Statistical Evaluation of a Glucose / Insulin Nonlinear Differential Equation Model with Classical and Bayesian Procedures.

Transcription

1 Recen Researches in Applied Compuers and Compuaional Science Saisical Evaluaion of a Glucose / Insulin Nonlinear Differenial Equaion Model wih Classical and Bayesian Procedures. SUTHAROT LUEABUNCHONG,, YONGWIMON LENBURY*,, SIMONA PANUNZI 3, ALICE MATONE 3 Deparmen of Mahemaics, Mahidol Universiy, Bango 0400, Thailand; Cenre of Excellence in Mahemaics, CHE, 38 Si Ayuhaya Road, Bango 0400, Thailand; 3 BioMaLab CNR-IASI, Fisiopaologia dello Shoc Universiy Caolica del SacroCuore Largo A. Gemelli Roma, Ialy *corresponding auhor: scylb@mahidol.ac.h Absrac: - In his paper, he Marov Chain Mone Carlo (MCMC) mehod and Generalized Leas Square (GLS) mehod are used o esimae he parameers in a glucose/insulin nonlinear differenial model wih GLP- DPP4 ineracion, describing he glucose-insulin meabolism. The model is used o generae he daa ha consiss of he ime-concenraion measuremens of plasma glucose and of insulin, which are imporan in Diabees Mellius (DM) reamen. Deails on our applicaion of MCMC and GLS o esimae parameers in he model are given in his paper. Our resuls sugges ha MCMC is beer able o esimae he parameers based on smaller bias and sandard deviaion. Alhough MCMC requires more calculaion ime han GLS, i offers a more appropriae mehod, in our opinion, for nonlinear model parameer esimaions wih no nowledge of he disribuion of he daa and when heerogeneiy of variance is eviden. ey-words: -DPP-4, GLP-, glucose/insulin meabolism, MCMC, GLS, nonlinear differenial equaions model. Inroducion The sudy of glucose and insulin meabolism is fundamenal o he undersanding of he mechanisms and he diagnosis of diabees mellius (DM), a widely spread disease affecing a growing number of people, especially in he developed counries. There are hree major forms of diabees: ype diabees mellius (TDM, also nown as insulin-dependen or juvenile-onse diabees), ype diabees mellius (TDM, also nown as noninsulin-dependen or mauriy-onse diabees), and gesaional diabees. Insulin is he principal hormone ha regulaes he upae of glucose from he blood ino mos cells (primarily muscle and fa cells, bu no cenral nervous sysem cells). Therefore, lac of insulin or he insensiiviy of is recepors plays a cenral role in all forms of DM. Gesaional diabees develops in a small percenage of pregnan women and usually resolves afer paruriion. TDM affecs less han 0% of diabeic people and is an auoimmune disease in which he desrucion of pancreaic beacells causes insulin deficiency. TDM is, insead, a progressive disease, ypically associaed wih obesiy. There are several mechanisms ha can be involved in TDM, such as defecs in insulin recepors which lead o a compromised insulin sensiiviy in peripheral issues (peripheral insulin resisance) or in he liver (cenral insulin resisance). In peripheral insulin resisance, issues do no absorb glucose correcly, while in cenral insulin resisance, hepaic glucose oupu is no correcly inhibied leading o unnecessary glucose producion. TDM can also exhibi insulin secreion insufficiency when he pancreaic bea-cells are compromised. All hese scenarios lead o an excess of glucose in he blood, a condiion called hyperglycemia, ha can lead o a number of dangerous consequences. Polyuria (frequen urinaion) causes loss of elecrolyes, dehydraion and polydipsia (excess of hirs). Loss of calories leads o polyphagia (excess of hunger), negaive nirogen balance, acidosis and hyperpnea (increased deph of breahing). These and oher hyperglycemia relaed problems can evenually cause coma and deah []. In order o sudy insulin and glucose meabolism, several mahemaical models [-4, 5, 6, 7] have been proposed, describing he effec of DM in he aemp o answer quesions relaed o physiological complicaions. In he presen sudy, we concenrae on he roles of Glucagon-lie pepide- (GLP-) and dipepidylpepidase-4 (DPP-4) on he glucose/insulin meabolism. GLP- is secreed from he L-cells of he inesinal mucosa (mosly of he ileum) afer ISBN:

