore during the dis- 14.99% fallen, 2.39% torn and 1.64% too small fragments. section. Because they were only partially transferred, the collected area is not measurable. A fragment was labeled Area loss and success rates “fallen” in the following two cases. The fragment fell Tables 2, 3 show the success rates of MaxFat and BiSect before the transferring process, which might be the case with regard to tissue yield, the results are also visualized if all connections to the surrounding membrane were in Fig. 15. We distinguish two different success rates. The already severed before the laser pulse. The other case cov - success rate of the microdissection (Table 2) represents ers fragments that fell back onto the slide (in the field of the ratio of the area of the fragments that was successfully Fig. 14 Distribution of labels assign during LCM. Comparison of the percentage of labels assigned to fragments during laser capture microdissection for MaxFat (A) and BiSect (B) Table 2 Comparison of microdissection success rates for MaxFat and BiSect sample 1 2 3 4 5 6 7 8 9 10 MaxFat 90.00 92.72 98.05 96.58 94.77 95.74 97.04 95.75 96.49 97.43 BiSect 80.39 76.61 73.02 81.79 81.79 79.19 81.47 81.09 85.10 95.80 The LCM success rate (in%) describes the amount of tissue area that was collected from the fragments by LCM S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 15 of 17 Table 3 Comparison of overall success rates for MaxFat and BiSect sample 1 2 3 4 5 6 7 8 9 10 MaxFat 82.28 72.85 92.93 73.54 91.53 84.53 93.74 82.77 86.41 91.52 BiSect 71.85 62.79 63.21 62.64 70.78 64.13 70.88 69.26 69.38 77.16 This success rate (in%) represents the overall tissue yield from as sample by combining the LCM success rate with the area loss during the decomposition of components, i.e. how much area of the original ROI is contained in the computed fragments at least 10% higher. In sample 3, the amount of lost tis- sue could potentially be decreased by 29.72% when using MaxFat rather than BiSect. On average, the tissue yield with the proposed decomposition approach is 17.55% higher. The practical evaluation confirms our conjecture that the proposed decomposition method performs better in practice than the heuristic bisection approach. The amount of successfully dissected fragments is consist- ently higher with the MaxFat approach. With BiSect, this rate varies more noticeably and the algorithm was not able to filter out fragments that did not fulfill the lower area constraint. Besides the quantity of successfully dis- sected fragments, the tissue area that was collected with MaxFat was larger as well. Together with the smaller area loss in our decomposition, which we observed in the computational assessment, the proposed method proved to minimize the tissue loss considerably. When used on Fig. 15 Comparison of success rates. Comparison of the amount of actual tissue samples, our decomposition method will tissue loss and the overall success rates of MaxFat and BiSect. With increase the tissue yield and thus the amount of protein 100% being the ROI area in the sample, the two lighter bar segments or DNA available for further analysis. correspond to the percentage of tissue loss in the decomposition and the microdissection, respectively. The darkest segment represents the percentage of the original area that was successfully collected during Conclusion the microdissection. This value represents the overall success rate In this paper, we presented a skeleton-based decompo- sition method for simple polygons as a novel approach to decompose disease-specific regions in tissue samples while aiming to optimize the amount of tissue obtained collected as computed by the LCM system. Over all ten by laser capture microdissection (LCM). The lack of samples, the values for MaxFat are higher than for BiSect. previous benchmark methods and results is somewhat The largest difference can be observed for sample 3 with remarkable. It indicates that previous studies utilizing a value of 25.03%. Overall, the success rate for the micro- LCM relied on manual decomposition of the regions to dissection averages at 95.46% for MaxFat and 80.99% for be dissected, which is clearly impractical in clinical study BiSect. Using these percentages, we calculated the overall settings involving dozens or hundreds of samples. As the success rate (Table 3) of both decomposition methods by first fully automated approach, we provide a conceptual combining the following factors: The ROI area contained contribution that may pave the way for making LCM in the samples (combined area of all components), the feasible in large clinical studies. Our