Building Mathematical Models for Vascular Growth and Inhibition.

Microvascular channel growth and inhibition, such as what occurs in vasculogenic mimicry, are generally represented in tables or shown in bar graphs. Although informative, those representations lack accurate predictions on dosage or the opportunity to report an unbiased metric when one wants to compare different signal dependence, for instance, the concentration of different drugs or enzymes or expression levels of particular genes.Mathematical model building is an exercise that makes you think of which are the key variables of a particular phenomenon and how they affect the targeting experimental output.Starting from early blood vessel formation and regression (number of vessels) due to an inducer/inhibitor effect, we show how a conceptual mathematical model may be built. As an example, the model was used to parameterize aloin bioactivity on a chick yolk sac membrane (YSM) assay with respect to its vasculogenic and vessel regression properties. A separable functional form where vessel formation and cell death occur as mutually exclusive concentration or signal-dependent functions showed that there was a good correlation with experimental data. Although an analytical solution for that simple case is presented, parameter determination and parametric analysis may be carried out numerically by solving the system of ordinary differential equations that represents the model using nonlinear regression for parameter determinations. Such model formulation thus allows for a more objective evaluation concentration dependence and is suggested as a novel method to evaluate blood vessel formation and inhibition as well as a general model for quantitative balance between chemical stimulation and toxicity.