2 Recen Researches in Applied Compuers and Compuaional Science meal ingesion and reduces pos-prandial glycaemia, enhancing insulin secreion and delaying gasric empying [8]. The enzyme DPP-4 is a pepidase ha inacivaes GLP- and rapidly reduces is circulaing levels [9]. Here, we perform and compare wo saisical mehodologies for parameer esimaion. The Marov Chain Mone Carlo mehod (MCMC), which is used in a Bayesian seing, implemened in he repeaed measuremen daa framewor, and he Generalized Leas Square mehod (GLS), which is used in he classical saisical formulaion, are used o esimae some parameers in a glucose/insulin model wih GLP-DPP4 ineracion. In his sudy, daa on a fixed number of subjecs are generaed wih errors from he glucose/insulin model wih GLP-DPP4 ineracion, when he model and is numerical inegraion are implemened in MATLAB. We focus on he comparison beween he wo procedures performed in erms of he poin and inerval parameer esimaors. We apply he Bayesian models for he hierarchical nonlinear framewor, which is regarded as an exension of he nonlinear regression models o handle daa from several individuals, o provide he inra- and iner- individual variaion. Finally, he resuls are compared and analyzed. Procedure. Glucose/insulin model wih GLP-DPP4 ineracion We here inroduce a model represening glucose and insulin homeosasis and accouning for he effecs of GLP- and DPP-4 pepides. The model is used o generae daa of individual glucose and insulin concenraions in ime. Glucose dynamics is described by hree sae variables, each one represening glucose in a differen sage during is meabolism, from enering he body wihin he meal, o is final absorpion ino he blood sream. Glucose firs appears in he somach wihin he meal, i is hen ransferred o he gu (here he duodenum is considered), where i simulaes increin (GLP-) secreion, and finally is absorbed from he blood sream, where i exers is acion in simulaing insulin ouflow from pancreaic bea-cells. Insulin, in urns, induces glucose absorpion from peripheral issues and inhibis hepaic glucose producion, hus reurning, in a healhy individual, plasmaic glucose concenraion o basal levels. Insulin secreion from he pancreas is increased (his is why he erm increin is used) from GLP-, which is released from he gu when glucose passes hrough i. GLP- is rapidly degraded from DPP-4, which is an ubiquious enzyme, presen in several forms and carrying ou differen acions in he body, depending on is locaion and specificiy. The degradaion of GLP- from DPP-4 is a safey mechanism which guaranees ha, when glucose is no in he gu and is plasma concenraion is no going o increase, increins sop simulaing insulin secreion. Figure gives a schemaic descripion of he glucose/insulin ineracion wih GLP-DPP4 ineracion. The referenced mahemaical model of his process is as follows. ds() dss(), S ( Sb () d dd() dss() D(), D ( 0 () d dg() D () G () IIG () () Tgl, G ( Gb (3) d Vg di() I xi () TG ig () TiGNNG () (), I ( Ib (4) d dn() TnDD () xnppn () () xn N () Tn, N ( Nb (5) d dp() xpp() Tp, P ( Pb (6) d where SDGIN,,,, and P are, respecively: he amoun of ingesed glucose appearing