Presented at the COMSOL Conferene 2010 Boston
Normal Human Blood Gluose and Insulin Levels In healthy humans, blood gluose levels have to be maintained in a relatively narrow range (3.5 7.0 mm, 60 130 mg/dl in fasting subjets) Mainly ahieved by adjusting insulin levels with the ells of panreati islets ating as gluose sensors and releasing insulin (mg/dl) 125 Total blood gluose (in 75 kg human) 5 g 60 Sukale, J.; Solimena, M. Front. Biosi. 2008, 13, 7156.
Islets of Langerhans Cellular aggregates of approx. 2,000 ells and diameters of about 150 m (range: 50 500 m) loated in the panreas and responsible for its endorine (hormone releasing) funtion Represent only 1 2% of the panreas Humans have approx. 1,000,000 islets ( 2 ml) Four major ell types sereting different hormones: ells (gluagon) ells (insulin) (somatostatin), and PP ells (panreati polypeptide) There are onsiderable speies differenes [~35%, human] [~60%, human] [~5%, human] Pileggi, A. et al. in Pro. Atlas of Diabetes 2005, p. 253. Insulin auses ells to take up gluose (from the blood) and store it as glyogen (liver, musle); it also stops the use of fat as energy soure Cabrera, O. et al. Pro. Natl. Aad. Si. USA 2006, 103, 2334.
Phases of Insulin Seretion Hedeskov, C. J. Physiol. Rev. 1980, 60, 442. Rorsman, P. et al. News Physiol. Si. 2000, 15, 72 (after Ma, Y. H.,, Grodsky, G.M. et al. Eur. J. Endorinol. 1995, 132, 370) Shemati illustration of lateny period and the two phases of insulin release. Depending on experimental onditions, seond phase of seretion may be of muh longer duration than shown here.
Quantitative Gluose-Insulin Models
Perifusion Devie with Flowing Media Routinely used to assess islet quality and funtion Allows the dynami measurement of the gluose-stimulated insulin release (GSIR) (and/or other metaboli produts) Sweet, I. R. et al. Diabetes Tehnol. Therap. 2002, 4, 661. Cabrera, O. et al. Cell Transplant. 2008, 16, 1039.
Insulin Release in Dynami Perifusion Model: Geometry and Mesh d = 150 m Fluid (gluose, O 2 ) in Insulin out d = 100 m Considering the size distribution of human islets (Buhwald, P. et al. Cell Transplant. 2009, 18, 1223), islets with diameters d = 100 and 150 μm are most representative.
Multiphysis Model Convetion and diffusion (3 ) t D R u Fluid dynamis (inompressible Navier-Stokes(onvetion and ondution): u 2 u t u u p F; u 0 1 : insulin; 2 : gluose; 3 : gen Parameter settings Hormone seretion and nutrient onsumption kinetis, whih form the essene of the model, are built into Rs Flow (aqueous media at room temperature): T 0 = 310.15 K, = 993 kg/m 3, = 0.7 10 3 Pa s, p = 4200 J/kg/K, k = 0.634 J/s/m/K, = 2.1 10 4 K 1 paraboli inflow profile on inlet 4v in (y/y max )(1 y/y max ); v in = 10-4 m/s Inoming gen: Inoming gluose: atm = 0.200 mol/m 3 (0.2 mm; po 2 140 mmhg; normal ulture 95% air, 5% CO 2 ; 37 C) hypoxia: in = 0.036 mol/m 3 (0.036 mm; po 2 25 mmhg), et. inoming glu inreased stepwise from 1 mm to 10 19 mm using sum of Heaviside funtions 2D ross-setion models with realisti geometries (islets with diameters of d = 100 and 150 m) Default extra fine mesh size used (mesh sizes of 5,000-9,000 elements) Solved as time-dependent (transient) problem with the Pardiso diret solver
Present Gluose-Insulin(-Oxygen) Model Insulin Gluose glu glu / t -ell Oxygen Sigmoid, Hill funtion type responses R R max n n C n Hf Buhwald, P. COMSOL Conf. Boston 2010.
Hill Funtion / Hill Equation (Generalized Mihaelis-Menten Kinetis) 1.00 R max 0.90 0.80 Mihaelis-Menten n = 1 0.70 0.60 f H () 0.50 0.40 R f H R max n n C n Hf 0.30 0.20 0.10 C Hf n = 1 n = 2 n = 3 n = 0.5 0.00 0 5 10 15 20 25 30 35 40
Main assumptions Oxygen Oxygen Dynamis Oxygen onentrations: in = atm = 0.200 mol/m 3 [140 mmhg; atmospheri, 21%] tissue = 0.050 mol/m 3 [35 mmhg; tissue & venous 40 mmhg] art = 0.130 mol/m 3 [90 mmhg; arterial] in,low = 0.036 mol/m 3 [25 mmhg; hypoxia for perifusion] Diffusion: D,w = 3.0 10-9 m 2 /s (O 2 in water) D,t = 2.0 10-9 m 2 /s (O 2 in islet tissue) D,Si = 2.0 10-9 m 2 /s (O 2 in silione rubber) Oxygen onsumption and ell viability: C R Rmax, o, g glu r, CHf, R max, = 0.034 mol/m 3 /s {i.e., 0.6 10-13 mol/s/ieq} (averaged best estimate) C Hf, = 1.0 10-3 mol/m 3 (Mihaelis-Menten onstant) [0.7 mmhg] C r, = 1.0 10-4 mol/m 3 (ritial for survival) [0.07 mmhg] () = fl1hs( 1.0 10-4, 0.5 10-4 ) COMSOL s smooth Heaviside funtion (step-down) o,g ( glu ) modulating fator to aount for inreased gen onsumption at high gluose due to inreased metaboli demand here, assumed to have a base omponent (50%) and a metaboli omponent that inreases in parallel with inreasing insulin seretion Buhwald, P. Theor. Biol. Med. Model. 2009, 6: 5.
General Hill-Type Conentration-Dependene of Oxygen Consumption in Mitohondria 0.8 0.7 Oxygen onsumption rate, R (μm/s) 0.6 0.5 0.4 0.3 0.2 R R max, Experimental Best fit Fit with n=1 Present model n n n CHf, n = 1.1; C Hf = 0.7 μm n = 1; C Hf = 0.8 μm n = 1; C Hf = 1.0 μm 0.1 0.0 0 2 4 6 8 10 12 14 16 18 C r, = 0.1 μm (μm) Fit of Hill type (generalized Mihaelis-Menten) type dose response for gen onsumption at low gen onentrations by allowing a variable Hill slope. Data from Wilson, D. L et al. J. Biol. Chem. 1988, 263, 2712. There seems to be no need for n > 1.
Islet Culture 2D Model Oxygen Conentrations in Nonvasularized Islets in Traditional Culture Calulated gen onentration for three islets (with diameters = 100, 150, and 200 m) in standard ulture onditions as stationary onditions are being reahed (h = 1 mm assumed). The olor-oded surfae represents the gen onentration (blue orresponding to higher and red to lower values). Areas with values below a ritial value (<10 4 mol m 3 ), where the lak of gen (hypoxia) is predited to ause ell death (nerosis) are left unolored (white). Beause this is a 2D ross-setion, it roughly orresponds to a 3D ulture density of about 1,600 IEQ/m 2. Buhwald, P. Theor. Biol. Med. Model. 2009, 5:6.
Gluose-Insulin(-Oxygen) Dynamis Main assumptions Gluose & Insulin Diffusion: D ins,w = 1.5 10-10 m 2 /s (insulin in blood) D ins,t = 0.5 10-10 m 2 /s (insulin in islet tissue) /somewhat lowered to aount for ellular release/ D glu,w = 9.0 10-10 m 2 /s (gluose in blood) D glu,t = 3.0 10-10 m 2 /s (gluose in islet tissue) /somewhat lowered to aount for ellular uptake/ Gluose onsumption: Insulin release: glu R glu Rmax, glu Cr, glu CHf, glu seond phase PR ins, ph2 PR first phase t PRins ph PR, 1 max, ins1 nins1, glu t total with gen modulation PR ins max, ins2 n glu ins 2, glu glu glu nins 2, glu glu nins 2, glu CHf, ins2, glu n ins1, glu Ct nins1, glu Hf, ins1, glu ins, PRins, ph1 PRins, ph2 nins nins, n C, Hf, ins, R max,glu = 0.028 mol/m 3 /s C Hf,glu = 10.0 10-3 mol/m 3 [10 M] PR max,ins2 = 3.0 10-5 mol/m 3 /s [~20 pg/ieq/min] n ins2,glu = 2; C Hf,ins2,glu = 7.5 mm (gluose) PR max,ins1 = 1.5 10-5 mol/m 3 /s n ins1,glu = 2; Ct Hf,ins1,glu = 0.01 mm/s (gluose hange) n ins, = 3; C Hf,ins, = 3.0 10-3 mol/m 3 [2.1 mmhg] Buhwald, P. COMSOL Conf. Boston 2010.
General Hill-Type Conentration-Dependene of Gluose-Indued Insulin Seretion in Perifused Human Islets 100 90 80 Insulin seretion rate (%) 70 60 50 40 30 20 10 PR ins, ph2 PR max, ins2 n glu ins 2, glu Experimental Best fit Fit with n=1 Present model nins 2, glu glu nins 2, glu CHf, ins2, glu n ins2,glu = 2.7; C Hf = 6.6 mm n ins2,glu = 1; C Hf = 4.9 mm n ins2,glu = 2; C Hf = 7.5 mm 0 0 5 10 15 20 25 30 glu (mm) Fit of Hill type (generalized Mihaelis-Menten) type dose response for insulin seretion rate allowing a variable Hill slope. Data from Henquin, J. C. et al. Diabetologia 2006, 55, 3470. There seems to be a lear need for n > 1.
Hill Type Conentration-Dependene of Insulin Seretion on Oxygen Conentration vs. Bilinear Version of Johnson, Colton et al. (MIT) 1.20 f = 1 if p > 5.1 mmhg; f = p/5.1 if p < 5.1 mmhg Fration of max. insulin release 1.00 0.80 0.60 0.40 0.20 i, o ins, nins nins, n C, Hf, ins, Bilinear MIT model Present model n ins, = 3; C Hf = 3 M (2 mmhg) 0.00 0 2 4 6 8 10 12 po 2 [mmhg] Loal gen-dependent limiting funtion for insulin release used in the present model ompared to the simple bilinear funtion used by Colton and o-workers at MIT (Johnson, A. S. et al. Chem. Eng. Si. 2009, 64,4470).
Insulin Release in Dynami Perifusion Model Fluid (gluose, O 2 ) in Insulin out Calulated insulin onentration shown as olor-oded from low (blue) to high (red) in two perifused islets shown at a time-point when the gluose onentration is dereasing abruptly (from 19 mm to 3 mm, olored ontour lines). Gray streamlines and arrows illustrate the veloity field of the flowing perifusion fluid.
Model-Calulated Insulin Release insulin outflow 10 mm 7 mm 5 mm 3 mm 1 mm gluose onentration Boundary surfae integral of total insulin flux out on outlet surfae as a funtion of time. Fully saled 2D ross-setion of islets in a hypothetial perifusion hamber. Finite element method (COMSOL Multiphysis 3.5) used for diffusion modeling with gluosedependent insulin release and gen onsumption rate. Aqueous media for flow; gen onentration: normal atm = 0.200 mol/m 3 (140 mmhg); fluid flow: v in = 1.0 10-4 m/s; extra fine mesh, Pardiso diret solver, transient solution.
Calulated vs. Experimental Insulin Release 0.07 G7 G10 10 0.06 G3 G5 5 Insulin seretion (% of ontent per min) G1 0.05 0.04 0.03 0.02 Human islets G1 0-5 -10-15 -20 Inoming gluose onentration (mm) 0.01-25 0.00-30 -20 0 20 40 60 80 100 120 Time (min) Boundary surfae integral of total insulin flux out on outlet surfae as a funtion of time. Fully saled 2D ross-setion of islets in a hypothetial perifusion hamber. Finite element method (COMSOL Multiphysis 3.5) used for diffusion modeling with gluosedependent insulin release and gen onsumption rate. Aqueous media for flow; gen onentration: normal atm = 0.200 mol/m 3 (140 mmhg); fluid flow: v in = 1.0 10-4 m/s; extra fine mesh, Pardiso diret solver, transient solution. Experimental data from Dufrane, D. et al. Diabetes Metabol. 2007, 33, 430.
