An Iterative Method for Estimating Nonlinear Elastic Constants of Tumor in Soft Tissue from Approximate Displacement Measurements

Maryam Mehdizadeh Dastjerdi; Ali Fallah; Saeid Rashidi

doi:10.1155/2019/2374645

An Iterative Method for Estimating Nonlinear Elastic Constants of Tumor in Soft Tissue from Approximate Displacement Measurements

Dastjerdi, Maryam Mehdizadeh;Fallah, Ali;Rashidi, Saeid 2019-01-06 00:00:00 Hindawi Journal of Healthcare Engineering Volume 2019, Article ID 2374645, 12 pages https://doi.org/10.1155/2019/2374645 Research Article An Iterative Method for Estimating Nonlinear Elastic Constants of Tumor in Soft Tissue from Approximate Displacement Measurements 1 1 2 Maryam Mehdizadeh Dastjerdi , Ali Fallah , and Saeid Rashidi Department of Biomedical Engineering, Amirkabir University of Technology (AUT), Tehran 15875-4413, Iran Faculty of Medical Sciences and Technologies, Science and Research Branch, Islamic Azad University, Tehran 1477893855, Iran Correspondence should be addressed to Ali Fallah; afallah@aut.ac.ir Received 24 March 2018; Revised 24 June 2018; Accepted 12 July 2018; Published 6 January 2019 Academic Editor: Zahid Akhtar Copyright © 2019 Maryam Mehdizadeh Dastjerdi et al. ,is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Objectives. Various elastography techniques have been proﬀered based on linear or nonlinear constitutive models with the aim of detecting and classifying pathologies in soft tissues accurately and noninvasively. Biological soft tissues demonstrate behaviors which conform to nonlinear constitutive models, in particular the hyperelastic ones. In this paper, we represent the results of our steps towards implementing ultrasound elastography to extract hyperelastic constants of a tumor inside soft tissue. Methods. Hyperelastic parameters of the unknown tissue have been estimated by applying the iterative method founded on the relation between stress, strain, and the parameters of a hyperelastic model after (a) simulating the medium’s response to a sinusoidal load and extracting the tissue displacement ﬁelds in some instants and (b) estimating the tissue displacement ﬁelds from the recorded/simulated ultrasound radio frequency signals and images using the cross correlation-based technique. Results. Our results indicate that hyperelastic parameters of an unidentiﬁed tissue could be precisely estimated even in the conditions where there is no prior knowledge of the tissue, or the displacement ﬁelds have been approximately calculated using the data recorded by a clinical ultrasound system. Conclusions. ,e accurate estimation of nonlinear elastic constants yields to the correct cognizance of pathologies in soft tissues. information related to the tissue acoustic impedance, vas- 1. Introduction cular ﬂow, and tissue mechanical characteristics or variables According to the World Health Organization report, cancer such as its stiﬀness or strain, respectively, have been ex- is one of the principal morbidity and mortality agents tensively utilized for medical diagnoses [1]. ,e US imaging throughout the world, with approximately 8.8 million deaths is recognized a noninvasive, safe, easy-to-use, low-cost, and widely accessible imaging modality for visualizing in vivo (nearly 1 in 6 deaths) in 2015 and 70% increase in the tissues. Elastography approach has currently been regarded number of new cases over the next two decades. ,e cancer a promising alternative to invasive medical procedures, for statistics imply the requisite to extend medical scrutiny to example, the biopsy, to characterize tissue abnormalities. improve cancer prevention, early correct diagnosis, metic- ,e wide variety of strategies that are being employed to ulous screening, and eﬀective treatment and reduce the quantify and image mechanical properties of biological invasiveness and costs of applied techniques. tissues are recognized as elastography or elasticity imaging Since the ﬁrst introduction of ultrasound (US) imaging techniques with reference to their similar premise [2]: in clinical practice in the 1970s, ultrasonography and other US modalities, for example, Doppler imaging and state-of- (1) ,e in vivo tissue is being deformed by a speciﬁed the-art elastography imaging methods, which provide the external or internal load or motion. 2 Journal of Healthcare Engineering (2) ,e response of tissue is being recorded by the use of constitutive equations, into consideration in order to provide a standard clinical imaging system, such as the US or the possibility to minutely describe the behavioral charac- magnetic resonance imaging (MRI) system. teristics of materials. Constitutive theories in continuum mechanics deal with formulating material models that are (3) ,e mechanical characteristics of tissue are esti- [18, 19] mated through the assessment of tissue displacement ﬁelds. (a) on the basis of some mechanical universal principles ,e alterations in the microstructure of tissue as a conse- (b) in accordance with experimental observations quence of pathophysiological phenomena would change the ,oughtful consideration of soft tissue’s model would result mechanical properties of tissue; for instance, the increase in the in the realistic prediction of its behavior such that it could be stromal density of cancerous tissue would cause the increment veriﬁed by experimental observations. Nonlinear constitu- in its Young’s modulus [3, 4]. ,e outcomes of numerous tive manners that have been observed from soft tissues in experimental research studies carried out by Krouskop et al. numerous in vivo and ex vivo experimental research studies [5], Samani et al. [6, 7], Lyshchik et al. [8], Soza et al. [9], Hoyt could be modeled by the use of hyperelastic models et al. [10], Schiavone et al. [11], O’Hagan et al. [12], and Moran [15, 16, 20]. et al. [13], to mention but a few, have conﬁrmed the relation ,e hyperelastic constitutive laws deal with modeling between tissue structures and macroscopic mechanical features materials with nonlinear elastic behaviors in reaction to large which are being evaluated, quantiﬁed, and/or imaged by strains. ,e nonlinearities that are the consequences of (a) employing the palpation, elastography, digital rectal exami- the material behavior and (b) the signiﬁcant change in the nation, and such like methods. shape of material are both regarded in the constitutive ,e precise determination of mechanical characteristics of theory of hyperelastic materials. Hyperelastic materials are understudy tissue by the use of an elastography technique generally described by speciﬁc forms of strain energy density would undoubtedly necessitate realistically appraising or (stored energy) functions. While characterizing the homo- modeling the tissue manners, speciﬁcally its nonlinear re- geneous material’s absorbed energy due to its deformation, sponse to the stimulation. Hyperelasticity theory is one of the the strain energy function, W, is deﬁned as a function of constitutive theories that have been manipulated to model the deformation gradient, F [21, 22] nonlinear constitutive demeanor demonstrated by biological soft tissues. A variety of hyperelastic models, for instance the W � W(F). (1) well-known Neo-Hookean, Mooney–Rivlin, Yeoh, and Polynomial models, have been recommended for this purpose It is considered that B and B represent, respectively, the [14–17]. In comparison with the studies involving the linear reference or undeformed conﬁguration, which refers to the elasticity imaging, the number of research studies conducted situations where no load is exerted to the material and the to image the nonlinear features of tissues is limited [2]. deformed conﬁguration, which is relevant to the situations With the aim of diagnosing a tumor inside the un- where the material is under load and therefore it may alter derstudy tissue correctly, we have utilized an iterative with time, t. In addition, it is assumed that X and x, re- method, as explicated in the next section, to accurately spectively, correspond to the position vectors of a material estimate the Mooney–Rivlin hyperelastic parameters of the point in the reference and deformed conﬁgurations, B and tumor inside the tissue. ,e displacement ﬁelds inside the B. ,e time-dependent deformation of material, that is, the tumor have been analyzed to extract its hyperelastic pa- motion of material point, from B to B could be described by rameters. An iterative technique has been employed since it the function χ, which, for each t, (a) is an invertible function has been assumed that no initial knowledge of the tumor and (b) satisﬁes proper regularity conditions as follows [23]: was accessible except the displacement ﬁelds inside the x � χ(X, t). (2) tumor. ,e displacement ﬁelds inside the tumor have been extracted from the simulated/recorded radio frequency ,e deformation gradient tensor, F, is deﬁned as (RF) signals using the cross correlation-based method. ,e response of an abnormal tissue to a sinusoidal load (with F � Gradx, (3) low frequency to negate the inertia) has been simulated by with Cartesian components applying the ﬁnite element (FEM) software package ABAQUS. ,e RF signals have been simulated by the use of zx F � i, α ∈ {1, 2, 3}, (4) the Field II US Simulation Program. In brief, in this paper, iα zX we scrutinize the diagnosis of tumor through its hypere- lastic parameters in the conditions where there is no pri- where Grad, x , and X refer to the gradient operator in the i α mary perception of the tumor and the displacement ﬁelds conﬁguration B and components of x and X, respectively, inside the tumor are estimated imprecisely. while the general convention, J ≡ det F > 0, (5) 2. Materials and Methods is satisﬁed. Due to the local invertibility of deformation, F 2.1. Hyperelasticity 3eory. Constitutive theories take the should be nonsingular. ,e unique polar decomposition of F improvement of mathematical models, also known as is deﬁned as Journal of Healthcare Engineering 3 F � RU � VR, (6) I � tr(F) � F + F + F , 1 11 22 33 where the tensor R is an appropriate orthogonal tensor and 1 I � 􏼐F F − F F 􏼑, (13) 2 ij ij ii jj the tensors U and V are symmetric positive-deﬁnite tensors known as the right and left stretch tensors, respectively. Equation (7) represents the spectral decompositions of I � det(F) � J, tensors U and V: for an unconstrained isotropic elastic material. ,e left Cauchy-Green deformation tensor, B, and its principal (i) (i) U � 􏽘 λ u ⊗ u , λ > 0, i ∈ 1, 2, 3 , { } invariants are calculated as follows: i i i�1 (7) T B � FF , (i) (i) V � 􏽘 λ v ⊗ v , λ > 0, i ∈ {1, 2, 3}, i i i�1 I � tr(B), (i) where each λ refers to one of the principal stretches, u and (14) 1 2 (i) B B 2 v are the unit eigenvectors of U and V known as the I � 􏼔􏼐I 􏼑 − tr􏼐B 􏼑􏼕, 2 1 Lagrangian and Eulerian principal axes, and ⊗ is the