A Mathematical Model of Maladaptive Inward Eutrophic Remodeling of Muscular Arteries in Hypertension

Abstract

We propose a relatively simple two-dimensional mathematical model for maladaptive inward remodeling of resistive arteries in hypertension in terms of vascular solid mechanics. The main premises are: (i) maladaptive inward remodeling manifests as a reduced increase in the arterial mass compared to the case of adaptive remodeling under equivalent hypertensive pressures and (ii) the pressure-induced circumferential stress in the arterial wall is restored to its basal target value as happens in the case of adaptive remodeling. The rationale for these assumptions is the experimental findings that elevated tone in association with sustained hypertensive pressure down-regulate the normal differentiation of vascular smooth muscle cells from contractile to synthetic phenotype and the data for the calculated hoop stress before and after completion of remodeling. Results from illustrative simulations show that as the hypertensive pressure increases, remodeling causes a nonmonotonic variation of arterial mass, a decrease in inner arterial diameter, and an increase in wall thickness. These findings and the model prediction that inward eutrophic remodeling is preceded by inward hypertrophic remodeling are supported by published observations. Limitations and perspectives for refining the mathematical model are discussed.

Keywords: endothelial dysfunction; phenotype switching; vascular smooth muscle and endothelial cells response; vascular solid mechanics.

Fig. 1

Illustration of postulated analytical expressions for relationships between model parameters in system (8). (a) Points marked by solid circles (●) indicate activation parameter values in maladaptive remodeling due to elevated pressure. (b) Points marked by open circles (○) indicate arterial CSA resulting from adaptive remodeling to elevated pressure and basal VSMCs tone; points marked by solid circles (●) indicate arterial CSA resulting from maladaptive remodeling to elevated pressure and correspondingly increased VSMCs tone.

Fig. 2

Outcomes of pressure-induced remodeling. (a) Normalized deformed arterial CSA, (b) normalized deformed inner radius, (c) normalized deformed wall thickness, (d) circumferential stretch, (e) normalized undeformed inner radius, and (f) normalized undeformed wall thickness. Depicted fictitious and maladaptive arteries were assigned a critical pressure of 160 mmHg; normalization was done with respect to the baseline values for arterial pressure of 80 mmHg. Note that CSA remains unchanged in the maladaptive case for hypertensive pressures higher than the critical pressure of 160 mmHg.

Fig. 3

Effects of critical pressure on arterial geometry upon completion of inward eutrophic remodeling. (a) normalized inner radius and (b) normalized wall thickness. Curves refer to remodeling outputs when the hypertensive pressure is equal to the critical pressure. The shaded regions indicate physiologically implausible remodeling response calculated for hypothetical critical pressure below 100 mmHg. Normalization was done with respect to the values for critical pressure of 80 mmHg.

Fig. 4

Outcomes of maladaptive inward remodeling after development of the eutrophic response. (a) Inner radius and (b) wall thickness. Normalization was done with respect to the baseline values for pressure of 80 mmHg.

Fig. 5

Arterial response before and upon completion of pressure-induced adaptive and maladaptive remodeling. (a) The pressure-radius response of the baseline and remodeled arteries due to hypertensive pressure of 120 mmHg; in the cases of fictitious and maladaptive remodeling, the assigned critical pressure is 160 mmHg; solid circles (●) denote deformed states under arterial pressure of 120 mmHg for remodeled arteries and 80 mmHg for the normotensive artery. (b) The pressure-radius response of arteries upon completion of maladaptive inward eutrophic remodeling under different critical pressures.