A coupling strategy for a 3D-1D model of the cardiovascular system to study the effects of pulse wave propagation on cardiac function

The impact of increased stiffness and pulsatile load on the circulation and their influence on heart performance have been documented not only for cardiovascular events but also for ventricular dysfunctions. For this reason, computer models of cardiac electromechanics (EM) have to integrate effects of the circulatory system on heart function to be relevant for clinical applications. Currently it is not feasible to consider three-dimensional (3D) models of the entire circulation. Instead, simplified representations of the circulation are used, ensuring a satisfactory trade-off between accuracy and computational cost. In this work, we propose a novel and stable strategy to couple a 3D EM model of the heart to a one-dimensional (1D) model of blood flow in the arterial system. A personalised coupled 3D-1D model of LV and arterial system is built and used in a numerical benchmark to demonstrate robustness and accuracy of our scheme over a range of time steps. Validation of the coupled model is performed by investigating the coupled system's physiological response to variations in the arterial system affecting pulse wave propagation, comprising aortic stiffening, aortic stenosis or bifurcations causing wave reflections. Our results show that the coupled 3D-1D model is robust, stable and correctly replicates known physiology. In comparison with standard coupled 3D-0D models, additional computational costs are negligible, thus facilitating the use of our coupled 3D-1D model as a key methodology in studies where wave propagation effects are under investigation.


Abstract
The impact of increased stiffness and pulsatile load on the circulation and their influence on heart performance have been documented not only for cardiovascular events but also for ventricular dysfunctions. For this reason, computer models of cardiac electromechanics (EM) have to integrate effects of the circulatory system on heart function to be relevant for clinical applications. Currently it is not feasible to consider three-dimensional (3D) models of the entire circulation. Instead, simplified representations of the circulation are used, ensuring a satisfactory trade-off between accuracy and computational cost. In this work, we propose a novel and stable strategy to couple a 3D EM model of the heart to a one-dimensional (1D) model of blood flow in the arterial system. A personalised coupled 3D-1D model of LV and arterial system is built and used in a numerical benchmark to demonstrate robustness and accuracy of our scheme over a range of time steps. Validation of the coupled model is performed by investigating the coupled system's physiological response to variations in the arterial system affecting pulse wave propagation, comprising aortic stiffening, aortic stenosis or bifurcations causing wave reflections. Our results show that the coupled 3D-1D model is robust, stable and correctly replicates known physiology.

Introduction
Computational modelling of cardiac EM is increasingly proposed and pursued as an effective tool for investigating cardiac physiology and numerous pathological conditions [1,2].
Computer models have been used for quantitative analysis, since they have the potential to assess valuable diagnostic information and represent a strong predictive tool for analysing the patient-specific response to a given treatment [3]. However, this is a very open and challenging field of research, since cardiac function relies on complex biophysical aspects that are strongly interlinked [4]. Indeed, the muscle contraction is stimulated by electrical activation and strongly interacts with intraventricular blood and the circulatory system to transport nutrients and clear waste products. As a consequence, efficient computational tools are required that account for the complex physiology and multiphysics of the heart and can be practically implemented and integrated into the clinic [5]. In this context, numerous mathematical models have been reported in the literature that aim to model the effect of the circulatory system on cardiac function. These range from simple lumped zerodimensional (0D) Windkessel-type models [6,7,8], to more refined 1D models [9,10,11].
While 0D models are suitable as global models of afterload, providing a simple way to evaluate global cardiovascular dynamics, they do not consider any effects due to pulse wave transmission and thus are unable to predict important markers such as pulse wave velocity. In contrast, 1D blood flow models enable an efficient representation of the effects of pulse wave propagation and reflection in the circulation. Consequently, 1D models may be preferred over 0D models when local vascular changes or distributed properties, e.g., vessel branching, tapering, stenoses, and their effect on central pressure waveforms are studied and when pulse wave transmission effects are under investigation [12]. To date, most 3D computational models of the heart proposed in the literature consider lumped 0D models as boundary conditions to model the circulation [13,14,15,1,16,17] for ease of implementation and calibration. To the best of our knowledge, no studies propose an effective and stable method to couple a 3D cardiac model to a 1D description of the circulation, which can provide new insights into the physiology and analyse the effects of disrupted wave propagation on cardiovascular dynamics. This is clinically relevant, since the impact of increased stiffness and pulsatile load on the circulation and cardiac performance have been clinically documented not only for cardiovascular events, but also for left and right ventricle dysfunctions [18]. In order to address this need, in this methodological paper we present a novel in silico model based on a multidimensional approach, namely a 3D EM model of cardiac function together with a 1D model of the human circulation. The building blocks of the coupled model and the main aspects of model parameterisation are thoroughly described in the text. The coupling strategy is inspired by recent methods proposed for 3D-0D models [13,19] and is based on the solution of a saddle-point problem for the volume and pressure in the cardiac cavity. A personalised 3D EM model of the LV [1] coupled to a 1D circulatory model [20] is built and employed in numerical test cases to demonstrate robustness and stability of the numerical scheme. The physiological response of the coupled model is investigated in different conditions affecting pulse wave transmission, such as aortic stiffening, aortic narrowing and complex vessel networks involving numerous bifurcations.
We show the ability of the coupled 3D-1D model to correctly replicate known physiological behaviours related to pulse wave propagation. The computational cost of our coupled 3D-1D model is comparable to that of standard 3D-0D models, suggesting the use of our 3D-1D model in clinical applications where wave transmission effects are investigated.

3D Electromechanical cardiac model
Cardiac tissue is modelled as a hyperelastic, anisotropic, nearly incompressible material endowed with a nonlinear stress-strain relationship [21]. Stresses associated with active contraction are characterised as orthotropic, and it is assumed that full contractile force acts along the myocyte fibre orientation, whereas 40% contractile force is directed along the sheet orientation [22]. Active stress generation is represented based on a simplified phenomenological contractile model [23]. Active stresses in this model are only lengthdependent, and dependence on fibre velocity is neglected. A recent reaction-eikonal (R-E) model [24] is considered for the generation of the electrical activation sequences that trigger active stress generation in the tissue. The hybrid R-E model is a combination of a standard reaction-diffusion model (based on the monodomain equation) and an eikonal model. Spatio-temporal discretisation of all PDEs and the solvers for the resulting systems of equations are based on the Cardiac Arrhythmia Research Package (CARPentry) [1,24,25], built upon extensions of the freely available openCARP EP framework (http: //www.opencarp.org). Further information on the EM model of cardiac function can be retrieved in Appendix A. For numerical details on the finite element discretisation, we refer the reader to Appendix E.1.

Circulatory system
The circulatory system imposes a load on the heart and, therefore, affects its mechanical activity. However, the interaction between the heart and the circulatory system is bidirectional, i.e., the outflow of blood from a heart cavity and pressure in the cavity depend on the current state of the circulatory system, and pressure and flow in the cardiovascular system are determined by the current state of the cavity itself. The full physics of this interaction is most accurately posed as a fluid-structure interaction (FSI) problem where pressure, p(X), and flow velocity, v(X), of the fluid are the coupling variables [26,27,28].
Any perturbation in blood flow velocity and pressure changes the state of deformation of the heart and attached vessels. As a result, this change in strain implies a change in stress within the myocardial muscle. Conversely, any change in strain or compliance of the attached vessel or the heart changes the pressure and flow of the fluid. Even though such a distributed PDE-based approach may accurately describe this interaction, at the level of the whole circulatory system it is hardly feasible, for computational and structural reasons.
In fact, a multi-beat simulation using a 3D FSI model solution would be inconvenient for clinical application. Moreover, the use of 3D models of haemodynamics would require the identification of a much larger number of parameters, that is not easily attainable within clinical constraints. A reduced-order approximation of the circulatory system is used instead in this work, which based on a 1D model for the circulatory system and is particularly suitable for simulating blood flow in the aorta and the major arteries. The parameters in the circulatory model are identified and constrained by imaging-based measurements of the heart and blood flow and invasive blood pressure measurements. In this framework the nonlinear PDE EM model of the heart is coupled to the 1D model of the circulatory system using the lumped hydrostatic pressure p cav in the cardiac cavity and the flow q cav out of the cavity into the circulatory system as coupling variables.

1D mathematical model of the blood flow circulation in the arterial system
The 1D equations of blood flow in the human circulation can be derived from the Navier-Stokes equations after assuming axisymmetric flow in a cylindrical tube with thin wall [9].
In this work, we only take into account the arterial circulation and we consider a Voigt-type visco-elastic constitutive law for the arterial wall [20]. A three-element Windkessel lumped parameter model is employed as a terminal boundary condition to simulate the effect of peripheral vessels on pulse wave propagation in larger arteries. The numerical discretisation of the resulting system of equations relies upon a FEM discretisation (discontinuous Galerkin) in space and finite difference (explicit second-order Adams-Bashforth) in time [20].
The numerical solution of the resulting hyperbolic system of equations is performed with the solver Nektar1D (http://haemod.uk/nektar). We refer the reader to Appendix B for more information on 1D blood flow modelling. The arterial network is connected to the left ventricle (LV) cavity through a model of aortic valve (AV) dynamics, based on Bernoulli's equation [29]. See Appendix C for further details on the valve dynamics model.

