Abstract Enhanced Parameter Identification for Complex Biomedical Models on the Basis of Fu

Enhanced Parameter Identification for Complex Biomedical Modelson the Basis of Fuzzy ArithmeticMichael Hanss,Oliver NehlsInstitute A of Mechanics,University of StuttgartStuttgart,GermanyM.Hanss,O.Nehls@mecha.uni-stuttgart.deAbstractWith the objective of future improvement of medical therapy,a method for the improvement of parameter identification for complex biomedical models on the ba-sis of fuzzy arithmetic is presented.The model param-eters to be identified are considered as uncertain,and are represented by fuzzy numbers with their member-ship functions quantifying an initial guess for the ac-tual value of the model ing the transfor-mation method as a special implementation of fuzzy arithmetic,the model can be analyzed with the inten-tion of determining the influence of each parameter on the variation of the overall model output.Finally,the parameter identification can be improved by reducing the high-dimensional identification procedure to a num-ber of lower-dimensional optimization problems with measurement data taken from those time intervals only where the parameters show significant influence.The enhanced performance of the model with the newly identified parameters is proved by a higher conformity of its predictions with reality.1.IntroductionIn order to improve medical therapy and to develop op-timal medication for human deseases which are caused by some physical deficiency,mathematical modeling of the biomedical systems has proved to be a very success-ful tool.By carrying out simulations of the mathemati-cal model,the human reactions on a special medication can be studied under various conditions.As an exam-ple which has been of great interest in recent years,di-abetes,or strictly speaking diabetes mellitus type I,is considered in the present paper.The effects of this dis-ease to the patients’everyday life are usually quite seri-ous,ranging from regular medication with injections up to being imperiled by bad risk of heart attacks. However,modeling of biomedical systems turns out to be a non-trivial problem since the models are usually rather complex,nonlinear and characterized by a huge amount of parameters which have to be identified from scratch for each single patient.The proper solution of this high-dimensional nonlinear optimization problem of identifying the model parameters can be considered as the crucial part of the entire modeling procedure.It does not only require plenty of computational effort,it very often leads to results which are reasonable only from a numerical point of view,but have no relation to the actual physical realities.The major reason for this failure of the global optimization problem can be seen in its high dimension and in the fact that the whole set of measured input-output data is utilized in an undiffer-entiated way to identify all model parameter at once. To avoid this effect,the model of the biomedical system has to be analyzedfirst with respect to a quantification of the influence of each model parameter on the mea-sured model output.By this,the time range for the sim-ulation of the model can be split into different phases where in each phase only a subset of the model param-eters shows major influence on the model output,and the significance of others can be considered as negligi-ble.The advantages of this way of proceeding are that on the one hand the overall high-dimensional identifi-cation procedure can be reduced to a number of lower-dimensional and much faster to solve optimization prob-lems.On the other hand,realistic values for the model parameters can be achieved since in the reduced opti-mization problem only measurement data from those specific time intervals are used for the identification of the model parameters where the parameters show sig-nificant influence.For the purpose of analyzing the physiological model, which is introduced in Section2of the paper,the param-eters to be identified are considered as uncertain,repre-sented by fuzzy numbers with their peak value given by some initial guesses.A special implementation of fuzzy arithmetic,namely the transformation method[5,6], is then applied to analyze the uncertain model.This method is outlined in Section3of the paper.It can be seen as an advanced and extended version of the so called vertex method[2,3]with the major advantage of not showing any overestimation effect[4]which usually occurs in conventional implementations of fuzzy arith-metic[9].As a part of this method,the evaluation of a measure of influence for each model parameter is pro-vided which allows practical analysis of the model even in the presence of a large number of model parameters. The results of the analysis of the model are presented Section4,followed by Section5where the results for the original,high-dimensional identification procedure on the one hand,and the enhanced model identification after fuzzy arithmetical analysis on the other hand,are compared for the model of human glucose metabolism.2.Physiological ModelBasically,the overall model of the human glucose metabolism for patients with diabetes mellitus type Ican be split into three parts:First,the model for the in-flow of insulin into the blood in consequence of the subcutaneous injected external insulin.Second,themodel for the inflow of glucose into the blood in consequence of the ingested food,which can again be subdivided into two parts representing the metabolisms in the stomach on the one side,and in the intestine on the other(Figure1).Finally,the outputs of the models are combined in a third model to predict the amount of in-blood glucose.For reasons of clearness,only the basic equations of the models are stated in the en-suing.Detailed listings of the actual parameter settings can be found in[7]or[8].Model for the inflow of insulin into the blood [10]After injection,insulin appears in two modifications in the subcutaneous depot which can be described by a hemisphere with the radial coordinate:as dimere in-sulin with the concentration and as hexamere insulin with the concentration.The uptake of insulin into the blood is only affected by dimere insulin. The injected external insulin,however,is a solution of pure hexamere insulin.