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Journal of Hypertension and Management





DOI: 10.23937/2474-3690/1510017



Advances in Low-Dimensional Mathematical Modeling of the Human Cardiovascular System

S Malatos1, A Raptis2 and M Xenos1*


1Department of Mathematics, University of Ioannina, Greece
2Cardiovascular Surgery Department, Sector of Surgery, Faculty of Medicine, University of Ioannina, Greece


*Corresponding author: M Xenos, Department of Mathematics, Section of Applied Mathematics and Engineering Research, University of Ioannina, Ioannina, 45110, Greece, E-mail: mxenos@cc.uoi.gr
J Hypertens Manag, JHM-2-017, (Volume 2, Issue 2), Review Article; ISSN: 2474-3690
Received: July 30, 2016 | Accepted: September 20, 2016 | Published: September 22, 2016
Citation: Malatos S, Raptis A, Xenos M (2016) Advances in Low-Dimensional Mathematical Modeling of the Human Cardiovascular System. J Hypertens Manag 2:017. 10.23937/2474-3690/1510017
Copyright: © 2016 Malatos S, et al. 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.



Abstract

The mathematical modeling of blood flow in the cardiovascular system has long history. Zero-dimensional (0D) and one-dimensional (1D) models contribute actively to the study of the human cardiovascular system. Usually, low-dimensional models consist of a system of time dependent equations that do not involve spatial derivatives, thus reducing the computational complexity compared to multi-dimensional models. Despite that more complex 3D cardiovascular models are available, there is a tendency of reintroducing the simpler 1D models, due to their capability of involving extensive segments of the cardiovascular system and providing boundary conditions for the advanced 3D models. The low-dimensional models can provide useful information to clinicians about the characteristics of blood flow at the level of individual incidents, patient-specific treatment, and can describe the general phenomena of circulatory physiology. The purpose of the current review is to discuss the advances and evolution of 0D and 1D models of human cardiovascular system.


Keywords

Mathematical modeling, Cardiovascular system, Blood flow, 0D models, 1D models, Computational methods


Introduction

The cardiovascular system is the blood transport mechanism that enables the nutrient transport to the tissues and organs of the body and the removal of various waste and toxic substances [1]. It includes the heart that pumps blood into circulation, the systemic blood vessels that drive blood to and from the peripheral organs, and the pulmonary vessels that enable the exchange of oxygen and carbon dioxide in the lungs. Blood is channeled from the aorta to the arteries, arterioles, capillaries, structuring an arterial tree with decreasing vessel diameters but increasing vessel luminal area overall, with the vessel wall becoming stiffer after consecutive branches.

Blood flow in the cardiovascular system obeys the laws of mass and momentum conservation and blood interaction with the arterial wall [2,3]. The selection of the appropriate model dimensionality, from 0D to 3D, depends on the aims and the desired accuracy. Lumped parameter models assume a uniform distribution of the fundamental variables, pressure, flow and volume, within any particular compartment, organ or vessel at any time instant. The higher dimensional models on the contrary account for the spatial variation of the fundamental variables as well. The distensibility and compliance of the arteries introduce additional constraints that strongly influence blood flow dynamics.

The 0D models give rise to a coupled system of ordinary differential equations (ODEs), suitable for the assessment of global distributions of pressure, flow and blood volume over a range of physiological conditions. For each vascular compartment included in the models, two ODEs are applicable, representing conservation of mass and momentum, complemented by an algebraic equilibrium equation that relates compartment volume to pressure. On the other hand, 1D models are based on simplified fluid flow equations that are solved in the frequency domain using Fourier or Laplace transformations [4-10]. The 1D models are well balanced with respect to modeling complexity and computational cost, rendered suitable for many biomedical applications. With the 1D models, it is possible in the near future to have fast, accurate, patient-specific and clinically applicable solutions for the dynamic flow prediction in the whole human cardiovascular system [11].

The mathematical models of blood flow, with their given noninvasive character, facilitate the study of pathological and physiological waveforms. It is strongly believed that the computation of pressure and flow profiles could be part of potential diagnostic tools in the future. On a patient specific basis, the profiles could be compared with physiological ones suggesting healthy or pathological blood flow.