3D-1D coupling
Following the approach for 3D-0D coupling proposed in Augustin et al. [30] and inspired by Gurev et al. [31], Kerckhoffs et al. [14], the coupling condition between an EM model of the heart cavities and a reduced-order circulatory model imposes that the volume change in each cavity (left ventricle LV, right ventricle RV, left atrium LA and right atrium RA) is balanced with the volume change in the attached circulatory system. For the sake of generality, we cast in what follows the general framework to couple a four-chamber heart model with a closed-loop circulatory system. Let us consider that approximately, at a given time-point, we have constant pressures in each cavity p c , with c ∈ {LV, RV, LA, RA}. Then, the following equations are the starting point of the method: where V heart c (u) is the cavity volume computed with the deformation using Eq. (2), and V CS c (p c ) is the volume as predicted by the blood flow model for the intra-cavitary pressure p c . We write p c = [p c ] c∈{LV,RV,LA,RA} for the vector of up to 1 ≤ N cav ≤ 4 pressure unknowns. Note, however, that in a purely EM simulation framework the fluid domain is not explicitly modelled. Therefore, the cavitary volume V heart c (u) is not discretised. Instead, only the surface enclosing the cavitary volume is known. Nonetheless, if we assume that the entire surface of the cavitary volume is available, including the faces representing the valves, then we can compute for each instant t of the cardiac cycle the enclosed volume V heart c (u) from the surface Γ c by means of the divergence theorem Then, the approach used to evolve the full system of equations of nonlinear elasticity Eq. (1) together with the coupling condition Eq.(2) is based on the resolution of a saddle-point problem in the variables (u, p), corresponding to the displacement and the pressure in the cavity: which is valid for all vector fields v smooth enough and satisfying the given boundary conditions, test functions q that are 1 for the cavity c and 0 otherwise, the duality pairing ·, · Ω 0 , and cavities c ∈ {LV, RV, LA, RA}. Note that A 0 (u), v Ω 0 can be physically interpreted as the rate of internal mechanical work, whereas F 0 (u, p c ), v Ω 0 takes into account the contribution of pressure loads. Using a Galerkin discretisation and the Newton method, the problem to be solved at each Newton-Raphson step becomes a block system in order to find δu ∈ R 3N and δp ∈ R Ncav such that: with the updates and the solution vectors u k ∈ R 3N and p k c ∈ R Ncav at the k-th Newton step. In particular, we can retrieve the expression of C (p k c ) and V CS (p k c ) from the equations of the circulatory model, whereas the term B u (u k ) only depends on the contribution of the EM model of the heart. In more detail, C (p k c ) is approximated by a discrete derivative (finite difference) of the volume V CS with respect to the cavity pressure. We refer the reader to Appendix D and Appendix E for further information on the coupling strategy and finite element formulation, respectively.

Numerical framework
The coupling scheme and a C++ circulatory system module have been embedded in CARPentry, based on the software Nektar1D, in order to preserve computational efficiency and strong scalability. A Schur complement approach (see Appendix F) is considered to cast the coupling problem in a pure displacement formulation, in order to use solver methods already implemented [1]. A generalised minimal residual method (GMRES) with an relative error reduction of = 10 −8 is employed. The library PETSc [32] and the incorporated solver suite hypre/BoomerAMG [33] are used for efficient preconditioning. A generalised-α scheme is considered for time integration, with spectral radius ρ ∞ = 0 and damping parameters β mass = 0.1, β stiff = 0.1, see [30] for further information.

Model parameterisation
In order to guarantee the stability of the coupled system, the solution of the arterial 1D model is initialised until a periodic solution is reached. This is achieved by performing 20 cycles of the circulatory model alone prior to coupling with the heart model, with a flow or pressure profile provided as an inlet boundary condition. For example, a flow or pressure waveform clinically measured at the aortic root can be used as a boundary condition, if available. Otherwise, analytical functions can be prescribed [34]. For the current study, we considered two different choices: an invasively-recorded pressure measurement at the aorta and an analytical function of flow waveform calibrated to match measured peak flow, opening, maximum and closing pressure at the aorta. The coupled system is solved using a fully converging Newton method with maximal number of steps k max = 10 and an absolute 2 norm error reduction of the residual of = 10 −6 . To illustrate the performance of the coupling method we consider a system consisting of a 3D EM model of the LV and for the 1D arterial model we analyse different geometries, either consisting of a single arterial segment or composed by more complex networks including the major vessels. To reproduce a realistic case, the LV computational domain is adapted from a patient-specific 3D-whole-heart-MRI scan collected and post-processed in a previous study, see parameters of the solid and fluid models under baseline conditions have been calibrated to match a patient-specific set of in vivo measurements at specific instants of the cardiac cycle for the same subject, see [35] for further information. The same pressure-volume data is used as for the initialisation of the 3D EM model. All parameters used for the baseline case are summarised in Table A.1.

Stability assessment of the coupling method
To analyse the stability of the proposed approach, we compared the results of simulations with a varying time resolution for the 1D blood flow scheme or the 3D LV model, respectively. Taking into account the intrinsic time-step limitation necessary to ensure stability of the 1D blood flow scheme, associated with a CFL condition [36], we did not consider a time step dt1D higher than 1e−3 s. In more detail, in the first test case we  As a second example, we performed two simulations considering a varying time resolution for the 3D heart model, dt3D ∈ {1e−3 s, 5e−4 s}, keeping the same temporal resolution dt1D = 1e−4 s for the 1D circulatory model. Figure 3 shows that, also in this case, solutions remain unchanged when considering a different time discretisation of the 3D heart model. Thus, we will consider dt3D = 1e−3 s in what follows. We refer to [1] for a more detailed convergence testing of the cardiac model alone.

Prediction of pulse wave propagation effects on cardiac function
To test the ability of the coupled model to predict the effect of pulse wave propagation on cardiac electromechanics we consider three test cases, ranging from an idealised aortic segment with a constant radius to a pathological condition called aortic coarctation and a network composed of 116 arteries.

Idealised aortic segment with constant radius
The first example illustrates the effect of vessel stiffening on haemodynamics and LV function. In particular, we considered a 3D LV solid model coupled to a vessel segment with constant radius and a length of 126 mm with lumped terminal boundary conditions.
For the baseline case we set the vessel wall stiffness described by the Young modulus to E = 0.25e6 Pa. For two test cases with increased stiffness we set the Young modulus E to 0.50e6 Pa and 0.75e6 Pa, respectively. The 1D blood flow model was initialised using an inflow boundary condition. For the simulations in this test case we fixed dt1D = 1e−4 s.
Simulation results of the coupled model are shown in Figure 4, where the pressure trace in the LV and at the inlet of the aortic segment is compared for the different stiffness values. It can be observed that an increase in E produced an increase in peak pressure and a substantial change in the shape of the pressure trace. This is due to the fact that the stiffening of the vessel wall affects pulse wave propagation in the vessel itself. In particular, the increase of aortic stiffness is associated with a premature rise and decay of the pressure wave and the well-known phenomenon of peak pressure augmentation [37].
The increased stiffness of the aortic vessel has a significant impact on LV function, see the PV loop in Figure 4 and

Stenotic aortic segment
As a second example, we considered a pathological configuration of aortic coarctation, that is a congenital defect consisting of a local narrowing of the aortic lumen (most commonly in the aortic arch). To this end, we considered an arterial segment of 126 mm endowed with a stenosis model [38], with a 30% stenosis halfway of its length. Terminal boundary conditions are given by an RCR Windkessel model. In order to explore the effect of stiffening of the vessel in this configuration, we considered a baseline case with E = 0.25e6 Pa and two test cases with E equal to 0.50e6 Pa and 0.75e6 Pa, respectively. For the simulations in this test case we set dt1D = 1e−5 s for stability considerations. Also in this example, an increase in aortic stiffness is reflected in an augmentation of peak pressure and a variation in the pressure profile, see Figure 6. The effect of vessel stiffness on LV function is shown in the PV loop of Figure 6 consists in a reduction in SV caused by an increase of the ESV.
For the sake of completeness we show the variation of pressure and flow signals along the distance of the 1D vessel for the case E = 0.25e6 Pa. In particular, we show the model predictions of the 3D-1D coupled model in Figure 8.

116 aortic segments
In order to explore the effect of bifurcations on pulse wave reflections and their impact on LV function the 3D EM model of the LV was coupled to a more complex model of the arterial system. To this end, as a third example we coupled the 3D model to a network consisting of 116 arterial vessels. Moreover, we explored two different physiological configurations for the 1D model, corresponding to a 25 year-old subject and a 65 year-old subject. We refer to [39] for further detail on the 1D networks considered. Concerning the 3D heart model, we considered the same parameter values as in the previous test cases, listed in Table A.1, with some exceptions. In particular, the heart cycle period was set to 800 ms in both configurations, and the parameters of the active stress law were adapted according to the new heart cycle period. We set dt1D = 1e−5 s in this test case as well. Also in this example, the increase in aortic stiffness and systemic vascular resistance associated with healthy ageing is reflected in an augmentation of the peak pressure and a variation in the pressure profile, see Figure 9. The effect of ageing on LV function is also striking, as shown in Figures 9 and 10. In particular, we can see a reduction in SV caused by an increase of the ESV. For this test case we can show the effect of pulse pressure amplification, i.e.
the amplification in pulse pressure signal towards more distal locations, associated with reflections at bifurcation sites. For the sake of illustration, in Figure 11 we consider the change in pressure trace from the aortic root to the right brachial artery. As expected, pulse pressure amplification was more pronounced in the younger subject than in the 65-yo subject.

Discussion
In this methodological study we have developed a numerical scheme that allows to couple an advanced 3D EM model of cardiac function to a 1D model of the human circulation.

Prediction of pulse wave propagation effects on cardiac function
The results in Section 5 show an augmentation of peak pressure in the LV and pulse pressure in the aorta and a substantial variation in the shape of the pressure trace as a consequence of stiffer vessel walls. Peak pressure in the LV increases with increasing E, since there is a higher afterload. In the aorta, the pulse pressure increases with increasing Young's modulus E, since the aortic model becomes less compliant. Physiologically, the pulse pressure trace is composed of two additive components: first, a forward propagating wave generated by LV contraction and second, a backward propagating wave from the periphery. From this, the behaviour observed in the simulations is to be expected as the increased aortic stiffness generates a faster pulse wave propagation. Hence, this causes a premature arrival of the forward propagating wave and, in turn, of the backward propagating wave, entailing peak pressure augmentation [37].
In addition, the stiffening of the arterial vessel is related to an increase in ESV. The main reason for this effect is that a lower compliance of the vessel, i.e., its ability to distend and increase volume with increasing transmural pressure, is responsible for a reduction of blood ejected by the heart chamber, thus producing a greater ESV. Since EDV is influenced by the cardiac passive behaviour and preload rather than by afterload, this volume does not vary when the aortic vessel properties change. Hence, the stiffening of the aortic vessel leads to a reduction in SV. We have shown that the proposed 3D-1D model is capable to reproduce the impact of increased stiffness and pulsatile load in the circulation and their effect on heart performance. These results confirm that such models are particularly suited to explore the effects of changes in pulse wave propagation, which can hardly be captured using lumped parameter models [20]. Also, effects of these interactions are very complex to study in vivo due to the need for high fidelity and simultaneous measurements of cardiac function and circulation at several locations, together with the capability to assess chamber interactions [40]. Hence, 3D-1D models are of high relevance to investigate interactions between vascular waves and cardiac chamber function. Overall, a coupling to 1D models should be preferred over a coupling to lumped parameter models for clinical applications that are profoundly influenced by disrupted blood flow propagation, e.g., aortic coarctation pathologies and pulmonary arterial hypertension.

Numerical aspects
Computationally, the 3D-1D approach is comparable to 3D-0D models, as the extra cost of solving the 1D model is essentially negligible compared to the cost of the 3D model.
As such, whenever wave transmission effects are to be investigated, the 3D-1D model is preferable as it can be used instead of a 3D-0D model as in [30] without any computational penalties. Further, we emphasise that the implementation of the coupling framework is not constrained to the specific choice of the 1D solver used for the solution of the blood flow equations in the circulatory system. In personalised applications the extra cost of identifying parameters of a distributed 1D system must be factored in though.

Limitations
An extension of the study is the use of more complex biventricular and four-chamber EM cardiac models [41], in order to consider the interactions among the heart chambers and thus provide more physiological simulations of the cardiac function. Also, it is well known that cardiac preload is mainly affected by venous return. As a consequence, in order to study more accurately the feedback of circulation (including venous return) on the cardiac function, the use of a 1D global model of the human circulation is more appropriate. Thus, another future perspective consists in coupling the 3D EM cardiac model with closed-loop global 1D blood flow models, e.g. [42], in order to explore the effect of circulation on preload as well. We emphasise that the coupling strategy holds for a general framework consisting of multiple cardiac cavities and is prone to different discretisations and geometries for the circulatory model, therefore such extensions are feasible. Third, this study did not specifically focus on the parameterisation of the coupled model to patient-specific data. However, the personalisation of cardiovascular models is of crucial importance for their predictive role, since the circulatory system has a complex network structure and shows a significant inter-individual variability. Owing to their distributed nature, 1D models of blood flow involve a larger amount of parameters that must be calibrated than lumped parameter models, related to geometric aspects, elastic properties and peripheral impedances of the network [10,43]. This poses serious limitations on the applicability of the use of detailed 1D circulatory models in clinical applications. Nonetheless, there are numerous contributions in the scientific literature to cope with this problem, in terms of simplification of the geometry and reduction of the parameter set, together with the application of optimisation methods and robust inverse problem strategies. For example, the topological complexity of the arterial tree can be optimised by effectively reducing the number of arterial segments included in the 1D model, still preserving the key characteristics of flow and pressure waveforms [44,45]. Common approaches also contemplate the rescaling of distal properties and vessel geometry based on allometric scales and global descriptors related to the geometry and properties of the network that are easy to measure, such as pulse wave velocity, body size, and biological age [46]. The level of detail of the network representing the circulatory system should be carefully considered taking into account the specific clinical application under study.
In addition, recent parameter estimation methods based on data assimilation techniques, i.e., algorithms combining mathematical models with available measurements to improve the accuracy of model predictions and estimate patient-specific parameters, have provided promising results in cardiovascular applications [47,48,49].

Conclusion
We developed a stable strategy to perform a coupling between a 3D EM model of the heart and a 1D blood flow model of the arterial system, based on the resolution of a saddle-point problem for the volume and pressure in the cavity. We built a personalised coupled 3D-1D model of LV and arterial system and showed robustness and accuracy of our scheme in a numerical benchmark. We demonstrated the ability of the coupled system to efficiently simulate physiological response to alterations in features of the circulatory system influencing pulse wave transmission, including aortic stiffening, aortic stenosis and bifurcations causing wave reflections. The additional computational costs associated with the use of 1D models instead of standard 0D models in the coupled system are negligible.
As a consequence, the use of our coupled 3D-1D model is beneficial for a broad spectrum of clinical applications where wave transmission effects are under study, such as aortic coarctation and pulmonary arterial hypertension.

Appendix A. Electromechanical PDE model
This section is devoted to the description of the mathematical models considered to describe the most fundamental aspects of cardiovascular function, comprising electrophysiology, passive and active mechanics and haemodynamics.

Appendix A.1. Electrophysiology
A recently developed R-E model [24] is considered for the generation of electrical activation sequences serving as a trigger for active stress generation in cardiac tissue. This hybrid R-E model combines a standard R-D model based on the monodomain equation with an eikonal model. In more detail, we consider the eikonal equation given as where (∇ X ) is the gradient with respect to the end-diastolic reference configuration Ω 0 , t a is a positive function that describes the wavefront arrival time at location X ∈ Ω 0 , and t 0 denote the initial activations at locations Γ * 0 ⊆ Γ 0,N . The symmetric positive definite 3 × 3 tensor V(X) contains the squared velocities (v f (X), v s (X), v n (X)) associated with the tissue's eigenaxes f 0 , s 0 , and n 0 . Then, the arrival time function t a (X) is used in a modified monodomain R-D model given as where an arrival time dependent foot current, I foot (t a ), is added, in order to mimic subthreshold electrotonic currents and produce a physiological foot of the action potential.
The key advantage of this R-E model is that it enables to compute activation sequences at much coarser spatial resolutions without being afflicted by the spatial undersampling artefacts leading to conduction slowing or even numerical conduction blocks that are observed in standard R-D models. The tenTusscher-Noble-Noble-Panfilov model of the human ventricular myocyte [50] is employed to model Ventricular EP.

Parameter Value Unit Description
Passive biomechanics

Appendix A.2. Passive mechanics of the heart
The deformation of the heart is governed by imposed external loads such as pressure in the cavities or from surrounding tissue and active stresses intrinsically generated during contraction. The cardiac tissue is characterised as a hyperelastic, nearly incompressible, anisotropic material with a nonlinear stress-strain relationship. Mechanical deformation is described by Cauchy's equation of motion leading to a boundary value problem We consider the following decomposition of the total stress S: where S pas and S act denote the passive and active stresses, respectively. Passive stresses are modelled according to the constitutive law with Ψ an invariant-based strain-energy function used to model the anisotropic behaviour of cardiac tissue. Numerous constitutive models have been developed in the literature, ranging from simpler transversely-isotropic models to more complex orthotropic laws. In this article, we consider a hyper-elastic and transversely isotropic strain-energy function Ψ proposed by Guccione et al. [21]. Here, the term in the exponent is

Appendix A.3. Active mechanics
Following the approach proposed by [22,54], we assume that stresses due to active contraction are orthotropic, with full contractile force along the myocyte fibre orientation f 0 and 40 % contractile force along the sheet orientation s 0 . In more detail, we define the active stress tensor S a as with S a the scalar active stress that describes the contractile force. Active stress generation is modelled using a simplified phenomenological contractile model [23]. Owing to its small number of parameters and its direct relation to clinically measurable quantities such as peak pressure and the maximum rate of rise of pressure, this model is easy to fit and thus suitable for being used in clinical EM modelling studies. In particular, the active stress transient is defined as and t s is the onset of contraction, φ(λ) is a nonlinear length-dependent function in which λ is the fibre stretch and λ 0 is the lower limit of fibre stretch below which no further active tension is generated, t a is the local activation time from Eq. (A.1), t emd is the EM delay between the onsets of electrical depolarisation and active stress generation, S peak is the peak isometric tension, t dur is the duration of active stress transient, τ c is time constant of contraction, τ c 0 is the baseline time constant of contraction, ld up is the length-dependence of τ c , τ r is the time constant of relaxation, and ld is the degree of length dependence. Thus, active stresses are only length-dependent in this simplified model, but dependence on fibre velocityλ is neglected.

Appendix B. Circulatory system: 1D PDE model
In what follows, the main mathematical aspects of 1D blood flow modelling in the human circulation are detailed.
Appendix B.1. Some features of 1D blood flow modelling The 1D equations of blood flow can be derived by integrating the Navier-Stokes equations of a generic section of the tube (after assuming axisymmetric flow in a cylindrical tube of radius R), obtaining where A(x, t) is the cross-sectional area of the lumen, Q(x, t) is the average flux and P (x, t) is the average internal pressure over the cross-section. Note that in the momentum balance a momentum flux correction factor α (also known as Coriolis coefficient) is introduced, defined as An energy inequality that bounds a measure of the energy of the hyperbolic system was derived in [55]. Alternatively, it is possible to rewrite the system in terms of the variables (A, u). In more detail, if we assume a flat velocity profile, i.e. α = 1, we obtain the system where A(x, t) is the cross-sectional area of the lumen, u(x, t) is the average axial velocity and P (x, t) is the average internal pressure over the cross-section. For a constant velocity profile satisfying the no-slip condition, the friction force per unit length is f ( where η is a coefficient depending on the blood viscosity and the average axial velocity. As System (B.2) is composed by two equations and three unknowns (A, u, P ), a closure condition is needed, namely the so-called tube law : where K and G are parameters depending on the wall stiffness and the wall viscosity, respectively.
A simple, purely elastic tube law consists in defining with E the Young modulus and h the wall thickness of the vessel. A more complex tube law is obtained when visco-elastic effects are considered, e.g., by setting with ϕ(x) the wall viscosity. In what follows we consider a visco-elastic tube law, see relative size of red blood cells to vessel diameter increases, therefore the assumption made in 1D modelling that blood is a continuum and a Newtonian fluid fails. Moreover, contrary to large arteries, fluid resistance dominates over wall compliance and fluid inertia in smaller vessels. For these reasons, linear lumped parameter models are commonly employed to simulate the effect of peripheral vessels on pulse wave propagation in larger arteries [20,56]. and two resistors [57]. In more detail, the first resistance Z corresponds to the characteristic aortic impedance. This is connected in series with a parallel combination of the peripheral arterial resistance R and compliance C. We define P out as the pressure at which flow in the microcirculation is equal to zero. The resulting model is governed by we used the solver Nektar1D [20], which is based on a FEM discretisation (discontinuous Galerkin) in space and finite difference (explicit second-order Adams-Bashforth) in time of Equations (B.2) and (B.3).

Appendix C. Valve dynamics models
The EM model of the heart is coupled to the arterial network via the cardiac valves.
Flow through these valves is basically triggered by a positive difference pressure between the two compartments. However, several models can be considered for modelling the dynamics of these valves, ranging from simple diode-like models to more sophisticated approaches, that are able to contemplate pathological conditions such as a stenosis or regurgitation. In the literature there are three basic approaches, ranging from simple diode-like models to more complex valve models [46], simulating a simplified valve dynamics without explicitly modelling the valve leaflets. However, when it is necessary to capture more complex flow dynamics inside the ventricles that are induced by valve pathologies, it may be required to explicitly model the shape and dynamics of the leaflets. Several approaches have been proposed [59], As a consequence, for this work we considered a more accurate model, namely a lumped parameter model of valve dynamics proposed in [29]. Its peculiarity consists in a smooth opening and closing dynamics of the valves using a simple approach. In particular, this model is derived from the Bernoulli equation, after neglecting Poiseuille-type viscous losses (proportional to Q), since they are small: The Bernoulli resistance B is associated with the pressure difference ∆P related to convective acceleration and dynamic pressure losses caused by the diverging flow field downstream to the vena contracta, and it reads where ρ is the blood density (usually set equal to 1060 kg m −3 ) and A eff is an effective cross-sectional area. The blood inertance L is related to the pressure difference associated with blood acceleration and it is governed by the following equation: with l eff an effective length, and it is here taken equal to the diameter of A eff . In order to consider valve dynamics, A eff is a state variable and depends on an index of valve state The rate of valve opening and closing only depends on two factors, i.e., the instantaneous pressure difference across the valve ∆P and the current state ξ of the valve. In particular, it is assumed that the valve opens when ∆P exceeds a threshold ∆P open , and that the rate of opening is governed by with K vo a rate coefficient for valve opening (in Pa −1 s −1 ). On the other hand, it is assumed that the valve closes when ∆P is lower than a threshold ∆P close , and that the rate of closing is governed by where K vc is a rate coefficient for valve closing (in Pa −1 s −1 ). Model parameters are depicted in • When one valve is open or regurging, there is a change in cavitary volume. In this configuration the pressure p n+1 is regulated by the state of the circulatory system or of a connected cavity. Therefore, p n+1 needs to be estimated by matching mechanical deformation and state of the system. In fact, pressure p n+1 in the cardiovascular system depends on flow, which depends in turn by cardiac deformation. As a consequence, the heart and circulatory models are tightly bidirectionally coupled.
From a mathematical perspective, this coupling can be performed in two ways. The simplest approach is to prescribe p n+1 explicitly. Then, the coupling is performed during the ejection phase by updating flow and flow rate based on the current prediction on the change in the state of deformation associated with the currently predicted pressure p n+1 . In particular, the pressure boundary condition in each nonlinear solver step is modified within each iteration ν. Note that the new prediction p n+1 ν+1 is then explicitly prescribed as a Neumann boundary condition. Therefore, this explicit and partitioned approach may introduce some inaccuracies during ejection phases and can lead to instabilities during isovolumetric phases.
Such instabilities arise from the difficulty of estimating the change in pressure necessary to keep the volume constant. In fact, a knowledge on cavitary elastance is required to know the p − V relation of the cavity at this given point in time, that is not available. Therefore, iterative estimates are needed to gradually inflate or deflate a cavity to its prescribed volume. However, since the elastance properties of the cavities are highly nonlinear, an overestimation may induce oscillations, whereas an underestimation may induce very slow convergence and prohibitively large numbers of Newton iterations. A more sophisticated approach consists in treating p n+1 as an unknown. Thus, an additional equation is required to close the system, leading to a saddle point formulation and a block system to be solved (monolithic approach).
Isovolumetric Phases. During the isovolumetric phases we can state that i.e., the volume of each cavity V heart c (u) equals an initial volume V 0 and does not change during the isovolumetric phase. Hence, in system (4) the matrix C (p k ) = 0 and V CS (p k ) = V 0 .