(2)(3) with the operator(6)and(8)and the parameters,,,,and.The input parameters,and denote the amount of carbohydrates,proteins and fat in the ingestedmeal.(11)(12)with the initial and boundary conditions(16)where reflects the degree of refinement of the dis-cretization(see Figure2).Thus,the fuzzy numbers canbe represented by a set of intervals,,of the formwith(17)(18)(19)Instead of applying standard interval arithmetic directlyto the intervals,,for each level ofmembership,,the intervals are nowtransformed into arrays of the form(20)with(21)Assuming the problem to be given by the arithmeticalexpression in the form(22)its evaluation is then carried out by evaluating the ex-pression separately at each of the positions of thearrays using the conventional arithmetic for crisp num-bers.Thus,if the result of the problem can be ex-pressed in its decomposed and transformed form by thearrays,,the-th element of the array is then given by(23) where denotes the-th element of the array. Finally,if simulation of the uncertain problem is the im-mediate object,the fuzzy-valued result of the problem can be achieved in its decomposed form(24)by retransforming the arrays–including a certain correction procedure–according to the recursive for-mulae(25) and(26) Some explanatory examples as well as a geometrical in-terpretation of the presented method can be found in[4] and[5].For the purpose of analysis,the coefficients,,,have to be calculated according to(29) satisfying the condition(30) Thus,the values quantify the influence of the-th varying parameter on the overall variation of the problem output,assuming every parameter to be var-ied relatively to the same percental extent.4.Model analysisIn order to analyze the model with respect to the influ-ence of the uncertain model parameters,the parameters are represented by fuzzy numbers which ex-press an initial guess for the actual values of the model parameters.The membership functions of the fuzzy numbers are defined to be symmetric and of(quasi-) Gaussian shape given bydimensionUsing the transformation method for the simulation andthe analysis of the fuzzy-parameterized model,the rela-tive degrees of influence can be determined.Exemplarily,the parameters and will be consid-ered in the ensuing.The resulting curves for the degreesare presented in Figure3.of influence andOne can see that the prediction of the in-blood glucose can be distinctly improved by applying the identification procedure in its enhanced version.Moreover,consider-ing the fact that biomedical systems are extremely hard to be modeled and to be identified,the presented results can be rated as very satisfactory.6.ConclusionsThe presented approach to improve the parameter iden-tification of complex biomedical models has turned out to be very promising and successful.Although the pro-cedure of model analysis,which has to be carried out beforehand,is rather time-consuming and requires quite high computational efforts,the advantages of the pre-sented method do clearly outweigh at the end.This is mainly due to the fact that the analysis has to be carried out only once for a particular biomedical model.The subsequent identification procedure,however,has to be performed anew for each single patient and every newly acquired set of data.Thus,the reduced dimension and the enhanced performance of the optimization compo-nentfinally entails a clear improvement of the overall method.References[1] C.Cobelli,G.Federspil,G.Pacini,A.Salvan,andC.Scandellari.An integrated mathematical model ofblood glucose and its hormonal control.Mathematical Biosciences,58:27–60,1982.[2]W.Dong and H.C.Shah.Vertex method for computingfunctions of fuzzy variables.Fuzzy Sets and Systems, 24:65–78,1987.[3]W.M.Dong and F.S.Wong.Fuzzy weighted averagesand implementation of the extension principle.Fuzzy Sets and Systems,21:183–199,1987.[4]M.Hanss.A nearly strict fuzzy arithmetic for solvingproblems with uncertainties.In Proc.of the19th Inter-national Conference of the North American Fuzzy Infor-mation Processing Society-NAFIPS2000,pages439–443,Atlanta,GA,USA,2000.[5]M.Hanss.The transformation method for the simula-tion and analysis of systems with uncertain parameters.Fuzzy Sets and Systems,(to appear)2001.[6]M.Hanss and L.Gaul.Simulation and analysis of afriction model with uncertain parameters using fuzzy arithmetic.In Proc.of the21st Iberian Latin American Congress on Computational Methods in Engineering-CILAMCE2000,Rio de Janeiro,Brazil,2000.[7]M.Hanss and O.Nehls.Simulation of the human glu-cose metabolism using fuzzy arithmetic.In Proc.of the 19th International Conference of the North American Fuzzy Information Processing Society-NAFIPS2000, pages201–205,Atlanta,GA,USA,2000.[8] B.H¨ofig.Physiologische Modellierung des mensch-lichen Glukose-Metabolismus f¨u r die simulations-gest¨u tzte Therapie des insulinabh¨a ngigen Diabetes mel-litus.Dissertation,Universit¨a t Stuttgart,1998.[9] A.Kaufmann and M.M.Gupta.Introduction to FuzzyArithmetic.Van Nostrand Reinhold,New York,1991.[10] E.Mosekilde,K.Jensen,C.Binder,S.Pramming,andB.Thorsteinsson.Modeling absorption kinetics of sub-cutaneous injected soluble insulin.Journal of Pharma-cokinetics and Pharmaceutics,17:67–87,1989. 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