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Modeling Oxygen Transport in a Cylindrical Bioartificial Pancreas

Thrash, Marvin

doi: 10.1097/MAT.0b013e3181e3f874
Liver/Pancreas Support

Encapsulated pancreatic islets in a cylindrical hollow fiber have been reported to reverse diabetes in test animals; however, for many of these animals, the effects have only been temporary. Oxygen deficiency within the fiber has been proposed as a cause of the observed loss of islet viability. A mathematical model of transport and reaction kinetics in a bioartificial pancreas (BAP) has been developed to calculate the oxygen concentration profiles in a cylindrical BAP. Simulation results indicate that hypoxic conditions will exist in large diameter fibers or fibers with islet concentrations >20,000 islets/ml. Moreover, our results show that a significant amount of oxygen is consumed in the tissue region surrounding the cylindrical BAP. Even if the islet survives in a low-oxygen environment, the insulin productivity will likely be reduced.

From the Department of Paper and Chemical Engineering, School of Engineering and Applied Science, Miami University, Miami, FL.

Submitted for consideration November 2009; accepted for publication in revised form March 2010.

Reprint Requests: Marvin Thrash, PhD, Miami University, Engineering Building, 650 East High Street, Oxford, OH 45056. Email: thrashme@muohio.edu.

The encapsulation of islets within a semipermeable membrane, called a bioartificial pancreas (BAP), was thought to hold promise as a treatment for Type 1 diabetes in the 1990's. Although the concept of the BAP was touted as a possible alternative treatment for diabetes, experimental results in test animals were not always successful. Examination of the BAPs removed1 from the test animals revealed that the islets in the center had died, whereas those closest to the fiber wall were still viable. The lack of oxygen within these devices has been hypothesized to be the cause of both islet mortality and loss of functionality. In previous reports,2 experimental oxygen measurements on implanted BAPs revealed the presence of oxygen in the fiber center. Despite this, the islets in the center were dead, whereas the islets at the wall were still viable. If a significant oxygen gradient existed in the initial stages after transplantation, then it stands to reason that the islets closest to the fiber wall would have the highest probability of surviving. Although this hypothesis is supported by islet viability studies,2 the experimental measurement of an oxygen gradient in an implanted fiber is challenging. Once the islets in the center die, pO 2 levels in the fiber center will increase and, thus, give the appearance of a minimal or nonexistent gradient. To test this hypothesis that a significant oxygen gradient existed in the initial stages after implantation, we have put together a two-compartment mathematical model to study oxygen transport in a cylindrical BAP and the impact of pO 2 levels on insulin secretion. Unlike prior modeling projects,3,4 this modeling effort includes the following processes: 1) oxygen transport between the blood supply and the interior wall of the hollow fiber, including diffusion through the surrounding tissue and membrane wall; 2) oxygen diffusion within the islet sphere (similar to that of Avgoustiniatos and Colton5); and 3) BAP insulin secretion with a pharmacokinetic model of insulin in the circulatory system. The results of this model are presented and compared qualitatively with experimental data from the literature.

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Materials and Methods

Model Development

BAP Oxygen Diffusion Model.

The BAP considered here is a cylindrically shaped device (fiber) that contains alginate seeded with a uniform distribution of islets, with each end of the fiber sealed (Figure 1). The transport of oxygen and glucose from the blood to the BAP is based on four sequential diffusion processes: 1) oxygen transport from the blood through the capillary wall into tissue; 2) oxygen transport through the tissue where the oxygen is partially consumed; 3) oxygen transport through the hollow fiber wall; and 4) oxygen transport through the alginate where it is partially consumed by the islets. The transport processes of insulin occur in the opposite direction.

Figure 1.

Figure 1.

Our model for the oxygen concentration profile is based on several assumptions. The oxygen concentration is at steady state, and thus, changing physiologic conditions brought about by extraordinary activities are not considered. Porcine islets with 150-μm diameter were used in our model, because the insulin molecule from porcine islets is similar to that produced by human islets. The effect of oxygen on porcine islet insulin kinetics is considered the same as that reported for rat islets.6,7 The partial pressure of oxygen in the tissue just outside the capillary wall is assumed to be equal that of the partial pressure just inside the capillary.

As seen in Figure 1, the BAP oxygen concentration model consists of three compartments—tissue, alginate, and islet. The governing equation for oxygen transport and reaction in the tissue compartment in cylindrical coordinates is shown in Equation 1. This equation was derived from mass conservation principles, assuming diffusive transport only and zero order oxygen consumption.

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where □is the radial coordinate, D O2−t is the diffusivity of oxygen in tissue, pO 2−t is the partial pressure of oxygen in tissue, σ is Henry's constant for oxygen solubility that converts concentration units into units of mm Hg, and ΓO2−t is the zero order consumption rate for oxygen in tissue.