Appendix E. Finite Element Formulation
Appendix E. 1

. Variational Formulation
For the sake of simplicity of formulation, we neglect here the acceleration term in Eq. The second term of (E.2) is computed using the 1D blood flow model, see Appendix B, for c ∈ {LV, RV, LA, RA}.

Appendix E.2. Consistent Linearisation
A Newton-Raphson scheme is employed to solve the nonlinear variational equations (E.1)-(E.2), with a finite element approach see [61]. Let us consider a nonlinear and continuously differentiable operator F : X → Y . Then, a solution to F (x) = 0 can be approximated by which is evaluated until convergence. In our approach, we assume X = H 1 (Ω 0 , Γ 0,D ) 3 ×R, Y = R 2 , ∆x = (∆u, ∆p c ) , x k = (u k , p k c ) , and F = (R u , R p ) . Thus, we obtain the following linearised saddle-point problem for each (u k , p k c ) ∈ H 1 (Ω 0 , Γ 0,D ) 3 × R, find with the updates u k+1 = u k + ∆u, (E.8) The Gâteaux derivative of (E.1) with respect to the displacement change update ∆u yields the first term in (E.6) Σ(u k , ∆u) : C k : Σ(u k , v) dX, (E.10) and the second term in (E.6), that is given by where we have defined, for the sake of simplicity: F k := F(u k ), J k := det(F k ), S k := S| u=u k , C k := C| u=u k .
The third term in (E.6) is defined using the Gâteaux derivative of (E.1) with respect to the pressure change update ∆p c : The right hand side of (E.6) depends on the residual R u , and it is defined as with the solution vectors u k ∈ R 3N and p k c ∈ R Ncav at the k-th Newton step. The tangent stiffness matrix A ∈ R 3N ×3N is computed from (E.10) as follows: (E.20) The mass matrix M ∈ R 3N ×3N is calculated from (E.11) and reads M (u k , p k c )[j, i] := ϕ i , F 0 (u k , p k c ) ϕ j Ω 0 . (E.21) The off-diagonal matrices B u ∈ R 3N ×Ncav and B p ∈ R Ncav×3N in (E.17) are assembled using (E.14) B u (u k , p k c )[i, j] = ϕ j , V heart c (u k )φ i Ω 0 , i = 1, . . . , N cav (E. 22) and (E.12) where the constant shape functions have valueφ j = 1 if τ j ∈ Γ 0,c andφ j = 0 if τ j / ∈ Γ 0,c for c ∈ {LV, RV, LA, RA}.
Considering the approach described in [63,Sect. 4.2], this assembling procedure can be simplified for closed cavities such that B p (u k , p k c ) = B u (u k , p k c ) .
Then, the circulatory compliance matrix C (p k A = a 1 · · · a m ∈ R n×m , B = b 1 · · · b m ∈ R m×n . We can write the Schur complement system as The realisation of (F.2) involves m + 1 solves and the inversion of an m × m matrix. As m is generally small, this can be done symbolically.