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

  • * Corresponding author: Alberto Zingaro

    * Corresponding author: Alberto Zingaro 
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  • 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.

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


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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]
    $ 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 $2.5 \cdot 10^{-4}$
     | Show Table
    DownLoad: CSV

    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]
     | Show Table
    DownLoad: CSV

    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 $
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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