The boundary conditions are given here:

  • At the tissue/capillary boundary:
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  • Flux continuity at tissue/fiber (r = R t,f) wall boundary:
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where pO 2−cap is the partial pressure of oxygen in the capillary, pO 2−t/cap is the partial pressure of oxygen at the tissue/capillary interface, A outer is external surface area of the fiber at the surrounding tissue/fiber boundary, V fiber is volume of the hollow fiber, and ΓO2−islets is the zero order oxygen consumption rate for islets encapsulated in the fiber, and ε is the void fraction. Using the Krogh tissue cylinder8 approach, the average distance between the capillaries and the fiber was estimated to be 0.003 cm. This value was used to calculate the distance from the center of the fiber to the outer capillary wall (R capillary) used in the boundary condition presented in Equation 2. The analytical solution for Equation 1 and the accompanying boundary conditions is presented in Equation 4.

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The oxygen balance equation in the alginate compartment is similar to that in the tissue:

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where D O2−a is the diffusivity of oxygen in alginate, pO 2−a is the partial pressure of oxygen in alginate.

The boundary conditions for the alginate compartment are shown in the following equation.

  • Flux continuity at alginate/fiber wall:
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  • Minimum oxygen concentration at the fiber center
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where pO2−t,f is the oxygen concentration at the tissue/fiber interface, pO2−a,f is the oxygen concentration at the alginate/fiber wall interface, A inner is internal surface area of the fiber at the alginate/fiber boundary, and kp is the permeability of the fiber. The governing equations and their boundary conditions were solved analytically to obtain the oxygen concentration profiles in the tissue and BAP. The analytical solution for Equation 5 and the accompanying boundary conditions are shown in Equation 8.

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The model described earlier gives the oxygen profile within the fiber assuming that the oxygen uptake rate is a continuous function within the alginate. In reality, an islet is a relatively large unit, and a considerable oxygen gradient may exist within each islet that is greater than the gradient within the islet-free regions of the alginate. Thus, the oxygen concentration profile within each 150-μm diameter porcine islet was calculated, as described here. The governing equation and boundary conditions for oxygen transport in a spherical islet are as follows:

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where pO 2−i is the partial pressure of oxygen within the islet, D O2−i is the diffusivity of oxygen in islet tissue and Γsingle−islet is the zero order oxygen consumption for a single islet. The first boundary condition results from assuming a minimum oxygen concentration at the islet center:

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The partial pressure of oxygen at all points on the islet surface was assumed to be constant to simplify the calculations. The partial pressure of oxygen at the radial location in the fiber (calculated from Equation 4), where the islet center is located was taken to be the average partial pressure on the surface of the islet, leading to the following boundary condition:

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where Ri is the islet radius.

The analytical solution to Equation 9 and the accompanying boundary conditions is presented in Equation 12.

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All modeling parameter values are shown in Table 1.

Table 1

Table 1

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Insulin Production Model.

The oxygen model equations presented earlier were solved, and the results were incorporated in the model equations for glucose and insulin reaction and transport in the BAP, and the pharmacokinetic model of insulin release into the circulatory system in response to a glucose challenge, originally presented by Buladi et al. 3 The complete model can be found in Buladi et al. 3; however, in this section, we will reproduce only the equation for insulin transport in the fiber, because the kinetics of insulin production in this equation will be modified to reflect dependency on oxygen. The governing equation describing insulin transport in the fiber, rewritten from Buladi et al.,3 is shown in the following equation:

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CG is the glucose concentration, CI is the insulin concentration, DI−a is the diffusivity of insulin through alginate, y islet is the islet concentration in the alginate, and RI (CG (t, r), t) is the rate of insulin production per islet. The equation was modified from the original source3 to reflect no axial variation. Equation 13 and the associated boundary conditions shown in Equations 14 and 15 were solved numerically using the method of finite differences specified in Buladi et al. 3

At the fiber center (r = 0), the assumption of symmetry is used, and as such, the change in insulin concentration is 0.

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At the fiber wall (r = R fiber), flux continuity is assumed.

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The overall mass transfer coefficient (K PI) for insulin transport accounts for the resistance of the fiber membrane, the resistance of the tissue, and the film mass transfer coefficient through a blood boundary layer. The initial insulin concentration in the fiber is assumed to be equal to the insulin concentration in bloodstream. Previous work had assumed that the oxygen supply was sufficient to yield maximal insulin secretion; however, Otha et al. 7 and Dionne et al. 6,11,12 have independently shown that the insulin production is also a function of the surrounding oxygen concentration.

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Modification of Insulin Production Kinetics.