in he somach; glucose concenraion in he duodenum; plasma glucose concenraion; plasma insulin concenraion; plasma GLP- concenraion; plasma DPP4 concenraion. In equaion (), he iniial condiion, S b, represens he amoun of ingesed glucose, and he only erm on he righ hand-side represens glucose eliminaion from he somach, where ds is he ransfer rae from he somach o he duodenum. Equaion () describes he dynamics of glucose concenraion in he duodenum. The firs erm represens he enry from he somach, while he eliminaion erm is plasma absorpion, being he ransfer rae consan from he duodenum o he blood. Equaion (3) represens plasma glucose concenraion dynamics. This is described by: insulin-independen glucose issue upae, where is he glucose-dependen eliminaion rae consan; insulin-dependen glucose issue upae, where I is he second order eliminaion rae consan, insulin and glucose-dependen; enry from he duodenum, where V g in he denominaor is he disribuion volume for glucose; and he consan ISBN:

3 Recen Researches in Applied Compuers and Compuaional Science enry depending on liver release, where T gl is he consan rae of hepaic glucose producion. Insulin dynamics is described by equaion (4), he firs erm represens physiological insulin eliminaion, where xi is he disappearance rae consan. The wo enries depend, respecively: on glucose concenraion, TiG being he producion rae consan of pancreaic release of insulin due o glucose, and on GLP- simulaory effec, where T ign is he rae of pancreaic release of insulin due o increin acion. Equaion (5) represens GLP- dynamics, where he firs erm corresponds o he release due o glucose concenraion in he duodenum, TnD being he D dependen consan producion rae. GLP- eliminaion due o DPP-4 cleavage is represened by he second righ erm, where is he xnp disappearance rae consan, GLP- and DPP-4 dependen. The physiological GLP- eliminaion is described by he hird erm where xn is he disappearance rae consan. A consan enry is also assumed, represened by he las erm, where T n is he consan producion rae. The las equaion (6) describes DPP-4 dynamics, where xp in he eliminaion erm is he disappearance rae, DPP-4 dependen, and he las erm corresponds o DPP-4 producion, where T is he appearance rae p consan.. Maerials and Mehods The objecive of his wor is o esimae he unnown parameers,,, I xi and V g in (3) and (4), describing he differences over ime of concenraion measuremens of plasma glucose G (), and of insulin I, () by using MCMC and GLS mehods. By using MATLAB, daa and errors for 30 subjecs were generaed from he glucose/insulin nonlinear differenial equaions model wih GLP- DPP4 ineracion. The daa consis of he ime concenraion measuremens of plasma glucose G (), and of insulin I, () every 0 minues, ranging from 0 o 300 minues, during which G () and I () of he subjec were found. Table repors he definiion of all he quaniies in (3) and (4). Le us consider yj fj (, j ) e, where j yj denoes he variable concenraion of he subjec ;,,3,...,30, a ime j; j,,3,...,30, while is equal o G for glucose or I for insulin. Then, f j (, j ) represens he predicion funcions a ime j for he -h subjec, which is derived from he numeric soluion f (, ) of he following differenial equaion model: Table. Definiion of he symbols in (7) and (8). Symbol Unis Definiion min Time G () Amoun of ingesed glucose in mmol he duodenum I () pm Plasma insulin concenraion N() pm Plasma GLP- concenraion D() mmol Amoun of ingesed glucose I min - min - /pm min - appearing in he duodenum The insulin-independen rae consan of issue glucose upae The insulin-dependen rae consan of issue