Simulation of the Human Glucose Metabolism Using Fuzzy Arithmetic

Transcription

1 Simulation of the uman Glucose Metabolism Using uy Aithmetic Michael anss live Nehls Institute A of Mechanics Univesity of Stuttgat Pfaffenwalding Stuttgat Gemany Abstact With the object of futue impovement of the medical teatment of diabetes the human glucose metabolism is simulated using fuy aithmetic to take into account the uncetainties in the physiological model o this pupose the uncetain model paametes the model inputs ae epesented by fuy numbes with thei shape deived fom expeimental data o expet knowledge To avoid the disadvantageous consequences of stad fuy aithmetic a new implementation of fuy aithmetic is used which guaantees a pope solution of the poblem inally as a bypoduct of the new implementation of fuy aithmetic a measue of influence is intoduced which allows a quantification of the pecental influence of each vaying facto in the physiological model on the vaiation of the oveall model output By this one can detect to what extent the model output is influenced by the diffeent model paametes which is of exteme impotance fo the pactical optimiation of medical theapy Intoduction Diabetes o stictly speaking diabetes mellitus type I is one of the most widely spead human diseases of ou time The effects of this disease to the patients eveyday life can be quite seious anging fom egula medication with injections up to being impeiled by bad isk of heat attacks In ode to impove theapy to develop optimal medication fo diabetes the main chaacteistics of human glucose metabolism need to be studied on the basis of simulations of an appopiate mathematical model In the last decade a pactical model fo the human glucose metabolism has been developed consisting basically of a numbe of nonlinea patial diffeential equations to be solved numeically [2] The conventional way of evaluating the model as a cisp model howeve does not pove to be easonable since biological models ae exceedingly subjected to uncetainties ist the paametes of the model ae uncetain they exhibit a athe lage vaiability depending on the individual physique of the patient as well as on the extent the duation of the disease And second the model input consisting of the nutitional contents of the ingested food can usually only be quantified with a high degee of impecision A pomising appoach to solve this poblem is the use of fuy aithmetic to take into account the uncetainties in the human glucose metabolism In detail the uncetain model paametes the model inputs ae epesented by fuy numbes with thei shape deived fom expeimental data o expet knowledge The conventionally applied stad fuy aithmetic as outlined in [] howeve shows some seious dawbacks poves to be completely unsuitable to be used fo the analysis of complex models with uncetainties [4 5] o this eason the model in this pape is evaluated using a special implementation of a nealy stict fuy aithmetic which avoids the wellknown oveestimation effect by accepting a much slighte side effect [4] Thus the fuyvalued esults ae nealy fee of any additional atificial uncetainties can be consideed as vey close to the pope solution of the uncetain poblem The new implementation of fuy aithmetic is pesented in Section following a pesentation of the physiological model of the human glucose metabolism with injection of extenal insulin in Section 2 In Section 4 a special measue of influence is intoduced which allows a quantification of the pecentile influence of each vaying model paamete on the oveall model output The simulation esults fo the physiological model with uncetainties ae finally pesented in Section 5 2 Physiological Model Basically the oveall model of the human glucose metabolism fo patients with diabetes mellitus type I can be split into two majo pats fist the model fo the inflow

2 / y y 7 of insulin into the blood in consequence of the subcutaneous injected extenal insulin second the model fo the inflow of glucose into the blood in consequence of the ingested food The second model is again subdivided into two pats epesenting the metabolisms in the stomach on the one side in the intestine on the othe (igue ) inally the outputs of the models ae combined in a simplified model to pedict the amount of inblood glucose at some points in time Model fo the inflow of insulin into the blood [] Afte injection insulin appeas in two modifications in the subcutaneous depot which can be descibed by a hemisphee with the adial coodinate as dimee insulin with the concentation! as hexamee insulin with the concentation! The uptake of insulin into the blood is only affected by dimee insulin The injected extenal insulin howeve is a solution of pue hexamee insulin! %'& (%*)+ /24 ()! & %*)+ /2 %+56 (2) 9;=> 7!8(5 D () 9 A@ B with the opeato the model paametes 7 2EA GINMPRSUT X&Y2 [Z] )^`_A aibnm T e SUd T Yf[Z Se SUg 5jlkm M bam M the initial bounday conditions!! on is GA SUd pq 4 ih S SUg SUg (4) (5) is is (6) Model fo the inflow tuv of glucose into the blood Model fo the concentation xw of cabohydates in the stomach of volume y ({ w y %'~y* w y %'~y Y% N (7) (8) with the initial conditions w ;ƒ y the paametes ~ 2y SU 2 Œ Ž dšg y S gc Ž k_k%+ Žhk' Mx Š N h k ;ƒ bhk ƒ fakk š ƒ y S gšc y S gšc (9) () () SU iz `A GEfhA (2) Y dšg 2A Zs;A[kfZ The input paametes ƒ ƒ denote the amount ƒ of cabohydates poteins fat in the ingested meal ž; u ŠªA«U stomach Ÿ intestine œa žœÿ ± œ ž Ÿ{ A igue The appaatus of digestion { Model fo the concentation = of cabohydates in the intestine of adius { length { 8µ %s 8µ % () %s (4) { A (5) (6) with the initial bounday conditions N N!¼ y S gšc Mµ½¾ w X { Š ` (7)