0D Models

Blood flow in the circulatory system and electrical conductivity in a circuit have shared characteristics and analogous abstract interpretation. Blood pressure in the circulatory system is the reason that blood flow overcomes the hydraulic resistance. Similarly, voltage in a circuit is the reason that current flow overcomes the electrical resistance. The hydraulic resistance represents the combined effect of the cohesive forces, due to the wall elasticity, and blood inertia, while the electric resistor represents the combined resistance in the circuit. Blood flow is described by the conservation of mass (continuity equation) and conservation of momentum (Navier-Stokes equations). Similarly, the electrical flow in the circuit is governed by Kirchhoff's and Ohm's law for the current and voltage, respectively. The equation in the mechanical system associated with the vessel wall distensibility is related to the equation for the capacitor. Figure 1 shows the mechanical model (Figure 1a), e.g. a straight elastic tube, the electrical model (Figure 1b), e.g. a resistance-capacitor (RC) circuit, and the associated unknown quantities that have uniform physical interpretation. The simplest 0D model of the cardiovascular system is based on the famous Windkessel model (Figure 1b).


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Figure 1: (a) Mechanical model utilizing fluid mechanics principles; (b) Electrical analogy of arterial resistances in the human circulatory system. View Figure 1



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One-compartment and multi-compartment 0D models

In one-compartment models, the whole systemic vasculature is treated as a whole, and thus the internal pressure distribution and flow velocity in different parts of the vascular network are not calculated separately. The 2-Elements Windkessel model (Figure 1b) is the simplest 0D model consisting of a capacitor (C) that represents the compliance of the large arteries, and a resistor (R) that represents the resistance of small peripheral arteries including arterioles and capillaries [12-18]. The introduction of an extra resistor in series with the existing RC extended the applicability of Windkessel models, accounting for the resistance of the aortic valve to blood flow [19]. The 3-Elements Windkessel model simulates the characteristic impedance of the proximal aorta and is widely used for assessing cardiovascular function in various pathophysiological states. Other studies further extended the RCR model by integrating the inertial effect of blood flow, forming a RLCR circuit. The 4-Element gives a more accurate representation of blood pressure with cardiac cycle time compared to the 2- and 3-Elements models.

Using one-compartment Windkessel models as building blocks, multi-compartment models provide more spatial details by including multiple vascular segments [20-25]. Other parts of the cardiovascular system may still be considered lumped, using less detail. For the construction of a multi-compartment model, appropriate RLC models are created for each represented part of the whole network [26-33]. Burattini and Natalucci [34] developed a different configuration introducing three RCR elements to describe the characteristics of the arterial system, accounting for the viscoelastic properties of the vessel wall. More complex models can also incorporate additional parameters, such as pulse in venous pressure and flow in the components.

A special case of 0D modeling is that of local circulation characteristics in important vascular subsystem, such as cerebral, coronary, renal or lower extremity vessels where often multiple compartment models have been designed to include features such as anastomoses, auto-regulation effects and occasionally collapsible vessels and internal valves. Most commonly, these local simulations do not include a heart model, and usually flow rates and pressures are directly applied as boundary conditions [35-43].

A model consisting of a series of 0D compartments is a representation of a 1D system. Milisic and Quarteroni have given a mathematical proof that 0D models for the vessel network can be regarded as first order discretizations of 1D linear systems [44]. Such models are readily interpreted in terms of electrical analogues. The biggest difference between multiple linear 0D compartment models and published 1D models is that the latter tend to include the nonlinear convective acceleration term whereas the former cannot [45].


