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_Acouplingstrategyfora3D

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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 心血管系统是人体内极为复杂而重要的循环网络，它不仅负责氧气和养分的输送，还是疾病发生发展的关键场所。随着计算科学的发展，利用模型来研究心血管系统的功能以及疾病影响成为可能。心血管系统的3D-1D模型耦合策略正是在这样的背景下提出的，它结合了心脏与血管系统的特性，通过高精度与高效率的模拟来分析脉搏波传播对心脏功能的影响。我们讨论心脏模型的构建。三维电机械（EM）模型能够详尽地描绘心肌的结构以及其与电生理特性之间的关系。这一模型通过模拟心肌纤维的方向、心室壁的厚度以及心脏瓣膜的运动等，能够高度复现心脏在电生理活动下产生的机械动作，如收缩和舒张。而血管系统三维模型则侧重于血流动力学，它能够模拟血液在血管内的流动情况，包括流速、压力和血管壁对血流的反应等。这两种模型在模拟心脏与血管功能时都有其独特的优势，但它们无法独立解释整个心血管系统的动态交互作用。因此，为了更好地理解脉搏波传播对心脏功能的影响，文章提出了一种将3D心脏模型和1D血管模型耦合起来的方法，从而形成一个完整的3D-1D模型。这个模型不仅能够考虑心脏本身的功能，还能够考虑到整个循环系统对心脏性能的影响，包括血管对脉搏波传播速度与心脏负荷的影响。这种耦合策略允许研究者从更加全面的视角来评估心血管系统的功能状态和病理变化。具体来说，该耦合模型通过在3D心脏模型和1D动脉模型之间建立动态交互，能够在保证计算效率的同时维持模型的精确度。研究者通过一系列数值基准测试，验证了模型的稳定性和准确性。其中，测试包括在不同的时间步长下模型的反应，以及模拟不同的生理条件下心脏功能的变化，如动脉硬化、主动脉瓣狭窄和血管分支引起的波反射等。动脉硬化会导致主动脉硬度的增加，进而加快脉搏波的传播速度，这将增加心脏的负荷。耦合模型能够捕捉到这种变化，并分析其对心脏功能的具体影响。主动脉瓣狭窄将导致血液流动受阻，模型模拟出这种状况下心脏的工作状态，为诊断和治疗主动脉瓣狭窄提供了有力的理论支持。同时，血管分支引起的波反射会影响回心血流，耦合模型能够演示这些反射如何改变心脏的充盈和收缩，从而影响心脏的整体功能。这种3D-1D耦合模型的研究，为心血管疾病的诊断与治疗提供了新的视角。通过个性化的建模，医生和研究人员可以预测不同病理条件下心脏的反应，从而为精准医疗提供依据。未来，随着研究的深入，该模型有望纳入更多血管细节，甚至包括微血管网络，进一步提高其精确性和实用性。此外，通过与临床数据的结合，这些模型有潜力成为评估药物疗效、预测疾病进程以及个性化治疗规划的有力工具。心血管系统3D-1D模型耦合策略的提出，标志着在心血管疾病研究与治疗方面迈出了重要的一步。它不仅为心血管系统的研究提供了先进的工具，也为未来心血管医学的发展指明了方向。通过不断的完善和拓展，耦合模型在心血管疾病的早期诊断、疾病风险评估以及个性化治疗方案的制定等方面，将发挥越来越重要的作用。

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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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