A geometric multiscale model for the numerical simulation of blood flow in the human left heart

Alberto Zingaro; Ivan Fumagalli; Luca Dede; Marco Fedele; Pasquale C. Africa; Antonio F. Corno; Alfio Quarteroni

doi:10.3934/dcdss.2022052

Article Contents

2022, Volume 15, Issue 8: 2391-2427. Doi: 10.3934/dcdss.2022052

This issue Previous Article Well-posedness of a hydrodynamic phase-field system for functionalized membrane-fluid interaction Next Article Correction to "Thermoelasticity with antidissipation" (volume 15, number 8, 2022, 2173–2188)

A geometric multiscale model for the numerical simulation of blood flow in the human left heart

1.
MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milan, Italy
2.
Children’s Heart Institute, Hermann Children’s Hospital, University of Texas Health, McGovern Medical School, Houston, TX, USA
3.
Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Station 8, Av. Piccard, CH-1015 Lausanne, Switzerland (Professor Emeritus)

^* Corresponding author: Alberto Zingaro
^* Corresponding author: Alberto Zingaro

Received: December 31, 2021

Published: August 2022

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Abstract

We present a new computational model for the numerical simulation of blood flow in the human left heart. To this aim, we use the Navier-Stokes equations in an Arbitrary Lagrangian Eulerian formulation to account for the endocardium motion and we model the cardiac valves by means of the Resistive Immersed Implicit Surface method. To impose a physiological displacement of the domain boundary, we use a 3D cardiac electromechanical model of the left ventricle coupled to a lumped-parameter (0D) closed-loop model of the remaining circulation. We thus obtain a one-way coupled electromechanics-fluid dynamics model in the left ventricle. To extend the left ventricle motion to the endocardium of the left atrium and to that of the ascending aorta, we introduce a preprocessing procedure according to which an harmonic extension of the left ventricle displacement is combined with the motion of the left atrium based on the 0D model. To better match the 3D cardiac fluid flow with the external blood circulation, we couple the 3D Navier-Stokes equations to the 0D circulation model, obtaining a multiscale coupled 3D-0D fluid dynamics model that we solve via a segregated numerical scheme. We carry out numerical simulations for a healthy left heart and we validate our model by showing that meaningful hemodynamic indicators are correctly reproduced.

Keywords:

Mathematics Subject Classification: Primary: 65-XX, 76-XX, 76-10, 76Z05, 92C50.

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Figure 1. The 0D circulation model

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Figure 2. Fluid domain in reference configuration (left), ALE map $ {\boldsymbol{x}} = \mathcal{A}_t ( {\widehat{{\mathit{\boldsymbol{x}}}}}) $ and domain in current configuration (right). In the current configuration, the domain $ \Omega_t $ is bounded by $ \Gamma_t = \Gamma_t^\mathrm{D} \cup \Gamma_t^\mathrm{N} $; $ \Sigma_k $ is an immersed surface modeled by means of the RIIS method

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Figure 3. The LH geometry: (A) the three subdomains $ \Omega_t = \overline \Omega_t^{{\rm{LA}}} \cup \overline \Omega_t^{_{\rm{LV}}} \cup \overline \Omega_t^{{\rm{AA}}} $; (B) the boundary portions of the LH geometry $ \Gamma_t = \left (\bigcup_{i=1}^5 \overline \Gamma^{{\rm{PVein}}_i} \right )\cup \overline \Gamma_t^{\rm{AA}} \cup \overline \Gamma_t^{{\rm{w}}} $; (C) in yellow, the immersed surfaces $ \Sigma_{\rm{MV}} $ and $ \Sigma_{\rm{AV}} $ (respectively in their open and closed configurations); in red, the Neumann data (for both inlet and outlet sections); in green the Dirichlet datum at wall

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Figure 4. Boundary portions of the LH geometry in reference configuration. This splitting of the domain is used to define the Laplacian problem (13)

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Figure 5. Displacement procedure. Boxes numbers are referred to lines in Algorithm 1

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Figure 6. EM simulation of the LV: (A) LV in its reference configuration; (B), (C) LV during systole and diastole colored by displacement magnitude

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Figure 7. LH geometry warped by $ \widehat{\boldsymbol{d}}_{\Gamma} $ at different times during a heart cycle

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Figure 8. Volumes of LA, LV and AA achieved applying the displacement – $ \widehat{\boldsymbol{d}}_{\Gamma} $ defined in Eq. (12) – to $ \widehat \Gamma $

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Figure 9. Immersed valve $ \Sigma_\mathrm k $ with upwind and downwind control volumes where average pressures are computed. $ Q_\mathrm k $ is the flowrate across $ \Sigma_\mathrm k $. This picture corresponds to a simple a two-dimensional fluid domain for the sake of simplicity

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Figure 10. Cardiac valves in their fully closed and fully open configurations

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Figure 11. The geometric multiscale model: coupling between the 3D CFD model of the LH and the 0D circulation model of the remaining cardiocirculatory system

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Figure 12. Sketch of the algorithm for the CFD simulation of the LH. The coupling between the CFD and the circulation problem is solved via a segregated numerical scheme

