Modeling of encapsulated cell systems

Abstract

Tissue engineered substitutes consisting of cells in biocompatible materials undergo remodeling with time as a result of cell growth and death processes. With inert matrices that do not directly influence cell growth, remodeling is driven mainly by the concentration of dissolved oxygen (DO). Insulinsecreting cell lines encapsulated in alginate-based beads and used as a pancreatic substitute represent such a case. Beads undergo remodeling with time so that an initially homogeneous distribution of cells is eventually replaced by a dense peripheral ring of primarily viable cells, whereas inner cells are mostly necrotic. This paper develops and analyzes a mathematical model of an encapsulated cell system of spherical geometry that tracks the viable and dead cell densities and the concentration of DO within the construct as functions of radial position and time. Model simulations are compared with experimental histology data on cell distribution. Correlations are then developed between the average intrabead DO concentration (AIDO) and the total viable cell number, as well as between AIDO and the radial cell and DO distributions in beads. As AIDO can be measured experimentally by incorporating a perfluorocarbon emulsion in the beads and acquiring 19 F nuclear magnetic resonance (NMR) spectroscopic data, these correlations can be used to track the remodeling that occurs in the construct in vitro and potentially in vivo. The usefulness of mathematical models in describing the dynamic changes that occur in tissue constructs with time, and the value of these models at obtaining additional information on the system when used interactively with experimental measurements, are discussed.

Oxygen Diffusion through Natural Extracellular Matrices: Implications for Estimating “Critical Thickness” Values in Tendon Tissue Engineering

“…Additionally, cell-seeded scaffold constructs that rely solely on diffusion for oxygen could eventually result in proliferation-dominated regions at the construct surface, thereby potentially resulting in oxygen levels at the interior that are well below the acceptable critical value. 16,[20][21][22] Not accounting for the effect of time in culture is a limitation of the current model system. A mathematical model that accounts for time-dependent culture transport limitations would be more realistic of oxygen kinetics and thereby improve on estimates of critical scaffold thickness values.…”

Monitoring of dissolved oxygen and cellular bioenergetics within a pancreatic substitute

“…The cellular growth and death at specific locales within the beads were dictated by an interplay between spatial constraints and the DO concentration at each radial position (Gross et al, 2007). To provide an initial validation, the model was used to compute the intrabead radial cellular and DO distributions versus time.…”

“…To provide an initial validation, the model was used to compute the intrabead radial cellular and DO distributions versus time. In these calculations, the model inputs were the intrinsic parameters for bTC-tet cells in APA beads (Gross et al, 2007) and the measured DOin during the in vitro experiment. The model output was the time-dependent cell and DO distributions, from which the total viable cell number and a volumeaverage intrabead DO were calculated and compared against experimental results.…”

“…This is also evidenced by the dramatic reduction in AIDO from the start to the end of the DOin ¼ 0.20 mM period. A regression line fitted to the points from the beginning of the DOin ¼ 0.16 mM to the end of the DOin ¼ 0.12 mM period indicates a specific cell growth rate of 0.21 day À1 , or a doubling time of 3.3 days, which is reasonable for encapsulated bTC-tet cells (Gross et al, 2007;Simpson et al, 2006;Stabler et al, 2001). However, as the DOin was further decreased to hypoxic levels, the b-NTP/P i also decreased, reaching a minimum of approximately 60% of the initial value.…”

“…Specifically, the inverse of the T 1 relaxation time of specific perfluorinated chemical groups are known to vary linearly with the DO concentration surrounding the emulsion (Fan et al, 2002;McGovern et al, 1993;Noth et al, 2004;Reid et al, 1985;Shukla et al, 1996) 19 F chemical shift imaging, long acquisition times are required, which are a caveat to in vivo applications due to the long anesthesia times required of experimental animals. To overcome this limitation, one can acquire a single DO concentration from the entire construct, representing a volume-average measurement of the oxygen profile within the volume-of-interest (VOI); a mathematical model can then be used to assess the cellular and oxygen spatial profiles within the construct, thus allowing for rapid measurements in vivo (Gross et al, 2007).…”