Multimodal cardiovascular model for hemodynamic analysis: Simulation study on mitral valve disorders

Dibyendu Roy, Conceptualization , Formal analysis , Software , Writing – original draft , Writing – review & editing , * Oishee Mazumder, Conceptualization , Formal analysis , Software , Writing – original draft , Writing – review & editing , Aniruddha Sinha, Conceptualization , Project administration , Supervision , Writing – review & editing , and Sundeep Khandelwal, Conceptualization , Validation , Writing – review & editing

Dibyendu Roy

TCS Research, Tata Consultancy Services Limited, Kolkata, India

Oishee Mazumder

TCS Research, Tata Consultancy Services Limited, Kolkata, India

Aniruddha Sinha

TCS Research, Tata Consultancy Services Limited, Kolkata, India

Sundeep Khandelwal

TCS Research, Tata Consultancy Services Limited, Kolkata, India

Competing Interests: Tata Consultancy Services Ltd. (TCS) (https://www.tcs.com/) provided support in the form of salaries for all the authors and provided infrastructural support for research and development, data analysis, decision to publish and preparation of the manuscript. This does not alter our adherence to PLOS ONE policies on sharing data and materials. There are no patents, products in development or marketed products associated with this research to declare.

This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Associated Data

Abstract

Valvular heart diseases are a prevalent cause of cardiovascular morbidity and mortality worldwide, affecting a wide spectrum of the population. In-silico modeling of the cardiovascular system has recently gained recognition as a useful tool in cardiovascular research and clinical applications. Here, we present an in-silico cardiac computational model to analyze the effect and severity of valvular disease on general hemodynamic parameters. We propose a multimodal and multiscale cardiovascular model to simulate and understand the progression of valvular disease associated with the mitral valve. The developed model integrates cardiac electrophysiology with hemodynamic modeling, thus giving a broader and holistic understanding of the effect of disease progression on various parameters like ejection fraction, cardiac output, blood pressure, etc., to assess the severity of mitral valve disorders, naming Mitral Stenosis and Mitral Regurgitation. The model mimics an adult cardiovascular system, comprising a four-chambered heart with systemic, pulmonic circulation. The simulation of the model output comprises regulated pressure, volume, and flow for each heart chamber, valve dynamics, and Photoplethysmogram signal for normal physiological as well as pathological conditions due to mitral valve disorders. The generated physiological parameters are in agreement with published data. Additionally, we have related the simulated left atrium and ventricle dimensions, with the enlargement and hypertrophy in the cardiac chambers of patients with mitral valve disorders, using their Electrocardiogram available in Physionet PTBI dataset. The model also helps to create ‘what if’ scenarios and relevant analysis to study the effect in different hemodynamic parameters for stress or exercise like conditions.

Introduction

Cardiovascular diseases (CVD) account for a massive rate of mortality all over the world. Recent statistics from World Health Organization (WHO) reported nearly 17.9 million death due to cardiovascular disease along with the economic burden of billions of dollars spent every year on screening, diagnosis, and related health-care-associated to CVD [1]. Out of various CVDs, valvular heart disease (VHD) on the left ventricle is a prominent cause of cardiovascular morbidity and mortality worldwide [2]. The aging population is mostly affected by degenerative valve disease, whereas in developing countries, rheumatic valve disease remains a public health problem affecting young adults [3].

Out of the valvular disease associated with the left ventricle, dysfunction of the mitral valve remains a leading medical problem worldwide [4]. This valve plays a fundamental role in the structural and functional integrity of the left ventricle, along with maintaining forward cardiac output. Dysfunction may arise due to structural defects in any of the valve structures, causing ‘stenosis’ or ‘regurgitation’ resulting in symptoms like ventricular hypertrophy, atrial enlargement, reduced cardiac output, pulmonary venous congestion, and atrial arrhythmia [5]. Computational modeling may provide an intuitive platform to understand the biomechanics of the human mitral valve along with changes in hemodynamic parameters with progression of the disease that could eventually lead to the development of new treatment, prevention, and diagnosis of mitral valve disease.

Recently, cardiovascular research has shown significant involvement in simulating cardiac behavior through in-silico models [6]. Powered by improvement in computing technologies, cardiac modeling based on physical principles are being used to simulate the hemodynamic properties of the cardiovascular system [7]. These models are imparting an increasingly important role in the diagnosis of cardiovascular diseases along with the development of medical devices [8]. Although substantial research exists on modeling mechanisms related to hemodynamics, electrophysiology, computational fluid dynamics, biomechanics, etc., researchers are now focusing on the multiscale mathematical framework to simulate the cardiac function at the whole-organ scale [9]. In the hemodynamics domain, many analytical representations of the cardiovascular system have been proposed since the first system-level dynamic cardiovascular model [10]. Depending on the purpose of the underlying scientific questions, hemodynamic analysis vary from simple lumped models, 0-1D multiscale cardiovascular model to complex 3D image-based models [11]. Some of the recent models are the fluid-structure interaction in specific vascular beds [12], the distributed impedance of the arterial and pulmonary trees [13], and lumped models of the integrated cardiovascular system [14]. A particular area in hemodynamics that has received substantial attention is one-dimensional reduced-order models. These models are commonly used to simulate blood flow regimes for which pulse waves propagate in large compliant arteries [15]. These models are computationally efficient and mostly linked to study the physiological hemodynamic phenomena under various pathological conditions [16, 17]. In the integrated multimodal modeling domain, there are some hemodynamic models based on cellular properties such as ion equations, myofilament structure, and fiber orientation [18]. However, computational load inhibits these models from use in clinical and educational applications of central hemodynamics [19]. Computational models related to the mitral valve mostly concentrate on the structural variation of the valve with fluid-structure interaction [20]. Such models might aid in surgical intervention but lack the holistic integration of pressure and flow dynamics in the heart chamber, caused due to structural defects of the valve [21]. Models focusing on mitral valve dynamics and heart remodeling [22] offer good examples of how cardiovascular simulation models can be validated in specific situations and used clinically.

In this paper, we present a closed-loop, real-time, lumped parameter cardiovascular simulation model, which contains the dynamics of four cardiac chambers, heart valves, and lumped pulmonic and systemic circulation. It is a reduced order model, where the pulsatile behavior of the heart chambers is triggered and modulated through cardiac electrophysiology. The forward electrophysiology pipeline (EP) has been implemented to generate a single lead electrocardiograph (ECG) signal from a cardiac source model. Hemodynamics is governed through time-varying compliance as derived from the EP model. A schematic workflow of the developed prototype is shown in Fig 1 . We hypothesize that the working of the cardiac hemodynamics can be approximated by a reduced-order lumped model using pressure-flow variation at different instances of the cardiac cycle. We aim to replicate the normal cardiac hemodynamics physiology and then incorporate pathological conditions pertaining to mitral valve disorders and correlate the parametric variation of hemodynamic indices simulated by our model against values reported in the medical literature.

Cardiac dynamics incorporate both active and passive filling phases resulting in more realistic changes of pressures with ventricular dilatation. Valve dynamics and precise adjustment of valvular properties allow realistic simulation of both stenotic and insufficient valves. The hemodynamics is coupled with central nervous system through sympathetic and parasympathetic control via baroreflex auto-regulation mechanism. The integrated EP-hemodynamics approach takes into account the cellular to organ level manifestation of cardiac dynamics, thus making the model multimodal and multiscale. Such prototype model can be used to simulate the cardiac dynamics for healthy physical condition as well as mitral valve disorders, namely ‘stenosis’ and ‘regurgitation’ with varying severity. The model is used to simulate the pressure-volume (PV) dynamics of the left ventricle and atrium. This in turn provides the information on the stroke volume, variations in aortic and pulmonary pressure and indications of hypertrophy/enlargement in cardiac chambers. Additionally, as part of modeling the systemic circulation, variations in the peripheral blood volume pulse are used to simulate the Photoplethysmogram (PPG) signals [23]. Such PPG signal, in turn, could be used to correlate with observed PPG signals of an individual, and hence be used for early screening applications for CVD [24].

We demonstrate that the proposed model generates cardiac parameters, for both normal and pathophysiological cases, which are consistent with earlier published clinical and experimental data. Additionally, the model has been tested using a limited set of measured ECGs of patients with mitral valve disorders. In such a scenario, the measured ECG serves as an input to the model to generate hemodynamic parameters. A subset of these parameters namely, the dimensions of the left cardiac chambers are validated with the left ventricular hypertrophy and left atrial enlargement score, obtained from such clinical data.

The rest of the paper is organized as follows. Section 2 comprises the methodology, which includes the layout of the cardiovascular hemodynamic model containing the four-chambered heart, followed by the EP model to generate ECG signal from a cardiac source model and subsequently deriving the chamber and valve dynamics, and PPG synthesis for a healthy adult. In section 3, we simulate the mitral valve stenosis and regurgitation of varying severity followed by section 4, where we present the simulation results for healthy, mitral stenosis and regurgitation condition and a hypertrophy or enlargement analysis based on real ECG data. In section 5, we discuss the capability of ‘what if’ simulation through a simulated stress condition and report its effect on mild mitral valve disorders followed by a brief discussion on the capability and limitations of our model. Section 6 concludes the paper.

Methodology

The developed cardiovascular model integrates two separate modalities, namely the hemodynamic model and a reduced electrophysiological model, sufficient to derive the compliance function for driving the hemodynamic model. In this section, the layouts of the cardiovascular hemodynamic model of the four-chambered heart along with the EP model, and PPG synthesis have been discussed.

Mathematical model of cardiovascular hemodynamic system

The heart is a muscular organ in which each half is composed of a pair of atrium and ventricle acting like a pulsatile pump. The left heart chamber, comprising the left ventricle (lv) and left atrium (la), pumps oxygenated blood to all the tissues of the body. This specific circulation is called systemic circulation [25]. On the other hand, the right heart, comprising right ventricle (rv) and right atrium (ra), drives deoxygenated blood to the lungs forming the pulmonic circulation. In addition, there are four cardiac valves namely, mitral (mi), aortic (ao) valves in the left heart and tricuspid (tr), pulmonic (pu) valves in the right heart respectively. These valves synchronously open and close based on the pressure difference within the heart chambers and ensures rhythmic unidirectional flow through the heart. The block-diagrammatic representation is shown in Fig 2 .

P_la, V_la, P_lv, V_lv, P_ra, V_ra, P_rv, V_rv are the pressure and volume in the left atrium and ventricle, right atrium and ventricle respectively. P_sa and P_pa are the pressures in the systemic and pulmonary artery respectively. C_ra, C_la, C_lv, and C_rv are the compliances across right atrium, left atrium, left ventricle, and right ventricle respectively, with associated delays of d_la and d. The resistances across pulmonic and systemic vessels are R_p, R_s respectively. R_mi, R_ao, R_tr, and R_pu are the valvular resistances for the mitral (MI), aortic (AO), tricuspid (TR) and pulmonary (PU) valves respectively.

To describe the hemodynamics of the cardiac system, we have considered the following assumptions [25]:

Assumption 1: Each of the heart chambers is triggered by an autonomous compliance function (C(t)), due to the elasticity of the cardiac walls. So, the volume (V(t)), at the time t, across any cardiac chamber can be defined as V(t) = C(t) × P(t) + V_s; where P(t) is the pressure at the t th time and V_s is the fixed unstressed volume of that particular chamber.

Assumption 2: The cardiac chambers are considered as compliance vessels. Hence, the rate of change of volume across a cardiac chamber at time t, can be defined as the difference between the inflow Q₁(t) and the outflow Q₂(t), so, d V ( t ) d t = Q 1 ( t ) - Q 2 ( t ) .

Assumption 3: Each vessel is considered as resistive vessel, as the blood flow is impeded due to the frictional forces, depending on the viscosity of the blood, diameter of the vessels etc. Thus, flow across a resistive vessel will be Q(t) = ΔP(t)/R; where ΔP(t) is the pressure difference in the successive compartments of the vessel, at the time t and R is the vascular resistance of that vessel.

Based on these assumptions, the pressure dynamics, replicating the left-heart hemodynamics can analytically be defined as given by Eqs (1)–(3) [26]).

P ˙ l v = 1 C l v ( t ) [ U m i × P l a - P l v R m i - U a o × P l v - P s a R a o - C ˙ l v ( t ) P l v ]

Similarly, the pressure dynamics describing the right-heart functionality can be interpreted by Eqs (4)–(6).

P ˙ r v = 1 C r v ( t ) [ U t r × P r a - P r v R t r - U p u × P r v - P p a R p u - C ˙ r v ( t ) P r v ]

Here P_la, P_lv, P_sa, P_ra, P_rv and P_pa are the pressure variables in the la, lv, systemic arteries (sa), ra, rv, and pulmonary arteries (pa) respectively, having the initial conditions of p l a 0 , p l v 0 , p s a 0 , p r a 0 , p r v 0 and p p a 0 . The valvular resistance across the mitral, aortic, tricuspid, and pulmonic valves are R_mi, R_ao, R_tr, and R_pu respectively. The vascular resistance and compliance pair, across the pulmonic and systemic vessels are R_p, C_pa and R_s, C_sa respectively. U_mi, U_ao, U_tr, and U_pu are the control inputs for opening and closing of the heart valves. The functionalities of these valves are defined by Eqs (7) and (8), where 1 represents the complete opening of the cardiac valves and δ_i;∀i ∈ mi, ao, tr, pu> represents the closing of the same. In healthy cardiac condition, δ_i = 0. In such a scenario, whenever the left-atrium pressure (P_la) is greater than the left-ventricle pressure (P_lv), then the mitral valve opens and U_mi is considered as 1. Similar is the case for other valves.

As per the Assumption 1, the heart chambers are activated sequentially, in a synchronized manner, by time-varying compliance functions. Typically, this activation starts from sinoatrial node [27], which is located inside ra, then, it traverses to the la with a time delay of d_la, causing them to contract for pumping the blood into the ventricles. After that, the activation traverses from the atrium to the ventricles via atrioventricular node [27] with a time delay of d ( Fig 3a ), allowing the ventricles to fill with blood. To replicate these phenomena, we map the output of the EP source model, described later, to the compliance of the cardiac chambers, using a non-linear function. Among many options for such a mapping function [28], we have considered a cosine function which is capable of modeling the activation of cardiac chambers in both diseased and healthy heart [29]. Using such a mapping function, we have defined the compliance functions as C_ra(t), C_la(t), C_lv(t) and C_rv(t) for actuating the ra, la, lv, and rv respectively. The compliance function across ra is given by Eqs (9) and (10), where C_min,ra, C_max,ra are the minimum and maximum values of the ra compliance and u(t) is the activation function. The time t is considered over a complete cardiac cycle. T_a is the start of the activation of ra and T is the end of the cardiac cycle. The same is repeated for every cardiac cycle, where the temporal characteristics of the electrical activation are utilized from the EP model.