sign of tensor product [23, 24]. B 2 I � det(B) ≡ (det F) . ,e tensor function H is regarded as the function of material response in the conﬁguration B with respect For incompressible materials, a slightly diﬀerent set of to the nominal stress, S, that is, the transpose of the principal invariants of B, as represented, is generally ﬁrst Piola-Kirchhoﬀ stress. ,e following equation for the employed: nominal stress, S, I � , 2/3 zW (8) S � H(F) � , zF B (15) I � , is validated for an unconstrained homogeneous hyperelastic 4/3 material. While the material is incompressible, the arbitrary 􏽰�� hydrostatic pressure, p, which is the Lagrange multiplier J � det(B). el associated with the material incompressibility, modiﬁes the relation for the nominal stress, S, as ,e right Cauchy–Green deformation tensor, C, and its principal invariants are computed similarly. zW −1 (9) ,e Cauchy stress tensor for an unconstrained isotropic S � − pF , det F � 1. zF elastic material is computed in terms of strain invariants, I , I , and I , as follows: With regard to the relation between the nominal stress 2 3 tensor, S, and Cauchy stress tensor, σ, σ � α I + α B + α B , 0 1 2 −1 S � JF σ, (10) zW 1/2 α � 2I , 0 3 zI the Cauchy stress tensor, σ, could be calculated through (11) in which the symmetric tensor function G denotes the (16) zW zW −1 2 function of material response in the conﬁguration B as- α � 2I 􏼠 + I 􏼡, 1 3 1 zI zI sociated with the Cauchy stress tensor, σ, 1 2 zW −1 zW −1/2 (11) σ � G(F) � J F . α � −2I , 2 3 zF zI as the result of the absence of α (because I � 1) and the With respect to the arbitrary hydrostatic pressure, p, presence of p for incompressible materials, the Cauchy stress deﬁned previously, for an incompressible material, the re- tensor changes to lation for the Cauchy stress tensor, σ, modiﬁes as [23–25] (17) σ � −pI + α B + α B , zW 1 2 (12) σ � F − pI, det F � 1. zF for the forenamed materials [21–24], which simpliﬁes to First, second, and third invariants of F, known as the zW zW σ � −pI + 2 B + 2 􏼐I B − B 􏼑. (18) strain invariants of deformation, which make provision for 1 zI zI 1 2 mapping the area and volume between the deformed con- ﬁguration, B, and reference conﬁguration, B , are computed A variety of stored energy functions have been in- through troduced in the literature that could be employed to model 4 Journal of Healthcare Engineering the nonlinear elastic behavior of soft tissues precisely to provide the accessibility to displacement values at some [26, 27], from the popular long-standing Neo-Hookean sequential moments, the US images or RF signals should be continuously saved for a period of time. model (originated by Treloar in 1943 [28]) and Mooney– Rivlin model (proposed by Rivlin et al. in 1951 [29, 30]) to the state-of-the-art models, for instance, the ones introduced 2.3. Simulation of US RF Signals and Images. When a tissue is by Limbert in 2011 [31], Nolan et al. in 2014 [32], and inspected with the US imaging system, the tissue is scanned Shearer in 2015 (for modeling ligaments and tendons) [33], with respect to the probe; in other words, when the tissue is although this is hitherto an active ﬁeld of study in the compressed with the probe, the presented features of the material and biomedical sciences. tissue in the image seemingly move upward, although in Being pertinent to model the behavior of a wide range of point of fact they would move downward. In furtherance of materials, for instance the soft tissues and polymers, the simulating the postcompression RF signals and B-mode Mooney–Rivlin model is one of the conventional hypere- images in the probe coordinate system, instead of the lastic models in the literature [34–39]. Two historical at- phantom’s surface in contact with the probe, the surface in tributes have made this model a distinguished one: (1) it is front of the probe was assigned to be moving [40–42]. one of the primarily introduced hyperelastic models; (2) it ,e RF signals and B-mode images of the simulated could meticulously predict the nonlinear demeanor ob- phantom while responding to the sinusoidal load have been served from some materials, speciﬁcally the isotropic simulated using the Field II US Simulation Program (A rubber-like materials. ,e most general version of this model MATLAB toolbox for US ﬁeld simulation) [43]. ,e nodal has been deﬁned based on (the linear combination of) the displacement measurements of the phantom in response to ﬁrst and second strain invariants of deformation, I and I . 1 2 the sinusoidal load have been applied to simulate the ,e Mooney–Rivlin strain energy function is expressed as postdeformation RF signals and B-mode images in varied 1 deformation states. With the view to simulating the RF W � C I − 3 + C I − 3 + (J − 1) , (19) 􏼁 􏼁 10 1 01 2 signals and B-mode image correlated with a particular de- formation state, the correspondent nodal displacement where C and C are the material hyperelastic constants, D 10 01 values achieved by the ﬁnite deformation analysis have been is the constant related to the material volumetric response linearly interpolated to compute the positions of scatterers (i.e., the material bulk modulus), and J is the determinant of corresponding to the speciﬁed deformation state. deformation gradient tensor, F. In addition, regarding the initial shear modulus of material, μ , the relation 2.4. Estimation of Displacement Field. In addition to the US (20) C + C � μ , elastography imaging techniques, motion tracking algo- 10 01 0 rithms have been applied in various US-based methods, for links the two hyperelastic parameters [26, 36]. For in- example, blood ﬂow imaging, thermal strain imaging, phase- compressible materials, the Mooney–Rivlin strain energy aberration correction, strain compounding, and tempera- function simpliﬁes to (since J � 1) ture imaging, to name a few. ,e prominence of clinical applications of motion tracking methods has contributed to W � C I − 3 􏼁 + C I − 3 􏼁 . (21) 10 1 01 2 the signiﬁcant accrual in the number of relevant in- vestigations and the proposal of a multitude of techniques including phase-domain tactics, time-domain (1D) or space- domain (2D) procedures, and spline-based methods [44]. 2.2. Soft Tissue Simulation. To estimate nonlinear elastic parameters of an unidentiﬁed tumor using the proposed Among the propounded techniques, the cross-correlation technique, we have simulated a simpliﬁed 3D breast tissue algorithm is known as the gold standard of motion estima- geometry utilizing the FEM software ABAQUS (Dassault tion. In this technique, the displacement quantities are Systemes ` Simulia Corp., Johnston, RI, USA). ,e breast computed through searching the locations of the maximums tissue is comprised of three partial tissues, fat, ﬁbro- of cross-correlation values between corresponding axial- glandular, and tumor, located consecutively from outside to lateral grids of small windows (with high overlap) in the inside. We have applied the Mooney–Rivlin hyperelastic pre- and postdeformation frames recorded from the medium. model to evaluate the deformation of simulated breast tissue ,e shifts between the pre- and postdeformation windows quantify the displacement ﬁeld in the medium [44, 45]. induced by the external excitation. For estimating hyperelastic parameters of the tumor, we ,e cross-correlation algorithm with guided search has applied a sinusoidal load with frequency of 0.1 Hz (the very been applied to initially estimate the displacement quantities low frequency to ignore the inertia eﬀects) to the simulated in the tumor at the selected step times from the simulated tissue and registered its response to extract the displacement pre- and postdeformation RF signals and B-mode images, ﬁelds inside the tumor. It is feasible to estimate displacement and afterwards evaluate the errors of displacement estimates. quantities inside the in vivo tissue by the use of some ,e calculated displacement values of proximate regions conventional medical imaging systems such as the US im- (i.e., previous samples and lines) have been exploited to aging or MRI system. We discuss the methods of calculating reduce the search area and therefore signiﬁcantly decrease the displacements inside the tissue from RF signals or images the computational expense of the cross-correlation algo- recorded by the US imaging system in Section 2.4. In order rithm. By the use of simulated RF signals, although we Journal of Healthcare Engineering 5 devoted more time to calculate the displacement values (2) Extract the displacement ﬁelds inside the tissue at (because of their higher sampling rate) compared with the some sequential step times (i.e., eight consecutive US images, we achieved more accurate displacement esti- instants) from the recorded US RF signals or images. mates with higher spatial resolution. (3) Simulate the tissue and its neighboring mediums by making use of one of the FEM softwares corre- 2.5. Estimation of Hyperelastic Parameters. Regarding the sponding to aforementioned explanation related to the ﬁnite strain (a) the recorded predeformation US images theory (also known as large deformation theory), when (b) the loading speciﬁcations a uniaxial stress, σ, is applied to the material, the de- (c) the boundary conditions formation gradient tensor, F, could be computed as It is assumed that just the tumor and its mechanical λ 0 0 ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ characteristics are unidentiﬁed. ,e mechanical ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ F � ⎢ 0 λ 0 ⎥, (22) ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎣ ⎦ parameters of almost all healthy soft tissues have 0 0 λ been reported in the literature. ,e parameters have been predominantly estimated by performing me- where λ , λ , and λ are the principal stretches with respect to 1 2 3 ticulous in vivo or ex vivo experiments. the set of coordinate axes (corresponding with x � λ X , i i i (4) Consider the elastic modulus, E, and Poisson’s ratio, i � 1,2,3). ,e principal invariants, I , I , and I , could be 1 2 3 υ, of the simulated tumor, named elastic tumor, stated further in terms of principal stretches as follows: equal 1 Pa and 0.5, respectively. 2 2 2 I � λ + λ + λ , (5) Compute the displacement ﬁelds inside the par- 1 1 2 3 ticular elastic tumor at the same consecutive step 2 2 2 2 2 2 (23) I � λ · λ + λ · λ + λ · λ , 2 1 2 2 3 3 1 times by dint of the selected FEM software. 2 2 2 (6) Calculate the real elastic modulus of the understudy I � λ · λ · λ . 