Multifidelity-CMA: a multifidelity approach for efficient personalisation of 3D cardiac electromechanical models

Abstract

Personalised computational models of the heart are of increasing interest for clinical applications due to their discriminative and predictive abilities. However, the simulation of a single heartbeat with a 3D cardiac electromechanical model can be long and computationally expensive, which makes some practical applications, such as the estimation of model parameters from clinical data (the personalisation), very slow. Here we introduce an original multifidelity approach between a 3D cardiac model and a simplified “0D” version of this model, which enables to get reliable (and extremely fast) approximations of the global behaviour of the 3D model using 0D simulations. We then use this multifidelity approximation to speed-up an efficient parameter estimation algorithm, leading to a fast and computationally efficient personalisation method of the 3D model. In particular, we show results on a cohort of 121 different heart geometries and measurements. Finally, an exploitable code of the 0D model with scripts to perform parameter estimation will be released to the community.

Appendices

Appendix A: Mechanical equations and haemodynamics

As described in Marchesseau et al. (2013a) our 3D electromechanical model is based on the Bestel–Clement–Sorine model (BCS) of sarcomere contraction as extended by Chapelle et al. (2012), in conjunction with a Mooney–Rivlin energy for the passive hyperelasticity. Haemodynamics are represented through global values of pressures and flows in the cardiac chambers and coupled to the mechanical equations with the Windkessel model of blood pressure for the afterload (aortic pressure).

1.1 The BCS model: active contraction and passive material

The BCS model describes the sarcomere forces as the sum of an active contraction force in the direction of the fibre, in parallel with a passive isotropic visco-hyperelastic component (see Fig. 1b). It is compatible with the laws of thermodynamics and allows to model physiological phenomena at the sarcomere scale which translate at the macroscopic scale (such as the Starling effect).

The active force in the sarcomere is modelled by the filament model of Huxley (1957), which describes the binding/unbinding process of the actin and myosin in the sarcomere at the nanoscopic scale. At the mesoscopic scale, it results (Caruel et al. 2014) in a differential equation which relates the active stress $\tau _\text {c}$, the stiffness $k_\text {c}$ and the strain $e_\text {c}$ of the filament within the sarcomere:

$$\begin{aligned} \left\{ \begin{array}{@{}l@{}} \dot{k_\text {c}} = -({|}u{|}_\text {+}+{|}u{|}_\text {-}+\alpha {|}\dot{e_\text {c}}{|})k_\text {c} + k_\text {0}{|}u{|}_\text {+},\\ \dot{\tau _\text {c}} =-({|}u{|}_\text {+}+{|}u{|}_\text {-}+\alpha {|}\dot{e_\text {c}}{|})\tau _\text {c} + \dot{e_\text {c}}k_\text {c} + \sigma _\text {0}{|}u{|}_\text {+}, \end{array}\right. \end{aligned}$$

(3)

where $\alpha$ is a constant related to the cross-bridge destruction during contraction, and $k_\text {0}$ and $\sigma _\text {0}$ are, respectively, the maximum stiffness and contraction. The values of ${|}u{|}_\text {+}$ and ${|}u{|}_\text {-}$ are, respectively, the rate of build-up $k_\text {ATP}$ and decrease $k_\text {RS}$ of the force during contraction and relaxation, which depends on the depolarisation and repolarisation times $T_{d}$ and $T_{r}$ of the sarcomere:

$$\begin{aligned} u=\left\{ \begin{array}{@{}ll@{}} k_\text {ATP} &{} \text {when}\ T_{d} \le t \le T_{r} \\ -k_\text {RS} &{} \text {otherwise} \\ {|}u{|}_\text {+} = \max (u,0),\\ {|}u{|}_\text {-} = -\min (u,0). \end{array}\right. \end{aligned}$$

(4)

This active force is applied in the direction of the fibre through the visco-elastic component, made of a spring $E_s$ and a dissipative term $\mu$ (see Fig. 1b). As derived in Caruel et al. (2014), the resulting stress $\sigma _\text {1D}$ in the fibre direction is given by:

$$\begin{aligned} \left\{ \begin{array}{@{}l@{}} \sigma _\text {1D} = E_s\frac{e_\text {1D}-e_\text {c}}{(1+2e_\text {c})^2},\\ (\tau _c + \mu \dot{e_c}) = E_s\frac{(e_\text {1D}-e_\text {c})(1+2e_\text {1D})}{(1+2e_\text {c})^3}, \end{array}\right. \end{aligned}$$

(5)

where $e_\text {1D}=\underline{\tau _1} \cdot \underline{\underline{e}} \cdot \underline{\tau _1}$ is the strain in the fibre direction $\tau _1$ ($\underline{\underline{e}}$ is the Green–Lagrange strain tensor).

Finally, for the passive component the isotropic Mooney–Rivlin model of hyperelastic material is used, driven by the following strain energy:

$$\begin{aligned} W_e=c_1(I_1-3)+c_2(I_2-3)+\frac{K}{2}(J-1)^2, \end{aligned}$$

(6)

where $I_1$, $I_2$ and J are the invariants of the Cauchy–Green deformation tensor, $c_1$, $c_2$ and K are the parameters of the material.

