A logistic-type curve fits pressure-diameter relationship for relaxed and contracted dog renal arteries

Abstract

In order to describe a possible effect of smooth muscle cell (SMC) activation on arterial wall distensibility, the present study derived a mathematical equation applicable to relaxed and contracted arterial walls. Pressure(P)-diameter(D) relationship of dog renal arteries was investigated in vitro under a cyclic loading and unloading process in the pressure range of 5–180 mmHg. Smooth muscle cells were activated by 10⁻⁵M norepinephrine. On the basis of the P-D curves obtained with fully contracted arteries, the vessel wall compliance dD/dP was assumed to be given by a second order polynomial of D, $${\displaystyle \frac{\mathrm{dD}}{\mathrm{dP}}}=-{\displaystyle \frac{\mathrm{(}\mathrm{D}-{\mathrm{D}}_{\max }\mathrm{)}\mathrm{(}\mathrm{D}-{\mathrm{D}}_{\min }\mathrm{)}}{\mathrm{E}\mathrm{(}{\mathrm{D}}_{\max }-{\mathrm{D}}_{\min }\mathrm{)}}}$$ The equation, including three parameters, D_min, D_max, and E, is integrated to yield the solution similar to the logistic curve as follows $$\mathrm{D}={\mathrm{D}}_{\min }+{\displaystyle \frac{{\mathrm{D}}_{\max }-{\mathrm{D}}_{\min }}{1-\mathrm{M}\mathrm{(}0\mathrm{)}exp\mathrm{(}-\mathrm{P}\mathrm{/}\mathrm{E}\mathrm{)}}}$$ where $\mathrm{D}\mathrm{(}0\mathrm{)}=\mathrm{(}{\mathrm{D}}_{\max }-\mathrm{D}\mathrm{(}0\mathrm{)}\mathrm{)}\mathrm{/}\mathrm{(}\mathrm{D}\mathrm{(}0\mathrm{)}-{\mathrm{D}}_{\min }\mathrm{)}$ , and D(0) is the diameter at the point P=0 The constant, E, has the same dimension as the modulus of elasticity. The calculated P-D relationships coincided well with the experimental data for contracted and relaxed arteries. The most significant change due to wall contraction took place in the magnitude of M. This result, therefore, suggests that the parameter M is a good index of the degree of SMC contraction.