August  2022, 15(8): 2391-2427. doi: 10.3934/dcdss.2022052

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

Received  January 2022 Published  August 2022 Early access  March 2022

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.

Citation: Alberto Zingaro, Ivan Fumagalli, Luca Dede, Marco Fedele, Pasquale C. Africa, Antonio F. Corno, Alfio Quarteroni. A geometric multiscale model for the numerical simulation of blood flow in the human left heart. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2391-2427. doi: 10.3934/dcdss.2022052
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show all references

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http://www.vmtk.org

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https://github.com/marco-fedele/vmtk

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https://lifex.gitlab.io/

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https://www.paraview.org/Wiki/PvPython_and_PvBatch

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P. C. Africa, R. Piersanti, M. Fedele, L. Dede' and A. Quarteroni, Life$^\mathrm x$ - heart module: A high-performance simulator for the cardiac function. Package 1: Fiber generation, arXiv preprint, arXiv: 2201.03303, 2022.

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L. AntigaM. PiccinelliL. BottiB. Ene-IordacheA. Remuzzi and D. A. Steinman, An image-based modeling framework for patient-specific computational hemodynamics, Medical & Biological Engineering & Computing, 46 (2008), 1097-1112.  doi: 10.1007/s11517-008-0420-1.

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C. P. Appleton, Evaluation of diastolic function by two-dimensional and doppler assessment of left ventricular filling including pulmonary venous flow, In Diastology, Elsevier (2008), 115–143. doi: 10.1016/B978-1-4160-3754-5.50016-0.

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Figure 1.  The 0D circulation model
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
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
Figure 4.  Boundary portions of the LH geometry in reference configuration. This splitting of the domain is used to define the Laplacian problem (13)
Figure 5.  Displacement procedure. Boxes numbers are referred to lines in Algorithm 1
Figure 6.  EM simulation of the LV: (A) LV in its reference configuration; (B), (C) LV during systole and diastole colored by displacement magnitude
Figure 7.  LH geometry warped by $ \widehat{\boldsymbol{d}}_{\Gamma} $ at different times during a heart cycle
Figure 8.  Volumes of LA, LV and AA achieved applying the displacement – $ \widehat{\boldsymbol{d}}_{\Gamma} $ defined in Eq. (12) – to $ \widehat \Gamma $
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
Figure 10.  Cardiac valves in their fully closed and fully open configurations
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
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
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
Figure 14.  Flowrates (top) and pressures (bottom) at the interfaces of the 3D-0D model
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
Figure 16.  Volume rendering of velocity magnitude during the whole heartbeat
Figure 17.  Pressure on a clip in the LV apico-basal direction during the whole heartbeat
Figure 18.  AV section warped by velocity during the ejection phase. In trasparency: the AV represented by the RIIS method
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}} $)
Figure 20.  Iso-contours of Q-criterion ($ \mathbb Q({\boldsymbol{u}}) = 40 \, \mathrm{s}^{-2} $) colored according to velocity magnitude during a whole heartbeat
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} $) coloured according to velocity magnitude. Bottom: projection of the vorticity on the normal direction (pointing towards the reader) of a slice in the LV apico-basal direction
Figure 22.  Vortex formation under the MV section at early diastolic peak: (A) Streamlines colored according to velocity magnitude ($ t=0.45 $ s); (B) Surface LIC visualization – with velocity as integrator – on a slice colored according to velocity magnitude ($ t=0.48 $ s)
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 $2.5 \cdot 10^{-4}$
$ \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 $2.5 \cdot 10^{-4}$
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]
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]
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} $
$ 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} $
$ R_\mathrm{max} $ Maximum resistance [$ \mathrm{mmHg}\cdot \mathrm{s} / \mathrm{ml} $] $ 75006.2 $
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} $
$ 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} $
$ R_\mathrm{max} $ Maximum resistance [$ \mathrm{mmHg}\cdot \mathrm{s} / \mathrm{ml} $] $ 75006.2 $
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
$ Q^\mathrm{PUL}_\mathrm{AR} $ Flowrate [ml/s] 69.44
Pulmonary venous system $ p^\mathrm{PUL}_\mathrm{VEN} $ Pressure [mmHg] 16.16
$ Q^\mathrm{PUL}_\mathrm{VEN} $ Flowrate [ml/s] 0.00
Systemic arterial system $ p^\mathrm{SYS}_\mathrm{AR} $ Pressure [mmHg] 91.68
$ Q^\mathrm{SYS}_\mathrm{AR} $ Flowrate [ml/s] 63.71
Systemic venous system $ p^\mathrm{SYS}_\mathrm{VEN} $ Pressure [mmHg] 23.99
$ Q^\mathrm{SYS}_\mathrm{VEN} $ Flowrate [ml/s] 65.40
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
$ Q^\mathrm{PUL}_\mathrm{AR} $ Flowrate [ml/s] 69.44
Pulmonary venous system $ p^\mathrm{PUL}_\mathrm{VEN} $ Pressure [mmHg] 16.16
$ Q^\mathrm{PUL}_\mathrm{VEN} $ Flowrate [ml/s] 0.00
Systemic arterial system $ p^\mathrm{SYS}_\mathrm{AR} $ Pressure [mmHg] 91.68
$ Q^\mathrm{SYS}_\mathrm{AR} $ Flowrate [ml/s] 63.71
Systemic venous system $ p^\mathrm{SYS}_\mathrm{VEN} $ Pressure [mmHg] 23.99
$ Q^\mathrm{SYS}_\mathrm{VEN} $ Flowrate [ml/s] 65.40
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