3 1 2 3 tumor, E , with the help of the MATLAB soft- realt If λ symbolizes the stretch parallel to the stress applied to ware (,e MathWorks, Inc., Natick, Massachusetts, the medium in line with the ﬁrst coordinate axis (that is USA) using equal to the λ set), two assumptions, (1) the equality of (a) the axial displacement quantities of several deformations in the two other coordinate axes and (2) the points of the tumor at some consecutive instants incompressibility of the medium (I � 1), result in simpli- (based on the achieved outcomes, the axial fying the relation of the Cauchy stress tensor (18), to [35, 46] displacement values of twelve points of the tu- zW 1 zW 2 −1 mor at eight step times), Y realt σ � 2􏼐λ − λ 􏼑􏼠 + 􏼡. (24) zI λ zI 1 2 (b) the axial displacement values of the identical points of the elastic tumor at the same moments, D As explicated in the previous sections, the displacement (c) the relation [47–49] ﬁeld inside the in vivo tumor could be measured in a non- D D invasive way. With the purpose of estimating Mooney– E � . (27) realt Rivlin hyperelastic parameters of the tumor just by using the D Y realt displacement quantities, we manipulate (25), that is, the stress-stretch relation of the hyperelastic model, (7) Specify a set of strains, ε, and compute the corre- 2 −1 −1 spondent stresses, σ, using (28) in accordance with σ � 2􏼐λ − λ 􏼑􏼐C + λ C 􏼑. (25) 10 01 (in the ﬁrst iteration): ,e uniaxial load applied to the medium in alignment (a) ,e strain ﬁeld in the tumor could be roughly with the ﬁrst coordinate axis practically generates the axial approximated from the displacement estimates. strain in the same direction; therefore, the strain and relation (b) ,e achieved results imply the selection of a set between the stress and strain could be expressed as [47, 48] of high strains in the ﬁrst iteration since the ε � λ− 1, strain values are being modiﬁed in some steps of (26) 2 −1 −1 the proposed iterative algorithm; therefore, the σ � 2􏼐(ε + 1) −(ε + 1) 􏼑􏼐C + (ε + 1) C 􏼑. 10 01 strain values could be reduced uniformly. (c) Slight changes in the stress values, for instance With respect to the stress-strain relation, (26), the iterative by the use of the normal distribution function, algorithm propounded for estimating Mooney–Rivlin might cause the stress and strain values to hyperelastic parameters of a completely unknown interior conform more eﬀectively to the Mooney–Rivlin tissue (i.e., tumor) could be described in the following steps: hyperelastic model assigned to the tumor, (1) Image the tissue and its adjacent mediums by the σ � E ε. (28) realt use of clinical US imaging system before/while applying a sinusoidal load with low frequency (to (8) Compute hyperelastic parameters of the Mooney– annul the inertia) to the exterior medium. In other words, record the relevant US RF signals or images. Rivlin model, C and C , using the stress and 10 01 6 Journal of Healthcare Engineering strain sets and the relation between stress and 3. Results strain, as represented, 2 −1 −1 3.1. Soft Tissue Simulation. ,e breast tissue (with the di- σ � 2􏼐(ε + 1) −(ε + 1) 􏼑􏼐C + (ε + 1) C 􏼑, 10 01 3 mensions of 100 × 60 × 20 mm ), simulated using the FEM (29) software ABAQUS, has been depicted in Figure 1. ,e breast tissue consists of three partitions, namely, fat, ﬁbroglandular, with the help of MATLAB algorithms, for instance, and tumor. ,e Mooney–Rivlin hyperelastic model has been the regression algorithms. applied to obtain the response of simulated breast tissue to the (9) Assign the estimated hyperelastic parameters to the external sinusoidal load. ,e Mooney–Rivlin material constants simulated tumor and compute the displacement of the named breast tissues have been presented in Table 1. Since ﬁelds inside the tumor at the determinate step times we have utilized the elastic parameter of the tumor for esti- by means of the FEM software. mating its hyperelastic parameters, we have additionally re- (10) Calculate the elastic constant of the simulated tu- ported the elastic parameters of the named breast tissues in mor, E , as explained previously in step 6, using Table 1. ,e linear and nonlinear elastic parameters have been estt the axial displacement quantities (of the appointed supposed to be constant throughout each tissue partition. ,is points at the selected step times) of the simulated set of hyperelastic parameters has been broadly utilized in the tumor, Y , and the elastic tumor, D (determined literature to simulate the breast tissue [38, 50–54]. estt in step 6), by applying the relation ,e mesh considered for the simulated phantom consists of 183783 second-order (quadratic) tetrahedral hybrid ele- D D E � . (30) estt ments (C3D10H) with 259813 nodes. ,e convergence ana- D Y estt lyses have warranted the accuracy of the simulation results. With reference to the explanations in Section 2.3, the number (11) Appraise the estimated hyperelastic parameters for of nodes in the simulated medium should signiﬁcantly be the tumor through considering the following: increased to precisely calculate the positions of scatterers after (a) ,e error of the computed axial displacement applying the load to the medium. With regard to the boundary values of the selected points at the speciﬁed conditions and the load applied to the tissue (represented in moments, Y , by comparing them with the estt Figure 1), the postcompression RF signals and B-mode images correspondent displacement quantities esti- have been simulated in the probe coordinate system. Two mated from the recorded US RF signals or snapshots of the response of the simulated breast tissue to the images, Y sinusoidal load have been demonstrated in Figure 2. realt (b) ,e error of the elastic constant calculated for the tumor, E , by comparing it with the real estt elastic modulus of the understudy tumor, E , realt 3.2. Simulation of US RF Signals and Images. ,e RF signals estimated in step 6 and B-mode images of part (with the dimensions of 50 × 60 × 10 mm ) of the simulated breast tissue which en- ,e errors of the hyperelastic parameters estimated circles the tumor, as illustrated in Figure 3, have been for the tumor, by comparing them with the real simulated using the Field II US Simulation Program. In the hyperelastic parameters of the tumor, could not be Field II US Simulation Program, considered because it has been assumed that the tumor is entirely obscure. (a) ,e properties considered to model the probe array (12) Alter the set of strains speciﬁed in step 7 (based on and simulate the US RF signals and images are as the above explanation, decrease them regularly) follows: and repeat steps 7 to 12. By reducing the strain (1) Linear array (with 64 active elements) values steadily, the error of the estimated elastic (2) Transducer center frequency of 3.5 × 10 Hz parameter for the tumor and the error of the (3) Sampling frequency of 100 × 10 Hz calculated displacement quantities in the tumor (4) Transmit focus of 70 mm (in depth) are decreasing below the deﬁned tolerance values, (5) Element’s width (the distance between the ele- as illustrated, ments or the pitch of the probe array) of 0.44 mm � � � � � � (equal to the wavelength) � � Y − Y ≤ e , � realt� displacement estt (6) Element’s height of 5 mm � � (31) � � � k � � � E − E ≤ e , (7) Element’s kerf of 0.05 mm � � estt realt elastic (8) Lateral spatial spacing of 0.08 mm (512 scan lines where e and e are the tolerance values in the image) elastic displacement and k represents the number of iterations of the (b) With regard to the elastic and hyperelastic param- algorithm. By decreasing the strain values chosen eters of the tumor, for scatterers which have resided with regard to the mentioned conditions, the strain within the tumor, the amplitudes are set to zero. and stress values successively adjust more to the Mooney–Rivlin stress-strain relationship of the ,e postdeformation RF signals and B-mode images in tumor. eight deformation states of the phantom (corresponding to Journal of Healthcare Engineering 7 Z X (a) (b) Figure 1: �e simulated breast tissue comprises three parts, namely, fat, �broglandular, and tumor (from outside to inside). Table 1: �e elastic and Mooney–Rivlin hyperelastic constants of 4. Discussion breast tissues [38]. Hyperelastic and In the majority of diversi�ed approaches pro’ered for esti- Fat Fibroglandular Tumor elastic parameters mating elastic parameters of soft tissues, particularly the C (Pa) 2000 3500 10000 nonlinear ones, for instance, the techniques proposed by C (Pa) 1333 2333.3 6667 MacManus et al. [55], Esmaeili et al. [56], Omidi et al. [57], E (kPa) 20 35 100 Roy and Desai [58], Liu et al. [59], Boonvisut and Çavu¸soglu [60], and Wang et al. [61], to mention but a few, the alter- ations (of precise values) of at least two deformation variables, eight sequential step times: 7.75 s, 8.00 s, 8.25 s, 8.50 s, 8.75 s, which are associated with the mechanical characteristics of soft tissues, have been exploited. �e assessment of recom- 9.00 s, 9.25 s, and 9.50 s, while responding to the sinusoidal load) have been simulated using the displacements of the mended techniques would reveal that the direct dependencies phantom’s nodes computed by the �nite deformation anal- of methodologies to deformation variables except the dis- ysis. �e nodal displacement values correlated with a partic- placement (and strain) have impelled the researchers to carry ular deformation state have been linearly interpolated to out experiments on ex vivo tissues, or perform invasive compute the positions of scatterers and simulate the corre- procedures to precisely measure the variables; consequently, spondent RF signals and B-mode images thereafter. Two the emphasis of recent investigations should be on advancing simulated postdeformation B-mode images (pre- and post- noninvasive methods with the capability to accurately esti- deformation images) associated with the step times of 8.00 s mate nonlinear elastic parameters of tissues. �e hyperelastic constitutive theory takes two types of and 9.00 s have been represented in Figure 3. nonlinearities perceived in responses of soft tissues, into consideration [62, 63]: 3.3. Estimation of Hyperelastic Parameters. After simulating (a) �e material nonlinearity of the stress-strain re- the pre- and postdeformation RF signals and B-mode images lation, known as the physical nonlinearity correlated with the de�ned deformation states, the cross- (b) �e nonlinearity of the strain-displacement relation, correlation algorithm with guided search, as brie…y de- called the geometrical nonlinearity scribed in Section 2.4, has been employed to initially esti- mate displacement quantities inside the tumor at the consequently, it has been regarded as one of the best selected step times and afterwards evaluate the errors of practical theories for formulating mechanical behaviors of displacement estimates. soft tissues. To the best of our knowledge, amongst the �e suggested iterative algorithm, comprehensively ex- strategies proposed for quantifying hyperelastic parameters plicated in Section 2.5, has been applied to extract the of materials, two methods founded on the