approach will also amount of area lost due to the decomposition algorithm facilitate systematic assessment of optimal size and mor- and lastly the success rate of the microdissection. For phology criteria for LCM experiments, which would be example, the MaxFat decomposition resulted in 8.58% difficult if not impossible to conduct based on manual area loss for sample 1 (see Table 1). Thus, 91.42% of the original area was contained in the fragments for LCM. shape decomposition. The microdissection showed a success rate of 90%, which Size and morphology of the fragments are assumed to results in an overall success rate of 82.28%. This means be the key factors that influence the success of dissec - that 82.28% of the tissue contained in sample 1 could be tions using LCM. Our approach is designed to minimize collected using the MaxFat decomposition approach. For tissue loss by utilizing a size constraint and optimizing all ten samples, the overall success rate of MaxFat was the shapes towards fat or circular fragments. As we Selbach et al. Algorithms Mol Biol (2021) 16:15 Page 16 of 17 Abbreviations demonstrated, this translates into practice when com- LCM: Laser capture microdissection; ROI: Region of interest; H[MYAMP: E-stain] paring our approach to a recursive bisection method Hematoxylin and eosin stain; FFPE: Formalin-fixed paraffin-embedded; M: that is currently used and only applies a size constraint. Arithmetic mean; SD: Standard deviation. Our approach is tailored towards complex morpho- Acknowledgements logical structures that are commonly found in can- We acknowledge support by the Open Access Publication Funds of the Ruhr- cerous tissue and are usually the most challenging to Universität Bochum. Moreover, we gratefully acknowledge Nina Goertzen for providing infrared microscopic imaging data for our study. dissect using LCM without major loss of tissue mate- rial. Not surprisingly, the algorithm does not perform Authors’ contributions as well when decomposing relatively large and fat AM proposed the project. LS, MB and AM developed the method. LS imple- mented the method. TK carried out the microdissection supervised by KG. LS shapes. However, such shapes do not occur frequently analyzed the experimental results. LS, MB and AM participated in writing the and the impact on the overall success seems to be mini- manuscript. All authors read and approved the final manuscript. mal. These shapes can be easily decomposed using Funding simple approaches, e.g. the bisection-based approach, Open Access funding enabled and organized by Projekt DEAL. This work was without major tissue loss. Thus, we plan to improve our supported by the Ministry for Culture and Science (MKW ) of North Rhine- method by distinguishing these shapes and compute Westphalia (Germany) through grant 111.08.03.05-133974 and the Center for Protein Diagnostics (PRODI). their decompositions separately. The implementation of our approach relies on a (dis - Availability of data and materials crete) skeletonization of the underlying polygons. Spe- Source code and all shapes from our validation samples as well as computa- tional results are available from https:// github. com/ Lahut ar/ Skele ton- based- cifically, we utilize the approach of Bai et al. [30], which Shape- Decom posit ion. git. uses a heuristic pruning approach. While other high- quality implementations of discrete skeletonization Declarations algorithms exist [38, 39], the approaches lack pruning strategies that are essential for our approach to pro- Consent for publication Not applicable. duce practically relevant results. It is likely that recent improvements for skeletonization and pruning will fur- Competing interests ther improve our results. For example, recent methods The authors declare that they have no competing interests. of Durix et al. [27, 28] promise to avoid short and other Author details spurious branches, which contributed to area loss in Department of Computer Science, Faculty of Mathematics, Ruhr University our decomposition. More broadly, concepts for robust Bochum, Bochum, Germany. Department of Biophysics, Faculty of Biology and Biotechnology, Ruhr University Bochum, Bochum, Germany. Bioinformat- skeletonizations have been proposed based on the ics Group, Faculty of Biology and Biotechnology, Ruhr University Bochum, -medial axis [25, 40], which are built on solid theoreti- Bochum, Germany. Center for Protein Diagnostics, Ruhr University Bochum, cal grounds and thus may provide useful concepts for Bochum, Germany. further improved shape decomposition approaches. Received: 3 February 2021 Accepted: 21 June 2021 Here, skeletonization is considered as an input to our decomposition method. While better and more robust skeletonization may further improve performance, a detailed investigation is beyond the scope of our pre- References sent study. 1. Emmert-Buck MR, Bonner RF, Smith PD, Chuaqui RF, Zhuang Z, Gold- stein SR, Weiss RA, Liotta LA. Laser capture microdissection. Science. The practical evaluation with LCM showed the 1996;274(5289):998–1001. advantage of using the proposed decomposition 2. Großerueschkamp F, Bracht T, Diehl HC, Kuepper C, Ahrens M, Kallenbach- method. Over the entire sampling, our decompositions Thieltges A, Mosig A, Eisenacher M, Marcus K, Behrens T, et al. Spatial and molecular resolution of diffuse malignant mesothelioma heterogeneity contained more successful fragments and we achieved by integrating label-free ftir imaging, laser capture microdissection and a success rate of 95% in the dissection. In combination proteomics. Sci Rep. 2017;7(1):1–12. with the lower area loss during the decomposition, we 3. Kondo Y, Kanai Y, Sakamoto M, Mizokami M, Ueda R, Hirohashi S. Genetic instability and aberrant DNA methylation in chronic hepatitis and were able to minimize the overall tissue loss for all sam- cirrhosis-a comprehensive study of loss of heterozygosity and microsatel- ples and increased the tissue yield by 10–30%. Over- lite instability at 39 loci and dna hypermethylation on 8 cpg islands in all, our work contributes to further optimization and microdissected specimens from patients with hepatocellular carcinoma. Hepatology. 2000;32(5):970–9. automation of LCM and thus promises to contribute to 4. Datta S, Malhotra L, Dickerson R, Chaffee S, Sen CK, Roy S. Laser capture the further maturing of the technology and enhancing microdissection: Big data from small samples. Histol Histopathol. its suitability for systematic use in larger scale clinical 2015;30(11):1255. 5. Ong C-AJ, Tan QX, Lim HJ, Shannon NB, Lim WK, Hendrikson J, Ng WH, Tan studies. JW, Koh KK, Wasudevan SD, et al. An optimised protocol harnessing laser capture microdissection for transcriptomic analysis on matched primary and metastatic colorectal tumours. Sci Rep. 2020;10(1):1–12. S elbach et al. Algorithms Mol Biol (2021) 16:15 Page 17 of 17 6. Vu QD, Graham S, Kurc T, To MNN, Shaban M, Qaiser T, Koohbanani NA, 24. Shaked D, Bruckstein AM. Pruning medial axes. Comput Vision Image Khurram SA, Kalpathy-Cramer J, Zhao T, et al. Methods for segmenta- Understand. 1998;69(2):156–69. tion and classification of digital microscopy tissue images. Front Bioeng 25. Chazal F, Lieutier A. The “ -medial axis’’. Graph Models. 2005;67(4):304–31. Biotechnol. 2019;7:188. 26. Sud A, Foskey M, Manocha D. Homotopy-preserving medial axis sim- 7. Wang S, Rong R, Yang DM, Fujimoto J, Yan S, Cai L, Yang L, Luo D, Behrens plification. In: Proceedings of the 2005 ACM symposium on solid and C, Parra ER, et al. Computational staining of pathology images to study physical modeling. 2005, p 39–50 the tumor microenvironment in lung cancer. Cancer Res. 2020;7:548. 27. Durix B, Chambon S, Leonard K, Mari J.-L, Morin G. The propagated skel- 8. Selbach L, Kowalski T, Gerwert K, Buchin M, Mosig A. Shape Decomposi- eton: a robust detail-preserving approach. In: International conference tion Algorithms for Laser Capture Microdissection. In: 20th International on discrete geometry for computer imagery. Berlin: Springer; 2019, p workshop on algorithms in bioinformatics ( WABI 2020). Leibniz interna- 343–354 tional proceedings in informatics (LIPIcs), 2020, p 13:1–13:17. 28. Durix B, Morin G, Chambon S, Mari J-L, Leonard K. One-step compact 9. Keil JM. Polygon decomposition. Handbook of computational geometry. skeletonization. Eurographics. 2019;5:21–4. Berlin: Springer; 2000. p. 491–518. 29. Bai X, Latecki LJ, Liu W-Y. Skeleton pruning by contour partitioning 10. Saha PK, Borgefors G, di Baja GS. A survey on skeletonization algorithms with discrete curve evolution. IEEE transactions on pattern analysis and and their applications. Pattern Recogn Lett. 2016;76:3–12. machine intelligence. 2007;29:3. 11. Drechsler K, Laura CO. Hierarchical decomposition of vessel skeletons for 30. Bai X, Latecki L.J. Discrete skeleton evolution. In: International workshop graph creation and feature extraction. In: 2010 IEEE international confer- on energy minimization methods in computer vision and pattern recog- ence On Bioinformatics and biomedicine (BIBM). 2010, p 456–61. nition. Springer; 2007, p 362–374. 