A mathematical model to investigate nutrient transport and metabolism using an indicator dilution technique
Thirty-three paired dilution/nutrient curves across the intact bovine mammary glands were obtained from four lactating cows following rapid injection of p-aminohippuric acid (PAH) and glucose into the external iliac artery and continuously sampling from the mammary vein for 125 s into 5-s fractions. An innovative and easy-to-apply mathematical simulation model, the compartmental capillary, convolution integration (CCCl) model, was developed to interpret multiple indicator dilution (MID) curves. The key achievements of the approach are that the mechanisms of target nutrient or drug transport and metabolisms occurring in compartmental capillary spaces can be conveniently expressed in ordinary differential equations (ODEs), that the heterogeneity of non-exchanging blood vessel transit delays in target system is used to integrate from single capillary outflow to whole system outflow, and that a specific simulation and optimization software CONVNLIN, written in C language, has been built to estimate parameters for any user-specified set of ODEs describing target nutrient compartmental capillary dynamics. Accordingly, the superiority of the CCCl model is that researchers can screen many candidate mathematical models and identify the best one for specific solute under study in the tissue of interest and for a given physiological status of the animal. By implementing the CCCl model to analyze the paired dilution curves across the bovine mammary glands, the mean heterogeneous non-exchanging vessel transit delays was 25% of the mean residence transit time in extracellular space and was accounted one of major dispersion factors for both indicators and nutrients. Glucose transport across the plasma basal membrane of bovine mammary epithelial cells involves rapid exchange into an intracellular cage accounting for approximately 23% of intracellular glucose space, which is not a rate-limiting step for glucose metabolism. The glucose sequestration rate is linearly related to extracellular glucose concentration. The success of the CCCl model could boost the implementation of the powerful MID technique to explore the dynamics of intact biological systems without disturbing physiological status.