glucose upae The rae consan of glucose absorbed from he duodenum ino he blood xi min - The disappearance rae consan for insulin V The disribuion volume for g L glucose T The increase in plasma gl pm/min glucose concenraion due o hepaic glucose release G b mm Glycemia a 0 I b pm Insulinema a 0 T pm/min/ The rae of pancreaic release ig mm of insulin due o glucose T pm /min/ The rae of pancreaic release ign pm/mm of insulin due o GLP- dg() D () G () IIG () () Tgl, G ( Gb (7) d Vg di() I() T G() T N() G(), I( ) I (8) xi ig ign 0 b d Hierarchical nonlinear models are ypically applied o biosaisical problems. These models arise frequenly in siuaions where several measuremens are made on a number of subjecs. Repeaed measuremens on a subjec may be aen over ime, a differen analye concenraions. The exisence of repeaed measuremens requires paricular care in characerizing he random variaion in he daa. In paricular, i is imporan o recognize explicily wo levels of variabiliy: random variaion among measuremens wihin a given subjec (inra-individual variaion) and random variaion among subjecs (iner- individual variaion). Normally, specificaion of a disribuion o characerize iner-individual variaion is oo ISBN:

4 Recen Researches in Applied Compuers and Compuaional Science difficul. There are many models o ae ino accoun his variaion. In he presen wor, we used a Bayesian model specificaion o represen he iner-individual variaion. A Bayesian nonlinear model involves 3 sages, according o Davidian and Gilinan [0], as in he following. Sage I Inra-individual variaion In his sage, we specify he mean response and variance-covariance srucure for a given subjec. yj fj (, j ) e, E( e ) 0, Cov( e ) R (, ) (9) j G fg(, ) 0 j where R(, ) ( 0 I fi (, j) ( G, I ) is he variance-covariance vecor parameer, and f G and f I are he mean responses for he glucose and insulin, respecively. Sage II Iner- individual variaion The second sage involves a model for variaion in he regression parameers. This variaion can be due o sysemaic dependence on subjec characerisics, such as gender, age or simply o biological variabiliy among differen individuals. In his wor, we consider he simples case of linear model: b, b~ N(0, D) () where log( ) log(,, I, xi, Vg ) ; D he join covariance marix for all random effecs; and ˆ is he individual esimae for Subjec. Sage III Hyperprior disribuion The model specificaion is compleed by an assumpion of a disribuion for all parameers in Sages and :, and D : * * * ~ N(, ), D ~ Wi(,[ D ] ),( ) ~ Gam(, ) where Wi and Gam represen Wishar and Gamma disribuions, respecively..3 MCMC Implemenaion By following he sudies of Davidian and Gilinan [0], Gelfand e al. [] and Waefield e al. [], i can be shown from sages, and 3 ha he full condiional disribuion-he disribuion of he parameers- given he remaining parameers and he daa, and for he parameers, D and, may be wrien explicily as ( y, *,..., m, D, G, I) NV ( ( md ), V) () ( D y,,,..., m, G, I ) m T * Wi([ ( )( ) D ], m ) (3) ( G y,,,..., m, D) Gam( ( 0G N), m [ ( ( )) T y fg ( y fg ( )) 0G0G ]) (4) ( I y,,,..., m, D) Gam( ( 0I N), m [ ( ( )) T y fi ( y fi ( )) 0I0I ]) (5) However, he full condiional disribuion of each, given he remaining parameers and he daa, canno be calculaed explicily bu can be wrien up as a densiy funcion ha is proporional o ( y,, j j i, G, I, D) R (, ) / T exp{ ( y f{ }) R (, )( y f{ })} T exp{ ( ) D ( )} The proceeding for MCMC mehod by using he Meropolis Hasings algorihm inside he Gibbs sampling o draw he samples of he parameers from Bayesian poserior disribuion is given as follows. ( ( ( ( () Sar