displacement Mooney–Rivlin hyperelastic parameters of the tumor from �elds inside and on the boundary of the medium (i.e., the axial displacement values of some points of tumor at the phantoms) which have been introduced by Mehrabian and speci�ed step times. As represented in Table 2, precise es- Samani [26, 27, 64] and Hajhashemkhani and Hematiyan timates of hyperelastic parameters of the tumor have been [47, 48], respectively, could be utilized to noninvasively achieved. �e automatic iteration of the algorithm would be reconstruct hyperelastic parameters of in vivo tissues. �e feasible through bilaterally connecting the MATLAB and displacement �eld inside the understudy medium could be FEM softwares. extracted from RF signals or images recorded by a clinical 8 Journal of Healthcare Engineering Y Y Z X Z X (a) (b) Figure 2: �e responses of the simulated breast tissue at (a) t 4.00 s and (b) t 8.00 s after starting to apply the sinusoidal load. Predeformation US image Postdeformation US image Predeformation US image Postdeformation US image 35 35 35 35 120 120 40 40 40 40 45 45 45 45 100 100 50 50 50 50 55 55 55 55 80 80 60 60 60 60 60 60 65 65 65 65 70 70 70 70 40 40 75 75 75 75 80 80 80 80 20 20 85 85 85 85 90 90 90 90 –20 –10 0 10 20 –20 –10 0 10 20 –20 –10 0 10 20 –20 –10 0 10 20 Lateral distance (mm) Lateral distance (mm) Lateral distance (mm) Lateral distance (mm) (a) (b) Figure 3: �e simulated pre- and postdeformation B-mode images of breast phantom, based on its states at (a) t 8.00 s and (b) t 9.00 s after starting to apply the sinusoidal load. Table 2: �e elastic and hyperelastic parameters estimated for the correlates the stress distribution computed for the tissue (with tumor. the help of a �nite element model of the tissue deformation) to its hyperelastic parameters. �e displacement values at some Estimated elastic and hyperelastic parameters boundary points of the understudy medium have been ma- 2 estimates nipulated by Hajhashemkhani and Hematiyan [47, 48] for E (kPa) 88908.41 realt characterizing its nonlinear material constants. Error of E (%) 11.09 realt On account of the explanations provided by Mehrabian C (Pa) 9426.98 10005.05 and Samani [26, 27, 64] and Hajhashemkhani and Hema- C (Pa) 6702.30 5992.82 tiyan [47, 48] and the results achieved through the imple- Error of C (%) 5.73 0.05 mentation of their methods (part of them published in our Error of C (%) 0.53 10.11 paper [65]), it has been realized that precise estimates of E (kPa) 88711.93 88658.27 estt Error of E (%) 0.22 0.28 hyperelastic parameters of the understudy medium could be estt Error of Y (%) 0.28 0.35 attained through the following: estt (a) Accurately calculating the displacement quantities, respectively, in a multitude of adjacent points of the US imaging system, for instance, by using the conventional medium and in several boundary points, which cross-correlation method. might not be possible using registered US images or In the technique recommended by Mehrabian and RF signals Samani [26, 27, 64], the displacement quantities of a large number of contiguous points inside the medium should be (b) Applying proper regularization techniques, for in- utilized to calculate the de�ned coe¦cient matrix which stance, the Tikhonov regularization, Truncated Axial distance (mm) Axial distance (mm) Axial distance (mm) Axial distance (mm) Journal of Healthcare Engineering 9 Table 3: ,e hyperelastic parameters estimated for the tumor using Singular Value Decomposition (SVD), and Wiener inaccurate displacement measurements. Filtering methods (c) Even considering appropriate initial guesses of the Inaccurate displacement measurements Estimated hyperelastic parameters, as indicated by Hajha- hyperelastic Error Error Error Error shemkhani and Hematiyan [47, 48], Aghajani et al. parameters 2% 5% 8% 10% [66], and Kim and Srinivasan [67] C (Pa) 9386.78 9484.46 9815.78 9858.99 C (Pa) 6889.50 6817.80 6437.30 6465.64 ,e limitations of the techniques propounded with the Error of C (%) 6.13 5.16 1.84 1.41 aim of reconstructing nonlinear elastic parameters of in vivo Error of C (%) 3.34 2.26 3.45 3.02 soft tissues persuade us to concentrate on developing a more Error of E (%) 0.53 0.39 1.89 2.29 estt practical method with consistent results. Our primary up- Error of Y (%) 1.77 4.59 6.28 7.96 estt shot represented in Section 3 conﬁrms that accurate esti- mates of hyperelastic parameters of the understudy tissue (i.e., tumor) could be obtained on the basis of the dis- calculated for the tumor, precise estimates of tumor’s placement values of some points inside the tissue, which has hyperelastic parameters could be obtained. With regard been excited by a low frequency sinusoidal load, even when to the outcomes summarized in Table 3, it is deduced that there is no prior knowledge of the tissue. the suggested method is strongly resistant to the dis- ,e displacement quantities of the selected points might placement errors. be computed approximately, for instance, by the use of the ,e US RF signals recorded by means of the Antares cross-correlation technique as a consequence of recording Siemens system (Issaquah, WA) at the center frequency of low-quality US images or RF signals or other attributes; 6.67 MHz from an elastography phantom (CIRS elastog- therefore, we have evaluated the consistency of calculated raphy phantom, Norfolk, VA) have been utilized to values for the hyperelastic constants by applying errors with evaluate the suggested method experimentally. ,e signals normal distribution to the measured displacement ﬁelds in were registered via a VF10-5 linear array at a sampling rate the tissue stimulated by the sinusoidal load. At this point, the of 40 MHz by Rivaz et al. to assess the performance of the average errors of the displacement values estimated for the proposed real-time static elastography techniques which selected points from the simulated US RF signals and images were based on the analytic minimization of regularized using the cross-correlation algorithms (without/with guided cost functions. Young’s moduli of the lesion and sur- search with respect to the calculated displacements of rounding medium have been reported 56 kPa and 33 kPa, previous lines or samples) have been regarded. ,e achieved respectively, while the phantom is under compression results have been demonstrated in Table 3. [68, 69]. It should be considered that the imprecise estimates of ,e enhanced cross-correlation algorithm, in that the displacement ﬁelds inside the tumor aﬀect all the computed search regions were minimized with respect to the estimated parameters and errors, even the real elastic modulus of the displacements of previous lines or samples, has been understudy tumor; therefore, the results presented in Table 3 employed to compute the axial displacement ﬁeld in the could not be compared. Similar to the case where the dis- compressed phantom. ,e Kalman ﬁltering, introduced by placement values of the appointed points are exact, Rivaz et al. [68], has been applied to calculate the strain ﬁeld in the compressed phantom from the displacement mea- (a) the error of calculated axial displacement values of surements. Minor diﬀerences between the displacement the selected points at the speciﬁed moments, Y estt ﬁelds estimated by the use of the enhanced cross-correlation (b) the error of elastic constant computed for the tumor, algorithm and analytic minimization method validate the estt results of the former technique. (as explained in step 11 of the proposed algorithm in Section ,e US images of the phantom constructed from the recorded RF signals and the estimated displacement and 2.5) have been considered except in the situations where the displacement errors are signiﬁcant. strain ﬁelds have been represented in Figure 4. Following the ,e convergence of the aforementioned errors to values, instructions in Section 2.5, the elastic and hyperelastic pa- which might not be small errors, speciﬁes the best estimates rameters of the lesion could be calculated from the estimated of hyperelastic parameters when the displacement values are axial displacement and strain ﬁelds in the lesion. ,e percent error of the elastic parameter computed for the lesion, on the highly inaccurate, as represented, � � � � basis of the explanations in step 6 of the algorithm, is 15.06%; � k k−1� � � ′ Y − Y ≤ e , � � estt estt displacement in other words, E has been estimated 47565.72 Pa. ,e realt � � (32) � � k k−1 � � relation between stress, strain, and the parameters of the � � ′ E − E ≤ e , � � elastic estt estt Mooney–Rivlin hyperelastic model, C and C , has been 10 01 manipulated to compute the hyperelastic parameters of the ′ ′ where e , e , and k are, respectively, the lesion. ,e values of 6871.65 Pa and 1020.00 Pa have been elastic displacement speciﬁed tolerance values and the number of iterations of obtained for the mentioned parameters. the algorithm. Provided that the set of strains (required On the basis of the achieved results summarized in in step 7 of the algorithm described in Section 2.5) is Sections 3 and 4, it is concluded that the main objectives that selected properly based on the displacement ﬁelds have been accomplished in this paper are as follows: 10 Journal of Healthcare Engineering Ultrasound image 1 Ultrasound image 2 �e displacement �elds inside the tissue could be 0 0 noninvasively computed from the data recorded by the 200 –0.5 200 –0.5 employment of conventional medical imaging modalities, –1 –1 400 400 for instance, the RF signals or images registered by the US –1.5 –1.5 600 600 imaging system. Even by processing approximate dis- –2 –2 800 800 –2.5 –2.5 placement measurements, accurate estimates of the material 1000 1000 –3 –3 constants could be obtained. �e competency of the pro- 1200 1200 –3.5 –3.5 posed method to estimate nonlinear elastic constants of 1400 –4 1400 –4 normal and abnormal in vivo tissues will be further ap- –4.5 –4.5 1600 1600 praised in the future research. 100 200 300 400 100 200 300 400 RF lines RF lines (a) Data Availability Axial strain Axial displacement 0 –0.02 Since the research is still in progress, the authors have –10 200 5 –0.03 decided to make data available upon request. –20 –0.04 –30 –0.05 –40 Disclosure –50 –0.06 –60 20 –0.07 �e funding had no role in the study design, data collection, –70 –0.08 25 or analysis. –80 –0.09 –90 –100 –0.1 0 20 40 60 100 200 300 400 500 Conflicts of Interest RF lines Width (mm) �e authors declare that there are no con…icts of interest (b) regarding the publication of this article. 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Publisher: Hindawi Publishing Corporation
Copyright: Copyright © 2019 Maryam Mehdizadeh Dastjerdi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ISSN: 2040-2295
eISSN: 2040-2309
DOI: 10.1155/2019/2374645
Publisher site: See Article on Publisher Site

Abstract

Hindawi Journal of Healthcare Engineering Volume 2019, Article ID 2374645, 12 pages https://doi.org/10.1155/2019/2374645 Research Article An Iterative Method for Estimating Nonlinear Elastic Constants of Tumor in Soft Tissue from Approximate Displacement Measurements 1 1 2 Maryam Mehdizadeh Dastjerdi , Ali Fallah , and Saeid Rashidi Department of Biomedical Engineering, Amirkabir University of Technology (AUT), Tehran 15875-4413, Iran Faculty of Medical Sciences and Technologies, Science and Research Branch, Islamic Azad University, Tehran 1477893855, Iran Correspondence should be addressed to Ali Fallah; afallah@aut.ac.ir Received 24 March 2018; Revised 24 June 2018; Accepted 12 July 2018; Published 6 January 2019 Academic Editor: Zahid Akhtar Copyright © 2019 Maryam Mehdizadeh Dastjerdi et al. ,is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Objectives. Various elastography techniques have been proﬀered based on linear or nonlinear constitutive models with the aim of detecting and classifying pathologies in soft tissues accurately and noninvasively. Biological soft tissues demonstrate behaviors which conform to nonlinear constitutive models, in particular the hyperelastic ones. In this paper, we represent the results of our steps towards implementing ultrasound elastography to extract hyperelastic constants of a tumor inside soft tissue. Methods. Hyperelastic parameters of the unknown tissue have been estimated by applying the iterative method founded on the relation between stress, strain, and the parameters of a hyperelastic model after (a) simulating the medium’s response to a sinusoidal load and extracting the tissue displacement ﬁelds in some instants and (b) estimating the tissue displacement ﬁelds from the recorded/simulated ultrasound radio frequency signals and images using the cross correlation-based technique. Results. Our results indicate that hyperelastic parameters of an unidentiﬁed tissue could be precisely estimated even in the conditions where there is no prior knowledge of the tissue, or the displacement ﬁelds have been approximately calculated using the data recorded by a clinical ultrasound system. Conclusions. ,e accurate estimation of nonlinear elastic constants yields to the correct cognizance of pathologies in soft tissues. information related to the tissue acoustic impedance, vas- 1. Introduction cular ﬂow, and tissue mechanical characteristics or variables According to the World Health Organization report, cancer such as its stiﬀness or strain, respectively, have been ex- is one of the principal morbidity and mortality agents tensively utilized for medical diagnoses [1]. ,e US imaging throughout the world, with approximately 8.8 million deaths is recognized a noninvasive, safe, easy-to-use, low-cost, and widely accessible imaging modality for visualizing in vivo (nearly 1 in 6 deaths) in 2015 and 70% increase in the tissues. Elastography approach has currently been regarded number of new cases over the next two decades. ,e cancer a promising alternative to invasive medical procedures, for statistics imply the requisite to extend medical scrutiny to example, the biopsy, to characterize tissue abnormalities. improve cancer prevention, early correct diagnosis, metic- ,e wide variety of strategies that are being employed to ulous screening, and eﬀective treatment and reduce the quantify and image mechanical properties of biological invasiveness and costs of applied techniques. tissues are recognized as elastography or elasticity imaging Since the ﬁrst introduction of ultrasound (US) imaging techniques with reference to their similar premise [2]: in clinical practice in the 1970s, ultrasonography and other US modalities, for example, Doppler imaging and state-of- (1) ,e in vivo tissue is being deformed by a speciﬁed the-art elastography imaging methods, which provide the external or internal load or motion. 2 Journal of Healthcare Engineering (2) ,e response of tissue is being recorded by the use of constitutive equations, into consideration in order to provide a standard clinical imaging system, such as the US or the possibility to minutely describe the behavioral charac- magnetic resonance imaging (MRI) system. teristics of materials. Constitutive theories in continuum mechanics deal with formulating material models that are (3) ,e mechanical characteristics of tissue are esti- [18, 19] mated through the assessment of tissue displacement ﬁelds. (a) on the basis of some mechanical universal principles ,e alterations in the microstructure of tissue as a conse- (b) in accordance with experimental observations quence of pathophysiological phenomena would change the ,oughtful consideration of soft tissue’s model would result mechanical properties of tissue; for instance, the increase in the in the realistic prediction of its behavior such that it could be stromal density of cancerous tissue would cause the increment veriﬁed by experimental observations. Nonlinear constitu- in its Young’s modulus [3, 4]. ,e outcomes of numerous tive manners that have been observed from soft tissues in experimental research studies carried out by Krouskop et al. numerous in vivo and ex vivo experimental research studies [5], Samani et al. [6, 7], Lyshchik et al. [8], Soza et al. [9], Hoyt could be modeled by the use of hyperelastic models et al. [10], Schiavone et al. [11], O’Hagan et al. [12], and Moran [15, 16, 20]. et al. [13], to mention but a few, have conﬁrmed the relation ,e hyperelastic constitutive laws deal with modeling between tissue structures and macroscopic mechanical features materials with nonlinear elastic behaviors in reaction to large which are being evaluated, quantiﬁed, and/or imaged by strains. ,e nonlinearities that are the consequences of (a) employing the palpation, elastography, digital rectal exami- the material behavior and (b) the signiﬁcant change in the nation, and such like methods. shape of material are both regarded in the constitutive ,e precise determination of mechanical characteristics of theory of hyperelastic materials. Hyperelastic materials are understudy tissue by the use of an elastography technique generally described by speciﬁc forms of strain energy density would undoubtedly necessitate realistically appraising or (stored energy) functions. While characterizing the homo- modeling the tissue manners, speciﬁcally its nonlinear re- geneous material’s absorbed energy due to its deformation, sponse to the stimulation. Hyperelasticity theory is one of the the strain energy function, W, is deﬁned as a function of constitutive theories that have been manipulated to model the deformation gradient, F [21, 22] nonlinear constitutive demeanor demonstrated by biological soft tissues. A variety of hyperelastic models, for instance the W � W(F). (1) well-known Neo-Hookean, Mooney–Rivlin, Yeoh, and Polynomial models, have been recommended for this purpose It is considered that B and B represent, respectively, the [14–17]. In comparison with the studies involving the linear reference or undeformed conﬁguration, which refers to the elasticity imaging, the number of research studies conducted situations where no load is exerted to the material and the to image the nonlinear features of tissues is limited [2]. deformed conﬁguration, which is relevant to the situations With the aim of diagnosing a tumor inside the un- where the material is under load and therefore it may alter derstudy tissue correctly, we have utilized an iterative with time, t. In addition, it is assumed that X and x, re- method, as explicated in the next section, to accurately spectively, correspond to the position vectors of a material estimate the Mooney–Rivlin hyperelastic parameters of the point in the reference and deformed conﬁgurations, B and tumor inside the tissue. ,e displacement ﬁelds inside the B. ,e time-dependent deformation of material, that is, the tumor have been analyzed to extract its hyperelastic pa- motion of material point, from B to B could be described by rameters. An iterative technique has been employed since it the function χ, which, for each t, (a) is an invertible function has been assumed that no initial knowledge of the tumor and (b) satisﬁes proper regularity conditions as follows [23]: was accessible except the displacement ﬁelds inside the x � χ(X, t). (2) tumor. ,e displacement ﬁelds inside the tumor have been extracted from the simulated/recorded radio frequency ,e deformation gradient tensor, F, is deﬁned as (RF) signals using the cross correlation-based method. ,e response of an abnormal tissue to a sinusoidal load (with F � Gradx, (3) low frequency to negate the inertia) has been simulated by with Cartesian components applying the ﬁnite element (FEM) software package ABAQUS. ,e RF signals have been simulated by the use of zx F � i, α ∈ {1, 2, 3}, (4) the Field II US Simulation Program. In brief, in this paper, iα zX we scrutinize the diagnosis of tumor through its hypere- lastic parameters in the conditions where there is no pri- where Grad, x , and X refer to the gradient operator in the i α mary perception of the tumor and the displacement ﬁelds conﬁguration B and components of x and X, respectively, inside the tumor are estimated imprecisely. while the general convention, J ≡ det F > 0, (5) 2. Materials and Methods is satisﬁed. Due to the local invertibility of deformation, F 2.1. Hyperelasticity 3eory. Constitutive theories take the should be nonsingular. ,e unique polar decomposition of F improvement of mathematical models, also known as is deﬁned as Journal of Healthcare Engineering 3 F � RU � VR, (6) I � tr(F) � F + F + F , 1 11 22 33 where the tensor R is an appropriate orthogonal tensor and 1 I � 􏼐F F − F F 􏼑, (13) 2 ij ij ii jj the tensors U and V are symmetric positive-deﬁnite tensors known as the right and left stretch tensors, respectively. Equation (7) represents the spectral decompositions of I � det(F) � J, tensors U and V: for an unconstrained isotropic elastic material. ,e left Cauchy-Green deformation tensor, B, and its principal (i) (i) U � 􏽘 λ u ⊗ u , λ > 0, i ∈ 1, 2, 3 , { } invariants are calculated as follows: i i i�1 (7) T B � FF , (i) (i) V � 􏽘 λ v ⊗ v , λ > 0, i ∈ {1, 2, 3}, i i i�1 I � tr(B), (i) where each λ refers to one of the principal stretches, u and (14) 1 2 (i) B B 2 v are the unit eigenvectors of U and V known as the I � 􏼔􏼐I 􏼑 − tr􏼐B 􏼑􏼕, 2 1 Lagrangian and Eulerian principal axes, and ⊗ is the sign of tensor product [23, 24]. B 2 I � det(B) ≡ (det F) . ,e tensor function H is regarded as the function of material response in the conﬁguration B with respect For incompressible materials, a slightly diﬀerent set of to the nominal stress, S, that is, the transpose of the principal invariants of B, as represented, is generally ﬁrst Piola-Kirchhoﬀ stress. ,e following equation for the