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Figure 13. The LH tetrahedral mesh made of three conforming meshes for the LA, LV and AA subdomains; a clip of the mesh showing the local mesh refinement near the MV and AV

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Figure 14. Flowrates (top) and pressures (bottom) at the interfaces of the 3D-0D model

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Figure 15. Flow properties to determine opening and closure of valves. Top: average pressure in LA, LV and AA; bottom: time derivative of LV volume. Average pressures in the chambers are computed in the control volumes in the LH displayed on the right

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Figure 16. Volume rendering of velocity magnitude during the whole heartbeat

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Figure 17. Pressure on a clip in the LV apico-basal direction during the whole heartbeat

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Figure 18. AV section warped by velocity during the ejection phase. In trasparency: the AV represented by the RIIS method

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Figure 19. Left: mean, first and third quartile of the space distribution of the velocity magnitude $ {\boldsymbol{u}}_\mathrm{MV} $ in a control volume $ \Omega_t^\mathrm{MV} $ downstream the MV section. Right: flowrate computed on a section downwind the AV ($ \int_{\Gamma_t^*} {\boldsymbol{u}} \cdot {\boldsymbol{n}} $)

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Figure 20. Iso-contours of Q-criterion ($ \mathbb Q({\boldsymbol{u}}) = 40 \, \mathrm{s}^{-2} $) colored according to velocity magnitude during a whole heartbeat

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Figure 21. Formation of ring shaped vortex during early diastole. Top: iso-contours of Q-criterion (with $ \mathbb Q({\boldsymbol{u}}) =1000\, \rm{s}^{-2} $) coloured according to velocity magnitude. Bottom: projection of the vorticity on the normal direction (pointing towards the reader) of a slice in the LV apico-basal direction

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Figure 22. Vortex formation under the MV section at early diastolic peak: (A) Streamlines colored according to velocity magnitude ($ t=0.45 $ s); (B) Surface LIC visualization – with velocity as integrator – on a slice colored according to velocity magnitude ($ t=0.48 $ s)

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Table 2. Parameters for the setup of the numerical simulations

$ \rho $			$ \mu $	$ {\boldsymbol{u}}_0 $	$ R_{\mathrm k} $		$ \varepsilon_\mathrm k $		$ T_f $	$ T_\mathrm{HB} $
$ [\mathrm{kg/m}^3] $			$ [\mathrm{kg/(m}\cdot\mathrm{s})] $	$ [\mathrm m / \mathrm s] $	[kg/(m$ ^2\cdot $s)]		[mm]		[s]	[s]
					MV	AV	MV	AV

$ 1.06 \cdot 10^3 $			$ 3.5 \cdot 10^{-3} $	$ {\boldsymbol{0}} $	$ 10^4 $	$ 10^4 $	0.6	0.6	2.0	1.0

$h$			cells	DOFs ( $ \mathbb{P}_1 - \mathbb{P}_1 $)					BDF	$\Delta t$
[mm]			[-]	[-]					[-]	[s]
min	avg	max		$\boldsymbol{u}$	$p$		total

0.4	1.2	4.1	1'627'795	806'295	268'765		1'075'060		1	$2.5 \cdot 10^{-4}$

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Table 3. Biomarkers: comparison between numerical results and clinical values acquired in healthy individuals

Biomarker	In-silico result	In-vivo measurements	Reference
LV stroke volume [ml]	82.6	$ 95 \pm 14 $	[49]
LV ejection fraction [%]	55.8	$ 57.5\pm 7.5 $	[47]
Peak AV flowrate [ml/s]	$ 493.3 $	$ \approx 489 $	[38]
LV peak pressure [mmHg]	121.2	$ 119 \pm 13 $	[69]
Peak E-wave velocity [m/s]	$ 0.96 $	$ 0.89 \pm 0.15 $	[77]
Peak A-wave velocity [m/s]	$ 0.71 $	$ 0.78 \pm 0.26 $	[77]
EA ratio $ [-] $	$ 1.35 $	$ 1.30 \pm 0.57 $	[77]

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Table 4. Parameters used in the circulation model