Buladi et al. 3 modified Pillarella's13 model to include a linear increase in insulin production during the first phase of insulin secretion, to yield the following:

First phase:

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Second phase:

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where

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CG is the glucose concentration. CG 1, CG 2, Vm 1, Vm 2, Km 1, and Km 2 are described as reaction rate parameters in the original reference, and the corresponding values for these parameters presented by Buladi et al. 3 were used in this application. The reaction rate parameters in Equations 17 to 19 were evaluated by fitting the model to experimental data presented in Inoue et al. 14 and are shown in Table 2. From the concentration data used by Buladi et al.,3 the slope was calculated at each point to get the rate of insulin production, and the parameters were calculated using nonlinear regression.3 The insulin production rates for both phases are essentially independent of the oxygen level when the partial pressure of oxygen within the islet exceeds 60 mm Hg. At lower levels partial pressure of oxygen, the first phase of insulin production rate was reduced approximately 67% of its nominal value when the partial pressure of oxygen in the fiber decrease between 10 and 60 mm Hg. The second phase of the insulin production rate was found to be proportional to the oxygen concentration less than 60 mm Hg. However, the oxygen level did not seem to affect any of the delay times or time constants.

Table 2

Table 2

This information was used to modify the expressions for the two phases of insulin secretion, given by Equations 16 to 19. When the oxygen pressure is >60 mm Hg, both phases are unaffected by oxygen, as stated earlier, so expressions for A and B remain unchanged. When oxygen pressure is <60 mm Hg within the islet, the expressions for both phases of insulin production can be represented as:

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where φ is ratio of the total amount of oxygen present in an encapsulated islet divided the total amount of oxygen present in the islet when the partial pressure of oxygen is at least 60 mm Hg everywhere in the islet.6,11,12 The quantity φ can be thought of as a pseudo effectiveness factor that accounts for insulin production dependency on oxygen.

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Results

The pO 2 profiles within the fiber, generated from Equations 1 to 8, are shown in Figure 2 for various islet concentrations. The data used to generate these plots are presented in Table 1. Figure 2A shows that as the cell concentration is reduced by a factor of eight from 40,000 to 5,000 islets/ml, the partial pressure of oxygen in the fiber center significantly increases, indicating the sensitivity of oxygen concentration to cell concentration. When the capillary pO 2 is 95 mm Hg (Figure 2A), the pO 2 level in a large portion of the fiber is <60 mm Hg for all islet concentrations. As previously stated, studies6,11,12 have shown that insulin secretion will be lower when the pO 2 level decreases below 60 mm Hg in the islet. When the islet concentration is 40,000 islets/ml, the pO 2 level in the fiber center is hypoxic. Partial pressure <20 mm Hg is of particular concern, because islets are roughly spherical-shaped organs; consequently, a considerable oxygen gradient can exist within the islet. For example, the approximate diameter of a porcine islet is 150 μm. Experimental results11,16 have shown that when the pO 2 level in the immediate vicinity of the islet is approximately 20 mm Hg, the islet center becomes necrotic with only a thin layer of cells remaining viable for a limited period of time on the exterior of the islet. Figure 2B shows that a capillary partial pressure of oxygen of 68 mm Hg worsens the situation. The results in Figure 2, A and B show that, at high islet concentrations, the islets closest to the fiber wall have the best chance to survive, because the pO 2 level is highest in this area. Our modeling results that indicate the presence of significant oxygen gradient will exist when 1) the islet concentrations are >5,000 islets/ml, 2) the pO 2 level in the blood decreases to 68 mm Hg, and 3) when the external layer of tissue thickens. Increasing the fiber wall thickness or internal fiber diameter will also reduce oxygen availability to the islets in the center.

Figure 2.

Figure 2.

As previously stated, the actual partial pressure of oxygen within each islet will be much lower than the values indicated in Figure 2, because an oxygen concentration gradient will exist within each islet. The theoretical oxygen partial pressure profiles in porcine islets at different locations in a hollow fiber were calculated using Equation 12. Figure 3 shows the theoretical oxygen partial pressure profiles within islets whose centers reside at radial locations 405, 235, and 75 μm within a 1.0-mm diameter fiber transplanted into a host with an arterial blood oxygen partial pressure of 95 mm Hg. The partial pressures of oxygen at the radial locations 405, 235, and 75 μm determined from Figure 2A were used as the average pressure at the islet surface. At 5,000 islets/ml (Figure 3A), the modeling results show that oxygen is present in all the islet centers. At 10,000 islets/ml, the oxygen concentration in the islets was slightly lower. At 20,000 islets/ml (Figure 3B), oxygen was present in the center of all the islets. At concentrations of 30,000 islets/ml and above, the only islets without hypoxic cores were those next to the fiber wall. The modeling data presented thus far support the original hypothesis that the lack of oxygen in the fiber center leads to loss of islet functionality and death in the BAP center.

Figure 3.

Figure 3.

