A hybrid Windkessel-Womersley model for blood flow in arteries

Journal of Theoretical Biology 462 (2019) 499–513

Contents lists available at ScienceDirect

Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/jtb

A hybrid Windkessel-Womersley model for blood flow in arteries Yasser Aboelkassem a,∗, Zdravko Virag b a b

Department of Bioengineering, University of California San Diego, La Jolla, CA 92093, USA Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia

a r t i c l e

i n f o

Article history: Received 3 May 2018 Revised 31 October 2018 Accepted 3 December 2018 Available online 5 December 2018 Keywords: Pulsatile blood flow Arterial system Windkessel model Womersley solution

a b s t r a c t A hybrid Windkessel-Womersley (WK-W) coupled mathematical model for the study of pulsatile blood flow in the arterial system is proposed in this article. The model consists of the Windkessel-type proximal and distal compartments connected by a tube to represent the aorta. The blood flow in the aorta is described by the Womersley solution of the simplified Navier-Stokes equations. In addition, we defined a 6-elements Windkessel model (WK6) in which the blood flow in the connecting tube is modeled by the one-dimensional unsteady Bernoulli equation. Both models have been applied and validated using several aortic pressure and flow rate data acquired from different species such as, humans, dogs and pigs. The results have shown that, both models were able to accurately reconstruct arterial input impedance, however, only the WK-W model was able to calculate the radial distribution of the axial velocity in the aorta and consequently the model predicts the time-varying wall shear stress, and frictional pressure drop during the cardiac cycle more accurately. Additionally, the hybrid WK-W model has the capability to predict the pulsed wave velocity, which is also not possible to obtain when using the classical Windkessel models. Moreover, the values of WK-W model parameters have found to fall in the physiologically realistic range of values, therefore it seems that this hybrid model shows a great potential to be used in clinical practice, as well as in the basic cardiovascular mechanics research. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction The heart is a muscular pump connected to the systemic and pulmonary circulation. The arterial tree of the systemic circulation is composed of complex branches of viscoelastic blood vessels. The primary job of the heart and its arterial system is to ensure an adequate supply of oxygenated blood and metabolic species to all cells of the body under a wide range of conditions. Blood flow is driven by the pressure gradient along the arterial tree, which serves primarily to regulate the flow in these branching vessels (Ottesen, 2004; Pedley, 1980). The exact three-dimensional modeling of the blood flow in the entire arterial system is difficult to obtain. Therefore, reduced order models are normally used to model blood flow motions in the cardiovascular system (Burattini and Di Salvia, 2007; Burattini and Gnudi, 1982; Burattini and Natalucci, 1998; Burkhoff et al., 1988; Formaggia et al., 2001; Grant and Paradowski, 1987; Heldt et al., 2002; Milisic and Quarteroni, 2004; O’Rourke and Avolio, 1980; Stergiopulos et al., 1992; Ursino, 1999; Ursino and Magosso, 2003; Wang et al., 2003).

∗

Corresponding author. E-mail address: [email protected] (Y. Aboelkassem).

https://doi.org/10.1016/j.jtbi.2018.12.005 0022-5193/© 2018 Elsevier Ltd. All rights reserved.

Mathematical modeling of blood flow in arterial network is very important in the fields of applied and computational cardiovascular medicine (Abram et al., 2007; Coleman and Randall, 1983; Guyton et al., 2006; Werner et al., 2002). Such models have shown to play a crucial role in understanding many of cardiovascular diseases such as; hypertension, atherosclerosis, and atrial fibrillation (Scarsoglio et al., 2016; 2014). The simplest mathematical models are Windkessel or lumped parameter models describe the function of the whole arterial tree in terms of simple parameters such as compliance, resistance, and inertance. Because of their simplicity they are attractive for many clinicians. When using Windkessel model it is very appropriate to use electrical analogy in which the inertance, compliance and resistance are modeled by inductor, capacitor and resistor, respectively. The flow rate corresponds to electrical current and pressure to the voltage. Electrical impedance is voltage to current ratio, and hydraulic impedance is pressure to flow rate ratio in the frequency domain. Arterial input impedance offers complete description of the arterial system and serves as the heart afterload. Moreover, these simplified models have found to be useful in defining the total arterial compliance and resistance, which can explain many of changes in arterial trees due to various vascular abnormalities. On one hand, cardiovascular Windkessel (lumped) models offer a relatively simple and computationally tractable method for

500

Y. Aboelkassem and Z. Virag / Journal of Theoretical Biology 462 (2019) 499–513

capturing the fundamental behavior of the arterial system (Beyar et al., 1987; Hardy et al., 1982; Olansen et al., 20 0 0; Shi et al., 2011; Sun et al., 1997; Virag and Lulic, 2008; Wesseling et al., 1993; Westerhof et al., 2009). These lumped models are considered to be the most common method of modeling blood flow in the arterial network. The main purpose of the Windkessel approach is to model the arterial input impedance (the pressure-to-flow ratio in the frequency domain) as an afterload to the left ventricle. Although the basic essence behind this simple approach is to calculate the blood flow rate as a function of pressure gradient and constant resistance, which often derived based on a wellaccepted electrical circuit analogy (Chung et al., 1997; Shim et al., 2004; Tsitlik et al., 1992; Vis et al., 1997). However, more advanced models that include the time-varying properties of the vascular bed, sinus vortices, and the inertial effects were also derived (Aboelkassem et al., 2015; Avanzolini et al., 1989; Melchior et al., 1992; Smith et al., 2003b). On the other hand, classical Windkessel models can not capture enough details about the blood flow behavior in the arterial system. For instance, pressure and flow wave transmission and reflection cannot be studied using lumped approach. In other words, because of the beating nature of the heart, the flow through the arterial system is pulsatile and governed by complex wave dynamics which is not normally resolved by the lumped models. Although the blood flow pulsatility depends on the blood inertia, the vast majority of the simplified lumped models neglects the inertial effects even under varying flow velocity and acceleration conditions. These simplified models typically assume a constant resistance to blood flow. Therefore, there is an emerging need for advanced, physiologically detailed, accurate, and computationally tractable blood flow models that expand the simplified lumped models to include some of the blood flow complexity. Examples of these possible attempts to advance the blood flow modeling beyond the classical lumped models are given in Aboelkassem et al. (2015), Smith et al. (2003a) and Azer and Peskin (2007). In this paper, we derive two models that describe the blood flow motions in arterial network. The first model belongs to the classical Windkessel models and is based on multi-compartments with 6-elements (WK6). More specifically, The WK6 model is consisting of a proximal viscoelastic and distal elastic compartment, connected by a tube of constant inductor and resistor to mimic the blood flow in the aorta and large arteries. The second model offers a new generation of Windkessel models and uses a hybrid Windkessel-Womersley (WK-K) mathematical formulation in which the classical Windkessel modeling approach of the aortic flow is approximated by the solution of Womersley pulsatile flow equations. This is equivalent to using time-varying resistance in modeling of blood flow motions in large arteries. Both models are applied to simulate the arterial blood flow motions in both healthy and diseased human and animal subjects with measured aortic valve flow rate and aortic root pressure waveform. A detailed comparison between the WK6 and WK-W models in terms of ability to: (i) describe the arterial input impedance; (ii) to mimic changes in the aortic root pressure profile and wave reflections in large arteries; and (iii) to predict physiological frictional and inertial pressure drops in large arteries is presented. The detailed mathematical derivations for both models and the results for each subject are given in the following sections. 2. Material and methods Consider the motion of blood flow from the left ventricle into the main arteries, Fig. 1A. The pulsatile flow rate into the aorta is denoted by Q av , and the aortic root pressure by Pa , Fig. 1B. The simplest mathematical equations that govern this particular

flow-motions can be derived based on the compartmental Windkessel approach. For example, in a Windkessel model with a single compartment, the pressure Pa represents the whole arterial tree pressure, and flow rate Q s represents an output capillary flow rate from the arterial tree into the systemic veins. The pressure in the veins is much smaller than the arterial pressure and usually it is assumed to be zero. It is known that some pressure and flow rate wave reflections can be observed at the aortic root, therefore it is reasonable to use a Windkessel model containing at least two mutually connected compartments, since the interplay of pressure and flow rate between the two compartments can mimics the wave reflections. Here we use a model with two compartments. The first compartment accounts for the proximal part of the aorta (aortic root and a part of ascending aorta) with characteristic pressure Pa . The second compartment represents the rest of the arterial tree, including small arteries, arterioles and capillaries. The pressure in the second compartment is denoted by Ps , and the outflow to the systemic veins is given by the Q s term. The two compartments are then connected by the aorta (represented by a tube of a certain length La and a radius ra ). The aorta and branching arteries delivering blood to the head and upper limbs. Since the majority of the blood flow rate passes through the aorta, we will simply denote this tube as the aorta. The flow rate through the aorta is denoted by Q a . In Fig. 1B, we show the hydraulic representation of the above described Windkessel model. Since the pressure-volume relationship in the aortic root shows hysteresis due to the visco-elastic properties of the arterial wall. Therefore, the proximal compartment in our model is given by the Kelvin-Voigt model. Specifically, the resistance R0 is introduced to account for the wall viscosity and the compliance C0 , is given to approximate the effects of the aortic wall elasticity. The second compartment represents the distal parts of large arteries together with small arteries, arterioles and capillaries. It is reasonable to assume that distal parts of the arterial tree are seen as an elastic compartment of compliance C1 . The total pressure drop within small arteries, arterioles and capillaries is modeled by a resistance Rs . The flow in the connection tube between the two compartments is assumed to be laminar. Thus, this flow motion can be approximated by two different approaches: (i) one-dimensional unsteady Bernoulli equation and (ii) quasi two-dimensional Womersley model. By using the first approach, the 6-element Windkessel (WK6) arises as shown in Fig. 1C. In the second approach, a hybrid Windkessel-Womersley (WK-W) approach is obtained and represented in Fig. 1D. Since both models can be resolved in the frequency domain. The pressure and flow rate are decomposed into a Fourier series, according to the following expressions:

P (t ) =

N 

Snp sin (nω0 t ) + Cnp cos (nω0 t )

(1)

n=0

and

Q (t ) =

N 

SnQ sin (nω0 t ) + CnQ cos (nω0 t ),

(2)

n=0

where n is the harmonic number, N is the total number of harmonics, ω0 is the fundamental frequency ω0 = 2π /T , T is the heart period. It should be noted that N depends on the sampling frequency (number K of divisions of period T). If the sampling time is defined by t = T/K, then N=(K-1)/2. The Fourier coefficients formulae for both the pressure and the flow rate phasors can be√given p p by pˆ n = Sn + iCn and Qˆ n = SnQ + iCnQ , respectively, where i = −1.

Y. Aboelkassem and Z. Virag / Journal of Theoretical Biology 462 (2019) 499–513

501

Fig. 1. The blood flow in arteries problem definition. A schematic that shows a simplified left ventricle-aortic-arteries sub-domains A, simplified aorta-arterial network model using multi-compartments approach B, the six-elements Windkessel (WK6) model C, and finally the proposed hybrid Windkessel-Womersley (WK-K) model D.

In both models the impedance Z0 of the first compartment is defined by

Z0 =

Pˆa 1 = R0 + i ω C0 ˆ ˆ Q av − Q a

(3)

valve flow rate Q av and aortic root pressure Pa , we can use Q av as an input data and then optimize model parameters to obtain the best fit of the measured Pa waveform. In other words, we minimize the RMS error between the calculated and the measured pressure waves as:



and the impedance Zs of the distal compartment is given by

Pˆs Rs Zs = = 1 + iωC1 Rs Qˆ a

(4)

where ω = nω0 . Note that for simplicity, we have omitted the subscript n which refers to the harmonic number, but keep in mind that Eqs. (3) and (4) hold for all harmonics n = 0, 1, . . . N. That will be a rule for all other equations defining solution in the frequency domain. It should be noted that, the two models (WK6 and WK-W) differ only in approximating the aortic flow impedance Za , which is defined by:

Za =

Pˆa − Pˆs Qˆ a

(5)

Modeling of Za for both WK6 and WK-W models are derived separately in the following subsections. Once we know Za , the input impedance Zin is calculated from the expression

Qˆ av 1 1 1 = = + Zin Z0 Za + Zs Pˆa

P-RMS =

N 2 1   model Pˆa − Pˆameasured  N

(7)

n=0

For a given set of model parameters, it is possible to calculate Zin from Eq. (6), Pamodel from Zin and prescribed Q av , and P-RMS by using Eq. (7). We have used a Matlab function lsqnonlin to find a set of model parameters which minimize P-RMS. Once we have the optimal set of parameters, the phasors of Q a , Ps and Q s are calculated. Moreover, the solutions for all quantities in the time-domain can also be obtained by applying the inverse Fourier transform, according to Eqs. (1) and 2. As stated above, our modeling strategy is based on using two models (WK6 and WK-W) to resolve the blood motion in arteries. These two models differ by how Za is formulated. In other words, they differ only by how the flow in the aorta (tube) is obtained as we show next. 2.1. The 6-element Windkessel (WK6) model

(6)

The parameters involved in both models are estimated using optimization technique which can be summarized as follows. In subjects (human, pigs and dogs) with simultaneously measured aortic

The WK6 model is represented using electrical analogue scheme as shown in Fig. 1C. In this model, the flow in the aorta is approximated using the one-dimensional unsteady Bernoulli equation. This governing equation applied to the flow in the tube con-

502

Y. Aboelkassem and Z. Virag / Journal of Theoretical Biology 462 (2019) 499–513

necting the two compartments reads:

 Pa − Ps =

La 0

ρ

∂u dx + PF ∂t

(8)

It should be noted that, we have neglected the difference in kinetic energy at the tube inlet and outlet. Herein, ρ is the blood density (ρ =1050 kg/m3 ), u is the axial velocity, x is axial coordinate measured from the first compartment, and PF is the frictional pressure drop (FPD). The unsteady term representing the inertial pressure drop (IPD)which can be rearranged in the form

 IP D =

La 0

ρ

dQ a ∂u dx = L1 ∂t dt

(9)

where L1 is the inertance (represented by an inductance) coefficient. Although the blood flow in the arterial tree is pulsatile, a common practice in modeling of the pressure drop is to assume a laminar flow and to use steady solution. Therefore, the frictional pressure drop can be defined as

