Reconstruction of Insulin Secretion under the Effects of Hepatic Extraction during OGTT: A Modelling and Convolution Approach

Transcription

1 Reconsrucion of nsulin Secreion under he Effecs of Hepaic Exracion during OGTT: A Modelling and Convoluion Approach KATTYOT JUAGWON,2, YONGWMON LENBURY*,2, ANDREA DE GAETANO 3, PASQUALE PALUMBO 3 Deparmen of Mahemaics, Faculy of Science, Mahidol Universiy, Bangkok 0400, Thailand; 2 Cenre of Excellence in Mahemaics, CHE, 328 Si Ayuhaya Road, Bangkok 0400, Thailand; 3 BioMaLab CNR-AS, Fisiopaologia dello Shock Universiy Caolica del SacroCuore Largo A. Gemelli Rome, aly *corresponding auhor: scylb@yahoo.com hp:// Absrac: - The reconsrucion of insulin secreion raes deriving from insulin and C-pepide concenraions is sudied. Two ses of daa simulaing profiles of insulin and C-pepide concenraions during glucose adminisraion are generaed, one of which is generaed using an exended combined model (Waanabe e al., 998 under he assumpion of consan fracional hepaic exracion, while he second daa se is from assuming non-consan fracional hepaic exracion. A deconvoluion approach based on he exended combined model wih he consan exracion is hen applied o esimae he rae of insulin secreion. The resul indicaes ha variaion in he fracional hepaic exracion significanly effecs he reconsrucion of insulin secreion. When he convoluion approach is modified o accommodae he assumpion of non-consan fracional hepaic exracion, i can provide more accurae esimaes for he raes of insulin producion. The modified approach discussed in his paper may hus offer a beer opion in he aemp o deermine an esimaion of insulin secreion crucial for he conrol of diabees mellius. Key-Words: - insulin secreion, insulin esimaion, hepaic exracion, deconvoluion, C-pepide nroducion Deeper undersanding of insulin kineics is crucial for he conrol and reamen of diabees mellius. is clinically imporan o be able o idenify pancreaic secreion of insulin hormone during oral glucose olerance es (OGTT because a defec in insulin secreion is characerisic of ype 2 diabees []. An accurae esimaion of endogenous insulin secreion will allow he physicians o monior he level of he defec in insulin secreion of β-cells in he pahogenesis of ype 2 diabees in order o beer conrol and rea diabeic paiens. An approach based on a wo-comparmen kineics model for C- pepide and wo-phase clearance of insulin kineic model for insulin, called an exended combined model (ECM, was proposed by Waanabe e al. [2, 3] o be an alernaive approach for he idenificaion of he rae of pancreaic insulin secreion and kineic parameers. This approach uses he plasma concenraions of insulin and C-pepide sampled from a single experimen proocol o esimae kineic parameers of insulin and C-pepide conrol sysem and o esimae he prehepaic insulin secreion rae wihou he separae experimenal proocol suggesed by Eaon e al. [4] and improved by Polonsky e al. [5] or he experimenal proocol fixing he values of he parameers [6]. n vivo experimen in conscious dogs was carried ou o examine he accuracy of idenificaion [3]. The approach appeared o be able o reconsruc fairly well he known equimolar inraporal infusion of insulin and C-pepide. However, he approach assumes ha he fracion of hepaic exracion, defined as he proporion of he amoun or rae of insulin clearance during firs pass ransi of liver o he amoun or rae of insulin secreion, is consan. However, here is evidence [7, 8] ha he exracion decreases during high concenraion of insulin alhough he oal hepaic exracion increases. A possible explanaion is ha mos exracion is a recepor-mediaed process and hence high concenraion of insulin in he poral vein leads o a reducion in clearance due o recepor downregulaion [9]. By a direc assessmen of fracional hepaic exracion, quanified by measured hepaic venous-arerial difference in insulin and C-pepide concenraions and calculaed rae of hepaic blood flow, carried ou by Brundin [7] and Tura e al. [8], i has been shown ha he fracion is no consan during oral glucose adminisraion. Therefore, in his work we modify he approach presened in [2] SBN:

2 and [3] so ha i is able o accommodae variaions in he fracional hepaic exracion. Thus, he firs aim of his sudy is o re-examine he accuracy of esimaions derived by he approach proposed in [2] and is numerical applicaion. The daa on concenraions of insulin and C-pepide are generaed by he ECM wih known parameer values and rae of secreion and he approach is hen used o esimae he parameers based on he generaed daa added wih various levels of random Gaussian error. The second aim is o sudy he efficiency of he approach when i is applied o esimae he parameers and secreion rae from he daa of insulin and C-pepide concenraions generaed by ECM wih non-consan fracional hepaic exracion erm in he kineic model for insulin. Finally, we modify he approach o reconsruc he secreory rae when insulin and C-pepide concenraions are given by ECM wih non-consan fracional hepaic exracion. To invesigae he accuracy of he esimaion, he modified approach is used o esimae he rae of secreion from he daa, generaed by ECM wih non-consan fracional hepaic exracion, wih differen levels of random Gaussian error. Esimaed resul is hen compared wih he known values. 2 Mehods 2. The model and daa generaion The exended combined model has been described in [2]. The model akes ino consideraion he kineics of endogenous insulin and C-pepide. These componens have a common rae of secreion because of pancreaic equimolar release of insulin and C-pepide. For he insulin kineics, i is assumed ha here is only one comparmen of insulin and here are wo phases of insulin clearance. The firs phase removal is he hepaic insulin exracion, depending on he rae of pancreaic insulin release, by he liver during he firs pass ransi. The second phase removal is sysemic insulin clearance, depending on he curren insulin concenraion. The model assumes ha he fracional hepaic insulin exracion is consan hroughou he period of ineres. For he kineics of C-pepide, he wocomparmen model is applied. The C-pepide clearance is assumed o occur only in he firs comparmen and hepaic C-pepide degradaion in he liver is negligible. The ordinary differenial equaions in ECM can be wrien as d( ( H R( K ( d dc ( C R( 2 K0C ( K2C ( K2 C2 ( d C C (2 dc2 ( C K2C2 ( K2C ( (3 d where ( is he insulin concenraion a ime. C ( is he C-pepide concenraion in he firs comparmen. C 2 ( is he C-pepide concenraion in he second comparmen. H is he fracion of hepaic insulin eliminaion by he liver and H represens he fracion of insulin ransferred ino he insulin comparmen afer surviving hepaic eliminaion. R( is he rae of prehepaic insulin secreion. K is he fracional eliminaion of insulin in he insulin comparmen. K 0 is he fracional eliminaion of C- pepide in he firs comparmen. K 2 and K 2 are he fracional C-pepide ransfer consans beween he firs and he second comparmens, respecively.,, and are he volume disribuion of C- C C 2 pepide in he firs comparmen, he second comparmen, and insulin comparmen, respecively. n his sudy, here are wo differen ses of daa of insulin and C-pepide concenraions. The daa in he firs se is generaed by ECM under he assumpion of consan hepaic exracion and he daa in he second se is generaed under he assumpion of non-consan hepaic exracion. To generae he daa in he second se, he shape of he fracional hepaic exracion ime course repored in [6] is used o derive he following funcion o replace he consan fracional hepaic exracion H in Eq. ( h ( abe ( e (4 Tha is, his funcion has been chosen o represen he fracional hepaic exracion since he graph of his funcion, shown in Fig., is able o closely mimic he shape of he plo of experimenally measured fracional hepaic exracion repored in Fig. 4 in [6]. The known values of kineic parameers and rae of secreion ime series for he daa generaion are aken from [] and shown in Table. and he figure capion of Fig. 2, respecively. Each se of daa conains 24 profiles of insulin and C-pepide concenraions added wih random Gaussian error wih coefficiens of 3% and 5%, 2 profiles a 3% error and 2 profiles a 5% error. Once he daa has been generaed, he Waanabe C2 ( SBN:

3 approach is hen used o esimae kineic parameers and secreory rae from he daa in he firs se and hen in he second se o access he accuracy of parameer idenificaion under condiions of consan and non-consan fracional hepaic exracion, respecively. Nex, he modified Waanabe approach is applied wih he daa in he second se and he esimaes are hen compared wih he known values. dy ( ( K ( Y2 ( ( Y ( d H dy2 ( ( K2 2 ( 2Y2 ( d wih C C( Y( ( Y2( ( H or (6 (7 (8 dz( ( H( K( Z2( C( KZ( (9 d dz2 ( ( 2 K2 C( K2Z2( (0 d wih Fig.. Graph of h( in Eq. (4 represening he fracion of hepaic exracion during OGTT , a 0.60, b Parameer idenificaions and numerical mehods Waanabe s approach was described in [2] for he esimaion of he kineic parameers and prehepaic secreion. The approach can accuraely esimae he values of parameers and rae of secreion under he assumpion of consan fracional hepaic exracion based only on he daa of concenraions of insulin and C-pepide during OGTT wihou any daa on exponenial decrease in concenraion of C-pepide when a bolus of C-pepide is injeced [3]. This approach based on ECM has wo seps. The firs sep is he esimaion of kineic parameers and he second sep is he esimaion of he secreory rae. n he firs sep, he sysem of equaions, Eq. (-(3, are ransformed from real-domain ino s-domain by Laplace ransforms leading o he simplified ransfer funcion in he form ( s ( s s H C s C 2 ( ( ( ( sk( sk2 (5 where and C are he Laplace ransforms of and C, respecively, 2 K 2 K 2 K 0 and k k (Please see ref. [3] for furher deail. This ransfer funcion in Eq. (5 is also he ransfer funcion of following sysem of differenial equaions. Z H Z C C ( ( ( ( 2( ( where Y (, Y 2 (, Z ( and Z 2 ( are sae variables in he equivalen sysems. Nex, using MATLAB funcion fminsearch, he values of he parameers in Eq. (6-(8 or Eq. (9- ( are esimaed by using he daa of insulin and C-pepide concenraions. For Eq. (6-(8, he parameers are esimaed by fiing he C-pepide concenraion while insulin concenraion is assigned o be he inpu, while he esimaion by using Eq. (9-( is he opposie. Then, he esimaed parameers are subsiued in he analyic soluion of Eq. (-(3 given by K 2 ( K 2 2 2( C ( R( [ e e ] d C (2 ( H K ( ( R( e d 0 Therefore, he work in he second sep is he esimaion of he funcion of secreory rae R( by using Eq. (2 and Eq. (3. (Please see ref. [2] for more deail for he derivaion of Eq. (2-(3. To modify he Waanabe approach, he fracional hepaic exracion H defined in Eq. ( is replaced by he funcion h( in Eq. (4, and he secreory raes are calculaed by he deconvoluion echnique [0] as follows. Eq. ( is now wrien as d( K ( ( h( R( (4 d ( (3 SBN:

4 Muliplying boh sides wih he inegraing facor Kd e and inegraing from 0 o, one obains Tha is, K K de ( ( e ( h( R( d 0 0 K K ( ( (0 ( ( ( e e h R d (6 0 n he case ha (0 = 0, we hen arrive a he following inegral expression for (, in place of Eq. (3. K ( ( ( e R d 0 (5 (7 wih R ( R ( (8 h ( When he esimaed K from he firs sep is subsiued, R ( is esimaed by fiing he insulin concenraion using funcion fminsearch in MATLAB. Tha is, deconvoluion allows us o derive he funcion R ( a differen ime, by which we can calculae he secreion rae R( from Eq. (8. 3 Resuls 3. Waanabe approach on daa from ECM assuming consan fracional hepaic exracion The abiliy of he approach o esimae he kineic parameers is shown in Table. The kineic parameers and rae of secreion do no differ significanly from he known values for all levels of error. Wih no error added, he approach is able o provide he correc values of kineic parameers and rae of secreion. The resuls a no error added, a 3%, and 5% added error indicae ha he approach provides efficien assessmen of insulin secreion when he fracional hepaic exracion is assumed o be consan. 