The effectiveness factor φ was calculated for the second phase insulin production with the oxygen profiles obtained for a 1.0-mm diameter fiber, for islets whose centers reside at r fiber = 405, 235, and 75 μm and at islet concentrations of 20,000 and 5,000 islets/ml (Table 3). The results indicate a significant reduction in insulin production for islets near the center of the fiber. At islet concentrations >20,000 islets/ml, second phase insulin production for islets in the fiber center approaches 0. At 5,000 islets/ml, a smaller decrease in the effectiveness factor is obtained in the center of the fiber compared with that near the wall.

Table 3

Table 3

The simulated insulin response to a glucose challenge was compared to in vivo measurements obtained from rats implanted with immunoisolated islets (Figure 4). Three thousand islets that contained microspheres were implanted into a group of rats.9 The simulated insulin response was obtained with the model presented in this article, but with the physiologic parameters that reflect the rat host and with the cylindrical fiber parameters with equivalent islet concentrations (Table 4). The results indicate that the model using the narrow fibers predicts the insulin output within 4 to 8 μU/ml of the in vivo data for the first 20 minutes (Figure 4). Within the first few minutes, the simulations underestimate the experimental insulin concentration, indicating that the model either overestimates the diffusion limitations or underestimates the magnitude of the first peak of insulin secretion. The difference between the simulations and the in vivo data after 20 minutes is likely due to insulin metabolism by the host, which is not considered in the model.

Figure 4.

Figure 4.

Table 4

Table 4

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Discussion

The work presented here shows the relationship between islet concentration and oxygen concentration in a cylindrical BAP. Hypoxic conditions have been shown experimentally to lead to the loss of cell viability and suppressed insulin production. Our work has corroborated previous findings that islet density, fiber diameter, and the surrounding tissue layer are critical components in designing a BAP. Increasing the fiber diameter to accommodate higher islets concentrations will not be productive because of oxygen limitations. Experimental results1 are consistent with these modeling results. These data15 showed that in large diameter fibers with high islet concentrations retrieved from dogs, the only viable islets were within 0.5 mm of the fiber's inner wall. The observation by many showed that the reversal of diabetes in test animals by the BAP is temporary, supports the possibility that many designs may have hypoxic centers leading to islet necrosis. Our results show that narrow bore fibers (≤1.0 mm) with small islet concentrations (approximately 5,000 islets/ml) provide optimum conditions for oxygen availability and insulin production. Similar results have also been observed experimentally using hollow fibers containing islet concentrations of approximately 5,000 islets/ml.1,15

The simulation results for oxygen transport within a single islet suggest that oxygen levels <20 mm Hg inside the fiber are not sufficient. Previous studies2 investigated the in vivo oxygen level in fiber centers for hollow fibers implanted in mice with varying islet concentrations. These data, indicating viabilities of 50% to 80% at 26 to 29 mm Hg, corroborate the model results. However, the measured oxygen at this condition was unusually high (40 mm Hg), most likely because the islets died at the fiber center soon after implantation because of the lack of oxygen, and with a reduced oxygen demand in the fiber center, the partial pressures of oxygen increased again.

Our model assumes that each fiber implanted as a part of the BAP is located at an equal distance from the nearest blood supply and further, that the fibers do not interact in terms of glucose and oxygen availability. The advantage of this work is that the oxygen profile is coupled to insulin secretion and the expected insulin concentration in the bloodstream in response to a glucose challenge, which is the ultimate metric for a successful BAP.

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Conclusion

The work presented here shows the impact of islet concentration on the oxygen concentration profile in a cylindrical hollow fiber seeded with an even distribution of islets. Our results show that not enough oxygen is present to support encapsulated islets when the islet concentration is >20,000 islets/ml. Moreover, our results show that a significant amount of oxygen is consumed in the tissue region surrounding the cylindrical BAP. At the higher islet concentration, the only islets with oxygen present in the center where those closest to the fiber wall. This may be a likely reason that the islets closest to the wall survived and those in the center died. Given the low level of oxygen in the fibers with 1.0-mm diameter, it is clear that any increase in fiber diameter would not be beneficial at any islet concentration used in this study. Even if the islets survive in a low oxygen environment, the insulin productivity will likely be reduced.

In summary, our model with oxygen-dependent insulin secretion kinetics can reasonably predict the effect of implant dimensions and islet concentration on oxygen concentration within the fiber and predict the in vivo insulin response of encapsulated islets in a hollow fiber to a glucose challenge. Our model results have shown that partial pressures of oxygen should be maintained above 35 mm Hg everywhere within the fiber as a safety precaution to assure that the islets do not develop necrotic cores. The analytical solutions for oxygen transport in a cylindrical hollow fiber and an individual islet can be used to design fibers for larger animals and experimental treatment plans.

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References

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