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A computational study of pressure wave reflections in the pulmonary arteries

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Abstract

Experiments using wave intensity analysis suggest that the pulmonary circulation in sheep and dogs is characterized by negative or open-end type wave reflections, that reduce the systolic pressure. Since the pulmonary physiology is similar in most mammals, including humans, we test and verify this hypothesis by using a subject specific one-dimensional model of the human pulmonary circulation and a conventional wave intensity analysis. Using the simulated pressure and velocity, we also analyse the performance of the P–U loop and sum of squares techniques for estimating the local pulse wave velocity in the pulmonary arteries, and then analyse the effects of these methods on linear wave separation in the main pulmonary artery. P–U loops are found to provide much better estimates than the sum of squares technique at proximal locations, but both techniques accumulate progressive error at distal locations away from heart, particularly near junctions. The pulse wave velocity estimated using the sum of squares method also gives rise to an artificial early systolic backward compression wave. Finally, we study the influence of three types of pulmonary hypertension viz. pulmonary arterial hypertension, chronic thromboembolic pulmonary hypertension and pulmonary hypertension associated with hypoxic lung disease. Simulating these conditions by changing the relevant parameters in the model and then applying the wave intensity analysis, we observe that for each group the early systolic backward decompression wave reflected from proximal junctions is maintained, whilst the initial forward compression and the late systolic backward compression waves amplify with increasing pathology and contribute significantly to increases in systolic pressure.

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Notes

  1. As is common for studies of cardiovascular dynamics, all pressures are given in mmHg. The conversion to SI units is 1 mmHg = 133.3 Pa.

  2. Wave speeds are 1.8–4.8 \(\hbox {ms}^{-1}\) in human large pulmonary arteries (Dwyer 2010; Milnor et al. 1969) and 3.5–14 \(\hbox {ms}^{-1}\) in human systemic arteries (Zambanini et al. 2005), under normal to hypertension conditions.

  3. The generation number, \(i+j\), is the number of bifurcations between the root vessel and a given vessel. Here \(i=0\dots n\) and \(j=0\dots m\), where \(n\) and \(m\) are maximum integer number of generations, along left and right sides of the structured tree.

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Acknowledgments

M. Umar Qureshi was funded by the National Institute of Health Virtual Physiological Rat Center under award # 5-P50-GM094503-02. He is also thankful for the award of a scholarship from the International Islamic University, Islamabad and the Higher Education Commission of Pakistan that funded his PhD studies at the University of Glasgow.

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Authors and Affiliations

  1. Department of Mathematics, International Islamic University, Sector H10, Islamabad, 44000, Pakistan

    M. Umar Qureshi

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

    M. Umar Qureshi & N. A. Hill

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  1. M. Umar Qureshi
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Correspondence to M. Umar Qureshi.

Appendices

Appendix A

In this appendix, we provide a brief description of the model and the boundary conditions used to compute the pressure \((p(x,t))\) and flow \((q(x,t))\) waveforms.

Boundary conditions

Since the system of Eqs. (1)–(3) is hyperbolic, for each vessel a boundary condition must be specified at each end. These are:-

  • (i) The inflow boundary condition: A time varying flow profile is specified at the inlet of MPA using an MRI-measured output from the right ventricle (Qureshi et al. 2014).

  • (ii) Bifurcation conditions: At each bifurcation, to ensure conservation of flow and continuity of pressure, two conditions are imposed to connect the outflow from parent vessel and the inflows to the daughter vessels, i.e.

    $$\begin{aligned} p_p(L,t)&=p_{d_1}(0,t)=p_{d_2}(0,t),\\ q_p(L,t)&=q_{d_1}(0,t)+q_{d_2}(0,t). \end{aligned}$$

  • (iii) Outflow conditions: At the outflow of large veins to the left atrium, a constant pressure of 2 mmHg is imposed.

  • (iv) Matching boundary conditions: As shown in Fig. 9, the outflow from each large artery is matched with the inflow to each large vein via a connected arterial-venous structured tree. The structure of each tree is governed by three relations, a radius relation \((r_p^\xi =r_{d_1}^\xi +r_{d_2}^\xi \) where \(2.33\le \xi \le 3)\), an asymmetry ratio \((\gamma = r_{d_2}^2/r_{d_1}^2)\) and an area ratio \((\eta =(r_{d_1}^2+r_{d_2}^2)/r_{d_p}^2)\), only two of which are independent. These relations yield two positive scaling factors, \(\alpha \) and \(\beta \) such that the radius \((r)\) of any vessel, bifurcating from a parent vessel, after \(i+j\) number of generationsFootnote 3 from the root vessel is equal to \(\alpha ^i\beta ^jr_0\), where \(r_0\) is the radius of root vessel and \(i\) and \(j\) are integers. Finally the length \((l)\) of each vessel is determined using an empirically determined length-to-radius ratio \(l_{rr}=l/r\). The pressure \((P(x,\omega ))\) and flow \((Q(x,\omega ))\) for each vessel within the structured tree is obtained by solving analytically the linearised versions of Eqs. (1)–(3) in the Fourier-transformed domain. Evaluating \(P(x,\omega )\) and \(Q(x,\omega )\) at the inlet \((x=0\), yielding \(Q_1\) and \(P_1)\) and the outlet \((x=L\) yielding \(Q_2\) and \(P_2)\) of a single vessel, a \(2\times 2\) admittance matrix \(\mathbf {Y}(i,j)\) can obtained for each arterial and venous vessel segment such that

    $$\begin{aligned} \mathbf {Q}(i,j)=\mathbf {Y}(i,j)\mathbf {P}(i,j),\quad \text {where}\quad \mathbf {Q}=[Q_1 \ \ Q_2]^{\dagger }\quad \text {and} \quad \mathbf {P}=[P_1 \ \ P_2]^{\dagger }. \end{aligned}$$

    Here the index \((i,j)\) indicates that the quantities are associated with the vessels having radius \(\alpha ^{i}\beta ^{j}r_0\). A total admittance matrix, \(\mathbf {Y}(\omega )\), for an entire tree may be computed by setting up a recursion algorithm to obtain the admittances for vessels joined in parallel \((\mathbf {Y}^{\Vert })\) and in series \((\mathbf {Y}^{\Leftrightarrow })\). An example of how the total admittance matrix for a simple network of connected vessels is obtained is shown in Fig. 10, and a general algorithm for trees with any number of generations can be found in Qureshi et al. (2014). Once \(\mathbf {Y}(\omega )\) is obtained for each connected tree, the haemodynamics on arterial and venous roots of these trees can be related in the time domain using

    $$\begin{aligned} q_k(t)=\sum _{l=1}^2\int \limits _0^T y_{kl}(\tau )\,p_l(t-\tau )\,d\tau ,\quad k=1,2, \end{aligned}$$
    (15)

    where \(q_1(t)\) and \(p_1(t)\) are the volume flux and pressure at the root of the arterial tree, and \(q_2(t)\) and \(p_2(t)\) are the volume flux and pressure at the root of the venous tree. \(y_{kl}(t)\) is the inverse Fourier transform of \(Y_{kl}(\omega )\) (components of \(\mathbf {Y}(\omega )\)). Equation (15) is the matching boundary condition to be applied at the end of each pair of terminal arteries and veins.

Fig. 9
figure 9

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 ‘.’ in the large arteries (Qureshi et al. 2014)

Full size image
Fig. 10
figure 10

Given the admittance matrix \(\mathbf {Y}(i,j)\) for a single vessel, the total admittance matrix \(\mathbf {Y}(\omega )\) can be obtained for a simple network, consisting of an arterial connected to a venous tree, each having one bifurcation only, with the radii of the branches scaled by factors \(\alpha \) and \(\beta \). \(\mathbf {Y}(\omega )\) for this network is given by \( \mathbf {Y}(\omega )=\mathbf {Y}^A(0,0)\Leftrightarrow [\{\mathbf {Y}^A(1,0)\Leftrightarrow \mathbf {Y}^V (1,0) \} \Vert \{\mathbf {Y}^A(0,1) \Leftrightarrow \mathbf {Y}^V(0,1)\}]\Leftrightarrow \mathbf {Y}^V (0,0). \) The symbols ‘\(\Leftrightarrow \)’ and ‘\(\Vert \)’ represent combining admittances in a series and in parallel, respectively, and the superscripts ‘A’ or ‘V’ specify if the vessel is an artery or a vein