3.2 Waanabe approach on daa from ECM assuming non-consan fracional hepaic exracion The kineic parameers esimaed according o Eq. (6-( are compared wih he known values in Table, and in Fig. 2 he esimaed rae of insulin secreion by he deconvoluion mehod based on Eq. (2-(3 when he esimaed kineic parameers are subsiued is compared wih he daa generaed upon he assumpion of non-consan exracion. We observe ha he esimaed mean of secreory rae is markedly lower han he known rae during he firs 90 min and is higher han he known rae afer he firs 90 min for all levels of error. This is o be expeced, since he profiles of insulin concenraion have been generaed under he assumpion of nonconsan fracion of hepaic exracion bu he processes in he Waanabe approach assume ha he fracion is consan, and hus he approach is no able o provide he correc rae of insulin secreion. 3.3 Modified Waanabe approach on daa from ECM assuming non-consan fracional hepaic exracion Afer modificaion is made on he expression involving he esimaed rae of secreion leading us o he new expression shown in Eq. (7 ogeher wih (8, he adjused rae of secreion is shown in Fig. 3. The resul appears o be able o provide accurae rae of secreion from he daa generaed by he exended combined model under he assumpion of non-consan fracion of hepaic exracion. Table. Esimaed kineic parameers (mean ± SE. of insulin and C-pepide kineics from Waanabe approach and he known values used o generae he insulin and C-pepide concenraions by ECM. n = 2. Parameer 2 K K 2 H Known alue Wih daa from ECM assuming consan fracion of hepaic exracion 0% Error % Error % Error Wih daa from ECM assuming non-consan fracion of hepaic exracion 0% Error % Error % Error SBN:

5 0% error secreion rae R( and he fracional eliminaion of insulin K in ECM. 0% error 3% error 3% error 5% error 5% error Fig. 2 Esimaed rae of insulin secreion from Waanabe approach wih daa from ECM assuming non-consan fracion of hepaic exracion. Squares represen he known rae of secreion (daa poins aken from [] by using Daahief program and circles indicae he esimaed rae of secreion. Lef panel shows esimaed rae of secreion for each profile of insulin and C-pepide concenraions and righ panel shows he mean of esimaed rae of secreion a 0%, 3% and 5% errors. 4 Conclusion To dae, many approaches for quanificaion of pancreaic insulin secreion have been proposed o provide accurae esimae of he rae of insulin producion. One of he approaches was suggesed in [4] and [5]. This classic approach requires wo separae ses of daa. The firs daa se is on he exponenial decrease in C-pepide concenraion for he esimaion of kineic parameers of C-pepide and he second daa se is on insulin and C-pepide concenraions during a period of glucose adminisraion for he esimaion of insulin secreion rae. Then, a more advanced approach was proposed by Waanabe [2, 3]. The advanage of his approach is ha i needs only he daa of insulin and C-pepide concenraions for he idenificaion of kineic parameers and he rae of secreion during he period of glucose adminisraion. The approach, proposed in [2], uses algebraic manipulaions o facor ou he insulin secreion rae R( in ECM in order o avoid high correlaion beween he Fig. 3 Esimaed rae of insulin secreion from modified Waanabe approach wih daa from ECM assuming non-consan fracion of hepaic exracion were given. Squares represen he known rae of secreion (daa poins aken from [] by using Daahief program and circles indicae he esimaed rae of secreion. Lef panel shows esimaed rae of secreion for each profile of insulin and C-pepide concenraions and righ panel shows he mean of esimaed rae of secreion a 0%, 3% and 5% errors. The resuls in [2] and [3] indicae ha he approach is able o reconsruc insulin producion. However, experimenal evidences indicae ha he fracion of hepaic exracion is probably no consan hroughou he period of glucose adminisraion. Hence, Waanabe s approach may no be sufficien for he esimaion of insulin producion because he approach is based on he assumpion of consan fracion of hepaic exracion. Therefore, he approach should be exended o also cover reconsrucion of insulin producion under he assumpion of non-consan fracion of hepaic exracion. n his work, he approach, based on ECM under he assumpion of consan fracional hepaic exracion, has been used o esimae he rae of secreion from he daa on insulin and C-pepide concenraions during OGTT, generaed by ECM assuming ha he fracion of hepaic exracion SBN:

6 varies as Eq. (4. The aim was o sudy he performance of Waanabe approach and, as expeced, he approach was no able o provide accurae rae of secreion when compared wih he known rae because he approach is based on he consan fracion of hepaic exracion assumpion bu he daa has been generaed under he assumpion of non-consan fracion. The esimaed rae is found o be higher han he known rae during he firs 90 min and lower han he known rae afer he firs 90 min. When he paern of error of esimaion is known, he Waanabe approach is modified by esimaing he rae of secreion using Eq. (4-(8 by deconvoluion. The resul indicaes ha he esimaed rae of secreion is quie close o he known rae. The key modificaion is in he expression for ( in Eq. (7. This funcion requires wo imporan inpus, an esimaed fracional eliminaion of insulin K calculaed by he Waanabe approach and he funcion h( of fracion of hepaic exracion given in Eq. (4. The erm K can be derived by esimaion, bu h( was aken from curve fiing of Eq. (4 o he hepaic exracion curve repored in [6]. n he presen day, he mechanism ha underlies such variaions in he fracion of hepaic exracion is sill no clearly undersood. This work demonsraes ha he modified approach may be a reliable alernaive ool for he esimaion of insulin secreion in he case ha he paern of fracional hepaic exracion during slow dynamics of glucose adminisraion is known. 5 Acknowledgmen The firs auhor has been suppored by a scholarship from he Developmen and Promoion of Science and echnology Talens projec (DPST and, ogeher wih he hird auhor, he BioMaLab CNR- AS, aly. The second auhor is suppored by he Cenre of Excellence in Mahemaics, CHE, Thailand. References: [] W.C. Duckworh, R.G. Benne, F.G. Hamel, nsulin degradaion: progress and poenial, Endocrine Reviews, ol.9, No. 998, pp [2] T. Brundin, Splanchnic and exrasplanchnic exracion of insulin following oral and inravenous glucose loads, Clinical Science (London, England: 979, ol.97, No.4, 999, pp [3] A. Tura, B. Ludvik, J.J. Nolan, G. Pacini, K. Thomaseh, nsulin and C-pepide secreion and kineics in humans: direc and model-based measuremens during OGTT, American Journal of Physiology Endocrinology and Meabolism, ol.28, No.5, 200, pp [4] L.L. Kjems, E. Chrisiansen, A. ølund, R.N. Bergman, S. Madsbad, alidaion of mehods for measuremen of insulin secreion in humans in vivo, Diabees, ol.49, No.4, 2000, pp [5] R.M. Waanabe, G.M. Seil, R.N. Bergman, Criical evaluaion of he combined model approach for esimaion of prehepaic insulin secreion, American journal of physiology, ol.274, 998, pp [6] R.M. Waanabe, R.N. Bergman, Accurae measuremen of endogenous insulin secreion does no require separae assessmen of C- pepide kineics, Diabees, ol.49, No.3, 2000, pp [7] P.P. Eaon, R. C. Allen, D. S. Schade, K. M. Erickson, J. Sandefer, Prehepaic insulin producion in man: kineic analysis using peripheral connecing pepide behaviour, The Journal of clinical endocrinology and meabolism, ol.5, No.3, 980, pp [8] K.S. Polonsky, J. Licinio-Paixao, B.D. Given, W. Pugh, P. Rue, J. Galloway, T. Karrison, B. Frank, Use of biosynheic human C-pepide in he measuremen of insulin secreion raes in normal voluneers and ype diabeic paiens, The Journal of clinical invesigaion, ol.77, No., 986, pp [9] M. Campioni, G. Toffolo, R. Basu, R.A. Rizza, C. Cobelli, Minimal model assessmen of hepaic insulin exracion during an oral es from sandard insulin kineic parameers, American Journal of Physiology Endocrinology and Meabolism, ol.297, No.4, 2009, pp [0] M. L. Johnson, J. D. eldhuis, Evoluion of deconvoluion analysis as a hormone pulse deecion mehod, Journal of Neuroscience Mehods, ol.28, 995, pp SBN: