## Journal of Fluid Mechanics

### Rarefaction and blood pressure in systemic and pulmonary arteries

a1 Department of Mathematics, North Carolina State University, Raleigh, NC 27502, USA

a2 School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QW, UK

a3 BHF Glasgow Cardiovascular Research Centre, University of Glasgow, Glasgow G12 8QQ, UK

a4 Scottish Pulmonary Vascular Unit, Golden Jubilee National Hospital, Glasgow G81 4HX, UK

Abstract

The effects of vascular rarefaction (the loss of small arteries) on the circulation of blood are studied using a multiscale mathematical model that can predict blood flow and pressure in the systemic and pulmonary arteries. We augmented a model originally developed for the systemic arteries by Olufsen and coworkers and Ottesen et al. (2004) to (a) predict flow and pressure in the pulmonary arteries, and (b) predict pressure propagation along the small arteries in the vascular beds. The systemic and pulmonary arteries are modelled as separate bifurcating trees of compliant and tapering vessels. Each tree is divided into two parts representing the ‘large’ and ‘small’ arteries. Blood flow and pressure in the large arteries are predicted using a nonlinear cross-sectional-area-averaged model for a Newtonian fluid in an elastic tube with inflow obtained from magnetic resonance measurements. Each terminal vessel within the network of the large arteries is coupled to a vascular bed of small ‘resistance’ arteries, which are modelled as asymmetric structured trees with specified area and asymmetry ratios between the parent and daughter arteries. For the systemic circulation, each structured tree represents a specific vascular bed corresponding to major organs and limbs. For the pulmonary circulation, there are four vascular beds supplied by the interlobar arteries. This paper presents the first theoretical calculations of the propagation of the pressure and flow waves along systemic and pulmonary large and small arteries. Results for all networks are in agreement with published observations. Two studies were done with this model. First, we showed how rarefaction can be modelled by pruning the tree of arteries in the microvascular system. This was done by modulating parameters used for designing the structured trees. Results showed that rarefaction leads to increased mean and decreased pulse pressure in the large arteries. Second, we investigated the impact of decreasing vessel compliance in both large and small arteries. Results showed that the effects of decreased compliance in the large arteries far outweigh the effects observed when decreasing the compliance of the small arteries. We further showed that a decrease of compliance in the large arteries results in pressure increases consistent with observations of isolated systolic hypertension, as occurs in ageing.

(Reviewed April 24 2012)

(Accepted May 03 2012)

(Online publication July 02 2012)

Key Words:

• blood flow

Correspondence:

c1 Email address for correspondence: N.A.Hill@maths.gla.ac.uk

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