Multi-scale modeling of the human cardiovascular system with applications to aortic valvular and arterial stenoses

Original Article

Abstract

A computational model of the entire cardiovascular system is established based on multi-scale modeling, where the arterial tree is described by a one-dimensional model coupled with a lumped parameter description of the remainder. The resultant multi-scale model forms a closed loop, thus placing arterial wave propagation into a global hemodynamic environment. The model is applied to study the global hemodynamic influences of aortic valvular and arterial stenoses located in various regions. Obtained results show that the global hemodynamic influences of the stenoses depend strongly on their locations in the arterial system, particularly, the characteristics of hemodynamic changes induced by the aortic valvular and aortic stenoses are pronounced, which imply the possibility of noninvasively detecting the presence of the stenoses from peripheral pressure pulses. The variations in aortic pressure/flow pulses with the stenoses play testimony to the significance of modeling the entire cardiovascular system in the study of arterial diseases.

Keywords

Multi-scale model Cardiovascular system Stenosis Wave propagation Pressure pulse 

1 Introduction

Arterial stenosis, a serious form of arterial disease, is frequently found to affect large and middle-sized arteries. The strongest influences of a stenosis are on local flows, such as increased pressure drop and reduced flow rate, which are major causes of ischemic events in the distal tissues, and have, accordingly, been studied extensively [3, 25, 33].

Stenosis influence on far-field flows has been relatively less well addressed. Associated abrupt changes in geometry and elastic properties induce abnormal wave reflections which modify the characteristics of arterial wave propagation and even affect cardiac dynamics. We call such influences global, as opposed to influences on local flows. In the bounded arterial system, the global hemodynamic influences of an arterial stenosis depend not only on its geometry but also on its location. Therefore, understanding the global hemodynamic influences of arterial stenoses in relation to their locations is necessary for improving the applicability of arterial-pulse-based stenosis diagnosis.

Several efforts have been made in the past to characterize the global hemodynamic influences of arterial stenoses with mathematical models [5, 7, 27]. These studies revealed many novel insights, but none systematically investigated the major arterial stenoses in relation to their locations, and, particularly, the models developed in the studies have various degrees of limitation when applied to such a systematic study. For instance, a one-dimensional (1-D) model of the arterial system requires a definition of flow conditions at the boundaries [27]. As the same set of boundary conditions is used to simulate different arterial stenoses, there is potential for a mismatch between the fixed boundary conditions and variations in arterial properties. Such a mismatch can be avoided by using an integrated model of the ventricular-arterial system [5, 7] which describes ventricular-arterial interaction thus generating aortic inflow conditions spontaneously. Nevertheless, this approach is limited, especially for problems where preload-afterload interaction is important, due to the fixed left ventricular preload.

The present study was therefore performed to develop a computational model of the entire cardiovascular system. The model integrates a 1-D arterial network model into a lumped parameter description of the entire system, and thus is able to account for global hemodynamics and arterial wave propagation simultaneously. In this study, the model is used to systematically investigate the global hemodynamic influences of aortic valvular (AV) and arterial stenoses in relation to their locations. The potential diagnostic values of the global hemodynamic changes associated with arterial stenoses, and the effects of stenoses on the heart are discussed.

2 Methodology

The arterial tree consisting of the 55 largest arteries was described by a 1-D model that has been proved suitable for describing wave propagation phenomena [1, 7, 20, 21, 24, 27], while the remaining system was described by a lumped parameter (0-D) model (see Fig. 1a). Coupling the 1-D model and the 0-D model yielded a closed-loop multi-scale model of the entire cardiovascular system.
Fig. 1

a Schematic description of 0–1-D multi-scale modeling of the human cardiovascular system; b electric circuit analogy of a lumped parameter network

2.1 Modeling of the arterial tree

2.1.1 Arterial blood flow model

1-D governing equations for blood flow in an artery can be obtained by integrating the continuity equation and Navier–Stokes equations over an artery cross-section [7]:
$$ \frac{\partial }{\partial t}\left( \begin{gathered} A \hfill \\ U \hfill \\ \end{gathered} \right) + \frac{\partial }{\partial z}\left( \begin{gathered} UA \\ \frac{{U^{2} }}{2} + \frac{P}{\rho } \\ \end{gathered} \right) = \left( \begin{gathered} 0 \\ - K_{R} \frac{U}{A} \\ \end{gathered} \right), $$
(1)
where t is the time, and z the axial coordinate along the artery; ρ is the blood density; A, U and P represent the cross-sectional area, mean axial velocity, and mean pressure, respectively; KR is the friction force per unit length, which is modeled to be 22πν with ν the dynamic viscosity of the blood [1]. This study used ρ = 1.06 g cm−3 for the density and a constant value ν = 4.43 s−1 cm2 for the dynamic viscosity.
The system of Eq. 1 is completed by a pressure-area relationship [1, 7, 24]:
$$ P - P_{0} = \frac{{Eh_{0} }}{{r_{0} (1 - \sigma^{2} )}}\left( {\sqrt {\frac{A}{{A_{0} }}} - 1} \right), $$
(2)
where A0 and h0 represent, respectively, the cross-sectional area and wall thickness at the reference state (U0, P0); r0 is the radius corresponding to A0; E is the Young’s modulus, and σ is the Poisson ratio, taken to be 0.5.
The pulse wave velocity (c) can be calculated as [1, 7, 24]:
$$ c = \sqrt {\frac{A}{\rho }\frac{\partial P}{\partial A}} = \sqrt {\frac{\beta }{{2\rho A_{0} }}} A^{{\frac{1}{4}}} ,\quad \beta = \frac{{\sqrt \pi Eh_{0} }}{{1 - \sigma^{2} }}. $$
(3)

2.1.2 Bifurcation model

Flow conditions at the bifurcations were described by mass conservation and total pressure continuity [1, 7, 24, 26]:
$$ A_{1} U_{1} = A_{2} U_{2} + A_{3} U_{3} , $$
(4)
$$ P_{1} + \frac{1}{2}\rho U_{1}^{2} = P_{2} + \frac{1}{2}\rho U_{2}^{2} = P_{3} + \frac{1}{2}\rho U_{3}^{2} , $$
(5)
where the subscript “1” denotes the mother artery, and “2”, “3” the two daughter arteries.

2.1.3 Stenosis model

The 1-D system of Eq. 1 cannot fully account for pressure drop through a stenosis. Therefore, the stenosis model proposed by Young et al. [33] was incorporated, where pressure drop across a stenosis is related to flow rate and geometrical parameters of the stenosis:
$$ \Updelta P = \frac{{K_{\text{v}} \mu }}{{A_{0} D_{0} }}Q + \frac{{K_{\text{t}} \rho }}{{2A_{0}^{2} }}\left( {\frac{{A_{0} }}{{A_{\text{s}} }} - 1} \right)^{2} Q\left| Q \right| + \frac{{K_{\text{u}} \rho L_{\text{s}} }}{{A_{0} }}\dot{Q}, $$
(6)
where ΔP and Q denote pressure drop and flow rate through the stenosis, respectively; \( \dot{Q} \) is the time derivative of Q, A0 and As refer to the cross-sectional areas of the normal and stenotic segments, respectively, Ls represents the stenosis length, and μ is the blood viscosity which was here taken to be 0.0047 Pa s. Further, Kv, Kt and Ku are empirical coefficients, with Kv = 32(0.83Ls + 1.64Ds) × (A0/As)2/D0, Kt = 1.52, and Ku = 1.2 [23], where D0 and Ds are the diameters corresponding to A0 and As. Stenosis severity is defined as the percentage reduction in cross-sectional area: (1 − As/A0) × 100%.

2.2 Modeling of the peripheral circulation, the heart and the pulmonary circulation

2.2.1 Simplified lumped parameter model of the peripheral circulation

For the purpose of simplicity, we assumed that blood flow leaving the peripheral arteries converged, respectively, to two regions (upper and lower body blocks) at the capillary level (Fig. 1a). Lumped parameter modeling of the blocks was implemented in a way similar to that in our previous studies [15, 16] and other studies [12, 13]. Here, each of the blocks consists of three series-arranged compartments (capillary, venular, and venous). The governing equations for each compartment are obtained by formulating mass and momentum conservation at V (C points) and Q (points in-between R and L) nodes (Fig. 1b), respectively.

At a V node
$$ \frac{{{\text{d}}V_{j} }}{{{\text{d}}t}} = Q_{j} - Q_{j + 1} , $$
(7)
and at a Q node
$$ L_{j} \frac{{{\text{d}}Q_{j} }}{{{\text{d}}t}} = P_{{j - 1}} - Q_{j} R_{j} - P_{j} , $$
(8)
where Qj and Qj+1 are the inflow and outflow of the Vj node, respectively, and Pj−1 and Pj are the blood pressures upstream and downstream of the Qj node, respectively; Pj = Vj/Cj.

2.2.2 Elastance model of the heart

Elastance-based modeling of the heart has been widely adopted since the first proposal by Suga [28]. In this study, we developed a heart model which describes each of the four cardiac chambers, respectively. Based on the elastance model, blood pressure (Ph) in each cardiac chamber is given by [29]:
$$ P_{\text{h}} (t) = E(t)(V - V_{0} ) + S\dot{V}, $$
(9)
where V is the cardiac volume and \( \dot{V} \) its time derivative, V0 refers to the dead volume, and S is the viscoelasticity coefficient of the cardiac wall, which is related linearly to the cardiac pressure [29]. E(t) is the time-varying elastance,
$$ E(t) = E_{\text{A}} e(t) + E_{\text{B}} , $$
(10)
where EA is the amplitude of elastance, EB is the baseline value of elastance, and e(t) is a normalized time-varying function of the elastance, which for ventricles is:
$$ e_{\text{v}} (t) = \left \{ {\begin{array}{*{20}l} {0.5[1 - \cos (\pi t/T_{\text{vcp}} )]} & {0 \le t \le T_{\text{vcp}} } \\ {0.5\{ 1 + \cos [\pi (t - T_{\text{vcp}} )/T_{\text{vrp}} ]\} } & {T_{\text{vcp}} < t \le T_{\text{vcp}} + T_{\text{vrp}} } \\ 0 & {T_{\text{vcp}} + T_{\text{vrp}} < t \le T_{0} } \\ \end{array} } \right., $$
(11)
and for atria is:
$$ e_{\text{a}} (t) = \left\{ {\begin{array}{*{20}l} {0.5\{ 1 + \cos [\pi (t + T_{0} - t_{\text{ar}} )/T_{\text{arp}} ]\} } & {0 \le t \le t_{\text{ar}} + T_{\text{arp}} - T_{0} } \\ 0 & {t_{\text{ar}} + T_{\text{arp}} - T_{0} < t \le t_{\text{ac}} } \\ {0.5\{ 1 - \cos [\pi (t - t_{\text{ac}} )/T_{\text{acp}} ]\} } & {t_{\text{ac}} < t \le t_{\text{ac}} + T_{\text{acp}} } \\ {0.5\{ 1 + \cos [\pi (t - t_{\text{ar}} )/T_{\text{arp}} ]\} } & {t_{\text{ac}} + {\text{T}}_{\text{acp}} < t \le T_{0} } \\ \end{array} } \right.. $$
(12)
Here, the subscript “v” denotes the ventricles, and “a” the atria, T0 is the duration of a cardiac cycle, Tvcp, Tacp, Tvrp and Tarp refer to the durations of ventricular/atrial contraction/relaxation, and tac, tar the times when the atria begin to contract and relax, respectively.
The pressure–flow relationship across a normal cardiac valve was assumed to follow the Bernoulli law [29]. Additionally taking into account the contribution of viscous resistance and blood inertia, the pressure drop written as a function of flow is
$$ \Updelta P_{\text{cv}} = R_{\text{cv}} Q_{\text{cv}} + B_{\text{cv}} Q_{\text{cv}} \left| {Q_{\text{cv}} } \right| + L_{\text{cv}} \dot{Q}_{\text{cv}} , $$
(13)
where Rcv, Bcv, Lcv are the coefficients of the viscous, flow separation, and inertial terms, respectively.
In the case of severe AV stenosis, the transvalvular pressure drop is dominated by the energy loss associated with a sudden flow expansion from the vena contracta to the ascending aorta. In this study, we adopted the model proposed by Garcia et al. [8, 9, 10], in which the pressure drop (∆Pav) is expressed as a function of the transvalvular flow rate (Qav), the cross-sectional area of the flow jet at the vena contracta [generally termed effective orifice area (EOA)], and the cross-sectional area of the ascending aorta (Aao):
$$ \Updelta P_{\text{av}} = \frac{1}{2}\rho Q_{\text{av}}^{ 2} \left( {\frac{1}{\rm EOA} - \frac{1}{{A_{\text{ao}} }}} \right)^{2} + 2\pi \rho \frac{{\partial Q_{\text{av}} }}{\partial t}\left( {\frac{1}{\rm EOA} - \frac{1}{{A_{\text{ao}} }}} \right)^{0.5} $$
(14)

Here, Aao is about 7.0 cm2 according to the ascending aortic diameter defined in the 1-D model, and EOA was assumed to be 4.0 cm2 in normal conditions [10, 29].

The pulmonary circulation plays an important role in mediating left-right heart interaction. This study used the model developed in [29].

2.3 Numerical methods

The 1-D system (Eq. 1) was solved with the two-step Lax–Wendroff method. Equations 4 and 5 describing the bifurcation conditions were solved by means of a ‘ghost-point’ method [11] coupled with a Newton–Raphson method (see “Appendix 1” for details). To deal with the impedance discontinuity induced by a stenosis, the artery was divided into two parts at the stenosis with the stenosis modeled as an interface into which the pressure drop (calculated by Eq. 6) is incorporated. Mass and momentum conservation across the interface was achieved through iterative computation. The 0-D system (Eqs. 7, 8) was solved using a fourth-order Runge–Kutta method. Solutions of the 0-D and 1-D systems at the interfaces were coupled for each time step (see “Appendix 2”).

2.4 Physiological data

The geometrical parameters of the 55 arteries were prescribed based on the data reported in [20, 27]. The elastic parameters of the arteries were estimated based on the pulse wave velocities given in [30] according to Eq. 3. The peripheral resistances (RT) were derived from [27]. The compliances (C1) of distal arteries were estimated to give a total arterial compliance of about 1.5 ml/mmHg [19]. The data for the arterial tree are given in Table 1.
Table 1

Physiological data of the arterial tree

No.

Arterial segment

L (cm)

r0 (cm)

r1 (cm)

c0 (m s−1)

R0 (mmHg s ml−1)

R1 (mmHg s ml−1)

C1 (ml mmHg−1)

1

Ascending aorta

2.0

1.525

1.420

5.11

2

Aortic arch I

3.0

1.420

1.342

5.11

3

Brachiocephalic

3.5

0.650

0.620

5.91

4

R.subclavian I

3.5

0.425

0.407

5.29

5

R.carotid

17.7

0.400

0.370

5.92

6

R.vertebral

13.5

0.200

0.200

9.64

6.10

27.87

0.0126

7

R.subclavian II

39.8

0.407

0.230

5.38

8

R.radius

22.0

0.175

0.140

10.12

14.21

18.34

0.0143

9

R.ulnar I

6.7

0.215

0.215

8.78

10

Aortic arch II

4.0

1.342

1.246

5.11

11

L.carotid

20.8

0.400

0.370

5.92

12

Thoracic aorta I

5.5

1.246

1.124

5.11

13

Thoracic aorta II

10.5

1.124

0.924

5.11

14

Intercoastals

7.3

0.300

0.300

7.13

2.00

6.04

0.0542

15

L.subclavian I

3.5

0.425

0.407

5.29

16

L.vertebral

13.5

0.200

0.200

9.64

6.10

27.87

0.0126

17

L.subclavian II

39.8

0.407

0.230

5.38

18

L.ulnar I

6.7

0.215

0.215

8.78

19

L.radius

22.0

0.175

0.140

10.12

14.21

18.34

0.0143

20

Celiac I

2.0

0.350

0.300

5.86

21

Celiac II

2.0

0.300

0.250

6.54

22

Hepatic

6.5

0.275

0.250

6.86

2.80

17.48

0.0208

23

Splenic

5.8

0.175

0.150

7.22

8.59

22.97

0.0139

24

Gastric

5.5

0.200

0.200

6.40

4.05

9.59

0.0325

25

Abdomainal aorta I

5.3

0.924

0.838

5.11

   

26

Sup.mensenteric

5.0

0.400

0.350

5.77

1.20

4.15

0.0810

27

Abdominal aorta II

1.5

0.838

0.814

5.11

   

28

R.renal

3.0

0.275

0.275

6.05

2.02

4.64

0.0667

29

Abdominal aorta III

1.5

0.814

0.792

5.11

   

30

L.renal

3.0

0.275

0.275

6.05

2.02

4.64

0.0667

31

Abdominal aorta IV

12.5

0.792

0.627

5.11

   

32

Inf.mesenteric

3.8

0.200

0.175

6.25

5.37

33.10

0.0110

33

Abdominal aorta V

8.0

0.627

0.550

5.11

34

R.com.iliac

5.8

0.400

0.370

5.50

35

R.ext.iliac

14.5

0.370

0.314

7.05

36

R.int.iliac

4.5

0.200

0.200

10.10

6.40

25.28

0.0136

37

R.deep femoral

11.3

0.200

0.200

7.88

4.99

14.40

0.0226

38

R.femoral

44.3

0.314

0.275

8.10

39

R.ext.carotid

17.7

0.200

0.200

8.26

5.23

23.53

0.0148

40

L.int.carotid

17.6

0.300

0.275

7.51

2.53

25.42

0.0148

41

R.post.tibial

34.4

0.175

0.175

11.98

9.90

32.78

0.0102

42

R.ant.tibial

32.2

0.250

0.250

9.78

3.96

15.11

0.0226

43

R.interosseous

7.0

0.100

0.100

15.57

39.43

424.01

0.0009

44

R.ulnar II

17.0

0.203

0.180

12.53

10.12

21.20

0.0143

45

L.ulnar II

17.0

0.203

0.180

12.53

10.12

21.20

0.0143

46

L.interosseous

7.0

0.100

0.100

15.57

39.43

424.01

0.0009

47

R.int.carotid

17.6

0.300

0.275

7.51

2.53

25.42

0.0148

48

L.ext.carotid

17.7

0.200

0.200

8.26

5.23

23.53

0.0148

49

L.com.iliac

5.8

0.400

0.370

5.50

50

L.ext.iliac

14.5

0.370

0.314

7.05

51

L.int.iliac

4.5

0.200

0.200

10.10

6.40

25.28

0.0136

52

L.deep femoral

11.3

0.200

0.200

7.88

4.99

14.40

0.0226

53

L.femoral

44.3

0.314

0.275

8.10

54

L. post. tibial

32.2

0.175

0.175

11.98

9.90

32.78

0.0102

55

L. ant. tibial

34.4

0.250

0.250

9.78

3.96

15.11

0.0226

Geometric data were derived from [20, 27], and wave velocities were based on the data in [30]

Here, r0 and r1 denote the proximal and distal radii of an arterial segment, respectively, c0 is the wave velocity at the middle of each arterial segment at the reference state (A0, U0); R0 is the characteristic impedance of each peripheral artery; and R1 is the arteriolar resistance corresponding to each peripheral artery

The parameters used in the 0-D model have been assigned or estimated based on the data reported in [12, 13, 15, 16, 29]. The values of the parameters used in the heart model, the cardiac valve model, and the peripheral circulation model are given in Tables 2 and 3, respectively.
Table 2

Values of the parameters used in the heart model and the cardiac valve model

 

Right atrium

Right ventricle

Left atrium

Left ventricle

Tricuspid valve

Pulmonary valve

Mitral valve

Aortic valve

EA (mmHg ml−1)

0.06

0.55

0.07

2.75

EB (mmHg ml−1)

0.07

0.05

0.09

0.08

Tcp (s)

0.17

0.30

0.17

0.30

Trp (s)

0.17

0.15

0.17

0.15

tc (s)

0.80

0.0

0.80

0.0

tr (s)

0.97

0.30

0.97

0.30

S (mmHg s ml−1)

Pra × 0.0005

Prv × 0.0005

Pla × 0.0005

Plv × 0.0005

B (mmHg s2 ml−2)

0.000016

0.000025

0.000016

0.000025

R (mmHg s ml−1)

0.001

0.003

0.001

0.003

L (mmHg s2 ml−1)

0.0002

0.0005

0.0002

0.0005

Those of Tcp, Trp, tc and tr are for a cardiac duration of 1 s. Pra, Prv, Pla, and Plv represent, respectively, the pressures in the four cardiac chambers. The values for the cardiac valve model were derived from [29] for normal conditions

Table 3

Values of the parameters used in the peripheral circulation model

 

Capillary

Venule

Vein

C, R, L

C, R, L

C, R, L

Upper body

0.03, 0.97, 0.003

0.5, 0.14, 0.001

15.0, 0.03, 0.0005

Lower body

0.1, 0.29, 0.003

1.5, 0.04, 0.001

75.0, 0.009, 0.0005

Super vena cava

5.0, 0.0005, 0.0005

Abdominal and inferior vena cava

15.0, 0.0005, 0.0005

The units of C, R and L are ml mmHg−1, mmHg s ml−1, and mmHg s2 ml−1, respectively. The values of C of the veins and the vena cava were based on the data reported in [12, 13], but with modifications

3 Results

Simulations started from early systole when the ventricles begin to contract. Cardiac duration was set to be 1.0 s and physiological conditions were fixed in resting conditions for all the simulations.

3.1 Normal case

Blood pressure/flow pulses at several typical sites of the cardiovascular system are shown in Fig. 2. The results clearly depict the marked changes in pressure/flow pulse contours when blood travels through the cardiovascular system. Some essential characteristics of arterial pulse transmission have been predicted, such as the increase in systolic pressure from the heart towards the periphery. The simulated flow pulses in the vicinity of the left heart reasonably characterize the normal dynamic operation of the heart as described by in vivo data [29].
Fig. 2

Blood pressure (a)/flow (b) pulses in different regions of the cardiovascular system

3.2 Aortic valvular stenosis and arterial stenoses

3.2.1 Simulation conditions

The model was applied further to study the individual influences of an AV stenosis, and another four stenoses located in the thoracic aorta II (No. 13), the abdominal aorta IV (No. 31), the right renal artery (No. 28), and the right femoral artery (No. 38). The severity of each stenosis was set to be 85%, a critical value beyond which the effect of a stenosis on mean flow becomes important [27]. The lengths of the four arterial stenoses were set uniformly to be 2 cm. The severity of an AV stenosis is generally characterized by the value of EOA rather than a percentage reduction of EOA [10]. In this study, for the purpose of a uniform comparison with the arterial stenoses, we define that an EOA of 0.6 cm2 (15% of the normal value of 4 cm2) represents an 85% AV stenosis.

3.2.2 Influences on arterial pressure pulse

Figure 3 shows the variations of pressure pulse along the arterial tree (from the aorta to the right tibial artery) in the presence of the stenoses. As expected, there are evident pressure drops across the stenoses. The increase in systolic pressure along the arterial tree is augmented in the proximal regions while diminished in the distal regions. This can be explained by the fact that an arterial stenosis reflects forward waves from the heart to enhance backward waves to the proximal regions whilst reducing forward waves transmitted through the stenosis to the distal regions.
Fig. 3

Pressure pulse variations along the aorta up to the right tibial artery (the path is denoted by the bold blue line) in the presence of a aortic valvular (AV) stenosis, b thoracic aortic (TA) stenosis, c abdominal aortic (AA) stenosis, d right renal arterial stenosis, e right femoral arterial stenosis. Blood pressures in the regions immediately proximal to the stenoses are compared with the normal values in f

Four typical regions were chosen for more detailed investigation (Fig. 4). A common characteristic of pressure pulse changes in the proximal regions is the appearance of an overshoot on the pressure pulse contour. The timings of the overshoots correlate well with the locations of the stenoses. In the distal regions (here, the tibial artery), a marked reduction in systolic pressure and flattening of the pulse contour are observed in the simulated results for all the stenoses except for the renal arterial stenosis. Nevertheless, the distal pressure pulses exhibit few characteristics corresponding to the locations of the stenoses.
Fig. 4

Blood pressure pulses in the left ventricle (a), the ascending aorta (b), the left brachial artery (c), and the right anterior tibial artery (d), and left ventricular pressure–volume (P–V) loops (e) and aortic flow waveforms (f) computed for the normal case and for the stenosis cases

The pressure pulses simulated for the 85% AV stenosis are compared with the normal pulses in Fig. 5. We observe that the buildup time of the aortic systolic pressure is prolonged to 0.23 s from a normal value of 0.12 s and the systolic pressure increase from the aorta to the femoral artery is reduced by 57%; both of which are in good agreement with clinical observations [2].
Fig. 5

Comparisons of arterial pressure pulses simulated for the 85% AV stenosis with the normal pressure pulses

3.2.3 Influences on the heart

The AV stenosis exerts the most pronounced impact on ventricular dynamics (Fig. 4e). Particularly, the stroke volume is reduced by about 10.5%. The aortic stenoses have fairly moderate influences on ventricular dynamics. The renal and femoral arterial stenoses have almost no influence on ventricular dynamics. For aortic flow, again the AV stenosis induces the most significant changes (Fig. 4f).

3.2.4 Ankle-brachial index

The ankle-brachial index (ABI = systolic pressure in the ankle arteries (here, the tibial artery)/systolic pressure in the brachial artery), is a simple noninvasive test for the assessment of arterial obstructive disease [4]. Most studies have used a cutoff of 0.9 to define a low ABI value [32]. The computed results show that the ABIs are not significantly different from the normal value in the cases of AV and renal arterial stenoses, but they are lower than 0.92 in the other arterial stenosis cases (Fig. 6). Comparing the ABIs smaller than 0.92 reveals that the ABIs have poor correlation with the stenosis locations.
Fig. 6

ABI for the normal case and for the stenosis cases. AV, TA, AA, RN, and FM denote the locations of the stenoses, with AV aortic valve, TA thoracic aorta, AA abdominal aorta, RN renal artery, and FM femoral artery

4 Discussion

In the closed-loop cardiovascular system, the hemodynamic influences of an arterial stenosis are closely related to the dynamics of the whole system. Particularly, the interaction between the heart and the arterial load is a key determinant of the characteristics of arterial pulses. Therefore, it is important to take into account cardiac-arterial interaction in the study of arterial stenoses. Indeed, Formaggia et al. [7] have demonstrated that imposing a fixed aortic inlet pressure wave leads to significant underestimations of stenosis-associated wave reflections. The present study improved their work by constructing a model of the entire cardiovascular system. A distinct advantage of the model over previous similar ones is that it places arterial wave propagation into a closed-loop global hemodynamic environment, allowing the influences of different arterial stenoses to be characterized and compared within a uniform framework.

Computed results for the five stenoses show that the global hemodynamic influences of the stenoses depend strongly on their locations (see Fig. 3). Particularly, the marked variation of aortic pressure pulse with stenosis location evidences the significance of the present modeling strategy.

Although all the arterial stenoses are predicted to be associated with specific changes in proximal arterial pressures, only the changes corresponding to the aortic stenoses are profound; those associated with the renal and femoral arterial stenoses are fairly insignificant. Moreover, the mechanical performance of the left ventricle (LV) is found to be less affected by the arterial stenoses.

The V3 model [embodying the AV stenosis model (Eq. 14)] proposed by Garcia et al. [8, 10] reasonably reproduced in vivo hemodynamic phenomena associated with a severe AV stenosis, such as prolonged LV ejection, delayed peak transvalvular flow, and increased LV pressure load. Such phenomena are also captured by our model when incorporating their AV stenosis model (see Fig. 4e, f). Our model further propagates the hemodynamic changes induced by the AV stenosis from the heart to the periphery along the arterial system, as manifested by the prolonged and blunted buildup of systolic pressure, and the diminished aortic-peripheral systolic pressure amplification (see Fig. 5). Similar results were obtained in a previous model study [27], but that study prescribed a measured aortic pressure pulse on the inflow boundary and hence could not give information about aortic flow and cardiac dynamics as presented in this study. Furthermore, we find that global hemodynamic interaction together with the Frank–Starling mechanism of the myocardium can in part compensate for stroke–volume (SV) reduction caused by the AV stenosis. The relatively small value predicted for SV reduction (10.5%) is due to an approximate 10% enlargement of the LV end-diastolic volume (EDV). Without the EDV enlargement, reduction in SV would have exceeded 16%. This passive compensatory mechanism is likely to remain valid in patients with chronic AV stenosis since a relatively large LV EDV is often observed preoperatively in this cohort, and which reduces significantly after valvular replacement.

The ABI as an index for assessing stenosis is effective for stenoses present in arteries located in series with the ankle arteries and, at the same time, in parallel with the brachial arteries (see Fig. 6). However, the value of the ABI cannot infer the stenosis location. This is because the relative degrees of brachial and tibial systolic pressure change irregularly with stenosis location (see Fig. 4c, d), and the irregular pressure changes to some extent counteract one another when being used to calculate the ABI.

4.1 Clinical implications

Current angiographic techniques permit a fairly accurate detection of arterial stenosis once clinical symptoms suggestive of the existence of stenotic disease are found. However, minor symptoms are usually difficult to identify and significant symptoms often would not develop (especially at rest) until the late stage of stenotic disease due to inherent cardiovascular compensatory responses. In this sense, analyzing arterial pulses may potentially provide additional insight. Indeed, several arterial- pulse-based indices are being used to assist in diagnosis, such as the pulsatility index [27], and the ABI [4]. The results of this study confirm the validity of the ABI in assessing certain arterial stenoses. Other methods may be explored to exploit the pronounced hemodynamic changes associated with aortic stenoses, such as wave intensity analysis [31] and wavelet transform [18], methods with which the timing and intensity of wave reflection associated with a stenosis may be highlighted. However, the results of arterial pulse analysis depend on measurement accuracy and are potentially affected by the differences between patients, and hence can only serve as a preliminary assessment. On the other hand, this study suggests that stenoses located in arterial branches induce less significant global arterial pulse changes and hence such stenoses would be difficult to detect by analyzing pulses measured in peripheral arteries.

The present study shows that an 85% aortic stenosis has a relatively small influence on cardiac output; however, it induces early wave reflection, leading to higher LV systolic pressure and aortic valve-closing pressure, which may in turn inversely affect the LV by triggering hypertrophic remodeling [14] and hampering myocardium relaxation [17].

The ABI as a valid stenosis indicator is not only limited to stenoses located in certain regions, its specificity for mild/moderate stenoses is low since other changes in arterial properties may also yield a low ABI. In fact, the ABI is also used in clinical practice for screening patients with suspected peripheral arterial disease [4]. A recent clinical study suggested that a low ABI value at rest may be an independent predictor of cardiovascular mortality [32]. In this sense, a quantitative study of the relation between the ABI and arterial stiffening in various regions would help explain clinical observations and explore underlying etiological factors. We believe that the present model will be a useful tool for such a study.

4.2 Limitations

The present findings are based on computations for the cardiovascular system of a young adult without concomitant cardiac or other arterial diseases. Some characteristics of the hemodynamic changes associated with a single arterial stenosis may be complicated by the presence of multiple stenoses or cardiac diseases. Changes in both the severity (here, 85%) and length (here, 2 cm) of the stenoses will alter, though not the characteristics, the degrees of the hemodynamic changes we presented. Furthermore, compensatory responses to the stenoses were not considered, which may have led us to somewhat underestimate the distal hemodynamic variations. These limitations point to areas of future study but do not challenge the fundamental aspects of our conclusions concerning the global influences of a single arterial stenosis.

5 Conclusions

A multi-scale model of the cardiovascular system has been developed and applied to study the global hemodynamic influences of an AV stenosis and arterial stenoses located in various regions. Results show a strong location dependence of the global hemodynamic influences of an arterial stenosis. The five stenoses studied here are among those most frequently found in clinical practice, and hence the model-based findings may serve as a theoretical basis for exploring novel diagnostic methods.

Future model studies should focus on the influences of combined cardiac and arterial diseases that are often found in the elderly. Although the many parameters involved make it difficult to be completely patient-specific, the model furnishes a useful tool for theoretically examining the hemodynamic influences of available patient-specific parameters.

Notes

Acknowledgments

This research was supported by Research and Development of the Next-Generation Integrated Simulation of Living Matter, a part of the Development and Use of the Next-Generation Supercomputer Project of MEXT, Japan. We sincerely thank the reviewers for the helpful comments. Special thanks to P. L. Wilson for help in manuscript preparation.

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Copyright information

© International Federation for Medical and Biological Engineering 2009

Authors and Affiliations

  • Fuyou Liang
    • 1
  • Shu Takagi
    • 1
  • Ryutaro Himeno
    • 1
  • Hao Liu
    • 2
  1. 1.Computational Science Research ProgramRIKENWakoJapan
  2. 2.Graduate School of EngineeringChiba UniversityChiba-ShiJapan

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