Journal of Theoretical Biology 365 (2015) 280–288

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Mathematical modeling of aortic valve dynamics during systole Yasser Aboelkassem n, Dragana Savic, Stuart G. Campbell Department of Biomedical Engineering, Yale University, New Haven, CT 06520, USA

H I G H L I G H T S







Mathematical modeling of aortic valve dynamics is given. The effects of sinus vortices are included. The effects of variable vascular resistance are included. Aortic valve opening–closing phases are captured.

art ic l e i nf o

a b s t r a c t

Article history: Received 9 July 2014 Received in revised form 17 October 2014 Accepted 23 October 2014 Available online 4 November 2014

We have derived a mathematical model describing aortic valve dynamics and blood flow during systole. The model presents a realistic coupling between aortic valve dynamics, sinus vortex local pressure, and variations in the systemic vascular resistance. The coupling is introduced by using Hill's classical semispherical vortex model and an aortic pressure–area compliance constitutive relationship. The effects of introducing aortic sinus eddy vortices and variable systemic vascular resistance on overall valve opening–closing dynamics, left ventricular pressure, aortic pressure, blood flow rate, and aortic orifice area are examined. In addition, the strength of the sinus vortex is coupled explicitly to the valve opening angle, and implicitly to the aortic orifice area in order to predict how vortex strength varies during the four descriptive phases of aortic valve motion (fast-opening, fully-opening, slow-closing, and fastclosing). Our results compare favorably with experimental observations and the model reproduces wellknown phenomena corresponding to aortic valve function such as the dicrotic notch and retrograde flow at end systole. By invoking a more complete set of physical phenomena, this new model will enable representation of pathophysiological conditions such as aortic valve stenosis or insufficiency, making it possible to predict their integrated effects on cardiac load and systemic hemodynamics. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Aortic valve Blood flow transport Sinus vortex Orifice area Systemic vascular resistance

1. Introduction Tricuspid, pulmonary, mitral and aortic valves of the heart regulate blood flow, transforming contraction of the heart chambers into bulk movement of blood through the circulatory system. These valves have complex biological structures that interact dynamically with the surrounding hemodynamic medium. They function in a cyclic manner over a billion times during the entire human lifespan. Valves experience diverse kinds of dynamic loading, such as large pressure gradients, pulsatile flow impingement, sinus vortices, and oscillatory shear stress. An overview of cardiac valve dynamics and recent research efforts in aortic valve mechanobiology can be found in Yoganathan et al. (2000) and Balachandran et al. (2011). The aortic

n

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

http://dx.doi.org/10.1016/j.jtbi.2014.10.027 0022-5193/& 2014 Elsevier Ltd. All rights reserved.

valve plays a critical role in left ventricular mechanics and blood flow. As blood leaves the contracting ventricle, the aortic valve is the first structure it encounters. As such, its properties have a direct bearing on the afterload applied to the left ventricle. Restriction of blood flow through the aortic valve is a serious pathological condition known as stenosis. Aortic valve stenosis and the accompanying increase in the afterload placed on the heart lead to cardiac hypertrophy, heart failure, and even sudden cardiac death (Pellikka et al., 2005; Briand et al., 2005). Another function of the aortic valve is to prevent retrograde flow from the aorta back into the left ventricle during the diastolic phase of the cardiac cycle. Aortic valve insufficiency or regurgitation is a condition that results when the valve fails to prevent retrograde flow due to congenital defects or other valve pathology. Chronic aortic regurgitation is associated with ventricular chamber enlargement and high mortality (Ishii et al., 1996). It is not clear why aortic stenosis and regurgitation trigger distinct ventricular remodeling responses. Part of