wih he iniial values (,,D, ( ) T where {,,,...,30} and choose ( (. () i () i () i () i () i T () Obain a new value (,, D, ) ( i ) from hroughou he proposal disribuion: ( i) ( i) ( i) ( j) ~ ( / y,, D, ) () i () i ( i) ( j) ~ ( / y,, D, ) ( i) ( i) ( i) ( j) D ~ ( D / y,,, ) () i (3) For, move a chain o a new value, which ( i ) is generaed from he proposal q( / ), from ( i ).Evaluae ( i) ( i) ( i) ( / y,,, D, ( ) { } ) i ( / ) min, ( i) () i () i ( i) ( / y,,, D, { } ) (4) Sample a uniform (0,) random variable U. If ( i ) U ( / ), hen se ( i ), oherwise se () i ( i ) and he chain does no move. (5) Increase i, and reurn o () unil convergence is reached..4 GLS Procedure We applied GLS, which is a wo-sage mehod, by following he sudies of Panunzi, Palumbo, and De Gaeano [3]. We used he firs run of MCMC o be ISBN:

5 Recen Researches in Applied Compuers and Compuaional Science an iniial value for his mehod. The proceeding is given as follows. Sage I () In separae esimaion procedures (where is he oal number of subjecs), obain preliminary ( ) esimaes ˆ p for each subjec,,,...,, by using Ordinary Leas Squares (OLS) esimaor. () Calculae residuals from () and esimae (, ) minimizing he following funcion: G I 30 ( p) (log (, ) ( p ) T ( p ) ( p ) y f R y f PL R [ ( )] (, )[ ( )]) (3) Consruc esimaed weigh marices which depend on he esimaed parameers ˆ and ( p) : ˆ ( ( p ) R, ˆ ). (4) Use he esimaed weigh marices from (3), reesimae he for each subjec,,,...,30 by T ( p) minimizing [ y f( )] R (, )[ y f( )] The resuling esimaes can be reaed as preliminary esimaes and i is possible o reurn o (). The algorihm should be ieraed a leas once and for each Subjec. Sage II We obain igls i GLS denoes he final esimaes. from Sage I, hen he populaion esimaors of he vecor and he variancecovariance marix D are obained by ˆ 30 i i GLS 30 and D ˆ ˆ ˆ T ( ) ( ˆ ˆ i i i i) ( ). m i GLS GLS. 3 Resuls The resuls of MCMC applicaion are obained in abou en days. The enire run involves 0,000 ieraions and he firs 5,000 ieraions are considered as burn-in period, while he GLS applicaion converges in ieraions. Table The rue values (TV), poserior means (PM), sandard deviaion (SD) and 95% credible inervals (CI) of populaion parameers based on MCMC. Parameer TV PM SD 95% CI (0.039,0.056) (0.00,0.004) I ( ,0.0009) xi (0.056,0.036) V g (.55,0.9) Table 3 The rue values (TV), poin esimaes(pm), sandard deviaion (SD) and 95% credible inervals (CI) of populaion parameers based on GLS. Parameer TV PM SD 95% CI (0.0460,0.063) (0.008,0.007) I ( ,0.0003) xi (0.056,0.037 V g (3.3,.45) Table 4 The rue values (TV), poin esimaes (PM) and bias of populaion parameers in comparison beween MCMC and GLS. Parameer TV MCMC Bias GLS Bias I xi V g Fig. Plo for Subjec. Glucose and insulin (circles) concenraions versus ime ogeher wih he prediced ime-curves from he glucose/insulin model wih GLP-DPP4 ineracion for Subjec.The solid and dashed lines represen esimaed subjec curves based on MCMC and GLS, respecively. The generaed values for Subjec are indicaed by he open circles. Fig. Plo for Subjec 9. Glucose and insulin (circles) concenraions versus ime ogeher wih he prediced ime-curves from he glucose/insulin model wih GLP-DPP4 ineracion for Subjec 9. The solid and dashed lines represen esimaed subjec curves based on MCMC and GLS, respecively. The generaed values for Subjec 9 are indicaed by he open circles. ISBN:

6 Recen Researches in Applied Compuers and Compuaional Science The iniial values for MCMC and GLS mehods 0 for he subjec parameers and populaion parameers were se o log[0.05,0.05, ,0.05,3.5].Table and Table 3 repor on a summary of he resuls of esimaed populaion parameers based on MCMC and GLS, respecively. In Table 4, we presen he esimaes of subjecspecific individual parameers. For comparison, he rue parameer values and esimaion bias are also presened for each parameer. Fig. and show he curves of esimaed value for Subjec and Subjec 9, respecively. 4 Conclusion The main aim of his sudy is o invesigae he applicaion of MCMC and GLS mehods o esimae parameers in he glucose/insulin nonlinear differenial model wih GLP-DPP4 ineracion. From our comparison beween MCMC and GLS resuls, we observe ha, in he esimaions of,,, I xi and V, he bias and sandard g deviaion for any parameers wih he use of MCMC are smaller han wih GLS. Thus, his indicaes ha MCMC performs beer han GLS in esimaing every parameer in he glucose/insulin nonlinear differenial model wih GLP-DPP4 ineracion. Based on he generaed daa, we sugges he use of MCMC insead of GLS for poin esimaion on he glucose/insulin nonlinear differenial model wih GLP-DPP4 ineracion because wihou any nowledge of he disribuion of he daa we can easily obain more accurae poserior means hrough MCMC mehod han GLS mehod. Alhough, MCMC aes more ime han GLS, MCMC would never give rise o such error as ha arising from GLS. 5 Acnowledgmen The firs auhor has been suppored by he Sraegic Scholarships (PhD) Fellowships Fronier Research Newor, he Office of Commission on Higher Educaion and, ogeher wih he hird auhor, BioMaLab CNR-IASI, Ialy. The second auhor is suppored by he Cenre of Excellence in Mahemaics, CHE, Thailand. [] R. N. Bergman, Y.Z. Ider, C.R. Bowden, C. Cobelli, Quaniaive esimaion of insulin sensiiviy, American Journal of Physiology, Vol.36, 979, pp [3] A. Bouayeb, A. Cheouani, A criical review of mahemaical models and daa used in diabeology, Biomedical Engineering OnLine, Vol.5, 006. [4] C. L. Chen, H.W. Tsai, S. S. Wong, Modeling hephysiological glucose-insulin dynamic sysem on diabeics, Journal of Theoreical Biology, Vol.65,00, pp [5] A. D. Gaeano, O. Arino, Mahemaical modeling of he inravenous glucose olerance es, Journal of Mahemaical Biology, Vol.40, 000, pp [6] A. Maroglou, J. Li, Y. uang, Mahemaical models and sofware ools for he glucose insulin regularory sysem and diabees: an overview, Applied Numerical Mahemaics, Vol.56, 006, pp [7] M. Chuedoung,W. Saria, Y. Lenbury, Dynamical analysis of a nonlinear model for glucose-insulin sysem incorporaing delays and - cells comparmen, Nonlinear Analysis, Vol.7, 009, pp [8] J. Schirra, B. Goe, The physiological role of GLP- in human: Increin, ileal brae or more, Regulaory Pepides, 005, pp [9] M. Bose, B. Oliván, J. Teixeira, F. X. Pi- Sunyer, B. Laferrère, Do Increins Play a Role in he Remission of Type Diabees afer Gasric Bypass Surgery: Wha are he Evidence?, Obes Surg, 009, pp [0] M. Davidian, D. M. Gilinan, Nonlinear models for Repeaed Measuremen Daa, Chapman & Hall, London, 995. [] A. L. Gelfand, S. E. Hills, A. Racine-Poon, A. F. M. Smih, Illusraion of Bayesian inference in normal daa models using Gibbs sampling, Journal of he American Saisical Associaion, Vol.85, 990, pp [] J. C. Waefield, A. F. M. Smih, A. Racine- Poon, A. E. Gelfand, Bayesian analysis of linear and non-linear populaion models using he Gibbs sampler, Applied Saisics, Vol.43, 994, pp.0-. [3] P. Simona, P. Pasquale, A. D. Gaeano, A discree single delay model for he inra-venous glucose olerance es, Theoreical Biology and Medical Modelling, Vol.6, 007, pp.-5. References: [] E. Rubin, H. M. Reisner, Essenials of Robin s Pahology, Ed Lippinco Williams & Wilins, 008. ISBN: