Computational and Mathematical Methods in Cardiovascular Physiology

Liang Zhong, Ru San Tan;Eddie Yin Kwee Ng;Dhanjoo N Ghista

• Publisher: WSPC
• Year: 2019
• ISBN:
  9789813270657
• Topic: Medicine
• Subtopic: Cardiology

Cardiovascular diseases (CVD) including heart diseases, peripheral vascular disease and heart failure, account for one-third of deaths throughout the world. CVD risk factors include systolic blood pressure, total cholesterol, high-density lipoprotein cholesterol, and diabetic status. Clinical trials have demonstrated that when modifiable risk factors are treated and corrected, the chances of CVD occurring can be reduced. This illustrates the importance of this book's elaborate coverage of cardiovascular physiology by the application of mathematical and computational methods.

This book has literally transformed Cardiovascular Physiology into a STEM discipline, involving (i) quantitative formulations of heart anatomy and physiology, (ii) technologies for imaging the heart and blood vessels, (iii) coronary stenosis hemodynamics measure by means of fractional flow reserve and intervention by grafting and stenting, (iv) fluid mechanics and computational analysis of blood flow in the heart, aorta and coronary arteries, and (v) design of heart valves, percutaneous valve stents, and ventricular assist devices.

So how is this mathematically and computationally configured landscape going to impact cardiology and even cardiac surgery? We are now entering a new era of mathematical formulations of anatomy and physiology, leading to technological formulations of medical and surgical procedures towards more precise medicine and surgery. This will entail reformatting of (i) the medical MD curriculum and courses, so as to educate and train a new generation of physicians who are conversant with medical technologies for applying into clinical care, as well as (ii) structuring of MD-PhD (Computational Medicine and Surgery) Program, to train competent medical and surgical specialists in precision medical care and patient-specific surgical care.

This book provides a gateway for this new emerging scenario of (i) science and engineering based medical educational curriculum, and (ii) technologically oriented medical and surgical procedures. As such, this book can be usefully employed as a textbook for courses in (i) cardiovascular physiology in both the schools of engineering and medicine of universities, as well as (ii) cardiovascular engineering in biomedical engineering departments worldwide.

Contents:

• Acknowledgements and Dedications
• Preface
• Heart Anatomy, Physiology and Imaging:
  • Anatomy and Physiology of the Heart (Liang Zhong, Ru San Tan and Dhanjoo N Ghista)
  • Computed Tomography: Applications in Imaging of Cardiac Structures (Shoen Choon Seng Low)
  • Coronary Artery Heart Disease: Phyiology, Stenosis Assessment, and Percutaneous Interventions (Jiang Ming Fam)
• Computational and Mathematical Methods in Cardiovascular Physiology:
  • Fluid Mechanics and the Cardiovascular System (Dhanjoo N Ghista, Foad Kabinejadian, and Joseph L Bull)
  • Cardiac Image Segmentation and Shape Modeling (Min Wan, Liang Zhong and Ru San Tan)
  • 2D FSI Simulation of Flow in Patient-specific Left Ventricle (Boyang Su, Siamak N Doost and Liang Zhong)
  • 3D Simulation of Flow in Patient-specific Left Ventricle (Boyang Su and Liang Zhong)
• Computational and Mathematical Methods in Vascular Physiology:
  • Coronary Artery Stenosis: Flow, Assessment, and Interventions (Foad Kabinejadian, Dhanjoo N Ghista, Mehul R Bhaljia, Owen N Mogabgab and Joseph L Bull)
  • Artery Buckling and Atherosclerotic Plaque Rupture under High Lumen Pressure (Seyed Saeid Khalafvand and Ali C Akyildiz)
  • Hemodynamics Simulation in the Left Anterior Descending Coronary Artery Tree (Boyang Su, Foad Kabinejadian, Yunlong Huo, Ghassan Kassab, Hwa Liang Leo and Liang Zhong)
  • Noninvasive Hemodynamic Assessment of the Significance of Coronary Artery Disease (Jun-Mei Zhang, Ru San Tan, Ris Low, Leok Poh Chua, Swee Yaw Tan, Aaron Sung Lung Wong, Terrance Siang Jin Chua, Tian Hai Koh, Soo Teik Lim and Liang Zhong)
• Computational and Mathematical Methods in Cardiovascular Devices:
  • Cardiac Devices and CFD: Current State and Challenges (Siamak N Doost, Liang Zhong and Yosry S Morsi)
  • Ventricular Assist Device: Hemodynamic Simulation and Design Analysis (Boyang Su and Leok Poh Chua)
  • The Percutaneous Mitral Valve Stents: Finite Element Based Design, Crimpability Features and Fatigue Performance (Fangsen Cui and Gideon Praveen Kumar)
  • Bileaflet Mechanical Heart Valves: In Vitro Study Based on Hemodynamic 3D Simulation (Yee Han Kuan, Vinh-Tan Nguyen and Hwa Liang Leo)
• Index

Readership: The book will attract a wide range of readers: academics, biomedical engineers, cardiologists, students, researchers, employees of funding agencies such as the National Institute for Health (NIH). Cardiovascular System;Fluid Mechanics;Physiology;Hemodynamics0 Key Features:

• This book provides a gateway for the new emerging scenario of (i) science and engineering based medical educational curriculum, and (ii) technologically framed medical and surgical procedures
• As such, this book can be usefully employed as a textbook for courses in (i) cardiovascular physiology in both the schools of engineering and medicine of universities, as well as (ii) cardiovascular engineering in biomedical engineering departments of universities world-wide

Section II

Computational and Mathematical Methods in Cardiovascular Physiology

4	Fluid Mechanics and the Cardiovascular System

Dhanjoo N. Ghista,* Foad Kabinejadian^†
and Joseph L. Bull ^†;

* University 2020 Foundation, Northborough, MA, USA
^†Department of Biomedical Engineering,Tulane University,
New Orleans, LA, USA

4.1Introduction

In the cardiovascular system, blood flow in the heart and the circulatory system governs the cardiovascular system physiology, and thereby provides the basis of cardiovascular system monitoring and diagnostics. Fluid mechanics plays a pivotal role in the quantitative formulation of the function of the cardiovascular system, for the purpose of its monitoring and diagnostics of system dysfunction due to diseases such as myocardial infarction and aortic arteriosclerosis.

In this chapter, we first present, in Section 4.2, the fluid mechanics governing the blood flow in the cardiovascular system. This involves equations on conservation of fluid mass and linear momentum. The momentum equation for a Newtonian fluid is known as the Navier–Stokes equation. These two sets of equations constitute a set of differential equations to be solved in order to describe the fluid flow. For inviscid flow, the momentum equation becomes known as the Euler equation. If we consider the flow to be steady, inviscid, and incompressible, and then integrate between two points on the same streamline, we obtain what is known as the Bernoulli equation. Streamlines are useful in visualizing the flow field. In the case of 2D flow, we can define a stream function, ψ, such that the continuity equation is satisfied by means of the flow velocities as derivatives of the stream function. Now, irrotational flow allows the requirement that the two points be on the same streamline, in order to use the Bernoulli equation. Additionally, we can define a velocity potential, ϕ, whereby the flow velocities are obtained as V_ϕ. The stream function and velocity potential represent alternative formulations from the pressure–velocity formulation of Navier–Stokes equation, which are applicable to certain flows. The equations for ψ and ϕ are solved subject to appropriate boundary conditions.

Next, in Section 4.3, we apply these fluid mechanics concepts, namely the Continuity equation and the Navier-Stokes equation, for incompressible Newtonian fluid, to model the steady flow profile in an artery. For this purpose, we employ the cylindrical coordinate system to obtain the relationship between the velocity field u_z(r) and the pressure field p(r, z). We solve the arterial flow model, to obtain the expressions for the pressure field in the z direction, the Poiseuille flow velocity profile for u_z(r), and the axial shear stress.

In the next Section 4.4, we then go within the left ventricle of the heart to analyze the intra-LV systolic velocity and pressure distributions, which are governed by the myocardial contraction patterns, and constitute tangible measures of left-ventricular pump performance. The intra-LV flow and pressure distribution are analyzed in a fluid-filled, ellipsoidal model of the left ventricle. The intra-LV fluid flow velocity is obtained in terms of a velocity potential ϕ, by solving the Laplace’s equation for the system in elliptic coordinates (ξ, η). The boundary conditions for the fluid velocity normal to the inner wall are expressed in terms of the ejection velocity v₀ (assumed uniformly distributed over the circular openings of radius R), which is in turn expressed in terms of the rate of change of LV chamber volume. The expression for the velocity components are obtained as

The expressions for the corresponding intra-LV pressure field p(ξ, η, t) are obtained from Bernoulli’s equation, which governs pressure and flow in an inviscid incompressible fluid of density ρ. We thereby obtain the expression for intracardiac pressure gradient vector field as

in elliptical coordinates, and therefrom the expressions for ∂p/∂ξ and ∂p/∂η. The functional pumping performance of the contracting left ventricle is directly related to this pressure gradient field. A uniform pressure gradient toward the aortic outflow tract will contribute to an efficient pumping function, whereas a nonuniform pressure gradient (caused by asynchronous myocardial contraction due to coronary lesions or infarcts) would give rise to inefficient pump performance and lead to heart failure.

We then proceed to Section 4.4.2 and determine the work done by the heart’s left ventricle (LV), as

wherein (i) P_ed and P_sy are end-diastolic and maximum systolic pressures, and (ii) V_s is the stroke-volume. For an average person, let us take P_sy = 120 mmHg, P_ed = 20 mmHg, and V_s = 75 ml. This then yields the value of the amount of work with each heartbeat to be 1 J. Now the heart power or work rate depends on the frequency of the heartbeat. So, for an approximate average frequency of 1 Hz (one beat per second), we have the heart power ΔW/Δt = 1 J/1 s = 1 W = 0.24 cal/s = 21 kcal/day.

Now in the next Section 4.5, we proceed out of the left ventricle and enter the aorta. First in Section 4.5.1, we develop the analysis to determine the aortic pressure profile in a segment of the aorta close to the aortic valve, which can be very useful for medical assessment. Based on the aortic blood control volume, we derive the governing differential equation representing the aortic pressure (P) response to outflow rate I(t) from the LV into the aorta as:

wherein m = volume elasticity of aorta (in Pa/m³), λ = (m/R) in s^–1, and I(t) is the LV outflow-rate. This governing equation is then solved for diastolic and systolic phases to obtain expressions for diastolic and systolic pressures p_d and p_s, in terms of the parameters of λ and m. Now we record systolic and diastolic pressures during the cardiac cycle by using cuff sphygmomanometry, and assume that they closely approximate the maximum aortic systolic pressure and the aortic end-diastolic pressure, respectively. We then solve the simultaneous algebraic equations involving these expressions for p_d and p_s equated to the corresponding sphygmomanometry pressures, along with the condition that dp_s/dt = 0 at t = t_m when the aortic flow rate I(t) is maximum. This enables us to obtain (i) the values of the parameters λ and m, and (ii) hence the expressions for p_d and p_s from which...