Gravitational effects on global hemodynamics in different postures: A closed-loop multiscale mathematical analysis

Abstract

We present a novel methodology and strategy to predict pressures and flow rates in the global cardiovascular network in different postures varying from supine to upright. A closed-loop, multiscale mathematical model of the entire cardiovascular system (CVS) is developed through an integration of one-dimensional (1D) modeling of the large systemic arteries and veins, and zero-dimensional (0D) lumped-parameter modeling of the heart, the cardiac-pulmonary circulation, the cardiac and venous valves, as well as the microcirculation. A versatile junction model is proposed and incorporated into the 1D model to cope with splitting and/or merging flows across a multibranched junction, which is validated to be capable of estimating both subcritical and supercritical flows while ensuring the mass conservation and total pressure continuity. To model gravitational effects on global hemodynamics during postural change, a robust venous valve model is further established for the 1D venous flows and distributed throughout the entire venous network with consideration of its anatomically realistic numbers and locations. The present integrated model is proven to enable reasonable prediction of pressure and flow rate waveforms associated with cardiopulmonary circulation, systemic circulation in arteries and veins, as well as microcirculation within normal physiological ranges, particularly in mean venous pressures, which well match the in vivo measurements. Applications of the cardiovascular model at different postures demonstrate that gravity exerts remarkable influence on arterial and venous pressures, venous returns and cardiac outputs whereas venous pressures below the heart level show a specific correlation between central venous and hydrostatic pressures in right atrium and veins.

Appendices

Appendix A: Cardiopulmonary model

Modeling of cardiac chambers

The heart is modeled as four chambers with cardiac contraction represented by a time-varying elastance E(t) [15],

$$\begin{aligned} E(t)=E _{\mathrm{A}} e(t)+E _{\mathrm{B}} , \end{aligned}$$

(A1)

where $E_{\mathrm {A}}$ denotes the elastance amplitude, $E_{\mathrm {B}}$ the elastance baseline, and e(t) the normalized time-varying elastance. The term e(t) can be further expressed for ventricles as

$$\begin{aligned} e_{v} (t)=\left\{ {\begin{array}{ll} 0.5\left[ {1-\cos \left( {{\uppi }t/t_\mathrm{vcp} } \right) } \right] ,&{} 0<t\le t _{\mathrm{vcp}} , \\ 0.5\left[ {1+\cos ({{\uppi }\left( {t-t _{\mathrm{vcp}} } \right) /t _{\mathrm{vrp}} }) } \right] , &{}t_{\mathrm{vcp}}<t\le t _{\mathrm{vcp}} +t _{\mathrm{vrp}} , \\ 0, &{}t _{\mathrm{vcp}} +t _{\mathrm{vrp}} <t\le t_{0} , \\ \end{array}} \right. \nonumber \\ \end{aligned}$$

(A2)

and for atria as

$$\begin{aligned} e_{a} (t)=\left\{ {\begin{array}{ll} 0.5\left[ {1+\cos \left( {{\uppi }\left( {t+t_{0} -t _{\mathrm{ar}} } \right) /t _{\mathrm{arp}} } \right) } \right] , &{}0\le t\le t _{\mathrm{ar}} +t _{\mathrm{arp}} -t_{0} , \\ 0, &{}t _{\mathrm{ar}} +t _{\mathrm{arp}} -t_{0}<t\le t _{\mathrm{ac}} , \\ 0.5\left[ {1-\cos \left( {{\uppi }\left( {t-t _{\mathrm{ac}} } \right) /t _{\mathrm{acp}} } \right) } \right] , &{}t_{\mathrm{ac}}<t\le t _{\mathrm{ac}} +t _{\mathrm{acp}} , \\ 0.5\left[ {1+\cos \left( {{\uppi }\left( {t-t _{\mathrm{ar}} } \right) /t _{\mathrm{arp}} } \right) } \right] , &{}t _{\mathrm{ac}} +t _{\mathrm{acp}} <t\le t_{0} , \\ \end{array}} \right. \nonumber \\ \end{aligned}$$

(A3)

where $t_{\mathrm {0}}$ represents the time period of a cardiac cycle (s), $t_{\mathrm {vcp}}, t_{\mathrm {acp}}, t_{\mathrm {vrp}}$, and $t_{\mathrm {arp}}$ the durations of ventricular/atrial contraction/relaxation (s), respectively, and $t_{\mathrm {ac}}$ and $t_{\mathrm {ar}}$ the time points at which the atria begin to contract and relax (s). The cardiac chamber pressure can be further obtained as

$$\begin{aligned} p_{h} (t)=E(t)(V-V_{0} )+S\frac{\text{ d }V}{\text{ d }t}, \end{aligned}$$

(A4)

where V is the cardiac chamber volume, $V_{\mathrm {0}}$ the dead chamber volume, and S the viscoelasticity coefficient of the cardiac wall.