Y. Aboelkassem et al. / Journal of Theoretical Biology 365 (2015) 280–288

the reason is that it is difficult to intuitively reason through the consequences that a valve problem might have on the closed-loop, feedback-regulated cardiovascular system. A multi-scale cardiovascular model that included a sufficiently detailed but computationally tractable representation of aortic valve function would enable new insights into questions such as the mechanisms underlying differential cardiac remodeling in aortic valve pathologies. Mathematical representations of the aortic valve range from simple diode-type models of unidirectional flow (Ottesen et al., 2004; Shi et al., 2011) to high-order fluid–solid interaction models with realistic geometry (Chandra et al., 2012). Diode models are not sufficiently detailed to represent aortic valve stenosis or regurgitation, while the most complex finite-element representations limit opportunities to model effects on the scale of the intact cardiovascular system. Hence, we sought a valve model of intermediate complexity that would capture the largest features of pressure-flow behavior while remaining computationally tractable enough for integrative models of the cardiovascular system. Two key physical components of aortic valve function are (a) sinus vortices and (b) valve geometry. Eddy currents (vortices) are formed in the sinuses of Valsalva and can be observed in both vivo and vitro experiments. They are typically generated behind the leaflets as a consequence of ejected blood flow transport from left ventricle to aorta and are accommodated by the anatomical structure of the aortic root region. Several studies indicate that eddy currents generated by those sinus vortices play an important role in the physiological closure of the aortic valve (Robicsek, 1991). Bellhouse and Talbot (1968) accounted for the presence of sinus vortices in their analysis of heart valve dynamics. An estimate of the local vortical pressure loading over a single leaflet using Hill's model (Milton et al., 1973) was also derived and related to mean aortic pressure. This relationship can be useful to integrate vortical effect on the dynamics of valve leaflets. Moreover, a model that studies specifically the closing behavior of the aortic valve for different closing frequencies or Strouhal numbers is given by Van Steenhoven and Van Dongen (1979). A classical summary for experimental, theoretical, and computational methods that involve the fluid dynamics of heart valve modeling is given by Peskin (1982). More recently, a mathematical model by Virag and Lulic' (2008) proposes an accurate relationship between the valve orifice area and the volume swept by the leaflets in order to model aortic valve function. Their model can

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predict the dynamical phases of opening–closing thereby generating physiological pressure and velocity waves. In addition, it can reproduce the dichrotic notch and predicts the amount of back flow during closing. However, the model does not account for the existence of sinus vortices and assumes fixed systemic vascular resistance. In this paper, we extended the work by Virag and Lulic' (2008) to derive a simplified, yet realistic aortic valve mathematical model during systole. The model considers the following physical features of ventricular–aortic coupling: (i) the existence of sinus eddy vortices behind leaflets, (ii) the variance in systemic vascular resistance owing to the fact that systemic arterial walls are elastic and their properties can change, and (iii) the strong relationship between the local pressure induced by sinus vortices and the downstream aortic pressure. The effects of these parameters on the left ventricular pressure, average aortic pressure, flow rate, and orifice area are given. In addition, the strength of the sinus vortex is coupled to the valve opening angle explicitly and to the orifice area implicitly. This coupling shows how the vortex strength varies during the four known valve opening–closing phases, namely fast opening, fully open, slow closing, and fast closing.

2. Method 2.1. Problem formulation Consider the motion of blood flow transport from the left ventricle to the aorta during the systolic cardiac phase. The blood is ejected through aortic valve leaflets that open and close according to the pressure gradient felt on both ventricular and aortic sides, Fig. 1(a). In normal conditions, current-eddies with coherent structures and variable strengths are formed behind these leaflets in semi-spherical anatomical regions known as sinuses. These eddy structures (vortices) oscillate and their local pressure interacts with the central luminal pressure located downstream in the aorta. The result is intense pressure loading on valve leaflets affecting valve dynamics and flow characteristics. Another important parameter which is also believed to greatly influence valve function is the peripheral systemic vascular resistance (SVR). SVR is a dynamic property that depends on aortic geometry and its instantaneous pressure. Therefore, accounting for SVR in a realistic manner rather

Fig. 1. Problem definition: (a) schematic that shows a simplified left ventricle-aortic sub-domains and shows the force induced by sinus vortex on leaflets. (b) The piece-wise left ventricle interpolation function used in the simulation and proposed by Virag and Lulic' (2008).

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than assuming constant aortic flow resistance should improve the accuracy and the predictive ability of aortic valve models. Here, we present a pressure–area relationship that accounts for the variations in the systemic vascular resistance and arterial wall compliance. The mathematical modeling that governs this particular flow-valve dynamical interaction problem will be derived based on the Bernoulli equations, orifice area–swept volume, pressure–volume, and pressure–area constitutive relationships (Wesseling et al., 1993). 2.2. Mathematical model A mathematical extension to the aortic valve model by Virag and Lulic' (2008) is derived below. The present analysis includes the effects of both sinus vortices and variations in SVR on aortic valve dynamics during systole. This is accomplished by coupling the conventional pulsed ventricular–aortic lumped model (typically derived based on the orifice area and Bernoulli equations) to the sinus eddies via Hill's half-vortex model (Bellhouse and Talbot, 1968; Milne-Thomson, 1960) and to SVR using the model of Wesseling et al. (1993). The blood flow is assumed to be Newtonian, incompressible, and isothermal with constant viscosity μ. Blood variables such as velocities, pressures, and flow rate in each sub-region (ventricular, aortic, and vascular) are hypothesized to be space-independent and vary only with the cardiac cycle. Following the model of Virag and Lulic' (2008), the governing equations for the above-mentioned problem can be given as follows. During systole the mitral valve is tightly closed, the flow rate transport or the rate of change of left ventricular volume can be given by dV lv ¼ Q L  Q av dt

ð1Þ

where Vlv is the left ventricular volume, QL is the flow rate swept by the motion of valve leaflets as described by Eq. (16), and Qav is the aortic flow rate across the valve. In a similar way, the time rate of change for the volume of the systemic arteries is dV sa ¼ Q av Q sc dt

ð2Þ

Lsa is the systemic inertial length for the flow acceleration between the aortic root and the systemic veins. Rsc(t) represents the overall time varying SVR which embodies arterial wall compliance. Psv is the pressure in the systemic veins which can be considered as a constant. It should be noted that both systemic vascular resistance Rsc(t) and lumped area Asa(t) are time dependent because they are coupled to the aortic pressure Psa(t) which varies in time. Since the blood flow through systemic segments (arterioles and capillaries) is assumed to be laminar, the pressure drop along systemic segments can be given using the Poiseuille formula

ΔP ¼ P sc  P sv ¼ Rsc ðtÞQ sc

ð6Þ

The flow through the systemic segments Qsc can be calculated using the pulsed wave velocity Vp as follows: Q sc ¼ Asc V p

ð7Þ

Now, using Eqs. (6) and (7), a relationship between pressure drop at two locations and systemic resistance, and pulsed flow velocity is obtained

ΔP ¼ P sc  P sv ¼ Rsc Asc V p

ð8Þ

Differentiate the above equation with respect to Asc, and recall that the systemic compliance per unit length can be given by C sc ¼ dAsc =dP sc , therefore 1 dP sc ¼ ¼ Rsc V p C sc dAsc

ð9Þ

Using Eq. (9), assuming that the systemic compliance equal to arterial compliance, an expression for the systemic resistance as a function of the pulsed flow velocity and compliance can be obtained from the above equation Rsc ¼

1 C sa V p

ð10Þ

According to Langewouters et al. (1984), the pulsed flow velocity can be approximated by sffiffiffiffiffiffiffiffiffi 1 ð11Þ Vp ¼ LC sa

where Vsa is the systemic volume and Qsc is the blood flow rate through the systemic capillaries. Since valve leaflets sweep the flow forward during the opening phase, the rate of change of the volume swept by the leaflets VL can be given as

where L is the systemic inertance, which is related to blood density ρ and area Asa as

dV L ¼ QL dt

Using Eqs. (10)–(12), an expression to time-varying systemic vascular resistance can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð13Þ Rsc ¼ ρ=ðAsa C sa Þ

ð3Þ

Now, if we assume that both left ventricle and systemic arteries have negligible kinetic energies and that the frictional losses induced by the pressure drop across the aortic valve are minor, the unsteady Bernoulli equation for flows across the aortic valve can be written as dQ av Aav 1 1 ¼ ðP  P Þ  Q2 dt ρLav lv sa 2 Aav Lav av

ð4Þ

where Aav is the root area of the aortic valve. Lav is an inertial length accounting for the blood volume and the mass of the leaflets is assumed to be constant. ρ is the blood density, Plv is the pressure in the left ventricle, and Psa is the pressure in the proximity of the aortic root. Similarly, the flow through the systemic segments Qsc can be also calculated using the Bernoulli equation dQ sc Asa ðtÞ ¼ ðP  P  R ðtÞQ sc Þ dt ρLsa sa sv sc

ð5Þ

where Asa(t) is a time-dependent lumped area that accounts for the possibility of remodeling and distension which might take place in the systemic capillaries as a result of pressure wave fluctuations.

L¼

ρ

ð12Þ

Asa

where Asa and C sa ðP sa Þ ¼ dAsa =dP sa are area and compliance which depend on the pressure at the proximity of the aortic root. In this paper, we used the static pressure–area relationship proposed by Langewouters et al. (1984) to relate both systemic vascular resistance and capillary area to the aortic pressure as    1 P sa  P o Asa ðP sa Þ ¼ Amax 0:5 þ arctan ð14Þ π P1 where Amax is the maximum cross-sectional area at very high aortic pressure. Po and P1 are constants (Langewouters et al., 1984) and listed in Table 1. The arterial wall compliance per unit length is the derivative of the area with respect to pressure C sa ¼

dAsa Amax ¼ dP sa π P 1



1

P sa  P o 1þ P1

2

ð15Þ

Since realistic valve leaflets do not operate instantaneously and its motion takes a considerable amount of total ejection time. Therefore, a transit expression for flow rate that smoothly swept

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Table 1 Parameters and values used in the computations. Parameter

Value

Units

δ β γ λp ηsa ρ θmax Kv K p;vort Kb Kf Ees

3.21 0.60 0.18 312.5 0.0911 1050 85 8 5500 2 5 1.7

(–) (–) (–) (–) (mmHg S/ml) (kg/m3) (deg.) (rad/s2 ml) (rad/s2 mmHg) (rad/s ml) (1/s) (mmHg/ml)

Ed Esa P0 ; P1 Psv P lv;ed V lv;ed V 0;sa V 0;d V 0;es Lav Lsa Aav Asa;max Usa Tp teivc tes tee τ αeivc α_ eivc αee – –

0.096 0.45 74,57 5 10 124 300 20  10 5 90 3.46 1.75 100 1000 45 336 362 38 0.328 7 0.85 – –

(mmHg/ml) (mmHg/ml) (mmHg) (mmHg) (mmHg) (ml) (ml) (ml) (ml) (cm) (cm) (cm2) (cm2) (cm/s) (ms) (ms) (ms) (ms) (ms) (–) (s  1) (–) – –

by the motion of leaflets is indeed required. In other words, valve leaflets move in transit phases that mark the beginning of blood ejection to the full valve opening, as well as the onset of rabid closing till fully closed phase. The flow rate swept by these leaflets transit motions QL can be given by multiple-definition function for specific opening and closing phases as described by Virag and Lulic' (2008) 8 V L o V L0 “opening” Q av ; > > " >  2 # > > V V > L L0 > > Q av 1  ; V L Z V L0 “opening” > > V L1 > > > >   < V L V L0 2 ð16Þ Q L ¼ Q av  Q av;co ; V L Z V L0 “closing” > V L1 > > >   > > > ðt  t cl ÞQ av;cl 2 > > ; V L r V L0 “closing” > Q av;cl exp > > V L0 > > : 0; “fully open or closed” where Q av;co , Q av;cl , and tcl are the crossover flow rate, coaptive flow rate, and coaptive time, respectively. These variables mark the onset of aortic valve closing and the flow reversal or “regurgitation” phase. More details about the valve closing–opening phases and the associated blood flow transport can be found in Virag and Lulic' (2008). Herein, both VL0 and VL1 are functions of the aortic root radius Rav, V L0 ¼ γπ R3av , V L1 ¼ ðβ  γ Þπ R3av , where β and γ are constants as listed in Table 1. If we assume that flow across the aortic root is uniform, the valve orifice area can be related to the displacing volume ratio of the leaflets as   A0 V L V L0 2 ¼ ð17Þ Aav V L1

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The left ventricular pressure Plv is coupled to left ventricular volume Vlv via a time-varying interpolation function αðtÞ, derived from the end-systolic and end-diastolic pressure–volume relationships as shown previously by Kass and Maughan (1988) P lv ¼ αP s þ ð1  αÞP d

ð18Þ

where end-systolic pressure–volume (ESPVR) is given by P s ¼ Ees ðV lv  V 0;es Þ where both the elastance slope Ees and the unloaded volume V 0;es are assumed to be constants. Similarly, the end-diastolic pressure–volume (EDPVR) is given by P d ¼ Eed ðV lv  V 0;ed Þ where both Eed and V 0;ed are constants. In this model, we used an accurate piecewise interpolation function (elastanceactivation) αðtÞ as proposed by Virag and Lulic' (2008). This elastance function adequately mimics isovolumic contraction, end-systole, end-ejection, and isovolumic relaxation phases, as shown in Fig. 1(b). The aortic pressure–volume relationship can be derived using the classical arterial wall constitutive relationships that relate normal wall stress to both strain and strain rates (Armentano et al., 1995). Since the aortic pressure is directly related to its wall elasticity and viscous effects, a relationship between the volume change and the flow rate change along the aorta can be given as P sa ¼ Esa ðV sa  V 0;sa Þ þ ηsa

dV sa dt

ð19Þ

where Esa and ηsa are constants accounting for aortic wall elasticity and resistance, respectively, and are given in Table 1. The above equation suggests that aortic pressure is related to its volumetric change, wall elasticity, and wall strain rate due to viscosity. It does not account for the existence of the sinus vortices, which can interact with aortic pressure significantly and can alter the valve opening–closing dynamics. Hence, we coupled the local pressure induced by the sinus vortices to the above aortic pressure–volume relationship to better estimate the pressure wave in the aortic root. A fundamental way to relate sinus vortex local pressure Pvort to the aortic pressure Psa is to use the Bernoulli equation. However, the pressure induced by sinus vortices requires an adequate vortex model that approximates well the flow field generated by flow separation from the leaflets. Here, we follow the work by Bellhouse and Talbot (1968) 1 2 P vort ¼ P sa þ ρU 2sa ð1  0:0672δ Þ 2

ð20Þ

where δ is constant. Usa is taken as the average mean flow velocity in the aorta. Now, one can use Eq. (20) to include the vortex effect on the aortic pressure and therefore Eq. (19) can be written as P sa ¼ Esa ðV sa  V0; saÞ þ ηsa

dV sa þ λp P vort dt

ð21Þ

where λp is a coefficient related to the vortex strength loading on valve leaflets. This loading can be estimated by assuming a jet-like flow through the valve (Korakianitis and Shi, 2006), therefore the load induced by a sinus vortex that acts perpendicular to the valve leaflets can be written as F vort ¼ K v Q av sin ð2θÞ ¼ K p;vort P vort cos ðθÞ

ð22Þ

where Kv and K p;vort are constants with values listed in Table 1 and given previously by Korakianitis and Shi (2006). The angle θ is the valve opening angle as illustrated in Fig. 1. This angle can be related to orifice area A0 =Aav ¼ ð1  cos ðθÞÞ2 . Using Eq. (22), an expression for the local pressure induced by sinus vortex can be found as P vort ¼

2K v Q sin ðθÞ K p;vort av

ð23Þ

Since the sinus vortex strength is expected to depend on the aortic valve opening orifice area, i.e., on the leaflets opening–

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closing angle θ, a relationship between the sinus vortex strength and the valve opening angle can be derived using Hill's vortex model. Recall the velocity stream function for the spherical vortex model by Hill (Milne-Thomson, 1960)

ψ¼

1 GðθÞða2  r 2 Þr 2 sin ðϕÞ2 10

ð24Þ

Now if we recall the velocity stream function distribution given by Bellhouse and Talbot (1968) using the Hill vortex model, Eq. (24). It is clear that the vorticity (curl of the velocity) has only one component in the direction normal to the planes passing through the axis of symmetry. Therefore, all vortex lines form a circle of radius r sin ðϕÞ, and the magnitude of the vortex strength (vorticity) can be given as

where r and ϕ are the planar spherical coordinate system and G ¼ GðθÞ is the vortex strength as a function of leaflets opening– closing angle θ. The loading exerted by the sinus vortex ring behind valve leaflets can be approximated by expression given by Batchelor (1967) and by Stamhuis and Nauwelaerts (2005)

Equating both Eqs. (26) and (27), an expression to the sinus vortex strength amplitude GðθÞ a function of the leaflets opening angle is given as

F vort ¼ ρQ av ζ

GðθÞ ¼

ð25Þ

where ζ ¼ Γ =I is the vortex ring strength, Γ is the circulation, and I is the moment of inertial of rotation. Now, if we use Eqs. (22) and (25), an estimate for the vortex strength ζ per unit mass as a function of valve opening angle θ can be given as

ζ ðθ Þ ¼

Kv

ρ

sin ð2θÞ

ð26Þ

ΩðθÞ ¼ GðθÞr sin ðϕÞ

K v sin ð2θÞ ρ r sin ðϕÞ

ð27Þ

ð28Þ

In summary, we have extended the aortic valve model of Virag and Lulic' (2008) to account for the existence of sinus vortices behind aortic valve leaflets and for the compliance of the systemic vascular walls. Relationships between the aortic pressure, local pressure induced by the sinus vortex, and with the systemic wall resistance are found and are then coupled with the main

Fig. 2. Activation function and the effect of both sinus vortices and variable systemic vascular resistance on the left ventricular–aortic pressure waves over single cardiac cycle. (a) Baseline vs. sinus vortex cases, (b) baseline vs. VSVR effect, (c) baseline vs. sinus vortex and VSVR effects and (d) variable systemic vascular resistance.

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governing equations. These modifications have allowed us to perform simulations examining the effects of both sinus vortex and variable systemic vascular resistance (VSVR) on the pressure of both left ventricle and aorta. Furthermore, their influence on blood flow rate during ejection and on the valve orifice area was also examined.

3. Results and discussion The above system of equations that describe ventricular–aortic valve dynamics during systole was solved numerically using a fourth-order Runge–Kutta method with an integration time step Δt ¼ Tp=5000. The initial conditions were chosen such that V vl ¼ V lv;ed , P vl ¼ P lv;ed , VL ¼0, QL ¼ 0, and Q av ¼ 0. All other initial variables were obtained by running the code with arbitrary physiological values for multiple cardiac cycles such that the solution to this particular system of equations converges to a periodic solution after multiple cycles. In all cases presented here, not more than 10 cycles were required to achieve periodic solutions. A list of all parameters along with their numeric values used in the simulation is given in Table 1. We obtained periodic solutions for a variety of different conditions by either including or excluding sinus vortices and VSVR in various combinations. This enabled an examination of the effect these two phenomena have individually or in combination on pressures, flow rate, and overall opening–closing valve dynamics. The four cases used were “Baseline”, in which neither vortex or VSVR were included, “Vortex”, in which only sinus vortex effects were added, “VSVR”, in which only variable systemic vessel compliance effects are added, and “VortexþVSVR”, in which both effects were added simultaneously. 3.1. Interpolation function αðtÞ The time-dependent interpolation function αðtÞ appearing in Eq. (18) is considered to be an important step in this simulation, and its value must be available at any instant of time during the cardiac cycle. The characteristics of this activation function are crucial in obtaining physiological responses in left ventricular– aortic pressures and blood flow rate ejection. Furthermore, it helps in capturing aortic flow features such as back flow events and the dicrotic notch as shown in Fig. 2(a–c). In this study, we used the piecewise expression proposed by Virag and Lulic' (2008) which satisfies the following identities and equations:

α ¼ A1 t 2 eA2 ðt  teivc Þ ; 

8 t r t eivc

t  t eivc t es  t eivc   t  t eivc eA4 ðt es  tÞ ;  A3 t es  t eivc

α_ II ¼ A3 þ 1 

α_ III ¼



8 t eivc o t r t es

ð30Þ

8 t es o t r t ee

ð31Þ



αee t  t es A5 ðt  tee Þ ; e τ t ee  t es

αIV ¼ αee eðtee  tÞ=τ ;

ð29Þ

 ðα_ eivc  A3 ÞeA4 ðt  teivc Þ

8 t 4 t ee

ð32Þ

where A1, A2, A3, A4, and A5 are constants to be determined using boundary and compatibility conditions. teivc, tes, and tee define the end-isovolumic contraction, end-systole, and end-ejection times, respectively. τ denotes the isovolumic relaxation time constant. The symbol ð UÞ implies time derivative. The above equations satisfy a set of conditions. For instance, at the beginning of isovolumic contraction t¼ 0, the interpolation function and its derivative vanish, i.e., α ¼0 and α_ ¼ 0. At the end of isovolumic phase when the aortic valve starts opening, i.e.,

285

t¼teivc, both function and its time derivatives must take a specific values of αeivc, and α_ eivc , respectively. At end-systole t ¼ t es , interpolation function reaches its maximum value α ¼1, α_ ¼ 0. In addition, pressure and volume in the left ventricle meet the end-systolic values i.e., P lv ¼ P lv;es , and V lv ¼ V lv;es . After this time point, the interpolation function decreases and at end-ejection time t ¼ t ej , aortic valve closes and isorelaxation phase takes place. During this phase, the left ventricle volume is constant, and pressure falls exponentially P lv ¼ ðP lv;ee  P d Þe  ðt  tee Þ=τ . The constants (A1 to A5) and additional two constants A6 and A7 appear after integrating (30) and (31) can be uniquely determined using the following continuity conditions: (i) when t ¼ t eivc , αI ¼ αII ¼ αeivc , and α_ I ¼ α_ eivc II III (ii) when t ¼ t es , αII ¼ αIII ¼ 1, and α€ ¼ α€ III (iii) when t ¼ t ee , α_ ¼  αee =τ Once the constants are evaluated, the value of interpolation function αðtÞ at instant of time during cardiac cycle will be available for use, as shown in Fig. 1(b). The parameters (t t eivc ; t es ; t ee ; τ; αeivc ; α_ eivc ; αee ) which are used to find these constants are list in Table 1. 3.2. Sinus vortex (SV) and variable systemic vascular resistance (VSVR) effects Baseline simulations were conducted using the original model by Virag and Lulic' (2008), where neither the sinus vortex pressure nor the time variations of the systemic vascular resistance effects are included. This was accomplished by setting λ ¼0 and using a constant vascular resistance Rsc ¼1.2161. The simulation outputs are repeated in Fig. 2(a–c) for comparison with each of the other cases, i.e., when both sinus vortex and variable systemic vascular resistance effects are included. The sinus vortex pressure on its own (i.e, λ a 0 and Rsc ¼1.2161, i.e., constant) is then coupled to baseline analysis using Eqs. (14)–(16). This “ Vortex” simulation case shows the effect of sinus vortex pressure on both left ventricular and aortic pressures as displayed in Fig. 2(a). Results indicate that during systole, the pressures in the left ventricle and in the aorta are higher than the pressures calculated from the baseline simulations. Pressures during the isovolumic, end-ejection, and isovolumicrelaxation periods of times are largely unchanged. The elevation in pressure versus baseline lasts throughout systole, and indicates the strong influence of sinus vortex pressure beginning with the onset of fast-opening and continuing until the onset of the slowclosing phase. However, during diastole, left ventricular pressure is similar to the baseline pressure but aortic pressure is slightly higher and decays faster to meet with the end-diastolic aortic pressure calculated from baseline simulation. Moreover, the aortic pressure Psa in the Vortex case is distinguished by two peaks that occur at the onset of both valve fastopening and slow-closing events. These features are not normally observed in aortic pressure measurements, which raise an interesting paradox, namely how is it that realistic representation of sinus vortices results in an unrealistic aortic pressure trace? The existence of this two-peak response is thought to be caused by the sinus vortices' strong wave reflections during opening and closing events. An alternative interpretation is that since a constant systemic vascular resistance model (i.e., Rsc ¼constant) has been assumed while the sinus vortex disturbance has been enabled, there will be a pressure pulse reflection backward to the aorta as observed in the aortic pressure signal. In other words, these pressure peaks might have appeared because of impedance mismatch reflection effect since we are using an aortic pressure– volume relationship with constant Esa.

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We next performed the “VSVR” simulation in which systemic resistance is considered to be variable and depends on the aortic pressure Psa. The sinus vortex pressure effect is not included (i.e., λ ¼0 and Rsc a constant). The variable resistance coupling occurs via Eqs. (6)–(8) which relate systemic vascular impedance to a pressure–area constitutive relationship as proposed previously by Langewouters et al. (1984). A comparison between baseline and VSVR pressure responses is shown in Fig. 2(b). Results show essentially no difference in left ventricular pressure traces obtained from these two cases specially during systole. In contrast, the aortic pressure matches the baseline aortic pressure wave only until the end of the dicrotic notch. Afterwards, the aortic pressure decays slightly more rapidly for the VSVR simulation and reached a different end-diastolic value relative to baseline. This discrepancy in the aortic pressure decay rate is apparently due to the compliance environment offered by using a variable systemic resistance. It is interesting to note that the two spikes in aortic pressure that were present in the Vortex case (Fig. 2(a)) are not present in the VSVR simulation. In the fourth case, we included both sinus vortex and VSVR effects (i.e., λ a 0 and Rsc a constant). Fig. 2(c) shows a comparison between pressure responses from Baseline and VortexþVSVR calculations. During systole, pressures obtained in this case were similar to those of VSVR, except that after the dicrotic notch aortic pressure decay had a steeper slope when both effects were operative. Interestingly, the two distinctive peaks seen in the Vortex case were significantly abolished by the presence of variable systemic resistance. It appears that the vortex-associated pressure peaks are smoothed by a compliance absorption effect once the systemic vascular resistance is made to vary with aortic pressure in a more realistic manner. Because the full model (VortexþVSVR) has a strong resemblance to physiological behavior, it seems likely that canonical waveforms for aortic pressure are actually the product of vortex and variable aortic flow resistance operating simultaneously, as shown in Fig. 2(c). The systemic vascular resistance Rsc as a function of cardiac cycle computed for all the above simulation cases is shown in Fig. 2(d). Rsc was set to be constant (¼1.2161) in both Baseline and Vortex simulations, while it was allowed to vary according to Eq. (8) during the rest of the above-mentioned cases. The systemic resistance adapts itself according to aortic pressure variations to satisfy Bernoulli, compliance, area, and pressure relationship equations (5)–(7). Hence, the systemic resistance is

tightly coupled with the aortic pressure and follows a similar time course trend. Systemic resistance calculated from “VSVR” and “VortexþVSVR” cases has similar behavior during systole. However, they decay with different rates during diastole and reach different values at the end-diastolic point. This explains why aortic pressure decay is steeper when compared with the Baseline simulation. Finally, it should be mentioned that, in order for us to make a fair comparisons between all the simulated cases, the constant value of Rsc ¼ 1.2161 for the baseline and Vortex simulations was chosen such that Z Tp Z Tp Rsc dt↓Baseline ¼ Rsc ðtÞ dt↓VSVR ¼ constant ð33Þ 0

0

3.3. Aortic orifice area and blood flow rate We also analyzed aortic valve orifice area A0 and blood flow rate Qav from the same four simulation cases (Fig. 3). These waveforms can be divided into phases relating to valve opening or closing events: during diastole and isovolumic systole, A0 is constant and Qav is zero. As soon as LV pressure exceeds aortic pressure, the onset of opening phase takes place. In this phase, leaflets start to move outward inside the aortic sub-domain but the orifice area remains zero. Rather, blood starts to accumulate in the aortic root region formed by the leaflets. As the pressure in the LV increases, the leaflets part in a rapid opening action (fastopening phase). This leads to sudden increase in both orifice area and aortic blood flow. The orifice area gradually increases until it becomes maximally opened (fully opened phase). Here, blood flow rate has reached its maximum value. The valve persists in this phase during most of the systole, until aortic pressure exceeds that of the ventricle. At that instant, the valve starts to close slowly (slow-closing phase) such that the orifice area falls. This causes a simultaneous lowering of blood flow rate. The fast-closing phase occurs as the ventricular pressure falls still further below aortic pressure. This causes a brief period of negative flow (i.e., some blood flows back into the left ventricle from the aorta), which is the driving force for fast closing and coaptation of the leaflets. These phases are readily seen in Fig. 3. The simulated orifice area distribution can be compared with experimental data from Bellhouse and Talbot (1968), in which they attempted to mimic the dynamics of the aortic valve using a

500

1

0.8

Baseline Vortex

400

Baseline Vortex

VSVR

VSVR Vortex + VSVR

Vortex + VSVR

Bellhouse & Talbot (1969)

A0 / Aav

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Q av [ ml / s]

300 200 100

0.4

0 0.2 -100 0 0

0.2

0.4 time [ S ]

//

-200 .0

0

0.2

0.4

//

time [ S ]

Fig. 3. The effects of both sinus vortex and variable systemic vascular resistance on the orifice area and aortic flow rate across the valve, respectively. (a) Orifice area and (b) aortic flow rate.

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Fig. 4. Snapshots of the sinus vortex induced by the aortic valve leaflets during opening–closing phases.

mechanical analogue, as shown in Fig. 3(a). Agreement between computational and experimental data, especially during the opening phases, is reasonable. Agreement in prediction of the area during closing events is not as close. The discrepancies between the simulation and the experiment counterpart might occur because the experimental setup was built primarily to mimic pulsatile-like flow in a channel and did not attempt to reproduce the physiological and anatomical features of the true system. Nevertheless, the experimental data serve as a useful qualitative way of validating the present model. As with the aortic pressure traces, the Vortex case had the largest differences in orifice area and flow relative to Baseline, Fig. 3. Hence, it is clear that sinus vortices can significantly influence the time-varying properties of blood flow between the LV and the aorta. It is also evident that, as before, the effects of VSVR mask the influence of sinus vortices. On the whole, the full model that includes both added mechanisms reproduces important physiological features of aortic valve behavior, including the presence of the dicrotic notch and the principal phases of leaflet dynamics. 3.4. Relationship between sinus vortex strength and valve opening–closing modes Model equations (17)–(20) allow the relationship between vortex strength and valve leaflet angle θ to be computed continuously during the cardiac cycle. Streamlines and the vortex strength distribution in the sinus area were calculated using Eq. (17) and are shown in Fig. 4. We found that sinus vortex strength increases during opening and decreases during closing in a nonlinear fashion. In other words, the development and decay of sinus vortices during valve opening–closing distinct phases, i.e., during the (A) fast-opening, (B) fully-open, (C) slow-closing, and (D) fast-closing, are captured. During the fast-opening phase blood is ejected in a jet-like fashion through a relatively small initial orifice area, creating a strong sinus vortex. The vortex weakens as the valve attains its fully opened position. It should be noted that during the fully opened phase, the sinus vortex becomes steady and there is no more vorticity shedding from the leaflet tips therefore it is less severe. This is due to vortex dissipation, and is consistent with the time course followed by the orifice area. During the closing-phases, sinus vortices recover their strength as leaflets move toward closed positions. Once the leaflets

coapt, there will be no vorticity shedding and presumably no more formation of eddy currents. Ideally, there will be no vortices in the aortic root area when the valve has completely closed. An important outcome of these calculations is that they demonstrate that orifice area and sinus vortex strength distributions follow an energy path during opening that is different from their closing counterparts. This suggests that there exists an energy asymmetry between opening and closing valve dynamics. Results from the present mathematical model are consistent with studies by Robicsek (1991) and Korakianitis and Shi (2006) which suggest that sinus vortices play an important role in the physiological valve opening–closing valve dynamics.

4. Conclusions We have derived a realistic yet simplified mathematical model describing the physical phenomena governing ventricular–aortic blood flow during systole. The model considers the coupling between aortic valve dynamics, sinus eddy vortical pressure, and accounts for variations in the systemic vascular resistance. The influence of this coupling on the overall valve opening–closing phases, as well as on the left ventricular and aortic pressures, blood flow rate, and aortic orifice area, is evident from our simulations. The results suggest that when the sinus vortex is included on its own, aortic pressure undergoes two unrealistic peaks appearing at the onset of fast-opening and slow-closing of the valve leaflets. These two peaks are then dissipated when variable systemic resistance is added to the model, which suggests that the majority of vortex effects on aortic pressure and flow are obscured by deformation of the aorta during systole. This interaction has the potential to be critical under pathological conditions such as aortic valve insufficiency or stenosis, as well as central artery stiffening. Our simulations suggest that any of these pathologies could disturb the compensatory balance between sinus vortices and aortic compliance, thereby introducing abnormal aortic pulse pressures. The effects of such changes on ventricular loading can now be examined in silico owing to the advances made by the model presented here.

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