F P D = PF = R1 Q a

(10)

where R1 is a constant resistance. After substituting Eqs. (9) and 10 into Eq. (8), an expression for the impedance Za using Eq. (5) can be obtained in the frequency domain as

Pˆa − Pˆs Za = = iωL1 + R1 Qˆ a

In this part, we derive a hybrid Windkessel-Womersley (WK-W) coupled mathematical model for the study of pulsatile blood flow in the arterial system. The model consists of the proximal and distal compartments connected by a tube to represent the aorta. The proximal compartment represents the aortic root and the ascending aorta is modeled as a viscoelastic wall chamber by using the Kelvin-Voigt model. The distal compartment represents the rest of the arterial network, and it is modeled by an elastic wall chamber as in the two element Windkessel model. The blood flow in the aorta is described by the Womersley solution of the Navier-Stokes equations. The WK-W model is represented using electrical analogue scheme as shown in Fig. 1D. In this model, the quasi twodimensional momentum equation is applied to the flow through the connection tube between the proximal and distal compartments

  ∂u ∂p ∂u μ ∂ = r − ∂t r ∂r ∂r ∂x

(12)

where μ = ρν is the blood viscosity and r is the radial coordinate. In the above equation the Newtonâ;;s law of viscosity is assumed. It is known that in general, blood does not follow the Newton’s law of viscosity, but at the high rate of deformation (is the case in large arteries), the Newton’s law properly describes the relationship between the shear stress and rate of deformation. Moreover, in this model it is assumed that axial velocity is not a function of x and depends only on both t and r. The pressure gradient can be then written as

∂ p Ps − Pa = ∂x La

(13)

Solution of the Eq. (12) in the frequency domain can be obtained using Womersley’s solution (Womersley, 1955). This is can be summarized as



uˆ (r ) =



Pˆa −Pˆs 4La μ Pˆa −Pˆs 1 iLa ρω





ω=0 for ω > 0

ra2 − r 2 for

−

∂ uˆ τˆw = μ | = ∂ r r=ra

J0 (λr/ra ) J0 (λ )



(14)



ˆ

ˆ

− Pa2−LaPs ra for Pˆa −Pˆs La



iν J1 (λ ) ω J0 (λ )

ω=0

for

ω>0

(15)

The frictional pressure drop (FPD) in the connection tube can be also expressed in the frequency domain as

F Pˆ D = PˆF =

2La τˆw ra

(16)

The inertial pressure drop in this case can be calculated in the time domain as

IP D = Pa − Ps − F P D

(17)

where Ps is the pressure in the systemic network of the WK-W model. Now, the impedance Za can be obtained as

Pˆa − Pˆs Za = = Qˆ a

(11)

2.2. Hybrid Windkessel-Womersley (WK-W) model

ρ

where J0 denotes the Bessel  function of the first kind (subscript denotes the order) and λ = ra −iω/ν . Moreover, the wall shear stress in the frequency domain (τˆw ) can be obtained by



8 μL a ra4 π

La λ2

ρ ω 2π μ 2

2

2

−

ω=0

λJ1 (λ ) −1 for

J0 (λ )

for

ω>0

(18)

In this model, two additional unknown parameters, namely; the tube length La and radius ra are introduced to mimic the aorta geometry. To increase the flexibility of the model, we also considered the blood viscosity as an unknown parameter. Thus, in the WK-W model, there is a total of seven model parameters: R0 , C0 , La , ra , ν , C1 , and Rs . 3. Results In this section, we show the results obtained for both human (aging and hypertension) and animal cases using both WK6 and WK-W models. The lumped parameters for both models are determined numerically using a nonlinear optimization technique. The total number of harmonics N used in the frequency domains for each case of study is listen in Table 1. The optimizer algorithm uses the measured aortic root flow rate as an input and compute the required model parameters which minimize the pressure root mean square error (P-RMS), i.e. the difference between measured and calculated aortic root pressure. In Fig. 2(A, B, and C), we show the experimental aortic valve flow rate Q av for each case of study, namely; human-aging, human-hypertension, and animals. The heart period T for all the subject, (Nichols et al., 1993; Segers et al., 2003) is also listen in Table 1. It should be noted that, the aortic flow rate distribution serves as an important input data in the optimization step for model parameter estimation. A list of the optimized model parameters along with their numerical values for both the WK6 and WKW models are given in Tables 2 and 3 respectively. These optimized parameters are then used in both WK6 and the hybrid WK-W models to calculate several hemodynamic results. These results include: the aortic pressure waveform (Pa ), friction pressure drop (FPD), inertial pressure drop (IPD), aortic flow rate (Q a ), input impedance (Zin ), pulsed wave velocity (PWV), and the axial velocity u distributions in the aorta. It should be noted that, the axial velocity u can be obtained only in the case of hybrid WK-W model and can’t be calculated when using the WK6 model. The detailed comparisons between the proposed mathematical models and their validation against several in-vivo data recorded from different species, including humans, dogs, and pigs are shown in the following subsections.

Y. Aboelkassem and Z. Virag / Journal of Theoretical Biology 462 (2019) 499–513

503

Table 1 The table includes; the cardiac beating period T for all the subjects used in the study, (Nichols et al., 1993; Segers et al., 2003), the total number of harmonics N used to construct the flow solution for both the WK6 and WK-K models, and the min zero-crossing frequency fmin of the phase angle for the WK-K model. Human-Hypertension

Animal-Healthy

Param.

Units

28-yr

Human-Aging 52-yr

68-yr

Normal

Mild

Sever

Dog

Pig

T N fmin

s – Hz

0.95 60 2.83

0.95 60 4.21

0.95 60 5.32

0.92 60 3.98

0.92 60 4.92

0.92 60 6.92

0.46 60 3.72

0.52 60 3.84

Fig. 2. The model input aortic-valve blood flow rate (Q av ) for each validation case of study. Human aging data sets digitized from Nichols et al. (1993) are given in A. Human hypertension data sets from Nichols et al. (1993) are shown B. The animal (Dogs and Pigs) data sets digitized from Segers et al. (2003) are shown in C. Table 2 The optimized parameters and corresponding P-RMS of the 6-element Windkessel (WK6) model for the human aging and hypertension cases, as well as for both the dogs and the pigs animal cases. WK6

Human-aging

Human-hypertension

Animal-healthy

Param.

Units

28-yr

52-yr

68-yr

Normal

Mild

Sever

Dog

Pig

R0 C0 L1 R1 C1 Rs P-RMS

mmHg s/ml ml/mmHg mmHg s2 /ml mmHg s/ml ml/mmHg mmHg s/ml mmHg

0.12 4.56 0.0016 0.036 1.81 0.62 3.01

0.26 3.51 0.0014 0.02 0.91 0.83 2.25

0.31 0.59 0.0015 0.03 0.56 1.02 2.51

0.26 1.85 0.0014 0.023 0.91 0.84 1.63

0.38 1.12 0.0013 0.0061 0.70 1.26 2.44

0.55 0.67 0.0012 0.045 0.38 1.79 2.70

0.43 0.072 0.0017 0.07 0.60 1.65 1.43

0.12 0.13 0.0016 0.063 0.50 2.25 1.10

Table 3 The optimized parameters and corresponding P-RMS, fmin , and cw of the hybrid Windkessl-Womersley (WK-W) model for the human aging and hypertension cases, as well as for both the dogs and the pigs animal cases. WK-W

Human-aging

Human-hypertension

Animal-healthy

Param.

Units

28-yr

52-yr

68-yr

Normal

Mild

Sever

Dog

Pig

R0 C0 La ra C1 Rs

mmHg s/ml ml/mmHg cm cm ml/mmHg mmHg s/ml cm2 /s mmHg

0.062 3.44 40.2 1.22 0.49 0.65 0.11 3.56

0.18 2.19 29.41 1.68 0.68 0.86 0.11 2.93

0.26 0.76 24.2 1.95 0.51 1.04 0.11 3.22

0.16 1.43 23.67 1.63 0.62 0.87 0.11 2.28

0.344 1.033 26.46 2.11 0.66 1.26 0.11 2.47

0.48 0.86 27.07 1.67 0.35 1.83 0.11 4.12

0.083 0.74 26.7 1.14 0.072 1.71 0.11 2.25

0.066 0.60 36.9 1.23 0.076 2.31 0.11 1.56

ν

P-RMS

3.1. Human-aging In this part, we use human data from an aging study (Nichols et al., 1993) to validate and to check the ability of both models in reconstructing the measured aortic pressure at different ages, namely 28, 52 and 68 years respectively. The cardiac beating period for all the three subjects is T = 0.95 s. For each data set, model parameters are optimized and identified using its measured aortic valve flow rate. The pressure, flow rate, and input impedance are computed and compared using both the standard WK6 and the proposed hybrid WK-W models. For example, the aortic pressure

(Pa ) is reconstructed and compared against the experimental data for each age, see Fig. 3(A, D, and G) respectively. It is clear that both models were able to capture the main features during both systolic and diastolic phases of the experimental data including, the dicrotic notch. The frictional pressure drop (FPD) distribution during the cardiac cycle for each age are also shown in Fig. 3(B, E, and H) for both WK6 and WK-W models. These particular results have shown that, there is a significant difference between the FPD calculated using the WK6 and the hybrid WK-W models. This difference in the FPD results is very consistent in all the tested cases of the ag-

504

Y. Aboelkassem and Z. Virag / Journal of Theoretical Biology 462 (2019) 499–513

Fig. 3. The pressure waveform validation results using in-vivo human aging data (Nichols et al., 1993). The results show a set of comparison between the WK6 and the WK-W models. This comparison includes the reconstructed aortic pressure waveform (Pa ), the frictional and inertial pressure drops FPD and IPD, respectively. Results of the 28-years old case validation are shown in A-C, for the 52-years old case in D-F, and for the 68-years old case in G-I.

ing study. The difference in the FPD distributions can be explained as; in the WK6 formulation we have assumed a laminar flow and there exist a steady solution to the problem, where in this case the pressure drop scales with the flow rate multiplied by a constant resistance R1 , Eq. (10). However, in reality the blood flow in an arterial tree is pulsatile and a variable resistance in connection with the arterial wall shear stress is indeed required (Aboelkassem et al., 2015; Smith et al., 20 03a; 20 03b), Eq. (16). Thus, the WK-K model is believed to be a better approach to approximate the frictional pressure drop distribution. Therefore, it seems like the WK6 model overestimates the frictional pressure drop of the blood flow in the arterial network when compared with the newly derived hybrid WK-W model. In addition to the aortic pressure and frictional pressure drop results, the inertial pressure difference (IPD) distributions for each age are also shown in Fig. 3(C, F, and I) respectively. These particular results indicate that, the predicted total pressure difference distributions in the arterial network using both models are within the same order of magnitude and only small differences between both models are observed. These differences are due to the ability of each model in estimating the pressure in the systemic network in addition to the original difference in the frictional pressure drop that we have previously shown in Fig. 3(B, E, and H). In Fig. 4(A–C), we show the blood flow rates in the aorta Q a for each age. The results are computed using both the WK6 and the hybrid WK-W models. These comparisons indicate that, the WK6

model predicts a higher flow rate in the aorta when compared with hybrid WK-W model for the 28-years old case of study. Moreover, the time-to-peak value of aortic flow rate during both systole and diastole are different between both models. These differences become smaller when the other cases, namely 52 and 68 years are tested. In other words, both the WK6 and the WK-W models predict similar aortic flow rates for the 52 and 68 years old cases. The estimated input impedance (Zin ) by each model and for each age case is shown in Fig. 5. Both the WK6 and the WK-W models were able to correctly predict the magnitude of the input impedance in the frequency range of (0–10 hz) for the three tested ages, see Fig. 5(A, C, and E). However, both models perform poorly when trying to match the phase angle of the input impedance for the 28-years case of study, especially at higher frequencies (i.e., hz > 5). Nevertheless, the hybrid WK-W performed better when tested the other ages (52 and 68 years) and it was able to better capture impedance phase angles, even at higher frequencies, see Fig. 5(B, D, and F). In Fig. 6, we show the axial velocity component of the blood flow in the aorta. That was only possible when applying our newly derived hybrid WK-W model and solving Eq. (12). Several axial velocity profiles as a function of radial coordinate direction are given at different time points. Specifically, we draw the velocity profile at four time snapshots (t1 , t2 , t3 and t4 ) during the cardiac cycle for each age, see Fig. 6(A–C). These time points are marked as (1,2,3, and 4) on the aortic pressure traces for illustration purpose.

Y. Aboelkassem and Z. Virag / Journal of Theoretical Biology 462 (2019) 499–513

505

Fig. 4. The aortic blood flow rate (Q a ) computed by the current models when using the human aging data sets (Nichols et al., 1993). The results show a set of comparison between the WK6 and the WK-W models. Results of the 28-years old case are shown in A, for the 52-years old case in B, and for the 68-years old case in C.

Fig. 5. The aortic input impedance validation results using in-vivo human aging data (Nichols et al., 1993). The results show a set of comparison between the WK6 and the WK-W models. This comparison includes the reconstructed impedance amplitude (presented in log-scale) as a function of the frequency and the impedance phase angle. Results of the 28-years old case validation are shown in A-B, for the 52-years old case in C-D, and for the 68-years old case in E-F.

These points are chosen such that, they represent the flow during peak systole and during the early and late diastolic phases of the cardiac cycle. The flow pulsatility is clearly shown where the axial flow profile can change from being a jet-like to be a wake-like profile and oscillates during the cardiac cycle. Moreover, the axial velocity contour lines are also shown in the same figure and at the time points. 3.2. Human-hypertension In this part, we use human data from hypertensive study (Nichols et al., 1993) to validate and to check the ability of both the WK6 and the WK-W models in reconstructing the measured

aortic pressure at different hypertension conditions, namely normotensive, mild-hypertension, and sever-hypertension respectively. The cardiac beating period for all the three subjects is T = 0.92 s. For each hypertension condition, model parameters are optimized and identified using its measured aortic valve flow rate. The pressure, flow rate, and input impedance are computed and compared using both models. For example, the aortic pressure (Pa ) is calculated and compared against the experimental data for each condition, see Fig. 7(A, D, and G) respectively. Although both models were able to capture the main features of the experimental data, including the dicrotic notch, the WK-W agrees better with the peak systolic value.

506

Y. Aboelkassem and Z. Virag / Journal of Theoretical Biology 462 (2019) 499–513

Fig. 6. The hybrid Windkessl-Womersley (WK-W) model prediction to the blood flow velocity distribution in the aorta. The axial velocity profiles as a function of the aortic radius are shown in the middle panels at four different time points during systole and diastole of the cardiac cycle. These time points are marked in the top panel of the aortic pressure waveform for illustration. In the right panels, we show the axial velocity contour plots at the same time points. Results of the 28-years old case are shown in A, for the 52-years old case in B, and for the 68-years old case in C.

The FPD distribution during the cardiac cycle for each hypertension condition is calculated and shown in Fig. 7(B, E, and H). There is a significant difference between the FPD calculated using the WK6 and the hybrid WK-W models. This difference appears in all the hypertension cases. These results are consistent with the aging study cases Fig. 3(B, E, and H). Therefore and as explained in the previous subsection, the WK6 model overestimates the frictional pressure drop of the blood flow in the arterial network when compared with the WK-W model. The IPD distribution for each case is also shown in Fig. 7(C, F, and I) respectively. These results suggest that, the total pressure difference in the arterial network using both models are within the same order of magnitude and only some differences between both models are observed. These differences are due to the ability of each model in estimating the pressure in the systemic tree. In Fig. 8(A–C), the blood flow rates in the aorta Q a for each hypertension condition are obtained. The results are compared using both the WK6 and the hybrid WK-W models. These comparisons indicate that, both the WK6 and the WK-W models predict almost similar aortic flow rates for the mild and sever hypertension cases. The models show some differences in the normotensive case of study. The input impedance (Zin ) by each model and for each age case is shown in Fig. 9. Both the WK6 and the WK-W models were able to predict the magnitude of the input impedance in the frequency range of (0–10 hz) for the three hypertension conditions, see Fig. 9(A, C, and E). However, both models perform poorly when trying to match the phase angle of the input impedance for the mild-hypertension case of study, especially at higher frequencies (i.e., hz > 6). Nevertheless, the hybrid WK-W performed better when tested the other hypertension conditions (normo and sever) and it was able to better capture impedance phase angles, even at higher frequencies, see Fig. 9(B, D, and F). In Fig. 10, we show the axial velocity component of the blood flow in the aorta. Again, this only possible when applying the hybrid WK-W model. We draw the velocity profile at four time

snapshots during the cardiac cycle for hypertension condition, see Fig. 10(A–C). The flow pulsatility is clearly shown where the axial flow profile can change from being a jet-like to be a wake-like profile and oscillates during the cardiac cycle. Moreover, the axial velocity contour lines are also shown in the same figure and at the time points. 3.3. Animal (dogs and pigs) The present analysis has kept general and it can be used to model the blood flow not only in the human arterial system but also the animal arterial networks. In this part, we use animal (dogs and pigs) data (Segers et al., 2003) to further validate and compare the efficiency of both the standard WK6 and the hybrid WKW models. The cardiac beating period for the dog case is T = 0.46 s, and for the pig subject is T = 0.52 s. Similar to the human aging and hypertension cases, the aortic pressure, frictional pressure drop, total pressure difference, blood flow rate, and the input impedance for both dogs and pigs are reconstructed using both the WK6 and the hybrid WK-W models. The results from both models are then compared and shown in Figs. 11–14. In Fig. 11(A and D), we show the reconstructed aortic pressure (Pa ) waveform for both dogs and pigs using both WK6 and WK-W models. The simulated pressure is then compared against the experimental data of both animals respectively. These results suggest that both models were able to capture the experimental pressure waveforms during both the systolic and the diastolic phases. In addition, both models were able to capture the dicrotic notch feature correctly. In The Fig. 11(B and E), the FPD distribution during the cardiac cycle for both the dogs and pigs is shown respectively using both models. Once again, there is a significant difference between the FPD calculated using the WK6 and the hybrid WK-W models. As explained previously when reconstructing the human aging and hypertension cases, the WK6 model overestimates the frictional pressure drop of the blood flow in the arterial network

Y. Aboelkassem and Z. Virag / Journal of Theoretical Biology 462 (2019) 499–513

507

Fig. 7. The pressure waveform validation results using in-vivo human hypertension data (Nichols et al., 1993). The results show a set of comparison between the WK6 and WK-W models. This comparison includes the reconstructed aortic pressure waveform (Pa ), the frictional and inertial pressure drops FPD and IPD, respectively. Results of the normotensive validation case are shown in A–C, for the mild-hypertension case in D–F, and for the sever-hypertension case in G–I.

Fig. 8. The aortic blood flow rate (Q a ) computed by the current models when using the human hypertension data sets (Nichols et al., 1993). The results show a set of comparison between the WK6 and the WK-W models. Results of the normotensive case are shown in A, for the mild-hypertension case in B, and for the sever-hypertension case in C.

508

Y. Aboelkassem and Z. Virag / Journal of Theoretical Biology 462 (2019) 499–513

Fig. 9. The aortic input impedance validation results using in-vivo human aging data (Nichols et al., 1993). The results show a set of comparison between the WK6 and the WK-W models. This comparison includes the reconstructed impedance amplitude (presented in log-scale) as a function of the frequency and the impedance phase angle. Results of the normotensive validation case are shown in A-B, for the mild-hypertension case in C-D, and for the sever-hypertension case in E-F.

Fig. 10. The hybrid Windkessl-Womersley (WK-W) model prediction to the blood flow velocity distribution in the aorta. The axial velocity profiles as a function of the aortic radius are shown in the middle panels at four different time points during systole and diastole of the cardiac cycle. These time points are marked in the top panel of the aortic pressure waveform for illustration. In the right panels, we show the axial velocity contour plots at the same time points. Results of the normotensive validation case are shown in A, for the mild-hypertension case in B, and for the sever-hypertension case in C.

Y. Aboelkassem and Z. Virag / Journal of Theoretical Biology 462 (2019) 499–513

509

Fig. 11. The pressure waveform validation results using in-vivo animals data (Segers et al., 2003). The results show a set of comparison between the WK6 and the WK-W models. This comparison includes the reconstructed aortic pressure waveform (Pa ), the frictional pressure drop (FPD) and the inertial pressure difference (IPD). Results of the dogs study are shown in A–C,and for the pigs case in D–F.

Fig. 12. The aortic blood flow rate (Q a ) computed by the current models when using the animal (dogs and pigs) data sets Segers et al. (2003). The results show a set of comparison between the WK6 and the WK-W models. Results of the dogs case are shown in A, for the pigs case in B.

when compared with the hybrid WK-W model. The IPD distributions for dogs and pigs are also shown in Fig. 11(C and F) respectively. These results indicate that, although the IPD distributions in the arterial network using the both models having similar order of magnitude. Yet, the IPD distribution obtained using the WK-W shows more oscillations during the cardiac cycle. In Fig. 12(A and B), we show the blood flow rates in the aorta Q a computed using both the WK6 and the hybrid WK-W models for dogs and pigs respectively. The results have shown that, the

WK6 model predicts a higher flow rate in the aorta when compared with hybrid WK-W model for both animals. The computed input impedance (Zin ) by both models is shown in Fig. 13. For the dogs and pigs subjects, both the WK6 and the WK-W models were able to correctly predict the magnitude of the input impedance in the frequency range of (0–10 hz), see Fig. 13(A and B). The WK6 model has performed better in capturing the impedance phase angle when compared with the WK-W model performance, see Fig. 13(C and D). Similar to the human aging and hypertension

510

Y. Aboelkassem and Z. Virag / Journal of Theoretical Biology 462 (2019) 499–513

Fig. 13. The aortic input impedance validation results using in-vivo animals data (Segers et al., 2003). The results show a set of comparison between the WK6 and the WK-W models. This comparison includes the reconstructed impedance amplitude (presented in log-scale) as a function of the frequency and the impedance phase angle. Results of the dogs study are shown in A and C,and for the pigs case in B and D.

Fig. 14. The hybrid Windkessl-Womersley (WK-W) model prediction to the blood flow velocity distribution in the aorta. The axial velocity profiles as a function of the aortic radius are shown in the middle panels at four different time points during systole and diastole of the cardiac cycle. These time points are marked in the top panel of the aortic pressure waveform for illustration. In the right panels, we show the axial velocity contour plots at the same time points. Results of the dogs case are shown in A and for the pigs case in B.

cases, the blood flow distributions in the aorta for the both the dogs and pigs are rendered using the axial velocity profiles as a function of the radial direction. In Fig. 14(A and B), we draw the velocity profile at four time snapshots during the cardiac cycle for both dogs and pigs using the WK-W model. The corresponding axial velocity contour lines are also shown in the same figure.

3.4. The aortic pulsed wave velocity In this part, we show the WK-W model prediction of the pulsed wave velocity for each simulated case of study. This is another important advantage than the standard Windkessel family of models included our WK6 model. Firstly, let’s recall that the distal compartment, which causes wave reflection, can be considered as

Y. Aboelkassem and Z. Virag / Journal of Theoretical Biology 462 (2019) 499–513

511

Fig. 15. The hybrid Windkessel-Womersley (WK-K) model prediction of the frequency at which the input impedance modulus is minimum and the corresponding pulsed flow velocity for each case of study. The minimum frequency is shown in A, and the pulsed wave velocity in B.

the effective reflection site. The length La of the tube connecting the two compartments can be considered as the effective length of the arterial network. According to the quarter wavelength rule (Nichols and O’Rourke, 2005), the effective length is defined by

La = PW V/4 fmin

(19)

where PWV is the pulse wave velocity and fmin is defined as the frequency at which the input impedance modulus is minimum and the zero-crossing of the phase angle occur. This rule is based on the assumption that zero-crossing of the phase angle and the minimum of the impedance modulus are at the same frequency. Secondly, we can calculate fmin (Table 1) using the frequency of zero-crossing of the phase angle from Figs. 5, 9 and 13 along with the optimized La values for each case (Table 3) to calculate PWV using Eq. 19. The results for fmin and PWV for all the human aging, hypertension and animal cases are given in Fig. 15. For example, in Fig. 15(A) we show the calculated fmin values for each case. These particular results indicate that, the frequency of zero-crossing of the phase angle is increasing in both human aging and hypertension cases and has almost the same value for both dogs and pigs. Moreover, the predicated PWV for each case is shown in Fig. 15(B) and suggested that the PWV increases and strongly correlated with aging and hypertension diseased conditions. These results are consistent with the experimental observations (Nichols and O’Rourke, 2005). The PWV in pigs has shown to be greater than dogs, this because the optimized La for pigs was greater than the one in dogs, see Table 3.

4. Discussion In Winkessel models, it is common to represent flow elements using its circuit analogous components. In other words, is convenient to use an electrical analogy when modeling, flow in arteries because the conversion from a hydraulic to the electrical scheme is straightforward. The problem of using mechanical analogies when dealing with hydraulic or electrical systems is in fact that the serial arrangement of hydraulic/electrical elements becomes a parallel arrangement of mechanical elements. For example, the serial arrangement of capacitor and resistor (Kelvin-Voigt model of the proximal chamber wall) in electrical scheme, becomes a parallel arrangement of spring and dashpot in the mechanical scheme. In this study, we formulated the proximal and the distal compartments in both the WK6 and WK-W models using electrical analogy as shown in Fig. 1. The blood flow motion in the tube (aorta) between the two compartments is described by the Womersley solution of the Navier-Stokes equations (WK-W), and by the one-dimensional unsteady Bernoulli equation (WK6).

Both the WK6 and hybrid WK-W models were able to reconstruct the measured aortic root pressure waveform very well. This is clearly shown by the results presented in the top panels of Figs. 3 and 7 (human aging and hypertension data) and left panels in Fig. 11 (dogs and pigs data). It is also important to point out the ability of both models to properly describe the inflection point in the pressure profile which comes from the pressure wave reflection from the periphery. The dicrotic notch associated with the aortic valve closure has been well captured by both models as well. This modeling capability is due to the use of two compartments in the model, and the interplay between compartments which can mimic the wave reflections in an arterial tree. Although both models performed very well in approximating the aortic pressure, the WK6 and the hybrid WK-W models show a great difference in predicting the friction pressure drop (FPD), see the middle row of panels in Figs. 3 and 7, and middle column of panels in Fig. 11. This difference in the FPD results are very consistence in all the tested aging and hypertension cases, and can be explained by a different nature of the friction term in the two models. In the WK6 formulation the pressure drop is modeled by the Hagen-Poiseuille formula which is valid for steady laminar flow, and in the WK-W model we used the Womersly solution which describes the nature of pulsatile flow in the aorta more realistically. For instance, the average pressure drop for the human subjects when using the WK-W model varies from 0.5811 mmHg (28-years) to 0.0722 mmHg (68-years), while in WK6 model this range is from 4.888 mmHg (28-years) to 0.3844 mmHg (68-years). These differences between both models are also observed in the human hypertension results. Similarly, the average pressure drop for the animals subjects when using the WK-W model varies from 0.1887 mmHg (Dogs) to 0.1975 mmHg (Pigs), while in WK6 model this range is from 3.216 mmHg (Dogs) to 2.967 mmHg (Pigs). Since it is commonly known that the friction pressure drop in large arteries is negligible with respect to pressure drop in small arteries, arterioles and capillaries. Therefore, we can state that the WK-W model shows more realistic results and agrees well with the physiological observations, and that the WK6 model overestimates the frictional pressure drop. The IPD from two models (see bottom panels in Figs. 3 and 7 and right columns in Fig. 11) are a bit different in the case of the young (i.e., 28 years) human aging data and the normotensive subject and animal data. While the other aging or hypertension cases the difference between both model solutions become smaller and smaller. The WK6 model shows the similar shape of IPD in all cases, while the WK-W model shows smooth oscillation in IPD in cases of the young subject and animal data. Similar behavior was observed in the human hypertension cases. The better agreement between two models in case of older subjects can be explained by the reduction in the proximal compliance

512

Y. Aboelkassem and Z. Virag / Journal of Theoretical Biology 462 (2019) 499–513

(i.e. by the more stiff aorta and higher pulse wave speed), when Q a becomes similar to Q av , i.e. the difference in Q a in the two models become very small (see Fig. 4B and C). In young subject and in animals (we can assume that animals were also young) the difference between Q a in the two models is significant (see Fig. 4A and Fig. 12A and B) and consequently the difference in IPD is great. It should be noted that, the flow Qav-Qa denotes the rate of volume accumulation in the proximal part of the arterial tree. The integral of this flow over a single heart period is zero. During systole this flow is positive, and during diastole it is negative. Figs. 5, 9 and 13 show comparison of the measured and calculated input impedance Zin (in terms of amplitude and phase angle). Both models show the minimum in the amplitude of input impedance and positive values of the phase angle. Since the positive phase angle denotes the dominant inertial effects, we can conclude that the inertial effects in the connection tube provide positive values of the phase angle. The WK6 model shows a bit smoother change in the phase angle. It is visible from Tables 1 and 2 that, the WK6 model shows a bit smaller P-RMS error, and if we take into account that this model is computationally cheaper it can be recommended as a good choice when only a proper description of the input impedance is required. Moreover, the obtained optimized parameters when using the WK-W suggests that the blood viscosity parameter ν remains constant (for the second significant digit). This reduces the number of parameters in the WK-W by one to have equal number of parameters as the WK6 model. Although both models predict the input impedance of the arterial tree quite well, it is important to see if this prediction is based on physiologically relevant values of the model parameters, and if the change of model parameters can explain changes for example in aging. It could be expected that in both models the compliance C0 of the proximal compartment is greater than C1 of distal compartment, since the majority of the arterial compliance is related to a part of the aorta close to the heart. It is visible from Tables 1 and 2 that in some cases WK6 model shows greater C1 than C0 , while WK-W model always shows C0 greater than C1 which is in a better agreement with expectation and arterial physiology. Both models show decrease in C0 (as well as in sum C0 +C1 ) with aging, and this is correct from physiological point of view as well. It should be noted that, the blood viscosity is considered as a free parameter in WK-W model and the obtained value of kinematic viscosity is about 0.11 cm2 /s, which is about 2.9 times greater than the commonly used viscosity (4 mPa s). This is however interesting because of, the WK-W model shows nearly a constant viscosity for all the tested cases. Also, the value of ra in WK-W model increase with aging which is in accordance with observations, and the obtained values for different subject is in reasonable range. Moreover, if we consider La as the effective length of the arterial network, the pulse wave velocity calculated from the fmin (frequency of zero-crossing of the phase angle) using the quarter wavelength rule shows pretty realistic values for both aging and hypertension disease conditions, see Fig. 15. It should be noted that, the capability to predicting the pulsed wave velocity is not possible to obtain when using the classical Windkessel models including the proposed WK6 model. Another advantage of the proposed hybrid WK-W model is the ability to predict the axial velocity component of the blood flow in the aorta, see Fig. 6, 10 and 14. This is expected to be a very valuable hemodynamic information in the arterial network. One can use this information to find the velocity gradient at the wall of the aorta, and hence use it to estimate the wall shear stress distribution over the cardiac cycle using Eq. (15). Eventually, the information obtained from the wall shear stress can be correlated to many of cardiovascular diseases. Certainly, this was not possible to obtain when using the classical WK6 model. The velocity distribution results clearly show the blood flow pulsatility in the aorta, where the

axial velocity profiles can change during the cardiac cycle from being jet-like to be wake-like profiles. The maximum velocity occurs at the aorta center line for all the times during the cardiac cycle. The peak value of axial velocity is the highest in young subject and starts to decrease with aging. This is very consistent with the input flow rate and pressure distributions in the aorta for the three simulated aging cases. The velocity contour lines clearly show this behavior. Similar results are obtained for the human hypertension and the animal cases. Finally, it should be noted that, we have focused our attention on presenting results such as; the pressure, flow rate, velocity profiles, and frictional pressure drop in the aorta only. Other flow information such as the pressure and flow rate in the second compartment (systemic circuit) for instance can also be obtained. Although, these systemic hemodynamic distributions are not physiological, yet they might provide a useful information about how the lumped systemic circuit behaves in such simplified arterial models.

5. Conclusion Based on comparison of hybrid WK-W and classical WK6 model, we can conclude: (i) Both models reconstruct the aortic root pressure, from the aortic valve flow with nearly same accuracy (WK6 model shows slightly smaller P-RMS error). Calculated input impedance from both models is in good agreement with the measured one (the amplitudes of the first few harmonics are very close to the measured ones and the phase angle changes the sign at a proper frequency). If one is interested only in the description of arterial input impedance, then the WK6 model is recommended. (ii) Both models properly show reduction in compliances in cases of ageing and hypertension. (iii) The hybrid WK-W model shows quite realistic friction pressure drop in the aorta (according to physiological observations, the friction pressure drop in large arteries is less than few mmHg, and it is negligible with respect to pressure drop in small arteries, arterioles and capillaries), while the WK6 model hardly overestimates this pressure drop, and cannot be related to any physiological data. Because of that the parameters in WK-W model have a clearer physiological meaning than in the WK6 model. (iv) WK-W model parameters related to aorta (radius and length) are consistent, and fall in the range of physiologically obtained values, and are sensitive with respect to spices. The viscosity is consistently over-predicted with respect to the values reported in literature. (v) Additionally, WK-W model predicts pulse wave velocity (another important parameter for estimation of the aortic stiffness). Again model shows changes of this velocity in the cases of ageing and hypertension which are quite in accordance to observations. Finally, results from WK-W model include a time variation of the axial velocity profile, the aortic wall shear stress, and more realistic description of wave reflection and it seems that this hybrid model could be well used for clinical applications and in the basic cardiovascular mechanics research.

Disclosure No conflicts of interest, financial or otherwise, are declared by the authors.

References Aboelkassem, Y., Savic, D., Campbell, S., 2015. Mathematical modeling of aortic valve dynamics during systole. J. Theor. Biol. 365, 280–288. Abram, S., Hodnett, B., Summers, R., Coleman, T., Hester, R., 2007. Quantitative circulatory physiology: an integrative mathematical model of human physiology for medical education. Adv. Physiol. Ed. 31, 202–210.

Y. Aboelkassem and Z. Virag / Journal of Theoretical Biology 462 (2019) 499–513 Avanzolini, G., Barbini, P., Cappello, A., Cevenini, G., Pohl, V., Sikora, T., 1989. Tracking time-varying properties of the systemic vascular bed. IEEE Trans. Biomed. Eng. 3, 37381. Azer, K., Peskin, C.S., 2007. A one-dimensional model of blood flow in arteries with friction and convection based on the Womersley velocity profle. Cardiovasc. Eng. 7, 51–73. Beyar, R., Hausknecht, M.J., Halperin, H.R., Yin, F.C., Weisfeldt, M.L., 1987. Interaction between cardiac chambers and thoracic pressure in intact circulation. Am. J. Physiol. 253, H1240H1252. Burattini, R., Di Salvia, P., 2007. Development of systemic arterial mechanical properties from infancy to adulthood interpreted by four-element Windkessel models. J. Appl. Physiol. 103, 66–79. Burattini, R., Gnudi, R., 1982. Computer identidcation of models for the arterial tree input impedance: comparison between two new simple models and first experimental results. Med. Biol. Eng. Comput. 20, 134–144. Burattini, R., Natalucci, S., 1998. Complex and frequency-dependent compliance of viscoelastic Windkessel resolves contradictions in elastic Windkessels. Med. Eng. Phys. 20, 502–514. Burkhoff, D., Alexander, J., Schipke, J., 1988. Assessment of Windkessel as a model of aortic input impedance. Am. J. Physiol. Heart Circ. Physiol. 255, 742–753. Chung, D., Niranjan, S., Clark, J., Bidani, A., Johnston, W., 1997. A dynamic model of ventricular interaction and pericardial influence. Am. J. Physiol. 272, H2924H2962. Coleman, T., Randall, J., 1983. Human: a comprehensive physiological model. Physiologist 26, 15–21. Formaggia, L., Gerbau, J.-F., Nobile, F., Quarteroni, A., 2001. On the coupling of 3d and 1d Navier-Stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Eng. 191, 561582. Grant, B., Paradowski, L., 1987. Characterization of pulmonary arterial input impedance with lumped parameter models. Am. J. Physiol. Heart Circ. Physiol. 252, 585–593. Guyton, A., Coleman, T., Granger, H., 2006. Circulation: overall regulation. Annu. Rev. Physiol. 34, 13–44. Hardy, H.H., E., C.R., E., C.R., 1982. A digital computer model of the human circulatory system. Med. Biol. Eng. Comput. 20, 55064. Heldt, T., Shim, E., Kamm, R., Mark, R., 2002. Computational modeling of cardiovascular response to orthostatic stress. J. Appl. Physiol. 92, 1239–1254. Melchior, F.M., Sriniwasan, R.S., Charles, J.B., 1992. Mathematical modeling of human cardiovascular system for simulation of orthostatic response. Am. J. Physiol. 262, H1920H1933. Milisic, V., Quarteroni, A., 2004. Analysis of lumped parameter models for blood flow simulations and their relation with 1d models. Math. Model. Numer. Anal. 38, 613–632. Nichols, W., O’Rourke, M., 2005. McDonald’s blood flow in arteries. theoretical, experimental and clinical principles. 5th ed.. Arnold, UK. Nichols, W.W., Avolio, A., Kelly, R., M.F., O., 1993. Effects of age and of hypertension on wave travel and reflections. In: O’Rourke, M., Safar, M., Dzau, V. (Eds.), Arterial Vasodilatation: Mechanisms and Therapy. Edward Arnold, London, pp. 23–40. Olansen, J., Clark, J., Khoury, D., Ghorbel, F., Bidani, A., 20 0 0. A closed-loop model of the canine cardiovascular system that includes ventricular interaction. Comput. Biomed. Res. 33, 26095. O’Rourke, M., Avolio, A., 1980. Pulsatile flow and pressure in human systemic arteries. studies in man and in a multibranched model of the human systemic arterial tree. Circ. Res. 46, 363–372.

513

Ottesen, J., 2004. Applied Mathematical Models in Human Physiology. Pedley, J.T., 1980. The Fuid Mechanics of Large Blood Vessels. Cambridge University Press, UK. Scarsoglio, S., Camporeale, C., Guala, A., Ridolfi, L., 2016. Fluid dynamics of heart valves during atrial fbrillation: a lumped parameter-based approach. Comput. Methods Biomech. Biomed. Eng. 19, 10601068. Scarsoglio, S., Guala, A., Camporeale, C., Ridolfi, L., 2014. Impact of atrial fbrillation on the cardiovascular system through a lumped-parameter approach. Med. Biol. Eng. Comput. 52, 905–920. Segers, P., Stergiopulos, N., Westerhof, N., Wouters, P., Kohl, P., Verdonck, P., 2003. Systemic and pulmonary hemodynamics assessed with a lumped-parameter heart-arterial interaction. J. Eng. Math. 47, 185–199. Shi, Y., Lawford, P., Hose, R., 2011. Review of zero-d and 1-d models of blood ow in the cardiovascular system. Biomed. Eng. Online 10, 33. Shim, E.B., Sah, J.Y., Youn, C.H., 2004. Mathematical modeling of cardiovascular system dynamics using a lumped parameter method. Jpn. J. Physiol. 54, H1920H1933. Smith, B.W., Chase, J.G., Nokes, R.I., Shaw, G.M., David, T., 2003. Velocity profle method for time varying resistance in minimal cardiovascular system models. Phys. Med. Biol. 48, 3375–3387. Smith, B.W., Chase, J.G., Nokes, R.I., Shaw, G.M., Wake, G., 2003. Minimal haemodynamic system model including ventricular interaction and valve dynamics. Med. Eng. Phys. 26, 131–139. Stergiopulos, N., Westerhof, B., Westerhof, N., 1992. Total arterial inertance as the fourth element of the Windkessel model. Am. J. Physiol. Heart Circ. Physiol. 276, 81–88. Sun, M., Beshara, Y., Lucariello, R.J., Chiaramida, S.A., 1997. A comprehensive model for right-left heart interaction under the infuence of pericardium and baroreflex. Am. J. Physiol. 272, H1499H1515. Tsitlik, J.E., Halperin, H.R., Popel, A.S., Shoukas, A.A., Yin, F.C., Westerhof, N., 1992. Modeling the circulation with three-terminal electrical networks containing special nonlinear capacitors. Ann. Biomed. Eng. 20, 595616. Ursino, M., 1999. A mathematical model of the carotid baroregulation in pulsating conditions. IEEE Trans. Biomed. Eng. 46, 382–392. Ursino, M., Magosso, E., 2003. Role of short-term cardiovascular regulation in heart period variability: a model study. Am. J. Physiol. Heart Circ. Physiol. 284, 1479–1493. Virag, Z., Lulic, F., 2008. Modeling of aortic valve dynamics in a lumped parameter model of left ventriculararterial coupling. Ann. Univ. Ferrara. Vis, M.A., Sipkema, P., N., W., 1997. Modeling pressure-flow relations in cardiac muscle in diastole and systole. Am. J. Physiol. 272, H1516H1526. Wang, J., O’Brien, A., Shrive, N., Parker, K., Tyberg, J., 2003. Time-domain representation of ven-triculararterial coupling as a Windkessel and wave system. Am. J. Physiol. Heart Circ. Physiol. 284, 1358–1368. Werner, J., Bohringer, D., Hexamer, M., 2002. Simulation and prediction of cardiotherapeutical phenomena from a pulsatile model coupled to the Guyton circulation model. IEEE Trans. Biomed. Eng. 49, 430–439. Wesseling, K.H., Jansen, J.C., Settels, J.J., Schreuder, J.J., 1993. Computation of aortic flow from pressure in humans using a nonlinear three-element model. J. Appl. Physiol. 74, 2566–2573. Westerhof, N., Lankhaar, J.W., Westerhof, E.B., 2009. The arterial Windkessel. Med. Biol. Eng. Comput. 47, 131–141. Womersley, J.R., 1955. Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. 127, 553–563.