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Diabetes mellitus is a group of metabolic diseases caused by malfunction of blood sugar regulatory processes and has been reported as related to 8.3% of adult population, i.e. nearly 400 million people worldwide. This paper provides a review of facts and principles important for understanding the regulation mechanisms and the role of insulin. The author relies on mathematical modeling of these mechanisms and provides few formulas and computer applications dedicated for use in diabetes. The modeling aims to find a correct dose of insulin as a response to a series of measurement results on glucose concentration. In conclusion, the author recommends selected methods for personal self-check of glucose level and stresses on the importance of regularly checking blood-related parameters. Keywords: diabetes; glucose; insulin. control system with negative feedback, where the system regulates blood glucose level . A healthy person has 70110 mg/dL glucose concentration in his or her blood. The glucose-insulin system is described in Figure 1. When the person eats a meal, the concentration of glucose increases. To stop this process, the pancreas produces insulin. During exercise, the glucose expansion is low; hence, a signal is sent to the pancreas to release glucagon. It is a simplistic way of explaining the metabolism, which will be presented in a mathematical model in this paper . Long-term consequences of hyperglycemia lead to a severe decrease in health status. The complications are very serious and expensive . 2 Testing Diabetes is the main reason that many mathematical models have been made over time to describe the glucoseinsulin dynamic system. The models and tests can help to improve the situation of diabetes. There are two kinds of tests that help one of the presented mathematical models (Bergman's minimal model). In the oral glucose tolerance test (OGTT), the patient fasts for an 8-h period, after which glucose and insulin levels in blood are measured. Then the patient ingests glucose in liquid solution, and after it is taken, new measurements are recorded for a 3-h period. The third test is very simple the fasting glucose concentration measurement. There is another kind of test, the intravenous glucose tolerance test (IVGTT) with a mathematical model, which can be used to estimate insulin sensitivity. This test procedure begins with an injection of a glucose bolus, containing 0.30 g glucose per kilogram of bodyweight. Then blood samples are collected every 3 h. These blood samples are analyzed. Then glucose and insulin levels in blood are measured. The third test is very easy. The patient has to fast for a period of 810 h. Then the measurement of the glucose is made. The results are shown in Table 1. 1 Introduction Diabetes is one of the most common diseases on the world, and the morbidity is growing. Diabetes is one the etiopathogenetically and clinically heterogenic group of the metabolic disorders due to impaired insulin production by pancreas or impaired insulin action or both. One of the main reasons is unhealthy food. This is a large problem. That is why many researchers try to find methods for diagnosing and treating the disease. One of the approaches is to design a mathematical model describing the glucose-insulin system. This disease is due to the improper operation of the pancreas, which results in incorrect glucose metabolism. The glucose-insulin system is a closed-loop physiological system. It is another example of a physiological *Corresponding author: Agnieszka wierkosz, AGH University of Science and Technology, 30 Mickiewicza Ave, 30-059 Cracow, Poland, E-mail: email@example.com 206wierkosz: Modeling of metabolic diseases a review of selected methods Figure 1:The blood glucose-insulin system . Table 1:Glucose level in blood . From 70 to 99 mg/dL From 100 to 125 mg/dL More than 126 mg/dL Normal glucose tolerance Prediabetes Diabetes is doing much exercise in the morning or gets too much insulin dosage. In this state, a diabetic can even lose consciousness. It is very important to avoid hypoglycemia . 3 Diabetes Many people are suffering from diabetes, and there will be more. The main problem in people with diabetes is the dysfunction in their blood glucose-insulin system. The common forms of diabetes are type 1 and type 2. In type 1 diabetes, the cells are destroyed by the autoimmune reaction in the body. This results in very low insulin production. At this production level, the insulin cannot decrease glucose in blood fast enough. In type 2 diabetes, which is the most common type of diabetes, insulin is produced sometimes like in normal people . 5 Model proposed by Stolwijk and Hardy in 1974 The total volume of blood and interstitial fluids is represented by a single large compartment (~15 L in a normal adult), and the steady-state concentration of glucose is x (mg mL-1). For x to remain constant, the total inflow of glucose must equal the total outflow. Glucose enters the blood through absorption from the gastrointestinal tract or through production from the liver. This input flow rate will be QL (mg h-1). There are three major ways through which glucose is eliminated from the blood : 1. When x is elevated beyond a certain threshold (), glucose is excreted by the kidneys: Renal loss rate = µ(x - ), = 0, x 2. x > (1) (2) 4 Hyperglycemia and hypoglycemia A patient has hyperglycemia when his or her blood glucose level is higher than 270 mg/dL. This can exist when a diabetic eats a large meal or has a low level of insulin in his or her blood. That state is very dangerous if not treated. A patient has hypoglycemia when his or her glucose level is lower than 60 mg/dL. This can happen when someone Glucose leaves the blood to enter most cells through facilitated diffusion: (3) Tissue utilization rate (insulin independent) = x 3. In certain types of cells (e.g. muscle), insulin helps to stimulate this facilitated diffusion process. The rate wierkosz: Modeling of metabolic diseases a review of selected methods207 at which glucose is taken up by these cells is proportional to x and to the blood insulin concentration, y: Tissue utilization rate (insulin dependent) = vxy (4) 6.1 The glucose minimal model The glucose minimal model describes how the glucose level behaves according to measured insulin data during an IVGTT. This model is divided into two parts. The first part is the main part. It describes the glucose clearance and uptake. The second part describes the delay in the active insulin I2, which is a remote interactor. Its level affects the uptake of glucose by the tissues and the uptake and production by the liver . These two parts are described in the following equations: dG( t ) =-( p1 + X ( t ))G( t ) + p1Gb dt dX ( t ) =- p2 X ( t ) + p3 ( I ( t ) - I b ) dt G( 0 ) = G0 X ( 0 ) =0 (9a) (9b) In these equations, , , and are constant proportionality factors. Equating the inflow to the sum of the three outflows, one can obtain the following mass balance equations for blood glucose: QL = x + vxy , x = x + vxy + µ( x - ) x > (5a) (5b) Insulin is produced by the pancreas at a rate dependent on the plasma glucose level . If x falls below a certain threshold (Ø), insulin production ceases: Insulin production rate = 0, x Ø = ( x - Ø), x > Ø (6a) (6b) Insulin is destroyed through a reaction involving the insulinase enzyme, at a rate proportional to its concentration in blood: Insulin destruction rate = y (7) The meaning of these equations is to show how they are derived. The derivation is based on the model by Steil et al.  and the rule of mass balances : accumulated = in- out + generated - consumed In the derivation of this model, the following parameters will be used : Parameter t G(t) Gb I2(t) X(t) I(t) Ib VG VI 2 QG1 QG2 QI 1 2 QI 2 2 w1 w2 Unit min mg/dL mg/dL mU/L 1/min mU/L mU/L Description Time Blood glucose concentration Steady-state blood glucose concentration (baseline) Active insulin concentration The effect of active insulin Blood insulin concentration Steady-state blood insulin concentration (baseline) Volume of the glucose compartment Volume of the remote pool Flow Flow Flow Flow Effect factor Effect factor Combining Equations (5) and (6), we obtain the following equations relating the steady-state level of y to x: y = 0, ( x - Ø), x Ø x >Ø (8a) (8b) The parameter values employed in this calculation correspond to the normal adult: =2.5 mg mL-1, =7200 mL h-1, =2470 mL h-1, =139,000 m U-1 h-1, Ø=0.51 mg mL-1, =mU mL mg-1 h-1, =7600 mL h-1, and QL=8400 mg h-1. This model is used to predict the steady-state operating levels of glucose and insulin, which is very important to diabetics . dL L dL/min dL/min L/min L/min dL2/min×mU dL2/min×mU 6 Bergman's minimal model There are many models to describe glucose-insulin system. Sometimes the simplest one is good enough and in many times the best. A simple method with few parameters was introduced in the 1980s by Richard R. Bergman and is called Bergman's minimal model . This model has been modified many times. The derivation of this model is presented by Friis-Jensen . 6.2 The insulin minimal model The model of glucose kinetics as a product of data insulin input is described. Bergman et al.  presented the minimal model of insulin kinetics: 208wierkosz: Modeling of metabolic diseases a review of selected methods dI ( t ) = p6 [ G( t ) - p5 ] + t - p4 [ I ( t ) - I b ], I ( 0 ) = I0 dt The parameters used in this model are as follows : Parameter I(t) Ib G(t) p5 VI QI (10) y(k ) = f (u(k - ), ..., u(k - nB ), y(u(k - 1), ..., u(k - nA )) n +n -+ 1 (11) Unit mU/L mU/L mg/dL mg/dL L L/min mUdL mgmin Description Blood insulin concentration Based blood insulin concentration Blood glucose concentration Threshold for blood glucose concentration Volume of insulin distribution pool Flow Flow where f : A B ,, nB . A feedforward MLP neural network with one hidden layer and linear output is used as the function f in Equation (11). The structure of the neural model is shown in Figure 2. The output of the model can be obtained as follows: 2 y( k ) = w0 + wi2 ( zi ( k )) i= 1 k (12) QI 7 Model predictive control (MPC) To describe diabetes, one can use several algorithms. The selection of the adequate process model is a major problem. The published control strategies, especially predictive control algorithms, were designed on the basis of a linear black box model of insulin-glucose dynamics . For example, Bergman's minimal model represents the dynamics of insulin and glucose concentration as well as the dynamics of insulin actions. This model has some drawbacks. The model is based on the assumption that the plasma glucose and insulin compartments are independent and can be identified independently. From the mathematical point of view, the minimal model can produce physiologically unreliable results . This section presents neural and fuzzy models for numerically efficient nonlinear predictive control of insulin dosage. For example, Havorka's model represents the input-output relationship between subcutaneous insulin infusion and intravenous glucose concentration. From the perspective of control algorithm, the insulin dose is the input of the whole dynamic system, and the glucose concentration is output of the system. Model predictive control (MPC) is an advanced control technique. MPC algorithms can take into account constraints imposed on both process inputs and outputs. Such control algorithms must solve a set of nonlinear differential equations comprising the model at each sampling instant online. It is computationally inefficient, and numerical problems are unavoidable . where zi(k) are sums of inputs of the ith hidden node, : is the nonlinear transfer function, and k is the number of hidden nodes. Recalling the input arguments of the general neuron model (Equation 11), one has the following: zi ( k ) = wi1,0 + wi1,j u( k - + 1- j ) j=1 IU (13) where Iu=nB+1. Weights of the network are denoted by wi1,j , j=1, ..., K, j=0, ..., nA+nB+1 and wi2 , I=0, ..., K, for the first and the second layers, respectively. Combining Equations (12) and (13), one can obtain the following : IU nA K 2 y( k ) = w0 + wi2 wi1,0 + wi1,j u( k - + 1- j ) + wi1,I + j y( k - j ) U i= 1 j=1 j=1 (14) 7.2 Fuzzy model The Takagi-Sugeno (TS) model is relatively easy to obtain. The wanted function can be chosen using expert knowledge, simulation experiments, or fuzzy neural networks. This model is described by following rules: 7.1 Neural network The single-input, single-output (SISO) neural model is described by the following nonlinear discrete-time equation: Figure 2:The structure of the neural model . wierkosz: Modeling of metabolic diseases a review of selected methods209 If yky is B1 y and ... and yky-n and uku is C1 j f , ju f ,j +1 j is Bn f , jy and ... f , ju and ... and uku-m +1 is Cm d d ,f m j m then ýkj +1 = aij ,m , f uk-i + a p,m , f uk- p + C j , f m=1 i=1 nu pd -1 (15) th where yky is a value of the jy output variable at the kth ju th sampling instant; uk is a value of the ju manipulated f ,j f ,j variable at the kth sampling instant; B1 y ,..., Bn y , P f , ju f , ju C1 ,...,Cm , are fuzzy sets; aij ,m , f , (i=1, ..., pd) is the P coefficient of step responses in the fth local model describing the influence of the mth input on the jth output; pd is equal to the number of sampling instants after which the coefficients of the step responses can be assumed as settled; cj,f is the constant value, jy=1, ..., ny, ju=1, ..., nu, f=1, ..., l; and l is the number of rules. For current sampling instant using current values of process variables, TS, and fuzzy reasoning, the following model is obtained: m y kj = a ij ,muk-i + a p,muk- p + c j ´ ´ ´j m ´ m=1 i=1 d d discussed. Section 5 is a mathematical description of the relationship between insulin and sugar in the body. It is based on physiological negative feedback control system. Section 6 describes the mathematical model that can be used to diagnose and to create simulators for different kinds of diabetes mellitus treatment. In Section 7, a model that can help in sugar self-monitoring is presented. It helps to provide, for example, how much insulin we have to dose. Self-monitoring blood glucose reduces the risk of serious secondary clinical complications (for the role and history of blood glucose meters, see Clarke and Foster ). Patients can measure their blood glucose level using a meter. They have to put a drop of blood on a test strip. This test can detect glucose level. Then they have to follow their physician's recommendations. However, the patients have to measure their glucose level still. This treatment is called self-monitoring blood glucose . Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission. Research funding: This study was supported by the AGH University of Science and Technology (grant/award no. 18.104.22.1688). Employment or leadership: None declared. Honorarium: None declared. Competing interests: The funding organization(s) played no role in the study design; in the collection, analysis, and interpretation of data; in the writing of the report; or in the decision to submit the report for publication. nu pd -1 (16) where ýkj is the jth output of the control plant model at the and f is the normalized weight. ^ ^ ´ ´ kth sampling instant, a ij ,m = f =1 w f aij ,m , f, c j = f =1 w f cij , f 8 Conclusions This paper is a review of several glucose-insulin models. It shows the importance of correct metabolism: on it depends if the patient is ill or healthy. When the glucose level is too high, there is a need to produce insulin. When the patient is ill, this relation is disrupted. The patient's organisms cannot produce the proper level of insulin. He needs to take it from outside. Diabetes complications such as angina, heart failure, amputation, and many others can be really bad. There are many options to keep organisms in homeostasis. The patient needs to choose one of the following models of glucose regulation: 1. Diagnosis 2. Analysis of the glucose level in blood 3. When glucose is too high, there is a need to dose insulin 4. Constant observation of glucose-insulin relationship and consult the physician responsible for the treatment In Section 2, the way of checking if we are ill or not is described. In Sections 3 and 4, diabetes mellitus is
Bio-Algorithms and Med-Systems – de Gruyter
Published: Dec 1, 2015
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