Transport of Free Fatty Acids from Plasma to the Endothelium of Cardiac Muscle: A Theoretical Study

Abstract

Fatty acids are transported in a multistep process from the plasma to the mitochondria, where they are oxidized in order to meet energy requirements of the myocardium. Some of those steps, mainly the crossing of the involved cells’ membranes are far from being understood. Here, by means of mathematical modeling we address the problem of the fatty acid transport from the microvascular compartment to the endothelium. Values of parameters that are incorporated in the model are deduced from relevant experimental work. Concentration profiles are established as solutions of diffusion–reaction equations both numerically and using an analytical asymptotic approximation. The analytical solution accurately determines the fatty acid flux for any set of parameter values in contrast to off-the-shelf numerical solvers that fail under quite a few circumstances due to the stiffness of the differential equation system. Sensitivity analysis indicates that in spite of few uncertain parameter values, most of our conclusions are expected to be valid throughout the physiological range of operation. We find that in order to have an adequate fatty acid uptake rate it is essential for the luminal endothelial membrane to have very fast fatty acid transporters and/or specific sites that interact with the albumin-fatty acids complex.

Appendix

The four independent variables of the differential system of equations, Eqs. 9–12, have two typical profiles: C _F is characterized by a moderate linear slope followed by a steep decline within a boundary layer near the membrane while the proteins are characterized by a linear slope followed by a much thinner boundary layer where they hardly change. This difference between the profiles is due to the different values of the diffusion constant of the two “species” and (mainly) due to the different boundary conditions imposed at the membrane (the fatty acids are evacuated but the proteins are confined within the region). Accordingly, establishing a solution using a singular perturbation approach necessitates expressing the variables near the membrane as series with two scaled coordinates (for the two types of boundary layers) and a solution of four simultaneous equations for each term that appear in each series. These formalistic steps mean a most formidable task. Instead, we expressed the protein concentrations by uniformly valid functional forms, Eqs. 13–15, that satisfy all boundary conditions and the differential equations after nullifying their left-hand sides. Any series that includes a deviation of ɛ(y − y ⁿ/n), n > 1 from the plasma concentrations will do for this purpose. Choosing n = 2 involves constant and minimal 2^nd derivatives (and errors). Outside the boundary layer the differential equations are satisfied up to O(D _A ɛ/x ²₁ ). Inside the boundary layer the inaccuracy is a bit higher. This approach induces normalized concentrations that differ at the membrane by less than one percent from the numerically computated ones for our range of parameter values.