Modeling of pulmonary circulation

Pulmonary circulation is modeled with three compartments: pulmonary artery, the pulmonary capillary, and the pulmonary vein. The elastance (E) of pulmonary circulation is defined by [21]

$$\begin{aligned} E=E_{0} \cdot e^{{V/\phi }}, \end{aligned}$$

(A5)

where $E_{\mathrm {0}}$ denotes the zero volume elastance, V the blood volume, and $\phi$ the volume constant. Pulmonary pressure is then obtained as

$$\begin{aligned} P=E\cdot \phi . \end{aligned}$$

(A6)

Fig. B1 17

Flowcharts of two numerical methods for 0D–1D coupling. a Characteristic method. b “Ghost-point” method

Appendix B: 0D–1D coupling algorithm

Here we present two numerical methods associated with coupling of a 1D vascular network and 0D lumped-parameter models. As depicted in Fig. 2, the 1D vessels that supply blood to the 0D lumped-parameter models are defined as feeding vessels and include terminal arteries, superior vena cava, and inferior vena cava, whereas the 1D vessels that collect blood from the 0D lumped-parameter models are called draining vessels and consist of ascending aorta and terminal veins.

Characteristic method

With respect to this method, the characteristic variants are used to set boundary conditions. The 1D governing equations may be transformed into a set of characteristic equations and, when the viscous resistance is ignored, can be reformulated as

$$\begin{aligned} \frac{\partial W}{\partial t}+\varLambda \frac{\partial W}{\partial x}=0, \end{aligned}$$

(B1)

with

$$\begin{aligned} W=\left[ {\begin{array}{l} W_{f} \\ W_{b} \\ \end{array}} \right] ,\quad \varLambda =\left[ {\begin{array}{l} \lambda _{f} \\ \lambda _{b} \\ \end{array}} \right] , \end{aligned}$$

(B2)

where $W_{f,b} =u\pm \int \limits _{A_{0} }^A {\frac{c(\tau )}{\tau }} \mathrm{d}\tau$ expresses the characteristic invariants of the system, and $\lambda _{f,b} =u\pm c$ denotes the propagation speeds of the characteristic invariants. The characteristic invariants $W_{f,b} (t^{n+1})$ at the time $n+1$ can be updated through an extrapolation from $W_{f,b} (t^{n})$ at the previous time step n.

For the first node of the draining vessel, the characteristic invariant at the next time step $(n+1)$ can be updated as

$$\begin{aligned} W_{b} \left( {t^{n+1},x_{1} } \right) =W_{b} \left( {t^{n},x_{1} -\lambda _{b}^{n} \cdot \Delta t} \right) , \end{aligned}$$

(B3)

and for the last node of the feeding vessel, we have

$$\begin{aligned} W_{f} \left( {t^{n+1},x_{m} } \right) =W_{f} \left( {t^{n},x_{m} -\lambda _{f}^{n} \cdot \Delta t} \right) , \end{aligned}$$

(B4)

where ${x}_{\mathrm {1}}$ denotes the first node, ${x}_{\mathrm {m}}$ the last node, and $\Delta {t}$ the time step. A flowchart of the characteristic variable method for 0D–1D coupling is described in detail in Fig. B1a. The limitation of this method is that it is only valid under the subcritical condition ($S<1$).

Fig. B1 18

Comparison of model-predicted pressures in superior vena cava in standing posture between two different numerical methods for 0D–1D coupling

Ghost-point method

For this method, a so-called ghost point is used to set boundary conditions. At the interface connecting the 0D and 1D models, we assume that the index of the last gridpoint of the vessel is m, and the index of the first gridpoint of the vessel is 1. Then the continuity equation at the ghost point ${m}-1/2$ of the feeding vessel can be discretized such that

$$\begin{aligned} \frac{\frac{A_{m-1}^{n+1} +A_{m}^{n+1} }{2}-\frac{A_{m-1}^{n} +A_{m}^{n} }{2}}{\Delta t}+\frac{(Q)_{m}^{n+1} -(Q)_{m-1}^{n+1} }{\Delta x}=0. \end{aligned}$$

(B5)

Similarly, the continuity equation at the ghost point 1/2 of the draining vessel can be obtained as

$$\begin{aligned} \frac{\frac{A_{1}^{n+1} +A_{2}^{n+1} }{2}-\frac{A_{1}^{n} +A_{2}^{n} }{2}}{\Delta t}+\frac{(Q)_{2}^{n+1} -(Q)_{1}^{n+1} }{\Delta x}=0. \end{aligned}$$

(B6)

Thus the flow rate ${Q}^{n+1}$ can be expressed in terms of ${A}^{n+1}$:

$$\begin{aligned} Q_{m}^{n+1}= & {} Q_{m-1}^{n+1} -\frac{\Delta x}{2\Delta t}\left( {A_{m-1}^{n+1} +A_{m}^{n+1} -A_{m-1}^{n} -A_{m}^{n} } \right) ,\nonumber \\ \end{aligned}$$

(B7)

$$\begin{aligned} Q_{1}^{n+1}= & {} Q_{2}^{n+1} +\frac{\Delta x}{2\Delta t}\left( {A_{1}^{n+1} +A_{2}^{n+1} -A_{1}^{n} -A_{2}^{n} } \right) . \end{aligned}$$

(B8)

A schematic description of the flowchart associated with the ghost-point method for the 0D–1D coupling is also given in Fig. B1b.

Figure B2 shows a comparison of the simulated pressures using the two methods. Obviously, there is almost no visible difference between the characteristic and ghost-point. It should be pointed out that the characteristic method is merely valid in the subcritical state, whereas the ghost-point method can be applied equally to the supercritical state as well, which makes it more robust and, hence, effective for 0D–1D coupling. Therefore, we used the ghost-point method to couple the 1D vascular network model with the 0D lumped-parameter model to close up the multiscale hemodynamic model for the entire CVS.