0D heart models

There have been numerous studies on the qualitative characterization of the heart as a pump. Due to double contraction and expansion of the left and right heart regions can be said that heart is a two pump system, connected together in series. Despite the similarity of these two pumps most studies examined the properties and functioning of one pump, usually the left one. Many researchers have discussed the characteristics of the left ventricle, to examine whether it should be described as a pressure or a flow source [46]. The atrial-ventricular interaction was also considered to be important in cardiovascular dynamic studies. In these studies, the effect of the pericardium on the cardiac dynamics was included in the analysis [20,22,26-33,47]. It is worth noting that the atrial-ventricular septum motion contributes about 10% of the cardiac output as it undergoes large displacements during the cardiac cycle. Korakianitis and Shi have derived detailed equations to model the motion of the septum [28,29]. Further developments in modeling heart dynamics in the cardiovascular network include multi-scale and multi-physics models of the cardiac function [48] and the coupling of lumped-parameter and distributed network models for cerebral pulse-wave hemodynamics [49]. Recently, there has been effort to develop 1D model of the entire cardiovascular system that includes anatomically realistic 1D vascular network in all major regions of the circulation, coupled to a 0D heart model that accounts for the main chamber interactions [50].

The simplest heart valve models used in 0D studies of cardiovascular dynamics described valves as a diode plus a linear or non-linear resistance. The valve has a low resistance to flow when fully open, and the flow is stopped completely when the valve is closed. This idealized description ignores more complex dynamics of the valve, but can be considered adequate for most studies [46].


Auto-regulation mechanism

When blood is pumped from the heart, and the pressure in the aorta is increased, then the aorta expands. When the pressure drops, the aorta is retracted so that the flow rate through the small peripheral vessels does not immediately go to zero. The acute increase in blood pressure causes an immediate increase in blood flow. However, in most tissues pressure returns to normal level, even if the blood pressure remains high. This restoration of flow to the normal mode is called automatic regulation. The detailed underlying mechanism of auto-regulation is unknown, but the metabolic requirement of the organ and the myogenic response of the vascular smooth muscle are considered to be two of the main causes [51]. Auto-regulation has an important influence on the blood flow in several local circulation loops, including the cerebral, renal, and hepatic circulations, and is an important component of any model of these subsystems [52-55]. Several mathematical models of auto-regulation mechanism have been proposed [41,43,55,56]. In these studies, Ursino and collaborators have proposed a comprehensive mathematical model of intracranial dynamics and cerebrovascular regulation. The results of this mathematical model were correlated with transcranial doppler (TCD) ultrasonography, an approach that is largely used today to achieve non-invasive assessment of cerebral autoregulation and cerebrovascular reactivity in neurosurgical patients.


1D Models

Propagation of pressure and flow waves in the vessel network is one of the most interesting problems in the study of cardiovascular physiology. It is generally believed that important information regarding cardiac function, the elastic properties of the vessels, and the pathophysiological conditions of the important organs, such as brain, liver and kidney, is implemented in these two wave forms and their relationship. Pulse wave studies have received extensive attention in cardiovascular research [57]. To this direction, 1D modeling can offer greater advantages in revealing the pressure and flow changes along the whole length of the studied vessel.

Due to the fact that the vascular system includes billions blood vessels, these 1D numerical models must be truncated and appropriate outflow boundary conditions have to be specified [58]. Additionally, these numerical models can be terminated at a level where the nonlinear losses are minor and linear models could describe flow velocity and pressure waves.


Numerical and analytical methods for 1D models

For the development of 1D mathematical models of the vascular system, the nonlinear 1D wave propagation method offers a good compromise between anatomic accuracy, inclusion of nonlinear losses and material properties. The 1D wave propagation method involves solving the governing equations of blood flow in a 1D domain and is based on the assumptions that the dominant component of blood flow velocity is oriented along the vessel axis and that pressure can be assumed constant over the cross-section of the vessel. The derived governing equations for the 1D pulse wave propagation have a hyperbolic nature [59]. A simplified system of equations that describes the 1D interaction of blood with the arterial tissue, as proposed by Olufsen et al. [9], is shown below:

Continuity Equation,

q x + A t =0                            (1)

Momentum Equation,

q t + x ( q 2 A )+ A ρ p x =F                  (2)

State Equation, (distensibility)

p( x,t ) p 0 = 4 3 Eh r 0 ( 1 A 0 A )                           (3)

where q is the flow through the vessel, A, is the cross-sectional area, F= 2πνR δ q A is the Poiseuille's term, ρ is the blood density, E is the Young's modulus, the term Eh r 0 is evaluated from an exponential expression.

Another approach is based on Womersley's solution for pulsatile flow in an elastic vessel [60]. Under the assumptions of axisymmetry, linear constitutive behavior, and small perturbations about a constant pressure and zero axial velocity reference state, a system of linear partial differential equations (PDEs) governing the fully developed pulsatile flow in an elastic vessel can be derived and solved analytically. For a prescribed input flow rate, the Fourier coefficients of the input pressure can be determined. An inverse Fourier transform can be used to compute the inlet pressure. Pressure and flow rates throughout the vascular tree can be computed by enforcing conservation of mass and continuity at branch points [61]. While impedance based linear methods can be used to describe pressure and flow wave propagation in vessels, these models do not incorporate non-linear convective losses or losses due to branching and stenoses [62-64]. This is the most significant limitation of Womersley's elastic vessel theory, especially as applied to blood flow in the main arteries and in the diseased state. Recent approaches, as the total variation diminishing (TVD) scheme have been proposed to solve 1D blood flow for human circulation. This method involves only a few modifications to existing shock-capturing TVD schemes. For 1D network simulations, the method has been shown to agree well with computational results [65]. Another approach to solve the 1D models is to linearize the system by neglecting the convection term and then find solutions in frequency domain [66,67].

The method of characteristics could provide a solution of the system of PDEs (continuity and momentum) transforming them into ordinary differential equations (ODEs) along the directions of the characteristic lines [67-71]. The governing equations have also been solved using finite difference methods such as the Lax-Wendroff and MacCormack schemes [72-76]. In recent years, various finite difference [9,10,63,72,76-78], finite volume [79,80], finite element [58,81-84], discontinuous Galerkin schemes [81,83,85-87], or combination of the above numerical methods [88] have been applied on this problem. The Godunov scheme was used to discretise the governing equations in a finite volume formulation [89,90]. For the finite element formulation, Formaggia, et al. adopted the Yoshida projection scheme for the solution of the equations [91]. Wan, et al. used a discontinuous Galerkin scheme [92]. Porenta, et al. [93] and Rooz, et al. [94] applied a Galerkin scheme and Sherwin, et al. solved the equations both with a discontinuous Galerkin scheme and with a Taylor-Galerkin scheme [95]. Wan, et al. [64] and Steele, et al. [96] have described a method to solve the 1D nonlinear equations of blood flow in elastic vessels utilizing a space-time finite element method with Galerkin/Least-Squares (GLS) stabilization technique. Numerical schemes have also been designed to solve the nonlinear equations in time domain [50,97]. Another class of models are described by power law [98] and more recently, models with viscous boundary layers have been proposed [9,99-101].

A small number of researchers have used spectral method for the solution of pulse wave equations. Bessems, et al. have applied a Galerkin weighted residual method to transform the governing equations into a spectral element space [100]. Sherwin, et al. applied the spectral/hp element method [102]. Ballarin and colleagues have developed a parameterized approach of the Navier-Stokes equations utilizing the POD-Galerkin approximation [103]. Additional studies used spectral method to perform 1D flow simulations [104,105]. Several researchers have proposed stochastic approaches for the human arterial network hemodynamics or for the boundary conditions of the developed model [106-109].

An effective way to describe the arterial structure that consists of thousands to millions of arteries is by using the concept of fractal tree models. Perdikaris, et al. have developed a flexible and effective model that accurately distributes flow in the downstream vasculature and can stably provide a physiological pressure outflow boundary condition [110]. They have scaled up a Discontinuous Galerkin solver that utilizes the MPI/Open MP hybrid programming to thousands of computer cores simulating blood flow in networks of millions of arterial segments. Linninger, et al. developed a computer fractal-like tree model that involves arterial, capillary and venous blood vessels of the cerebral microvascular bed as well as brain tissue occupying extra vascular space [111].

Finally, there is a small number of researchers [112,113] who specifically included the inertial forces in their 1D model derivation, and arrived at a wave dynamics equation in the form of the Korteweg-de Vries (KdV) equation. The corresponding wave dynamics is governed by a special kind of wave called solitary wave. The solitary wave is a better description of the arterial pulse wave since it matches the experimental results better than the hyperbolic wave equation that is currently used by the majority of researchers. This seems to be an interesting area that requires further exploration [112-116].


Boundary conditions

Due to the hyperbolic nature of the 1D pulse wave equation, boundary condition (BC) has to be imposed to each end of the vessel [57]. At the inlet of the vessel, the pressure or flow rate can be applied based on derivations or experimental data. For the outlet side, the boundary condition requires more consideration. Many researchers have directly specified a combination of the pressure and flow rate at the inlet and outlet. Other researchers used constant or varying resistance or a three-element Windkessel to specify the pressure/flow relation at the outlet [64,67,74,75,93,94]. Others have chosen to specify the wave reflection coefficient at the outlet [78,83,88] or used the non-reflecting boundary condition and compatibility condition [117].

The most popular downstream conditions in recent studies are to use a Windkessel model as an after-load or directly to specify the wave reflection coefficient. Olufsen proposed the structured tree model in which the impedance of smaller arterioles was estimated with linearised Navier-Stokes equations further improving the accuracy of downstream boundary condition [62]. Smith, et al. studied the blood flow in coronary network, and a pressure dependent vessel network for the coronary arteriole, capillary and venules was used as terminal load to each generation of coronary artery branches [76]. A special kind of boundary condition internal to a vessel segment is that of vessel branching, in which a parent vessel is branched into several daughter vessels [9,10,118]. Many researchers directly applied equal static pressures and conservation of flow rates at the branching points [62,64,93]. Sherwin, et al. proposed an improved description applying continuity of total pressure [95]. Recently, Morbiducci, et al. proposed inflow boundary conditions for image-based computational hemodynamics [119]. They study the impact of idealized versus measured velocity profiles in the human aorta and concluded that idealized velocity profiles as inlet BCs can lead to misleading representations of the aortic hemodynamics both in terms of disturbed shear and bulk flow structures.


Problems of 1D models

There are several complications in the study of 1D pulse wave propagation, including the tapering of the vessel branching, nonlinear pressure/cross-sectional area relationships for the vessel wall, axial tension and bending in the vessel wall, collapse of veins and pulmonary vessels. In deriving the governing equations, most researchers have considered the nonlinear pressure/cross-sectional area relation of the vessel wall by incorporating individually adapted constitutive relations. An additional factor that influences the validity of 1D models is the blood rheology as the Newtonian approach seems to be restrictive when looking for an accurate estimation of wall shear stress [120]. In vessels with an internal diameter of less than 1mm, the apparent viscosity is dependent significantly on temperature and shear rate [18]. However, in vessels smaller than this value, we can change the friction coefficient to account the altered viscosity. Researchers have developed non-Newtonian models, such as the Casson viscoplastic model, to describe arterial blood flow [121]. Another approach is the Dissipative Particle Dynamics (DPD) in combination with 1D global models of blood circulation [122]. Since blood is considered as a heterogeneous fluid composed of plasma and blood cells, DPD is a novel multi-scale approach to describe blood dynamics.

Lateral leakage and tapering can be modeled by the 1D models. The aorta is tapering from the proximal to the distal end. The tapering wall increases the pulse pressure by continuous reflection. It is also observed that there are tiny side branches at the aorta. Some studies show that the effect of tapering wall is compensated by the blood loss to the side branches. In net effect, it is more like a flow in a straight tube with no lateral leakage [18]. Brook and Pedley [90], Porenta, et al. [93], Formaggia, et al. [91], Rooz, et al. [94], Sherwin, et al. [95,102] included the effect of vessel tapering by considering a varying initial cross-sectional area of the vessel.

Since the mean Reynolds number could take a value of 4000 during the cardiac cycle [123], the flow should be described as turbulent. In unsteady case the flow is at least transitional to turbulence. In small arteries, the Reynolds number drops substantially due to the fact that the velocity and the diameter of the vessels become smaller. Given that the Womersley number is also much smaller in small arteries, laminar flow is likely to appear in small arteries. That gives justification to Poiseuille theory in estimating the peripheral resistance. In medium-sized arteries, there may be transition conditions between turbulent and laminar flow. The 1D models neglect the velocity components in the circumferential direction and thus a laminar flow is an implicit assumption. In the aorta, the wave speed is about 5 m/s and the radius is about 10 mm [18]. If we assume the period of one pulse is 1s, the ratio between the wave length and the radius is 500. In smaller arteries, the wave speed increases and the radius decreases, the ratio becomes even bigger. Thus the long wave assumption is fully justified. One crucial step in deriving the 1D model is to prescribe a proper axial velocity profile. The fluid friction coefficient and the correction factor of the convection term in the momentum balance equation are dependent on this profile.

In clinical applications, the blood pressure is measured when the subject is in supine or sitting position. In those cases, the pressure difference through the whole body caused by gravity is much smaller than the pressure at the arterial system. Curvature and bifurcation will dissipate blood flow energy; an experimental study shows that the energy loss at bifurcations is very small [81]. Since the network of vessels is very large, we usually have to truncate it at some levels. The truncated sub-networks have to be described properly. Reflection coefficient, structured tree [9], 0D models [124] and other kind of generalized methods for specific numerical schemes [58] could lead to problems in 1D cardiovascular models.

With some modifications, the 1D models can be used in simulation of venous flow. There are three special features of venous system that need to be treated, the inflow from venules bed, muscular contraction and valves. The inflow can be considered as a source term in the mass conservation equation, the muscular contraction can be described by an external pressure in the constitutive equation of the vessel wall; the valves, which in fact allow very small reverse flows, can be modeled by very large resistances when blood's pressure gradient to the heart is adverse [125].

The constitutive equation of the wall is mainly derived from thin shell theory. The wall shows nonlinear elasticity properties in high pressure and the inertia of the wall may affect the pulse wave as well. Usually linear elastic models are integrated into the 1D fluid models for simplicity. Recent studies indicate that the viscoelasticity may have significant influence on the pulse waves [63,81,100,126-130].


Applications of 1D models

Several 1D models of the vessel network have been proposed for the study of pulse wave transmission in various applications [62,64,67-71,73,75,89-91,93-95,131]. The models mainly differ in the boundary conditions applied and the solution methods used, and whether nonlinear effects were taken into account.

Stergiopulos and collaborators solved the nonlinear 1D blood flow equations in a comprehensive model of the arterial system using a lumped parameter model of the vasculature downstream of each branch to account for the capacitive and resistive vasculature effects [63]. Earlier work had been done on modeling flow in a single tapering tube where the 1D nonlinear equations were coupled to Windkessel models [132,133]. Although the geometry was simple, these studies conveyed most of the ideas such as the effect of nonlinearity, tapering, stenosis and dilatation of the distal beds on pressure and flow waveforms. Wang, et al. studied the coupling of the 1D linear equations with terminal resistances using simplified input data [67]. Sherwin, et al. compared the 1D linear model to the equivalent non-linear one [95,102]. Several groups have mathematically analyzed and developed the coupling of 1D non-linear equations with lumped models [124,134,135].

In contrast to methods coupling the 1D equations of blood flow to lumped parameter models, Olufsen, et al. developed a distributed model based on calculating the input impedance of an asymmetric binary structured tree using Womersley's linear wave theory [60] and an algorithm for computing the impedance of a vascular network initially proposed by Taylor [136]. Olufsen's distributed model of the downstream vasculature enabled the representation of more realistic flow and pressure waveforms than those obtained with lumped parameter models [62]. Steele, et al. used a modified version of Olufsen's impedance boundary condition to model blood flow at rest and during simulated exercise conditions [96]. In this case, vascular networks were assigned to the outlets of a model of the abdominal aorta, modified to represent the resting flow distribution of eleven different subjects and then dilated to simulate the effects of lower limb exercise.

Overall, the 1D models are well balanced between complexity and computation cost, thus they are very suitable for many biomedical applications [127,137]. Most of the 1D pulse wave transmission models have been applied to study the pulse wave dynamics in arterial segments [68-70]. Wang, et al. [71], and Wang and Parker [67] later extended the study to investigate the pulse wave dynamics in a complete arterial network, including ventricular-arterial coupling. Li and Cheng [73] studied the pulse wave features in the pulmonary arterial network. A model of the human cardiovascular-respiratory control system with one and two transport delays in the state equations describing the respiratory system was developed by Kappel, et al. [138,139]. The model can be used to study the interaction between cardiovascular and respiratory function in various situations as well as to consider the influence of optimal function in physiological control system performance. The model was applied to the simulation of the orthostatic stress phenomenon [140-142], which was also considered by Olufsen, et al. [143].

In one class of 1D model applications, we simulate the diseased blood flow in various surgery plans hoping that optimized guidance to surgeons may be given successfully. Even though there are many difficulties on setting the parameters of patient-specific models and usually only qualitatively accurate predictions can be given, we still can extract clinically relevant information. Huberts, et al. show that the mean pressures and flows after an arteriovenous fistula surgery can be simulated with 1D models with satisfactory accuracy [144]. Another successful example is the simulation of blood flow after bypass surgery [145]. Porenta, et al. [93] and Rooz, et al. [94] have studied the pulse wave features in arteries with stenosis. Wan, et al. [64] and Steele, et al. [92] calculated the pulse wave dynamics in diseased arterial vessels with bypass grafts. Surovtsova [131], Sherwin, et al. [95], and Pontrelli and Rossoni [74] studied the pulse wave transmission in stenotic arteries with implanted stents. Formaggia, et al. studied wave propagation and reflection due to stenting, solving the 1D nonlinear equations of blood flow with a prescribed pressure at the inlet and a non-reflecting outlet boundary condition representing a tube of infinite length [146]. They proposed an approach for the coupling with myocardial contraction [147,148].

Another development in 1D pulse wave transmission modeling is the wave intensity analysis proposed by Parker and Jones [69], in which they defined the product of pressure and velocity changes over a small interval as the evaluation of rate of energy flux per unit area in a profile of vessel segment. This indicator accurately describes the wave intensity accompanying the pulse wave transmission, and can be used to distinguish the forward transmission wave from the backward one. The wave intensity analysis has been applied for the study of pulse wave transmission in the left ventricle [149,150], coronary vessels [151], systemic arteries [152] and pulmonary arteries [153]. Some researchers have studied the pulse wave transmission in collapsible vessels. Elad, et al. studied the unsteady fluid flow through collapsible tubes [72].

The most advanced 1D cardiovascular models are taking under consideration the coupling of the blood with the arterial or venous wall. The properties of the vascular wall, elastic or viscoelastic, are governed by the wall equilibrium equations. Most researchers use simple linear or nonlinear constitutive equations to describe the pressure/cross-sectional area relationship [62,64,67-73,75,76,89-91,93-95,102,131]. However, more complex models are also available [74]. Pontrelli and colleagues studied the modeling of the wall using the coupling of simple lumped models with a six-compartment lumped model, representing the whole system as a closed loop [154,155].


Multi-scale models

Low-dimensional models (0D and 1D) are suitable for the characterization of blood flow along extended parts of the cardiovascular systems but, due to their design, fail to account for the complexity of the local vascular geometry. Multi-scale models for cardiovascular hemodynamics have been recently developed to combine the efficiency of the continuum models with the higher fidelity of the atomistic or mesoscopic models leading to hybrid type computations [156-159]. A representation of a multi-scale cardiovascular model is shown in figure 2.


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Figure 2: Schematic representation of a multi-scale approach, the heart is modeled as a 0D model, the arterial tree is modeled as a 1D model and the specific artery, such as the abdominal aorta is represented with a 3D patient-specific model. View Figure 2



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Higher dimensional models (2D and 3D) are able to predict the hemodynamic properties in more detail as they can account for the local vascular geometry, especially the patient-specific 3D models that derive from reconstructions of medical screening data, according to the workflow of figure 3. However, due to the intense computational requirements, the 3D models are normally restricted to smaller arterial segments [160]. Additionally, the predictability of 3D models largely depends on the accurate description of the boundary conditions. It is apparent therefore that the utilization of 3D models does not guarantee the accurate simulation of the in vivo blood flow conditions. The limitations of the various models are neutralized when combined in unified studies where low-dimensional models are attached at the inlets and outlets of the 3D computational domains [161]. Every sub-model lends different characteristics to the unified model as it is governed by different type of equations. The 0D lumped parameter models are governed by ODEs, the 1D models are mostly described by hyperbolic PDEs, while the 2D/3D models are based on the Navier-Stokes equations [45].


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Figure 3: Patient-specific workflow combining computed tomography (CT) data, image reconstruction of the 3D aortic structure and computational fluid dynamics (CFD) for predicting hemodynamic properties. View Figure 3



.




Methods to couple higher and lower dimensional models were initially outlined by Formaggia, et al. where the proposed iterative coupling was tested initially in simplified computational domains [91]. Pant, et al. introduced a framework for the estimation of lumped parameters based on sensitivity analysis tools and the sequential estimation approach of the unscented Kalman filter aiming to increase the levels of patient-specific analysis of hemodynamic flow [162]. Kim, et al. demonstrated the coupling of a lumped parameter heart model to a 3D finite element model of the aorta [2]. Extending the applicability of the coupling, Chandra, et al. applied boundary conditions that capture the wave propagation, in 3D fluid-structure interaction models of arterial blood flow [163]. Smith, et al. generated an anatomically-based coronary geometrical model [147], and then performed 1D simulations on six generations of vessels, using physiologically relevant lumped boundary conditions [76].


Validation

At the beginning, low-dimensional models agreed quantitatively with experimental observations when accurate parameters of the models were available [78,81]. In several cases the numerical results only meet qualitatively with in vivo data [9,75,92,164-166]. To know the sensitivity of the output to the uncertainties of each parameter of the 1D models, sensitivity analysis has been done [11,49,167]. To estimate the parameters, we can compare the model output with measurements and minimize the difference by tuning the parameters [116,168]. Many applications of the 1D models rely on the solution of the reverse problems. However, in these studies, the computations have to run millions of times, which contribute to the development of a fast numerical code.

From the above analysis is imperative to highlight that all these studies have to be validated against experimental measurements. Researchers have done substantial effort to validate the low-dimensional approaches [92,106,144,145,169]. Reymond, et al. extended and improved a previous 1D model of the systemic circulation by including a heart model, a detailed description of the cerebral arterial tree, viscoelasticity, and a Witzig-Womersley theory-based formulation for the friction and convective acceleration terms [75]. Morbiducci, et al. developed inflow boundary conditions for image-based computational hemodynamics. Their findings indicate that assumptions upon the shape of the inlet velocity profile in subject-specific in silico studies of the aortic hemodynamics might have completely different impact on the aortic bulk flow topology and on disturbed shear [119]. They also developed a model for atherosclerosis at arterial bifurcations. Arterial bifurcations can be exposed to disturbed shear stress and there by affected by local plaque formation, which reflects a complex interplay among hemodynamics, biological and systemic risk factors [170]. Van der Horst, et al. presented two disease types models, coronary stenoses located in the epicardial vessels and left ventricular hypertrophy with an aortic valve stenosis, affecting the coronary microvasculature [171]. Guala, et al. presented a detailed subject-specific validation against in vivo measurements from a population of six healthy young men. Several quantities of heart dynamics such as the mean ejected flow, ejection fraction, and left-ventricular end-diastolic, end-systolic and stroke volumes and the pressure waveforms are compared with measured data [172].


Conclusions

The purpose of this review is to discuss recent advances and evolution of low-dimensional models of human cardiovascular system and its applications. Configurations of the 0D models are becoming more sophisticated and advanced, and the various developed models have wide and successful applications in the study of cardiovascular physiology, evaluation of new devices and more. The 1D models were mainly confined to the study of arterial hemodynamics, where their ability to capture wave propagation effects is imperative, with minor extensions to venous dynamics. These models have been successfully applied in the context of clinical diagnosis of pathological changes in the cardiovascular system, such as hypertension, atherosclerosis, and in stent design. With the development of computers and numerical analysis techniques, higher dimensional hemodynamic analysis (2D/3D) using computational fluid dynamics is no longer prohibitive. So, in order to obtain very accurate results and to simulate the interaction among cardiovascular organs, it is required to couple the 0D models and the 1D/2D/3D models to build multi-dimensional and multi-scale models. It is possible in the near future to have accurate, real-time, patient-specific, clinically applicable cardiovascular models for the simulation of the whole human body cardiovascular dynamics.


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