3 ~ ¾ ¾ ~ P c S P c S the paametes ¼ 2A Z `A G S N 8µ e ^ S h h SUd =S 8µ 2 (8) v^ Z S i7k k7 SUd (9) >= A is Simplified model fo the amount of inblood glucose t ; i ¾ f D ¾ ¾ v ¾ 2 ƒ whee the sensitivity paametes (2) can be consideed as piecewise constant duing a multiple of the time inteval In pactice the time inteval is selected the sensitivity paametes ae consideed as constant fo about one hou ie `hf Special implementation of fuy aithmetic Given in geneal a poblem with a numbe of independent paametes which ae assumed to be uncetain the paametes can be epesented by fuy numbes ^Š7 P As illustated in igue 2 each of the fuy numbes can be decomposed into a set & of intevals of the fom (see also [4]) i fp &! ' µ % & with (2) )( +* (22) f7np i fp (2) Instead of applying stad inteval aithmetic diectly to the intevals Xf7NP fo each level of membeship fp the intevals ae now tansfomed into fields of the fom / gšv 4 9 ; ~ ; ~ ; (24) with igue B 2 Implementation of a fuy numbe ED by decomposition into intevals µ G / W ; v / W (25) Assuming the poblem to be given by the aithmetical expession I in the fom I µ M (26) its evaluation is then caied out by evaluating the expession sepaately at each of the 7 positions of the fields using the conventional aithmetic fo cisp numbes Thus if the esult of the poblem can be expessed in its decomposed N tansfomed fom by the fields fp N the th element is then given by of the field IP µ fš7 PŠ7 denotes the th element of the field if the fuyvalued esult its decomposed fom N R (27) inally of the poblem can be achieved in S' ( i fp (28) N including a cetain co by etansfoming the fields ection pocedue accoding to the ecusive fomulas A )UTWV % XY 42AP % )UTZV % % % ( (29) ()

4 7 % 4 Measue of Influence Using the pesented implementation of fuy aithmetic the pope o at least nealy pope esult fo a poblem with uncetainties can be achieved oweve the fuyvalued esult of the poblem only shows the oveall influence of all the uncetain paametes togethe As one can guess the degees of influence of the diffeent uncetain model paametes on the final esult of the poblem ae cetainly not equal in geneal In this egad the inteesting question may come up if one can detemine the pecentages to which the uncetain paametes of the poblem contibute to the oveall uncetainty of the poblem esult Indeed this tuns out to be possible esulting as a bypoduct fom the special implementation of fuy aithmetic as pesented in Section o this pupose the coefficients have to be fš7 P fp calculated accoding to M % whee stictly speakingo / / 2 () leads to the tivial case denote the lowe uppe bound of the inte ae the the elements of which ae defined fo i7 aap by the ecusive fomulas > fo evey f7nx The values M val the fields N 5 N 5 if N N M / / µ M M v /2 M fš7 PŠ7 M is odd N M v / M 5 N / µ /2 M M (2) () if fš7 PŠ7 M is even inally fo the index is always equal to unity N Š 5 N ae given by N v 2 5 N 2 µ (4) The coefficients can be intepeted as gain factos which expess the effect of the uncetainty of the th paamete on the uncetainty of the output of the poblem at level inally as an oveall measue of influence the values to can be detemined fo satisfying the condition Š7Nµ accoding (5) I (6) quantify the influence of the th vaying Thus the values on the oveall vaiation of the poblem output assuming evey paamete to be vaied elatively to the same pecentile extent 5 Simulation Results As can be seen fom Eqs (5) (2) the paamete in 5 in the insulin model the gastopaese facto dšg the glucose model can only be defined with a athe high degee of uncetainty Additionally the nutitional contents of the ingested food like the amount of cabohydates ;ƒ is quite difficult to be detemined pecisely Theefoe these paametes will be expessed by the fuy numbes 5 dg ƒ (7) with thei membeship functions deived fom expeimental data o expet knowledge as illustated in igue! % & 7 9 / ;=>A@6B 7 ED ƒ [bead units] (') * GI bead unit cabohydates igue Uncetain model paametes