1.2 Haemodynamic model

To model the influence of blood dynamics during the cardiac circle, the mechanical equations are coupled with a basic circulation model implementing the 4 phases of the cardiac cycle. For a given ventricle, if we note $P_\text {at}$ the pressure in the atrium, $P_\text {ar}$ the pressure in the artery and $P_\text {V}$ the pressure in the ventricle, the phases are the following:

• Diastolic filling when $P_\text {V} \le P_\text {at}$, the atrial valve is open and the ventricle fills up with blood.

• Isovolumetric contraction when contraction starts, $P_\text {V}$ rises. $P_\text {at} \le P_\text {V} \le P_\text {ar}$ and all the valves are closed.

• Systolic ejection when $P_\text {V} \ge P_\text {ar}$, the arterial valve opens and the blood is ejected into the artery.

• Isovolumetric relaxation when the contractile forces disappear, $P_\text {V}$ finally decreases. $P_\text {at} \le P_\text {V} \le P_\text {ar}$ again, and all the valves are closed.

We use the haemodynamic model introduced by Chapelle et al. (2012) which links the blood flow q to the ventricular, atrial and arterial pressures with the following equations:

$$\begin{aligned} q=\left\{ \begin{array}{@{}ll@{}} K_\text {at}(P_\text {V}-P_\text {at}) &{} \text {for}\ P_\text {V} \le P_\text {at} \\ K_\text {iso}(P_\text {V}-P_\text {at}) &{} \text {for}\ P_\text {at} \le P_\text {V} \le P_\text {ar} \\ K_\text {ar}(P_\text {V}-P_\text {at}) + K_\text {iso}(P_\text {ar}-P_\text {at}) &{} \text {for}\ P_\text {V} \ge P_\text {ar} \end{array}\right. \end{aligned}$$

(7)

Here the atrial pressure $P_\text {at}(t)$ (cardiac preload) is imposed at a constant value $P_{\text {at}\_\text {lower}}$ except for a pressure bump up to $P_{\text {at}\_\text {upper}}$ at the beginning of cardiac cycle, to account for the contraction of the atrium before the ventricular contraction. Finally, the pressure of the artery $P_\text {ar}$ (cardiac afterload) is modelled with the 3-parameter Windkessel model (Westerhof et al. 1969) and coupled to the ventricular outflow q through the equation:

$$\begin{aligned} R_pC\dot{P_\text {ar}}+P_\text {ar}-P_\text {ve}=(R_p+Z_c)q+R_pZ_cC\dot{q}, \end{aligned}$$

(8)

where $R_p$ is the Peripheral resistance, $Z_c$ is the characteristic impedance, C is the arterial compliance, and $P_{Ve}$ is the central venous pressure.

1.2.1 Implementation

The passive Mooney–Rivlin energy is discretised on the 3D mesh with the MJED (multiplicative Jacobian energy decomposition) method described in Marchesseau et al. (2010), and the BCS fibre stress and stiffness are computed at each node, separately from the positions and velocities. This allows a fast assembly and a good conditioning of the system of mechanical equations. A Rayleigh damping is then added to account for the viscous global dissipation, and finally, the ventricular pressure is computed using a prediction–correction approach, performed after solving the first system of mechanical equations. This efficient algorithm and all the details of the mechanical equations and their 3D discretisations are fully discussed in Marchesseau et al. (2013a).

Table 7 Mechanical equations of the 0D model

Table 8 Electrical activation in the 0D model

Appendix B: Reduced equations of the 0D model

1.1 Mechanical equations

The list of simplified equations of our 0D model is reported in Table 7. Equations (a), (b), (c) and (f) are the same sarcomere and visco-elastic equations than Eqs. 3 and 5, which are calculated once for the whole sphere. C in equations (d), (e), (g) and (h) denotes a component of the simplified Cauchy–Green deformation tensor which depends only on $y=R-R_0$. $\sigma _\text {passive}$ in equation (g) is the stress due to the passive law, and $\sigma _\text {viscosity}$ in equation (h) is the stress due to an additional viscous damping $\eta$, both expressed as a simple function of C (see Caruel et al. (2014) for the full derivations). In equation (i), $\varSigma _\text {sph}$ is the sum of all the stresses applied to the sphere. Equation (j) is the resulting equation of motion which, coupled with the haemodynamic model (k) and the Windkessel equation (l), gives the full system of 3 equations to be solved at each iteration.

1.2 Electrophysiology equations

Assuming synchronous and homogeneous electrical activation (and thus sarcomere force) means that all of the ventricle is depolarised simultaneously. This leads to a rate of ventricular pressure rise during the isovolumetric contraction (respectively, isovolumetric relaxation) which is very close to the rate of build-up $k_\text {ATP}$ (respectively, decrease $k_\text {RS}$) of the active stress $\tau _c$. However in 3D, this rate is also very dependent on the time for the ventricle to be fully depolarised, which is roughly the QRS duration.

In order to correct this discrepancy between the models, we adapted the electrical parameter u to take into account the QRS duration. We model the fraction $f_\text {depo}$ of the ventricle which is currently depolarised as a piecewise linear function of time which depends on $T_\text {d,global}$, $T_\text {r,global}$ and $QRS_\text {duration}$. Then, the values of ${|}u{|}_+$ and ${|}u{|}_-$ in Equation (a) are adapted to depend on the value of $f_\text {depo}$ as described in Table 8.