employed: nominal stress, S, I � , 2/3 zW (8) S � H(F) � , zF B (15) I � , is validated for an unconstrained homogeneous hyperelastic 4/3 material. While the material is incompressible, the arbitrary 􏽰�� hydrostatic pressure, p, which is the Lagrange multiplier J � det(B). el associated with the material incompressibility, modiﬁes the relation for the nominal stress, S, as ,e right Cauchy–Green deformation tensor, C, and its principal invariants are computed similarly. zW −1 (9) ,e Cauchy stress tensor for an unconstrained isotropic S � − pF , det F � 1. zF elastic material is computed in terms of strain invariants, I , I , and I , as follows: With regard to the relation between the nominal stress 2 3 tensor, S, and Cauchy stress tensor, σ, σ � α I + α B + α B , 0 1 2 −1 S � JF σ, (10) zW 1/2 α � 2I , 0 3 zI the Cauchy stress tensor, σ, could be calculated through (11) in which the symmetric tensor function G denotes the (16) zW zW −1 2 function of material response in the conﬁguration B as- α � 2I 􏼠 + I 􏼡, 1 3 1 zI zI sociated with the Cauchy stress tensor, σ, 1 2 zW −1 zW −1/2 (11) σ � G(F) � J F . α � −2I , 2 3 zF zI as the result of the absence of α (because I � 1) and the With respect to the arbitrary hydrostatic pressure, p, presence of p for incompressible materials, the Cauchy stress deﬁned previously, for an incompressible material, the re- tensor changes to lation for the Cauchy stress tensor, σ, modiﬁes as [23–25] (17) σ � −pI + α B + α B , zW 1 2 (12) σ � F − pI, det F � 1. zF for the forenamed materials [21–24], which simpliﬁes to First, second, and third invariants of F, known as the zW zW σ � −pI + 2 B + 2 􏼐I B − B 􏼑. (18) strain invariants of deformation, which make provision for 1 zI zI 1 2 mapping the area and volume between the deformed con- ﬁguration, B, and reference conﬁguration, B , are computed A variety of stored energy functions have been in- through troduced in the literature that could be employed to model 4 Journal of Healthcare Engineering the nonlinear elastic behavior of soft tissues precisely to provide the accessibility to displacement values at some [26, 27], from the popular long-standing Neo-Hookean sequential moments, the US images or RF signals should be continuously saved for a period of time. model (originated by Treloar in 1943 [28]) and Mooney– Rivlin model (proposed by Rivlin et al. in 1951 [29, 30]) to the state-of-the-art models, for instance, the ones introduced 2.3. Simulation of US RF Signals and Images. When a tissue is by Limbert in 2011 [31], Nolan et al. in 2014 [32], and inspected with the US imaging system, the tissue is scanned Shearer in 2015 (for modeling ligaments and tendons) [33], with respect to the probe; in other words, when the tissue is although this is hitherto an active ﬁeld of study in the compressed with the probe, the presented features of the material and biomedical sciences. tissue in the image seemingly move upward, although in Being pertinent to model the behavior of a wide range of point of fact they would move downward. In furtherance of materials, for instance the soft tissues and polymers, the simulating the postcompression RF signals and B-mode Mooney–Rivlin model is one of the conventional hypere- images in the probe coordinate system, instead of the lastic models in the literature [34–39]. Two historical at- phantom’s surface in contact with the probe, the surface in tributes have made this model a distinguished one: (1) it is front of the probe was assigned to be moving [40–42]. one of the primarily introduced hyperelastic models; (2) it ,e RF signals and B-mode images of the simulated could meticulously predict the nonlinear demeanor ob- phantom while responding to the sinusoidal load have been served from some materials, speciﬁcally the isotropic simulated using the Field II US Simulation Program (A rubber-like materials. ,e most general version of this model MATLAB toolbox for US ﬁeld simulation) [43]. ,e nodal has been deﬁned based on (the linear combination of) the displacement measurements of the phantom in response to ﬁrst and second strain invariants of deformation, I and I . 1 2 the sinusoidal load have been applied to simulate the ,e Mooney–Rivlin strain energy function is expressed as postdeformation RF signals and B-mode images in varied 1 deformation states. With the view to simulating the RF W � C I − 3 + C I − 3 + (J − 1) , (19) 􏼁 􏼁 10 1 01 2 signals and B-mode image correlated with a particular de- formation state, the correspondent nodal displacement where C and C are the material hyperelastic constants, D 10 01 values achieved by the ﬁnite deformation analysis have been is the constant related to the material volumetric response linearly interpolated to compute the positions of scatterers (i.e., the material bulk modulus), and J is the determinant of corresponding to the speciﬁed deformation state. deformation gradient tensor, F. In addition, regarding the initial shear modulus of material, μ , the relation 2.4. Estimation of Displacement Field. In addition to the US (20) C + C � μ , elastography imaging techniques, motion tracking algo- 10 01 0 rithms have been applied in various US-based methods, for links the two hyperelastic parameters [26, 36]. For in- example, blood ﬂow imaging, thermal strain imaging, phase- compressible materials, the Mooney–Rivlin strain energy aberration correction, strain compounding, and tempera- function simpliﬁes to (since J � 1) ture imaging, to name a few. ,e prominence of clinical applications of motion tracking methods has contributed to W � C I − 3 􏼁 + C I − 3 􏼁 . (21) 10 1 01 2 the signiﬁcant accrual in the number of relevant in- vestigations and the proposal of a multitude of techniques including phase-domain tactics, time-domain (1D) or space- domain (2D) procedures, and spline-based methods [44]. 2.2. Soft Tissue Simulation. To estimate nonlinear elastic parameters of an unidentiﬁed tumor using the proposed Among the propounded techniques, the cross-correlation technique, we have simulated a simpliﬁed 3D breast tissue algorithm is known as the gold standard of motion estima- geometry utilizing the FEM software ABAQUS (Dassault tion. In this technique, the displacement quantities are Systemes ` Simulia Corp., Johnston, RI, USA). ,e breast computed through searching the locations of the maximums tissue is comprised of three partial tissues, fat, ﬁbro- of cross-correlation values between corresponding axial- glandular, and tumor, located consecutively from outside to lateral grids of small windows (with high overlap) in the inside. We have applied the Mooney–Rivlin hyperelastic pre- and postdeformation frames recorded from the medium. model to evaluate the deformation of simulated breast tissue ,e shifts between the pre- and postdeformation windows quantify the displacement ﬁeld in the medium [44, 45]. induced by the external excitation. For estimating hyperelastic parameters of the tumor, we ,e cross-correlation algorithm with guided search has applied a sinusoidal load with frequency of 0.1 Hz (the very been applied to initially estimate the displacement quantities low frequency to ignore the inertia eﬀects) to the simulated in the tumor at the selected step times from the simulated tissue and registered its response to extract the displacement pre- and postdeformation RF signals and B-mode images, ﬁelds inside the tumor. It is feasible to estimate displacement and afterwards evaluate the errors of displacement estimates. quantities inside the in vivo tissue by the use of some ,e calculated displacement values of proximate regions conventional medical imaging systems such as the US im- (i.e., previous samples and lines) have been exploited to aging or MRI system. We discuss the methods of calculating reduce the search area and therefore signiﬁcantly decrease the displacements inside the tissue from RF signals or images the computational expense of the cross-correlation algo- recorded by the US imaging system in Section 2.4. In order rithm. By the use of simulated RF signals, although we Journal of Healthcare Engineering 5 devoted more time to calculate the displacement values (2) Extract the displacement ﬁelds inside the tissue at (because of their higher sampling rate) compared with the some sequential step times (i.e., eight consecutive US images, we achieved more accurate displacement esti- instants) from the recorded US RF signals or images. mates with higher spatial resolution. (3) Simulate the tissue and its neighboring mediums by making use of one of the FEM softwares corre- 2.5. Estimation of Hyperelastic Parameters. Regarding the sponding to aforementioned explanation related to the ﬁnite strain (a) the recorded predeformation US images theory (also known as large deformation theory), when (b) the loading speciﬁcations a uniaxial stress, σ, is applied to the material, the de- (c) the boundary conditions formation gradient tensor, F, could be computed as It is assumed that just the tumor and its mechanical λ 0 0 ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ characteristics are unidentiﬁed. ,e mechanical ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ F � ⎢ 0 λ 0 ⎥, (22) ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎣ ⎦ parameters of almost all healthy soft tissues have 0 0 λ been reported in the literature. ,e parameters have been predominantly estimated by performing me- where λ , λ , and λ are the principal stretches with respect to 1 2 3 ticulous in vivo or ex vivo experiments. the set of coordinate axes (corresponding with x � λ X , i i i (4) Consider the elastic modulus, E, and Poisson’s ratio, i � 1,2,3). ,e principal invariants, I , I , and I , could be 1 2 3 υ, of the simulated tumor, named elastic tumor, stated further in terms of principal stretches as follows: equal 1 Pa and 0.5, respectively. 2 2 2 I � λ + λ + λ , (5) Compute the displacement ﬁelds inside the par- 1 1 2 3 ticular elastic tumor at the same consecutive step 2 2 2 2 2 2 (23) I � λ · λ + λ · λ + λ · λ , 2 1 2 2 3 3 1 times by dint of the selected FEM software. 2 2 2 (6) Calculate the real elastic modulus of the understudy I � λ · λ · λ . 3 1 2 3 tumor, E , with the help of the MATLAB soft- realt If λ symbolizes the stretch parallel to the stress applied to ware (,e MathWorks, Inc., Natick, Massachusetts, the medium in line with the ﬁrst coordinate axis (that is USA) using equal to the λ set), two assumptions, (1) the equality of (a) the axial displacement quantities of several deformations in the two other coordinate axes and (2) the points of the tumor at some consecutive instants incompressibility of the medium (I � 1), result in simpli- (based on the achieved outcomes, the axial fying the relation of the Cauchy stress tensor (18), to [35, 46] displacement values of twelve points of the tu- zW 1 zW 2 −1 mor at eight step times), Y realt σ � 2􏼐λ − λ 􏼑􏼠 + 􏼡. (24) zI λ zI 1 2 (b) the axial displacement values of the identical points of the elastic tumor at the same moments, D As explicated in the previous sections, the displacement (c) the relation [47–49] ﬁeld inside the in vivo tumor could be measured in a non- D D invasive way. With the purpose of estimating Mooney– E � . (27) realt Rivlin hyperelastic parameters of the tumor just by using the D Y realt displacement quantities, we manipulate (25), that is, the stress-stretch relation of the hyperelastic model, (7) Specify a set of strains, ε, and compute the corre- 2 −1 −1 spondent stresses, σ, using (28) in accordance with σ � 2􏼐λ − λ 􏼑􏼐C + λ C 􏼑. (25) 10 01 (in the ﬁrst iteration): ,e uniaxial load applied to the medium in alignment (a) ,e strain ﬁeld in the tumor could be roughly with the ﬁrst coordinate axis practically generates the axial approximated from the displacement estimates. strain in the same direction; therefore, the strain and relation (b) ,e achieved results imply the selection of a set between the stress and strain could be expressed as [47, 48] of high strains in the ﬁrst iteration since the ε � λ− 1, strain values are being modiﬁed in some steps of (26) 2 −1 −1 the proposed iterative algorithm; therefore, the σ � 2􏼐(ε + 1) −(ε + 1) 􏼑􏼐C + (ε + 1) C 􏼑. 10 01 strain values could be reduced uniformly. (c) Slight changes in the stress values, for instance With respect to the stress-strain relation, (26), the iterative by the use of the normal distribution function, algorithm propounded for estimating Mooney–Rivlin might cause the stress and strain values to hyperelastic parameters of a completely unknown interior conform more eﬀectively to the Mooney–Rivlin tissue (i.e., tumor) could be described in the following steps: hyperelastic model assigned to the tumor, (1) Image the tissue and its adjacent mediums by the σ � E ε. (28) realt use of clinical US imaging system before/while applying a sinusoidal load with low frequency (to (8) Compute hyperelastic parameters of the Mooney– annul the inertia) to the exterior medium. In other words, record the relevant US RF signals or images. Rivlin model, C and C , using the stress and 10 01 6 Journal of Healthcare Engineering strain sets and the relation between stress and 3. Results strain, as represented, 2 −1 −1 3.1. Soft Tissue Simulation. ,e breast tissue (with the di- σ � 2􏼐(ε + 1) −(ε + 1) 􏼑􏼐C + (ε + 1) C 􏼑, 10 01 3 mensions of 100 × 60 × 20 mm ), simulated using the FEM (29) software ABAQUS, has been depicted in Figure 1. ,e breast tissue consists of three partitions, namely, fat, ﬁbroglandular, with the help of MATLAB algorithms, for instance, and tumor. ,e Mooney–Rivlin hyperelastic model has been the regression algorithms. applied to obtain the response of simulated breast tissue to the (9) Assign the estimated hyperelastic parameters to the external sinusoidal load. ,e Mooney–Rivlin material constants simulated tumor and compute the displacement of the named breast tissues have been presented in Table 1. Since ﬁelds inside the tumor at the determinate step times we have utilized the elastic parameter of the tumor for esti- by means of the FEM software. mating its hyperelastic parameters, we have additionally re- (10) Calculate the elastic constant of the simulated tu- ported the elastic parameters of the named breast tissues in mor, E , as explained previously in step 6, using Table 1. ,e linear and nonlinear elastic parameters have been estt the axial displacement quantities (of the appointed supposed to be constant throughout each tissue partition. ,is points at the selected step times) of the simulated set of hyperelastic parameters has been broadly utilized in the tumor, Y , and the elastic tumor, D (determined literature to simulate the breast tissue [38, 50–54]. estt in step 6), by applying the relation ,e mesh considered for the simulated phantom consists of 183783 second-order (quadratic) tetrahedral hybrid ele- D D E � . (30) estt ments (C3D10H) with 259813 nodes. ,e convergence ana- D Y estt lyses have warranted the accuracy of the simulation results. With reference to the explanations in Section 2.3, the number (11) Appraise the estimated hyperelastic parameters for of nodes in the simulated medium should signiﬁcantly be the tumor through considering the following: increased to precisely calculate the positions of scatterers after (a) ,e error of the computed axial displacement applying the load to the medium. With regard to the boundary values of the selected points at the speciﬁed conditions and the load applied to the tissue (represented in moments, Y , by comparing them with the estt Figure 1), the postcompression RF signals and B-mode images correspondent displacement quantities esti- have been simulated in the probe coordinate system. Two mated from the recorded US RF signals or snapshots of the response of the simulated breast tissue to the images, Y sinusoidal load have been demonstrated in Figure 2. realt (b) ,e error of the elastic constant calculated for the tumor, E , by comparing it with the real estt elastic modulus of the understudy tumor, E , realt 3.2. Simulation of US RF Signals and Images. ,e RF signals estimated in step 6 and B-mode images of part (with the dimensions of 50 × 60 × 10 mm ) of the simulated breast tissue which en- ,e errors of the hyperelastic parameters estimated circles the tumor, as illustrated in Figure 3, have been for the tumor, by comparing them with the real simulated using the Field II US Simulation Program. In the hyperelastic parameters of the tumor, could not be Field II US Simulation Program, considered because it has been assumed that the tumor is entirely obscure. (a) ,e properties considered to model the probe array (12) Alter the set of strains speciﬁed in step 7 (based on and simulate the US RF signals and images are as the above explanation, decrease them regularly) follows: and repeat steps 7 to 12. By reducing the strain (1) Linear array (with 64 active elements) values steadily, the error of the estimated elastic (2) Transducer center frequency of 3.5 × 10 Hz parameter for the tumor and the error of the (3) Sampling frequency of 100 × 10 Hz calculated displacement quantities in the tumor (4) Transmit focus of 70 mm (in depth) are decreasing below the deﬁned tolerance values, (5) Element’s width (the distance between the ele- as illustrated, ments or the pitch of the probe array) of 0.44 mm � � � � � � (equal to the wavelength) � � Y − Y ≤ e , � realt� displacement estt (6) Element’s height of 5 mm � � (31) � � � k � � � E − E ≤ e , (7) Element’s kerf of 0.05 mm � � estt realt elastic (8) Lateral spatial spacing of 0.08 mm (512 scan lines where e and e are the tolerance values in the image) elastic displacement and k represents the number of iterations of the (b) With regard to the elastic and hyperelastic param- algorithm. By decreasing the strain values chosen eters of the tumor, for scatterers which have resided with regard to the mentioned conditions, the strain within the tumor, the amplitudes are set to zero. and stress values successively adjust more to the Mooney–Rivlin stress-strain relationship of the ,e postdeformation RF signals and B-mode images in tumor. eight deformation states of the phantom (corresponding to Journal of Healthcare Engineering 7 Z X (a) (b) Figure 1: �e simulated breast tissue comprises three parts, namely, fat, �broglandular, and tumor (from outside to inside). Table 1: �e elastic and Mooney–Rivlin hyperelastic constants of 4. Discussion breast tissues [38]. Hyperelastic and In the majority of diversi�ed approaches pro’ered for esti- Fat Fibroglandular Tumor elastic parameters mating elastic parameters of soft tissues, particularly the C (Pa) 2000 3500 10000 nonlinear ones, for instance, the techniques proposed by C (Pa) 1333 2333.3 6667 MacManus et al. [55], Esmaeili et al. [56], Omidi et al. [57], E (kPa) 20 35 100 Roy and Desai [58], Liu et al. [59], Boonvisut and Çavu¸soglu [60], and Wang et al. [61], to mention but a few, the alter- ations (of precise values) of at least two deformation variables, eight sequential step times: 7.75 s, 8.00 s, 8.25 s, 8.50 s, 8.75 s, which are associated with the mechanical characteristics of soft tissues, have been exploited. �e assessment of recom- 9.00 s, 9.25 s, and 9.50 s, while responding to the sinusoidal load) have been simulated using the displacements of the mended techniques would reveal that the direct dependencies phantom’s nodes computed by the �nite deformation anal- of methodologies to deformation variables except the dis- ysis. �e nodal displacement values correlated with a partic- placement (and strain) have impelled the researchers to carry ular deformation state have been linearly interpolated to out experiments on ex vivo tissues, or perform invasive compute the positions of scatterers and simulate the corre- procedures to precisely measure the variables; consequently, spondent RF signals and B-mode images thereafter. Two the emphasis of recent investigations should be on advancing simulated postdeformation B-mode images (pre- and post- noninvasive methods with the capability to accurately esti- deformation images) associated with the step times of 8.00 s mate nonlinear elastic parameters of tissues. �e hyperelastic constitutive theory takes two types of and 9.00 s have been represented in Figure 3. nonlinearities perceived in responses of soft tissues, into consideration [62, 63]: 3.3. Estimation of Hyperelastic Parameters. After simulating (a) �e material nonlinearity of the stress-strain re- the pre- and postdeformation RF signals and B-mode images lation, known as the physical nonlinearity correlated with the de�ned deformation states, the cross- (b) �e nonlinearity of the strain-displacement relation, correlation algorithm with guided search, as brie…y de- called the geometrical nonlinearity scribed in Section 2.4, has been employed to initially esti- mate displacement quantities inside the tumor at the consequently, it has been regarded as one of the best selected step times and afterwards evaluate the errors of practical theories for formulating mechanical behaviors of displacement estimates. soft tissues. To the best of our knowledge, amongst the �e suggested iterative algorithm, comprehensively ex- strategies proposed for quantifying hyperelastic parameters plicated in Section 2.5, has been applied to extract the of materials, two methods founded on the displacement Mooney–Rivlin hyperelastic parameters of the tumor from �elds inside and on the boundary of the medium (i.e., the axial displacement values of some points of tumor at the phantoms) which have been introduced by Mehrabian and speci�ed step times. As represented in Table 2, precise es- Samani [26, 27, 64] and Hajhashemkhani and Hematiyan timates of hyperelastic parameters of the