^{1}

^{*}

^{1}

Because insulin released by the β-cells of pancreatic islets is the main regulator of glucose levels, the quantitative modeling of their glucose-stimulated insulin secretion is of obvious interest not only to improve our understanding of the processes involved, but also to allow better assessment of β -cell function in diabetic patients or islet transplant recipients as well as the development of improved artificial or bioartificial pancreas devices. We have recently developed a general, local concentrations-based multiphysics computational model of insulin secretion in avascular pancreatic islets that can be used to calculate insulin secretion for arbitrary geometries of cultured, perifused, transplanted, or encapsulated islets in response to various glucose profiles. Here, experimental results obtained from two different dynamic glucose-stimulated insulin release (GSIR) perifusion studies performed by us following standard procedures are compared to those calculated by the model. Such perifusion studies allow the quantitative assessment of insulin release kinetics under fully controllable experimental conditions of varying external concentrations of glucose, oxygen, or other compounds of interest, and can provide an informative assessment of islet quality and function. The time-profile of the insulin secretion calculated by the model was in good agree- ment with the experimental results obtained with isolated human islets. Detailed spatial distributions of glucose, oxygen, and insulin were calculated and are presented to provide a quantitative visualization of various important aspects of the insulin secretion dynamics in perifused islets.

Quantitative models describing the dynamics of glucosestimulated insulin secretion are of obvious interest [

• The development of “artificial pancreas” systems [

• The development of “bioartificial pancreas” systems [

• The improvement of existing models developed to describe the glucose-insulin regulatory system via organism-level concentrations that are widely used to estimate glucose effectiveness and insulin sensitivity from intravenous glucose tolerance tests (IVGTT) such as curve-fitting models (e.g., the “minimal model” [

• The identification of relatively simple indices that 1) are useful indicators of b-cell function in diabetic patients or islet transplant recipients and 2) can be derived from standard readouts such as C-peptide and A1C (e.g., b-score, HOMA-index of b-cell function, secretory unit of islets in transplantation SUIT, Cpeptide/glucose ratio CP/G, or transplant estimated function TEF-see [

We have recently developed a complex, finite element method (FEM)-based computational insulin secretion model that focuses on the quantitative modeling of local, cellular-level glucose-insulin dynamics and is particularly well suited for the modeling of the insulin secretion of isolated, avascular islets [

While there has been considerable work on modeling insulin secretion (especially for encapsulated islets, e.g., [19-23]), the present model has the unique advantage that allows the coupling of both convective and diffusive transport with reactive rates for arbitrary geometries with no symmetry restrictions. Furthermore, the present model also incorporates a comprehensive approach to account not only for firstand second-phase insulin responses, but also for both the glucoseand the oxygendependence of insulin release.

Here, we compare insulin secretion values calculated by the present model with experimental results obtained from two different dynamic glucose-stimulated insulin release (GSIR) perifusion studies performed in our Institute following standard procedures [26,27]. Such perifusion studies (

Islets were isolated at the Diabetes Research Institute’s Cell Processing Facility or other centers participating in the National Institute of Health Islet Cell Resources (ICR) as described before [26,31,32]. Briefly, human pancreata were obtained from multi-organ cadaveric donors and processed with the automated Ricordi method [

pancreas is “cannulated” through the duct to allow for the enzyme to distend the organ. Then, it is placed into the Ricordi chamber and continuously digested in order to obtain fragments of progressively decreasing sizes [_{2} in non-tissue treated flasks with Miami-defined media 1 (MM1; Mediatech) containing human serum albumin [27,34]. After 24 h of culture, islets are collected and divided in different culture conditions (1500 - 2000 IEQ per condition) for incubation with test compounds (here, loteprednol etabonate [

The dynamic glucose-stimulated insulin release (GSIR) perifusion experiments were performed using a custom built apparatus (BioRep Technologies, Inc., Miami, FL). Islets aliquots were loaded in Perspex microcolums, between two layers of acrylamide-based microbead slurry (Bio-Gel P-4, Bio-Rad Laboratories, Hercules, CA). Perifusing buffer containing 125 mM NaCl, 5.9 mM KCl, 1.28 mM CaCl_{2}, 1.2 mM MgCl_{2}, 25 mM HEPES, 0.1% bovine serum albumin, and 3 mM glucose at 37˚C with selected glucose (low = 3 mM; high = 11 mM) or KCl (25 mM) concentrations was circulated through the columns at a rate of 100 mL/min. After 45 - 60 minutes of washing with the low glucose solution for stabilization, the islets were stimulated with the following sequence: 5 min of low glucose, 10 min of high glucose, 15 min of low glucose, 5 min of KCl, and 5 min of low glucose. Serial samples (100 mL) were collected every minute from the outflow tubing of the columns in an automatic fraction collector designed for a multi-well plate format. The sample container harboring the islets and the perifusion solutions were kept at 37˚C in a built-in temperature controlled chamber. The perifusate in the collecting plate was kept at <4˚C to preserve the integrity of the analytes in the perifusate. Insulin concentrations were determined with a commercially available ELISA kit (Mercodia Inc., Wiston Salem, NC); values used here are averages across the different conditions used in the two different experimental settings.

For computational modeling, our previously developed local concentration-based insulin secretion model has been used here; a detailed description of its implementation and parameterization has been published [_{gluc}, c_{oxy}, c_{insL}, c_{ins}). Diffusion is assumed to be governed by the generic diffusion equation in its nonconservative formulation (incompressible fluid) [

where, c denotes the concentration [mol×m^{−3}] and D the diffusion coefficient [m^{2}×s^{−1}] of the species of interest, R the reaction rate [mol×m^{−3}×s^{−1}], u the velocity field [m×s^{−1}], and Ñ the standard del (nabla) operator . The following diffusion coefficients are used: oxygen, D_{oxy}_{, w} = 3.0 ´ 10^{−9 m2}×s^{−1 in aqueous media and D}_{oxy}_{, t} = 2.0 ´ 10^{−9 m2}×s^{−1 in islet tissue; glucose, D}_{gluc}_{, w} = 9.0 ´ 10^{−10 m2}×s^{−1} and D_{gluc}_{, t} = 3.0 ´ 10^{−10 m2}×s^{−1}; insulin, D_{ins}_{, w} = 1.5 ´ 10^{−10 m2}×s^{−1} and D_{ins}_{, t} = 5 ´ 10^{−11 m2}×s^{−1}.

As an important part of the model, all consumption and release rates are assumed to follow Hill-type dependence (generalized Michaelis-Menten kinetics) on the local concentrations because this provides a convenient and easily parameterizable mathematical function for biological/pharmacological applications:

Parameters here are R_{max}, the maximum reaction rate [mol×m^{–3}×s^{–1}], C_{Hf}, the concentration corresponding to half-maximal response [mol×m^{–3}], and n, the Hill slope characterizing the shape of the response. The parameter values used for the different release and consumption functions (i.e., insulin, glucose, oxygen; e.g., C_{Hf}_{, gluc}C_{Hf}_{, oxy}, etc.) are different; their values used in the model are summarized in

For avascular islets, oxygen availability is a main limiting factor because the solubility of oxygen in culture media or in tissue is much lower than that of glucose; hence, available oxygen concentrations are usually much more limited (e.g., around 0.05 - 0.2 mM for oxygen vs 3 - 15 mM for glucose assuming physiologically relevant conditions). In this context, it is important to note that, to account for the increased metabolic demand of insulin release and production at higher glucose concentrations, the model also includes a dependence of the oxygen consumption (R_{oxy}) on the local glucose concentration via a modulating function j_{o}_{, g} (c_{gluc}):

Undeniably, the most important part of the model is the functional form describing the glucose-dependence of the insulin secretion rate, R_{ins}. For this purpose, a Hill equation providing a sufficiently abrupt sigmoid response proved adequate; the shape of its dependence on c_{gluc} is shown in

with n_{ins}_{2, gluc} = 2.5, C_{Hf}_{, ins2, gluc} = 7 mM, and R_{max, ins2} = 3.0 ´ 10^{−5} mol×m^{−3}×s^{−1}. For the first-phase response, a Hill-type sigmoid component that depends on the glucose time-gradient (c_{t} = ¶c_{gluc}/¶t) is incorporated into the model. This is non-zero only when the glucose concentration is increasing, i.e., only when c_{t} > 0. For a correct time-scale of insulin release, an additional “local” insulin compartment had to be added (

Finally, to also incorporate media flow, these convection and diffusion models are coupled to a fluid dynamics model. For this purpose, the incompressible NavierStokes model for Newtonian flow (constant viscosity) is used to calculate the velocity field u resulting from convection [

The first equation is the momentum balance, and the second one is the equation of continuity for incompressible fluids. Standard notation is used with r denoting density [kg×m^{−3}], h viscosity [kg×m^{−1}×s^{−1} = Pa×s], p pressure [Pa, N×m^{−2}, kg×m^{−1}×s^{−2}], and F volume force [N×m^{−3}, kg×m^{−2}×s^{−2}]. The flowing media is assumed to be an essentially aqueous media at body temperature (37˚C). Incoming media is assumed to be in equilibrium with atmospheric oxygen and, thus, to have an oxygen concentration of c_{oxy}_{, in} = 0.200 mol×m^{−3} (mM) corresponding to ≈ 140 mmHg.

All models were implemented in COMSOL Multiphysics 4.2 (COMSOL Inc., Burlington, MA) and solved as time-dependent (transient) problems allowing intermediate time-steps for the solver as described before. Two spherical islets of 100 and 150 mm diameter are used for modeling and they are placed in a 2D crosssection of a cylindrical tube with fluid flowing from left to right. Our recent analysis of the size distribution of (isolated) human islets based on data from more than 200 isolations [^{6} mm^{3}, which corresponds to the volume of an islet with d = 133 mm; hence, the two sizes used for modeling here (d = 100 and 150 mm) can be considered as representative for human islets used in perifusion assays.

The experimental insulin secretion data of isolated human islets used here were obtained from dynamic glucose-stimulated insulin release (GSIR) perifusion experiments performed using a custom built apparatus as described in the Materials and Method section. Such perifusion experiments are regularly used in our Institute for the quantitative assessment of islet quality and function [_{oxy} = 0.2 mM; ≈ 140 mmHg).

As can be seen in

high-glucose stimulation used in these standard islet assessment protocols is of only a 10-min duration, the firstand second-phases of insulin secretion are not well separated, but the overall shapes of the insulin responses are in good agreement (calculated amplitude, which is a function of the actual islet mass present, was adjusted for best fit). The model seems to account well for both the rapid increase in insulin secretion following the glucose step-up (3 mM ® 11 mM) and the rate of its decline following the glucose step-down.

An important advantage of the present computational model is that it not only makes possible detailed simulations for arbitrary inflow conditions and islet arrangements, but also the easy generation of various corresponding 2D or 3D visualizations. For example,

In conclusion, our local concentration-based multiphysics insulin secretion computational model can account well for the insulin secretion profile of isolated,

avascular human islets obtained in standard GSIR perifusion assays. Detailed spatial distributions of all concentrations of interest can be obtained and visualized making it possible to investigate the quantitative aspects of the insulin secretion dynamics in a much more convenient manner and allowing user-friendly computational modeling for arbitrary conformations of cultured, perifused, transplanted, or encapsulated islets.

The financial support of the Diabetes Research Institute Foundation (www.diabetesresearch.org) that made this work possible is gratefully acknowledged.