12. Wang C, Gui C-P, Liu H-K, Zhang D, Mosig A. An image skeletonization- 31. Buchin M, Mosig A, Selbach L. Skeleton-based decomposition of simple based tool for pollen tube morphology analysis and phenotyping F. J polygons. In: Abstracts of 35th European workshop on computational Integr Plant Biol. 2013;55(2):131–41. geometry. 2019. 13. Narro ML, Yang F, Kraft R, Wenk C, Efrat A, Restifo LL. Neuronmetrics: Soft- 32. Efrat A, Katz MJ, Nielsen F, Sharir M. Dynamic data structures for fat ware for semi-automated processing of cultured neuron images. Brain objects and their applications. Comput Geomet. 2000;15(4):215–27. Res. 2007;1138:57–75. 33. Damian M. Exact and approximation algorithms for computing optimal 14. Schmuck MR, Temme T, Dach K, de Boer D, Barenys M, Bendt F, Mosig A, fat decompositions. Comput Geomet. 2004;28(1):19–27. Fritsche E. Omnisphero: a high-content image analysis (HCA) approach 34. Sui W, Zhang D. Four methods for roundness evaluation. Phys Proc. for phenotypic developmental neurotoxicity (DNT ) screenings of orga- 2012;24:2159–64. noid neurosphere cultures in vitro. Archiv Toxicol. 2017;91:4. 35. Alewijnse S.P.A, Buchin K, Buchin M, Kölzsch A, Kruckenberg H, Westen- 15. Reniers D, Telea A. Skeleton-based hierarchical shape segmentation. In: berg M.A. A framework for trajectory segmentation by stable criteria. In: SMI’07. IEEE international conference on shape modeling and applica- Proceedings of the 22nd ACM SIGSPATIAL international conference on tions, 2007, p 179–188. advances in geographic information systems. SIGSPATIAL ’14. 2014, p 16. Leonard K, Morin G, Hahmann S, Carlier A. A 2d shape structure for 351–360 decomposition and part similarity. In: 2016 23rd international conference 36. Buchin M, Selbach L. Decomposition and partition algorithms for tissue on pattern recognition (ICPR). IEEE. 2016, p 3216–3221 dissection. In: Computational geometry: young researchers forum. 2020, 17. Papanelopoulos N, Avrithis Y, Kollias S. Revisiting the medial axis for p 24–27 planar shape decomposition. Comput Vision Image Understand. 37. Buchin M, Selbach L. Partitioning algorithms for weighted cactus graphs, 2019;179:66–78. 2020;.arXiv: 2001. 00204 v2 18. di Baja GS, Thiel E. (3, 4)-weighted skeleton decomposition for pattern 38. Cacciola F. A CGAL implementation of the straight skeleton of a simple representation and description. Pattern Recogn. 1994;27(8):1039–49. 2D polygon with holes. In: 2nd CGAL user workshop. 2004, p 1 19. Tănase M, Veltkamp RC. Polygon decomposition based on the straight 39. Fabri A, Giezeman G.-J, Kettner L, Schirra S, Schönherr S. On the design line skeleton. Geometry morphology, and computational imaging. Berlin: of CGAL a computational geometry algorithms library. Softw Pract Exp. Springer; 2003. p. 247–68. 2000;30(11):1167–202. 20. Blum H. A transformation for extracting new descriptors of shape. Models 40. Chaussard J, Couprie M, Talbot H. Robust skeletonization using the Percep Speech Visual Forms. 1967;1967:362–80. discrete -medial axis. Pattern Recogn Lett. 2011;32(9):1384–94. 21. Blum H. Biological shape and visual science (part I). J Theor Biol. 1973;38(2):205–87. Publisher’s Note 22. Choi H.I, Choi S.W, Moon H.P. Mathematical theory of medial axis trans- Springer Nature remains neutral with regard to jurisdictional claims in pub- form. Pac J Math. 1997;181(1):57–88. lished maps and institutional affiliations. 23. Wolter F-E. Cut locus and medial axis in global shape interrogation and representation. Sea Grant College Program: Massachusetts Institute of Technology; 1993. Re Read ady y to to submit y submit your our re researc search h ? Choose BMC and benefit fr ? Choose BMC and benefit from om: : fast, convenient online submission thorough peer review by experienced researchers in your ﬁeld rapid publication on acceptance support for research data, including large and complex data types • gold Open Access which fosters wider collaboration and increased citations maximum visibility for your research: over 100M website views per year At BMC, research is always in progress. Learn more biomedcentral.com/submissions

Journal

Algorithms for Molecular Biology – Springer Journals

Published: Jul 8, 2021

Keywords: Laser capture microdissection; Shape decomposition; Skeletonization

Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Shape decomposition algorithms for laser capture microdissection

Shape decomposition algorithms for laser capture microdissection

Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Shape decomposition algorithms for laser capture microdissection

Shape decomposition algorithms for laser capture microdissection

References (40)

Abstract

Journal

Recommended Articles

There are no references for this article.

Our policy towards the use of cookies