Dynamic matrix composition in engineered cartilage with stochastic supplementation of growth factors

Abstract

Dynamic extracellular matrix (ECM) synthesis is explored in a hypothesized engineered cartilage construct. Growth (alpha) and decay (beta) rate parameters are developed from a previous engineered cartilage model. The presented mathematical model was constructed from the parameterized experimental data using a deterministic and stochastic examination of ECM synthesis based on a negative feedback control mechanism. A growth factor supplementation is incorporated in a probabilistic mathematical approach. The growth factor component modified an initial deterministic model through a Gaussian white noise fluctuation. As the primary constituents of ECM, the mathematical tool is intended to characterize the probable steady state distribution of glycosaminoglycan (GAG) and collagen molecules as well as mean mass accumulation at homeostasis. Computer simulation of the models is applied to reported data from four similar chondrocyte-polymer construct culture systems. The range in rate ratios reflect the differing nature of GAG and collagen synthesis (alphaGAG/betaGAG = 4.2 to 148.6; alphacollagen/betacollagen = 8.1 to 2590.4). This technique reduced the influencing synthesis factors to a few key descriptive parameters. Additional anabolic and catabolic factors may further be built into the models.

Figure 1

Simulated time-dependent ECM accumulation leading to steady state levels using the Deterministic Control Model, equations (7) through (12). Input parameter values were chosen from reported cartilage engineering experimental data (Table 1): a) Freed et al.; b) Vunjak-Novakovic et al.; c) Wilson et al.; and d) Vunjak-Novakovic et al..

Figure 2

Distribution plots produced from the Probability Density Function describing mean ECM accumulation at the steady state condition for each of the modelled constituents, equations (24) and (25). Here, the growth and density rates for both GAG and collagen were incorporated from representative published cartilage engineering experiments (Table 1): a) Freed et al.; b) Vunjak – Novakovic et al.; c) Wilson et al.; and d) Vunjak - Novakovic et al.. For all distributions, the relative growth factor effect is the same, ρ_GAG = ρ_collagen = 2.0.

Figure 3

A comparison of the mean steady state ECM accumulation with supplemented growth factor effects. Steady state values of growth factor-influenced accumulation are calculated from the means of the probability functions, ${\overline{P}}_{\infty }(ECM)$, applied to equations (24) and (25) and shown for each constituent in equation (26). ECM growth and density rates were incorporated from representative published cartilage engineering experiments (Table 1): A) Freed et al.; B) Vunjak-Novakovic et al.; C) Wilson et al.; and D) Vunjak-Novakovic et al.. For each ECM constituent, mean accumulation values are determined at distinct relative growth factor levels within the Stochastic Treated Model, ρ_GAG = ρ_collagen = 0.3, 0.5, 0.8, 1.0, and 2.0. Accumulation at ρ₀ = 0 represents the mean steady state level from the Deterministic Control Model.