5 [] YNZN[6>PNRS^]_9a`MbcG9dePf9E[Mg) hèpikj_mnlh nmsm o pq; 2sut Mv xw^[xebrmgfm RbG[=MB5WgRgfRPNE9 EWb6>W9bG[Ey gfrg[w9agfel@msb{yc={g MSgRRb6uEb4Nt E~ERbR6uEgfNRMS9Ebcy Wbct 5NVPN [PGDRƒ5Wb f9ebgrr9 [t 9 W9 f PNPb 6>WgRgfZNEbW_ fbb{gnrms9h9erš@pnbzt e NE nh WmSmŒ o q;ž =MSbPl5RgfEh/;5SP9bW9he f9epnh 5w^G6>6>f9 h59 5 [MNbcNWR9bbMS9 =MB5WgRR9 bmndenrms9albr9pnrwb_m4b 5y WEG9WMb R9 IWPNn>bMgfE grr9ebgrr9_ B B { S cšœ _E Gž> SŸ n7 {P E {ž; { 5P h ªB«Sªn BŒSª WmSŒSm o q =s 9bbW 9nNgRXbcNRP 47EVWVPX'NZN[6>PNRU47MUbMSgRŠBR9 DENMS EgRW6>b i ZN[9EWPGf9NRWbWuI9 œ c ± 5@@ Išu ;Yݵ^ c h Ng)9G5hA h p G G6G o q =s9bbwh/;^krgfgr9epnhy9¼ AEfNk½9¾DEDgZX5R9 47VPVPX>NRN[E6>PNR NM v9ernœpgfp6>w9_d5nms EgfP6>bWI9> œ c ± cš UYݵ^ c SÀ@ÀSÁhwÂW9EbNWMg) ng[ht ÃÄh nmsmœ h Afte implementing the uncetain paametes as fuy numbes in decomposed fom (see igue 2) the fuyvalued amount of inblood glucose is simulated accoding to Eqs () to (2) with the initial condition ƒ k!a Both the ingestion of food the injection of insulin ae assumed to take place at the time ae expessed by contou lines The esults fo fo the diffeent levels µ of membeship ( Z i [7 ) in igue 4 Additionally the membeship functions of two meaningful esults fo I at Z Y `az ae pesented in igue 5 I SUd 9 7 ` ^ igue 4 Simulation of the uncetain amount of inblood glucose t I I influence of which is vey high at the beginning but monotonically deceases with time Ÿ! 2 4 %'&)( * igue 6 Degees of influence B + D fo the uncetain paametes D 6 onclusions The application of fuy aithmetic to simulate the human glucose metabolism with uncetainties in the model paametes poves to be a vey successful appoach Amongst othe things one can daw the inteesting hadly expected conclusion that the evolution of the inblood glucose is significantly moe effected by uncetainties in the paamete of the insulin model than by those elated to the ingested food ence to impove medical theapy effots should athe be focussed on the identification of the individual insulin paametes than on the exact quantification of the nutitional contents of the ingested meal SUd igue 5 Amount of inblood glucose at t As can be seen the maximum ange of uncetainty in the amount of inblood glucose is about 7 when uncetainty in the model paametes is assumed Although this value seems to be athe high it can still be consideed as acceptable fom a medical point of view It also shows that thee is an enomous incease of uncetainty within the fist thee hous In the time afte howeve the uncetainty deceases again igue 6 povides evidence that this effect is mainly due to the uncetainty of paamete 5 the Refeences