Compartment	Parameter	Description	Unit of measure	Value
Right atrium	$ E_\mathrm{A} $	Active elastance	[mmHg/ml]	0.06
	$ E_\mathrm{B} $	Passive elastance	[mmHg/ml]	0.07
	$ d_\mathrm{c} $	Duration of contract.	relative w.r.t. $ T_\mathrm{HB} $	0.335
	$ d_\mathrm{r} $	Duration of relax.	relative w.r.t. $ T_\mathrm{HB} $	$ 1.45 \cdot 10^{-2} $
	$ t_\mathrm{c} $	Initial time of contract.	relative w.r.t. $ T_\mathrm{HB} $	0.80
	$ V_{0, \mathrm{RA}} $	Resting volume	[ml]	4.00
Right ventricle	$ E_\mathrm{A} $	Active elastance	[mmHg/ml]	0.65
	$ E_\mathrm{B} $	Passive elastance	[mmHg/ml]	0.05
	$ d_\mathrm{c} $	Duration of contract.	relative w.r.t. $ T_\mathrm{HB} $	0.335
	$ d_\mathrm{r} $	Duration of relax.	relative w.r.t. $ T_\mathrm{HB} $	$ 1.45 \cdot 10^{-2} $
	$ t_\mathrm{c} $	Initial time of contract.	relative w.r.t. $ T_\mathrm{HB} $	0.00
	$ V_{0, \mathrm{RV}} $	Resting volume	[ml]	10.00
Pulmonary arterial system	$ R^\mathrm{PUL}_\mathrm{AR} $	Resistance	$ [\mathrm{mmHg}\cdot\mathrm{s}/\mathrm{ml}] $	0.25
	$ C^\mathrm{PUL}_\mathrm{AR} $	Capacitance	$ [\mathrm{ml}/\mathrm{mmHg}] $	5.00
	$ L^\mathrm{PUL}_\mathrm{AR} $	Inductance	$ [\mathrm{mmHg}\cdot \mathrm{s}^2/\mathrm{ml}] $	$ 5.00 \cdot 10^{-4} $
Pulmonary venous system	$ R^\mathrm{PUL}_\mathrm{VEN} $	Resistance	$ [\mathrm{mmHg}\cdot\mathrm{s}/\mathrm{ml}] $	0.02
	$ C^\mathrm{PUL}_\mathrm{VEN} $	Capacitance	$ [\mathrm{ml}/\mathrm{mmHg}] $	100.00
	$ L^\mathrm{PUL}_\mathrm{VEN} $	Inductance	$ [\mathrm{mmHg}\cdot \mathrm{s}^2/\mathrm{ml}] $	$ 5.00 \cdot 10^{-5} $
Systemic arterial system	$ R^\mathrm{SYS}_\mathrm{AR} $	Resistance	$ [\mathrm{mmHg}\cdot\mathrm{s}/\mathrm{ml}] $	1.00
	$ C^\mathrm{SYS}_\mathrm{AR} $	Capacitance	$ [\mathrm{ml}/\mathrm{mmHg}] $	2.00
	$ L^\mathrm{SYS}_\mathrm{AR} $	Inductance	$ [\mathrm{mmHg}\cdot \mathrm{s}^2/\mathrm{ml}] $	$ 5.00 \cdot 10^{-3} $
Systemic venous system	$ R^\mathrm{SYS}_\mathrm{VEN} $	Resistance	$ [\mathrm{mmHg}\cdot\mathrm{s}/\mathrm{ml}] $	0.24
	$ C^\mathrm{SYS}_\mathrm{VEN} $	Capacitance	$ [\mathrm{ml}/\mathrm{mmHg}] $	60.00
	$ L^\mathrm{SYS}_\mathrm{VEN} $	Inductance	$ [\mathrm{mmHg}\cdot \mathrm{s}^2/\mathrm{ml}] $	$ 5.00 \cdot 10^{-4} $
Tricuspid valve	$ R_\mathrm{min} $	Minimum resistance	[$ \mathrm{mmHg}\cdot \mathrm{s} / \mathrm{ml} $]	$ 7.5 \cdot 10^{-3} $
Tricuspid valve	$ R_\mathrm{max} $	Maximum resistance	[$ \mathrm{mmHg}\cdot \mathrm{s} / \mathrm{ml} $]	$ 75006.2 $
Pulmonary valve	$ R_\mathrm{min} $	Minimum resistance	[$ \mathrm{mmHg}\cdot \mathrm{s} / \mathrm{ml} $]	$ 7.5 \cdot 10^{-3} $
Pulmonary valve	$ R_\mathrm{max} $	Maximum resistance	[$ \mathrm{mmHg}\cdot \mathrm{s} / \mathrm{ml} $]	$ 75006.2 $

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Table 5. Initial state of the circulation model

Compartment	Parameter	Description	Unit of measure	Value
Right atrium	$ V_\mathrm{RA} $	Volume	[ml]	78.95
Right ventricle	$ V_\mathrm{RV} $	Volume	[ml]	154.00
Pulmonary arterial system	$ p^\mathrm{PUL}_\mathrm{AR} $	Pressure	[mmHg]	33.50
Pulmonary arterial system	$ Q^\mathrm{PUL}_\mathrm{AR} $	Flowrate	[ml/s]	69.44
Pulmonary venous system	$ p^\mathrm{PUL}_\mathrm{VEN} $	Pressure	[mmHg]	16.16
Pulmonary venous system	$ Q^\mathrm{PUL}_\mathrm{VEN} $	Flowrate	[ml/s]	0.00
Systemic arterial system	$ p^\mathrm{SYS}_\mathrm{AR} $	Pressure	[mmHg]	91.68
Systemic arterial system	$ Q^\mathrm{SYS}_\mathrm{AR} $	Flowrate	[ml/s]	63.71
Systemic venous system	$ p^\mathrm{SYS}_\mathrm{VEN} $	Pressure	[mmHg]	23.99
Systemic venous system	$ Q^\mathrm{SYS}_\mathrm{VEN} $	Flowrate	[ml/s]	65.40

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