Full size image

Appendix B

Under the standard assumptions, we solve Eqs. (1) and (2) using the method of characteristics. For any point \((x,t)\), there exists a pair of characteristic paths C\(_\pm :\frac{\text {d}x}{\text {d}t}=u\pm c\) on which the Riemann invariants

$$\begin{aligned} R_\pm =u-u_0\pm \int \limits _{p_0}^p\frac{\text {d}p*}{\rho c(p*)} \end{aligned}$$
(16)

are constant. Here \(c(p)\) is the PWV given in (7). Assuming that at the end of diastole, \(A=A_0\) and \(u_0=0\), we get

$$\begin{aligned} R_{\pm }=u\pm 4\,(c_0-c(p)),\quad \text {where}\quad c_0=\sqrt{\frac{2Eh}{3\,\rho r_0}} \end{aligned}$$
(17)

is the Moens–Korteweg PWV at \(A=A_0\) (Korteweg 1878; Moens 1877). From (16),

$$\begin{aligned} \text {d}R_{\pm }=\text {d}u\pm \frac{\text {d}p}{\rho \,c(p)}. \end{aligned}$$
(18)

Solving for d\(p\) and d\(u\), we obtain

$$\begin{aligned} \text {d}p=\frac{\rho c(p)}{2}(\text {d}R_+-\text {d}R_-)\quad \text {and}\quad \text {d}u=\frac{1}{2}(\text {d}R_+ + \text {d}R_-). \end{aligned}$$
(19)

Since the Riemann invariant along a backward characteristic, \(\hbox {C}_-\), that intersects any two forward characteristics, must be preserved for all times, the changes in pressure and velocity in the absence of changes in \(R_-\), between the two points of intersection are defined as ‘forward’, and vice versa, so that

$$\begin{aligned} \text {d}p_\pm =\text {d}p\,|_{\text {d}R_\mp =0}\quad \text {and}\quad \text {d}u_\pm =\text {d}u\,|_{\text {d}R_\mp =0}, \end{aligned}$$
(20)

From Eqs. (18)–(20), we get

$$\begin{aligned} \text {d}p_{\pm }=\frac{1}{2}\left( \text {d}p\,\pm \,\rho \,c(p)\,\text {d}u\right) \quad \text {and}\quad \text {d}u_{\pm }=\frac{1}{2}\left( \text {d}u\,\pm \,\frac{\text {d}p}{\rho \,c(p)}\right) . \end{aligned}$$
(21)

The corresponding wave intensity profiles are then given by

$$\begin{aligned} \text {W}^{\text {I}_{\pm }}\,=\,\frac{\text {d}p^{\pm }}{\text {d}t}\,\frac{\text {d}u^{\pm }}{{\text {d}t}}=\pm \frac{\rho \,c(p)}{4}\,\left( \frac{\text {d}R_{\pm }}{{\text {d}t}}\right) ^2. \end{aligned}$$
(22)

Finally, the separated waveforms of pressure and velocity may be obtained by integrating the corresponding wavefronts over the complete cardiac cycle \(T\) i.e.

$$\begin{aligned} p_{\pm }(t)\,=\,p_0 \,+\, \int \limits _{t-T}^{t}\text {d}p_{\pm }\quad \text {and}\quad u_{\pm }(t)\,=\,u_0 \,+\, \int \limits _{t-T}^{t}\text {d}u_{\pm }. \end{aligned}$$
(23)

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Qureshi, M.U., Hill, N.A. A computational study of pressure wave reflections in the pulmonary arteries. J. Math. Biol. 71, 1525–1549 (2015). https://doi.org/10.1007/s00285-015-0867-2

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