Insulin Release in Dynami Perifusion Model Movie from InsulinCon Movie1.wmv Calulated insulin onentration shown as olor-oded from low (blue) to high (red) in two perifused islets shown at hanging gluose onentrations (inreasing stepwise from 3 mm to 19 mm than dereasing bak to 3 mm, olored ontour lines). Gray streamlines and arrows illustrate the veloity field of the flowing perifusion fluid.
Gluose-Insulin(-Oxygen) Perifusion Model Insulin Conentration Gluose Conentration Insulin Prodution Oxygen Conentration Calulated onentrations shown olor-oded in two perifused islets at a time-point when the gluose onentration is dereasing abruptly (from 19 mm to 3 mm, olored ontour lines).
Calulated Insulin Release at Hypoxia insulin outflow at normoxia (po 2 = 140 mmhg) 10 mm p O2 (mmhg),in (mm) F ins,exp F ins,pred 142 0.206 1.00 1.00 60 0.087 0.99 0.98 25 0.036 0.48 0.46 15 0.022 0.17 0.15 5 mm 7 mm insulin outflow at hypoxia (po 2 = 25 mmhg) Experimental vs. predited fration of insulin seretion rate (F ins ) at hypoxi onditions. 3 mm gluose onentration 1 mm Boundary surfae integral of total insulin flux out on outlet surfae as a funtion of time. Fully saled 2D ross-setion of islets in a hypothetial perifusion hamber. Finite element method (COMSOL Multiphysis 3.5) used for diffusion modeling with gluosedependent insulin release and gen onsumption rate. Aqueous media for flow; gen onentration: normal atm = 0.200 mol/m 3 (140 mmhg); fluid flow: v in = 1.0 10-4 m/s; extra fine mesh, Pardiso diret solver, transient solution.
Gluose-Insulin(-Oxygen) Perifusion Model at Hypoxia Insulin Conentration Gluose Conentration Insulin Prodution Oxygen Conentration Calulated onentrations shown olor-oded in two perifused islets at a time-point when the gluose onentration is dereasing abruptly (from 19 mm to 3 mm, olored ontour lines).
Insulin Release and Oxygen Normoxi vs. Hypoxi Conditions A. po 2 = 140 mmhg B. po 2 = 25 mmhg Loal insulin onentration (as height data) olored by loal gen onentration during a hange in the perifusing gluose onentration (ontour plot). Fully saled 2D ross-setion of islets in a hypothetial perifusion hamber. Normal gen onentration in = atm =0.200 mol/m 3 (140 mmhg) (A), hypoxi gen onentration in = 0.036 mol/m 3 (25 mmhg) (B); fluid flow: v in = 1.0 10-4 m/s (0.1 ml/min).
A. po 2 = 140 mmhg Insulin Release and Oxygen Movie from InsulinConHeight Lighted.wmv Loal insulin onentration (as height data) olored by loal insulin prodution during hanging perifusing gluose onentrations (ontour plot). Fully saled 2D ross-setion of islets in a hypothetial perifusion hamber. Normal gen onentration atm = 0.200 mol/m 3 (140 mmhg); fluid flow: v in = 1.0 10-4 m/s.
Insulin Release and Oxygen (Hypoxia) B. po 2 = 25 mmhg Movie from InsulinConHeight Lighted 25 mmhg.wmv Loal insulin onentration (as height data) olored by loal insulin prodution during hanging perifusing gluose onentrations (ontour plot). Fully saled 2D ross-setion of islets in a hypothetial perifusion hamber. Normal gen onentration atm = 0.036 mol/m 3 (25 mmhg); fluid flow: v in = 1.0 10-4 m/s.
Conlusions Exploratory insulin seretion model for avasular panreati islets has been implemented using Hill-type sigmoid response funtions Model was parameterized to fit experimental data and good fit ould be obtained both for gluose- and for gen-dependene (exept time-sale of first-phase release) With COMSOL Multiphysis it is relatively straightforward o o to ouple arbitrarily omplex hormone seretion and nutrient onsumption kinetis with diffusive and even onvetive transport and run simulations with realisti geometries without symmetry or other restritions problems that seriously limited previous gluose insulin modeling attempts
Aknowledgments Camillo Riordi Norma Kenyon Finanial Support: Diabetes Researh Institute Foundation (www.diabetesresearh.org)