Numerical simulation of blood flow and pressure drop in the pulmonary arterial and venous circulation

Abstract

A novel multiscale mathematical and computational model of the pulmonary circulation is presented and used to analyse both arterial and venous pressure and flow. This work is a major advance over previous studies by Olufsen et al. (Ann Biomed Eng 28:1281-1299, 2012) which only considered the arterial circulation. For the first three generations of vessels within the pulmonary circulation, geometry is specified from patient-specific measurements obtained using magnetic resonance imaging (MRI). Blood flow and pressure in the larger arteries and veins are predicted using a nonlinear, cross-sectional-area-averaged system of equations for a Newtonian fluid in an elastic tube. Inflow into the main pulmonary artery is obtained from MRI measurements, while pressure entering the left atrium from the main pulmonary vein is kept constant at the normal mean value of 2 mmHg. Each terminal vessel in the network of 'large' arteries is connected to its corresponding terminal vein via a network of vessels representing the vascular bed of smaller arteries and veins. We develop and implement an algorithm to calculate the admittance of each vascular bed, using bifurcating structured trees and recursion. The structured-tree models take into account the geometry and material properties of the 'smaller' arteries and veins of radii ≥ 50 μm. We study the effects on flow and pressure associated with three classes of pulmonary hypertension expressed via stiffening of larger and smaller vessels, and vascular rarefaction. The results of simulating these pathological conditions are in agreement with clinical observations, showing that the model has potential for assisting with diagnosis and treatment for circulatory diseases within the lung.

Fig. 1

Schematic of the pulmonary circulation model arranged in a sequence of larger arteries, arterioles, venules and large veins. The large pulmonary arteries and veins are specified explicitly, while the small vessels are represented by structured trees. The main pulmonary artery (MPA) is the root vessel within the pulmonary arterial tree. A flow waveform measured using MRI is specified at the inlet to this vessel. The MPA bifurcates into the right (RPA) and left (LPA) pulmonary arteries. The RPA bifurcates into the right interlobular artery (RIA) and the right trunk artery (RTA), and the LPA bifurcates into the left interlobular artery (LIA) and left trunk artery (LTA). The RIA, RTA, LIA and LTA are the terminal vessels of the large pulmonary arterial model, and it is to the outlet of these vessels that the structured-tree matching conditions are applied to join the arterial and venous systems. The outlet of the RIA is matched with the inlet of right inferior pulmonary vein (RIV), the RTA with the right superior vein (LSV), the LIA with the left inferior vein (LIV), and the LTA with the inlet of left superior vein (LSV). The pulmonary veins open into left atrium in pairs draining blood from left and right lungs and therefore at the outlet of each vein a constant pressure condition is applied. Continuous-pressure and flow-conservation conditions are used at each bifurcating junction, marked by a ‘.’ for large arteries.

Fig. 2

Inflow profile for the main pulmonary artery. This profile was interpolated from MRI measurements sampled at 45 points per period and averaged over 5 cardiac cycles.

Fig. 3

Linking an arterial tree with a venous tree. For each vessel in the arterial tree there is a mirror vessel in the venous tree which may have different compliance and length-to-radius ratio. The radii are defined as functions of root vessel radius via scaling factors α and β. Starting from terminals of structured trees, both trees are connected by joining the pairs of vessels in series and in parallel. Flows $q_2^A$ and $q_1^V$, denote q^A (L) (flow at distal end of large terminal arteries) and similarly q^V (0) (flow at proximal end of large terminal veins), respectively, and pressures $p_2^A$ and $p_1^V$, represent p^A (L) and p^V (0), respectively. The flows are related to the pressures by a 2 × 2 admittance matrix Y (ω). Note that the labels of branches are the ordered pairs, which refer to generation of vessel in the tree and are powers of scaling factors α and β, i.e. the label (i, j) indicates that the radius of the vessel is $r_0^{{\alpha }^i{\beta }^j}$.

Fig. 4

Relations between flow and pressure via admittance Y, Y^‖ and Y^⇔ for a single vessel (a), vessels connected in parallel (b), and vessels connected in series (c). Q₁ and Q₂ are positive when the flows are into the vessel.

Fig. 5

Predicted pressure (first two columns) and flow (last two columns) at three locations along the large arteries (MPA, RPA, and LPA) and veins (RIV, RSV and LIV). For each vessel, flow and pressure are evaluated at the vessel inlet (solid blue), at the midpoint (dashed magenta), and at the end (solid cyan).

Fig. 6

Pressure profiles in a typical vascular bed. The first two graphs give pressures at the roots of the vascular bed connecting the RIA and RIV. The last graph shows mean pressure changes along the α and β pathways, together with the mean over all vessels of the same radius. The mean (time-averaged) pressure is plotted against vessel radius on a linear-log scale. The pressure drop across the α branch is marked by a dashed red line, the β branch pressure drop is marked by a dashed-dot magenta line, and across all the branches of the same radius by a solid blue line.

Fig. 7

Effect of hypertension on pressure and flow at midpoints of the MPA and RSV. The first two columns show pressure and flow waveforms. The third column gives peak (solid blue), mean (dashed magenta), MPA pulse (dashed-dot red) and trough (solid cyan) pressures.

Fig. 8

Effects of pulmonary hypertension associated with hypoxic lung disease on mean pressure in vessels connecting the RIA with the RIV. Mean pressure is plotted against vessel radius on a linear log scale. The curves correspond to a reduction in vascular density: normal (solid blue), 10% decrease (dashed dot red), 20% decrease (solid cyan) and 30% decrease (dashed magenta). The first column shows the pressure averaged over all vessels of the same radius, the α branch is plotted in the second column, and the β branch in third column. The pressures at the the roots of the RIA and RIV (solid blue) in the case of HLD, c.f. Figure 7(b), are imposed as boundary conditions.