tumor have been [47, 48], respectively, could be utilized to noninvasively achieved. �e automatic iteration of the algorithm would be reconstruct hyperelastic parameters of in vivo tissues. �e feasible through bilaterally connecting the MATLAB and displacement �eld inside the understudy medium could be FEM softwares. extracted from RF signals or images recorded by a clinical 8 Journal of Healthcare Engineering Y Y Z X Z X (a) (b) Figure 2: �e responses of the simulated breast tissue at (a) t 4.00 s and (b) t 8.00 s after starting to apply the sinusoidal load. Predeformation US image Postdeformation US image Predeformation US image Postdeformation US image 35 35 35 35 120 120 40 40 40 40 45 45 45 45 100 100 50 50 50 50 55 55 55 55 80 80 60 60 60 60 60 60 65 65 65 65 70 70 70 70 40 40 75 75 75 75 80 80 80 80 20 20 85 85 85 85 90 90 90 90 –20 –10 0 10 20 –20 –10 0 10 20 –20 –10 0 10 20 –20 –10 0 10 20 Lateral distance (mm) Lateral distance (mm) Lateral distance (mm) Lateral distance (mm) (a) (b) Figure 3: �e simulated pre- and postdeformation B-mode images of breast phantom, based on its states at (a) t 8.00 s and (b) t 9.00 s after starting to apply the sinusoidal load. Table 2: �e elastic and hyperelastic parameters estimated for the correlates the stress distribution computed for the tissue (with tumor. the help of a �nite element model of the tissue deformation) to its hyperelastic parameters. �e displacement values at some Estimated elastic and hyperelastic parameters boundary points of the understudy medium have been ma- 2 estimates nipulated by Hajhashemkhani and Hematiyan [47, 48] for E (kPa) 88908.41 realt characterizing its nonlinear material constants. Error of E (%) 11.09 realt On account of the explanations provided by Mehrabian C (Pa) 9426.98 10005.05 and Samani [26, 27, 64] and Hajhashemkhani and Hema- C (Pa) 6702.30 5992.82 tiyan [47, 48] and the results achieved through the imple- Error of C (%) 5.73 0.05 mentation of their methods (part of them published in our Error of C (%) 0.53 10.11 paper [65]), it has been realized that precise estimates of E (kPa) 88711.93 88658.27 estt Error of E (%) 0.22 0.28 hyperelastic parameters of the understudy medium could be estt Error of Y (%) 0.28 0.35 attained through the following: estt (a) Accurately calculating the displacement quantities, respectively, in a multitude of adjacent points of the US imaging system, for instance, by using the conventional medium and in several boundary points, which cross-correlation method. might not be possible using registered US images or In the technique recommended by Mehrabian and RF signals Samani [26, 27, 64], the displacement quantities of a large number of contiguous points inside the medium should be (b) Applying proper regularization techniques, for in- utilized to calculate the de�ned coe¦cient matrix which stance, the Tikhonov regularization, Truncated Axial distance (mm) Axial distance (mm) Axial distance (mm) Axial distance (mm) Journal of Healthcare Engineering 9 Table 3: ,e hyperelastic parameters estimated for the tumor using Singular Value Decomposition (SVD), and Wiener inaccurate displacement measurements. Filtering methods (c) Even considering appropriate initial guesses of the Inaccurate displacement measurements Estimated hyperelastic parameters, as indicated by Hajha- hyperelastic Error Error Error Error shemkhani and Hematiyan [47, 48], Aghajani et al. parameters 2% 5% 8% 10% [66], and Kim and Srinivasan [67] C (Pa) 9386.78 9484.46 9815.78 9858.99 C (Pa) 6889.50 6817.80 6437.30 6465.64 ,e limitations of the techniques propounded with the Error of C (%) 6.13 5.16 1.84 1.41 aim of reconstructing nonlinear elastic parameters of in vivo Error of C (%) 3.34 2.26 3.45 3.02 soft tissues persuade us to concentrate on developing a more Error of E (%) 0.53 0.39 1.89 2.29 estt practical method with consistent results. Our primary up- Error of Y (%) 1.77 4.59 6.28 7.96 estt shot represented in Section 3 conﬁrms that accurate esti- mates of hyperelastic parameters of the understudy tissue (i.e., tumor) could be obtained on the basis of the dis- calculated for the tumor, precise estimates of tumor’s placement values of some points inside the tissue, which has hyperelastic parameters could be obtained. With regard been excited by a low frequency sinusoidal load, even when to the outcomes summarized in Table 3, it is deduced that there is no prior knowledge of the tissue. the suggested method is strongly resistant to the dis- ,e displacement quantities of the selected points might placement errors. be computed approximately, for instance, by the use of the ,e US RF signals recorded by means of the Antares cross-correlation technique as a consequence of recording Siemens system (Issaquah, WA) at the center frequency of low-quality US images or RF signals or other attributes; 6.67 MHz from an elastography phantom (CIRS elastog- therefore, we have evaluated the consistency of calculated raphy phantom, Norfolk, VA) have been utilized to values for the hyperelastic constants by applying errors with evaluate the suggested method experimentally. ,e signals normal distribution to the measured displacement ﬁelds in were registered via a VF10-5 linear array at a sampling rate the tissue stimulated by the sinusoidal load. At this point, the of 40 MHz by Rivaz et al. to assess the performance of the average errors of the displacement values estimated for the proposed real-time static elastography techniques which selected points from the simulated US RF signals and images were based on the analytic minimization of regularized using the cross-correlation algorithms (without/with guided cost functions. Young’s moduli of the lesion and sur- search with respect to the calculated displacements of rounding medium have been reported 56 kPa and 33 kPa, previous lines or samples) have been regarded. ,e achieved respectively, while the phantom is under compression results have been demonstrated in Table 3. [68, 69]. It should be considered that the imprecise estimates of ,e enhanced cross-correlation algorithm, in that the displacement ﬁelds inside the tumor aﬀect all the computed search regions were minimized with respect to the estimated parameters and errors, even the real elastic modulus of the displacements of previous lines or samples, has been understudy tumor; therefore, the results presented in Table 3 employed to compute the axial displacement ﬁeld in the could not be compared. Similar to the case where the dis- compressed phantom. ,e Kalman ﬁltering, introduced by placement values of the appointed points are exact, Rivaz et al. [68], has been applied to calculate the strain ﬁeld in the compressed phantom from the displacement mea- (a) the error of calculated axial displacement values of surements. Minor diﬀerences between the displacement the selected points at the speciﬁed moments, Y estt ﬁelds estimated by the use of the enhanced cross-correlation (b) the error of elastic constant computed for the tumor, algorithm and analytic minimization method validate the estt results of the former technique. (as explained in step 11 of the proposed algorithm in Section ,e US images of the phantom constructed from the recorded RF signals and the estimated displacement and 2.5) have been considered except in the situations where the displacement errors are signiﬁcant. strain ﬁelds have been represented in Figure 4. Following the ,e convergence of the aforementioned errors to values, instructions in Section 2.5, the elastic and hyperelastic pa- which might not be small errors, speciﬁes the best estimates rameters of the lesion could be calculated from the estimated of hyperelastic parameters when the displacement values are axial displacement and strain ﬁelds in the lesion. ,e percent error of the elastic parameter computed for the lesion, on the highly inaccurate, as represented, � � � � basis of the explanations in step 6 of the algorithm, is 15.06%; � k k−1� � � ′ Y − Y ≤ e , � � estt estt displacement in other words, E has been estimated 47565.72 Pa. ,e realt � � (32) � � k k−1 � � relation between stress, strain, and the parameters of the � � ′ E − E ≤ e , � � elastic estt estt Mooney–Rivlin hyperelastic model, C and C , has been 10 01 manipulated to compute the hyperelastic parameters of the ′ ′ where e , e , and k are, respectively, the lesion. ,e values of 6871.65 Pa and 1020.00 Pa have been elastic displacement speciﬁed tolerance values and the number of iterations of obtained for the mentioned parameters. the algorithm. Provided that the set of strains (required On the basis of the achieved results summarized in in step 7 of the algorithm described in Section 2.5) is Sections 3 and 4, it is concluded that the main objectives that selected properly based on the displacement ﬁelds have been accomplished in this paper are as follows: 10 Journal of Healthcare Engineering Ultrasound image 1 Ultrasound image 2 �e displacement �elds inside the tissue could be 0 0 noninvasively computed from the data recorded by the 200 –0.5 200 –0.5 employment of conventional medical imaging modalities, –1 –1 400 400 for instance, the RF signals or images registered by the US –1.5 –1.5 600 600 imaging system. Even by processing approximate dis- –2 –2 800 800 –2.5 –2.5 placement measurements, accurate estimates of the material 1000 1000 –3 –3 constants could be obtained. �e competency of the pro- 1200 1200 –3.5 –3.5 posed method to estimate nonlinear elastic constants of 1400 –4 1400 –4 normal and abnormal in vivo tissues will be further ap- –4.5 –4.5 1600 1600 praised in the future research. 100 200 300 400 100 200 300 400 RF lines RF lines (a) Data Availability Axial strain Axial displacement 0 –0.02 Since the research is still in progress, the authors have –10 200 5 –0.03 decided to make data available upon request. –20 –0.04 –30 –0.05 –40 Disclosure –50 –0.06 –60 20 –0.07 �e funding had no role in the study design, data collection, –70 –0.08 25 or analysis. –80 –0.09 –90 –100 –0.1 0 20 40 60 100 200 300 400 500 Conflicts of Interest RF lines Width (mm) �e authors declare that there are no con…icts of interest (b) regarding the publication of this article. 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An Iterative Method for Estimating Nonlinear Elastic Constants of Tumor in Soft Tissue from Approximate Displacement Measurements

An Iterative Method for Estimating Nonlinear Elastic Constants of Tumor in Soft Tissue from Approximate Displacement Measurements

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An Iterative Method for Estimating Nonlinear Elastic Constants of Tumor in Soft Tissue from Approximate Displacement Measurements

An Iterative Method for Estimating Nonlinear Elastic Constants of Tumor in Soft Tissue from Approximate Displacement Measurements

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