心血管系统3D-1D模型的耦合策略，用于研究脉搏波传播对心脏功能的影响

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心血管系统3D-1D模型的耦合策略，用于研究脉搏波传播对心脏功能的影响_A coupling strategy for a 3D-1D model of the cardiovascular system to study the effects of pulse wave propagation on cardiac function.pdf 在心血管系统的模拟研究中，3D（三维）和1D（一维）模型的耦合策略扮演着关键角色。这种耦合方法旨在更好地理解和分析脉搏波传播如何影响心脏功能。脉搏波的传播是血液循环中的重要现象，它与心血管健康紧密相关，尤其在疾病如高血压、动脉硬化等情况下，其影响更为显著。文章提出了一种新的、稳定的方法来将心脏的三维电机械（EM）模型与血管系统的三维模型耦合，以形成一个个性化的3D-1D模型。心脏的三维模型能够详尽地模拟心肌的结构和功能，而一维模型则可以高效地表示大血管的血流动力学。这种结合考虑了整个循环系统对心脏性能的影响，从而提高了模型的临床相关性。在实现上，耦合策略通过在3D心脏模型和1D动脉模型之间建立动态交互，确保了计算效率和精度之间的平衡。为了验证该耦合模型的有效性，研究者进行了数值基准测试，使用不同的时间步长来检验模型的稳定性。此外，他们还通过模拟各种生理条件下的变化，如动脉硬化、主动脉瓣狭窄或血管分支导致的波反射，来验证模型的准确性。当主动脉硬度增加时，脉搏波的传播速度会加快，可能导致心脏负荷增加，而耦合模型能够捕捉到这些变化对心脏功能的影响。主动脉瓣狭窄会导致血液流动受阻，模型也能够反映出这种病理状况下心脏的工作状态。血管分支引起的波反射则可以影响回心血流，模型能够模拟这些反射如何改变心脏的充盈和收缩。这种3D-1D耦合策略为理解心血管疾病和心脏功能之间的复杂关系提供了一个强大的工具。通过个性化建模，医生和研究人员能够预测不同病理条件下的心脏反应，为诊断和治疗决策提供依据。未来的研究可能进一步扩展这种耦合模型，纳入更多血管细节，甚至包括微血管网络，以提高模型的精确性和实用性。

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A coupling strategy for a 3D-1D model of the cardiovascular system to

study the eﬀects of pulse wave propagation on cardiac function

Federica Caforio

a,b,∗

, Christoph M. Augustin

, Jordi Alastruey

c,d

, Matthias A. F. Gsell

Gernot Plank

Institute of Mathematics and Scientiﬁc Computing, NAWI Graz, University of Graz, Graz, Austria

Gottfried Schatz Research Center: Division of Biophysics, Medical University of Graz, Graz, Austria

Department of Biomedical Engineering, Division of Imaging Sciences and Biomedical Engineering,

King’s College London, King’s Health Partners, St. Thomas’ Hospital, London SE1 7EH, UK

World-Class Research Center “Digital Biodesign and Personalized Healthcare”, Sechenov University,

Moscow, Russia

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. Valida-

tion 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.

∗

Corresponding author. Email: federica.caforio@uni-graz.at.

arXiv:2111.05683v1 [math.NA] 10 Nov 2021

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.

Keywords: 3D-1D coupling, multiphysics modelling, cardiovascular modelling, cardiac

electromechanics, pulse wave propagation.

1. Introduction

Computational modelling of cardiac EM is increasingly proposed and pursued as an ef-

fective 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, nu-

merous 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 zero-

dimensional (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 veloc-

ity. 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 imple-

mentation 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 physiologi-

cal 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.

2. Methodology

2.1. 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 length-

dependent, 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.

2.2. 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 bidirec-

tional, 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 at-

tached 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 in-

stead 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.

2.2.1. 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 discretisa-

tion 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

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