Cartilage Tissue Engineering Calculator

1. Diffusivities

This study was  mainly based on an alginate gel of concentration of 1.2% alginate  (Guo et al. 1989).  Mehmetoglu measured oxygen diffusivity in alginate gels and found when alginate concentration was from 1-3%, oxygen diffusion coefficient varied between 9.1 and 9.3 mm2/hr and was almost independent of alginate concentration  (Mehmetoglu et al. 1996).  This value is about 0.84 of that in aqueous solution and was used here.  We also assumed that diffusion coefficients of glucose and lactate in alginate gel have the same ratio to those in aqueous solution (diffusion coefficients of oxygen, glucose and lactic acid in water are 10.8, 3.3 and 5.0 mm2/hr)  (Haselgrove et al. 1993; Malda et al. 2004a; Selard et al. 2003; Sengers et al. 2005).  Thus  Dglu=2.8 mm2/hr and Dla=4.2 mm2/hr.  In this study, we have also considered the effect of change in diffusivities.

2. Metabolic rates

Data on metabolism kinetics were from our study and followed the Michaelis-Menten kinetics, ie.

…….. (1)

where g  is the consumption/production rate (nmol/106 cells/hr), Vmax the maximum consumption/production rate (nmol/106 cells/hr), Km the Michaelis-Menten constant, and C the concentration of substrate.

Parameters for lactate production are:

(nmol/million cells/hr);     Km(la)=0.198  (mM)

where std means standard culture condition (5.56 mM glucose, 380 mOsm, pH 7.4, 21% Oxygen)

Parameters for glucose consumption are:

(nmol/million cells/hr);         Km(glu)= Km(la);

Parameters for oxygen consumption are:

(nmol/million cells/hr);   Km(ox)=2.33  (% oxygen).

la, glu and ox stand for lactate, glucose and oxygen respectively.

The following interactions between metabolism rate and extracellular nutrient-metabolite environment exist:

…….. (2)

……..(3)

……..(4)

The relationship between lactate production and oxygen tension was not fitted to continuous function;    when oxygen tension was above 5%, lactate production rate was treated as constant:

(Cox ³ 5%)

When oxygen tension was below 5%, lactate production rate was assumed to be linear to oxygen tension and thus:

……..(5)

For normal DMEM (Invitrogen 22320-022), the relationship between pH and lactic acid was

pH=7.4-0.0406×Cla……..(6)

The relationship between metabolism and osmolarity was neglected since the effect of osmolarity was found to be insignificant.

It should be emphasized that the inclusion of these inter-dependent relationships introduces non-linearities into the numerical models.