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Med Biol Eng ComputDOI 10.1007/s11517-014-1228-9
ORIGINAL ARTICLE
Derivation of a simplified relation for assessing aortic root
pressure drop incorporating wall compliance
Hossein Mohammadi Raymond Cartier Rosaire Mongrain
Received: 3 May 2014 / Accepted: 12 November 2014 International
Federation for Medical and Biological Engineering 2014
List of symbolsAd Flow cross-sectional area downstream of
the
stenosis, cm2Au Flow cross-sectional area upstream of the
stenosis, cm2a Vessel radius, cmC Vessel compliance, cm/BaryeE
Youngs modulus of the vessel, BaryeEOA Flow effective orifice area,
cm2F Body and surface force vector, g cm/s2g( AdEOA ) Function of
the areas ratio, dimensionlessh Vessel thickness, cmkc Empirical
constant in the convective pressure
loss term, dimensionlesskp Empirical constant in the pressure
loss term due
to the vessel compliance, Baryekv Empirical constant in the
viscous pressure loss,
dimensionlessL23 Distance between two referenced positions
downstream and upstream of the stenosis, cmn Outward pointing
normal unit vector to the body
surface, dimensionlessp Blood flow pressure, dyn/cm2Q Volume
flow rate, cm3/su Blood flow velocity, cm/sV Fluid velocity vector,
cm/s Parameter defined as L23Ad +
x2x1
dxA , cm
1
Empirical constant, dimensionless Blood density, g/cm3 Dynamic
viscosity of blood, g/cm sv Kinematic viscosity of blood, cm2/s
Heart frequency, Hz Vessel wall shear stress, dyn/cm2p Pressure
gradient across stenosis, dyn/cm2
Abstract Aging and some pathologies such as arterial
hypertension, diabetes, hyperglycemia, and hyperinsu-linemia cause
some geometrical and mechanical changes in the aortic valve
microstructure which contribute to the development of aortic
stenosis (AS). Because of the high rate of mortality and morbidity,
assessing the impact and progression of this disease is essential.
Systolic transval-vular pressure gradient (TPG) and the effective
orifice area are commonly used to grade the severity of valvular
dys-function. In this study, a theoretical model of the transient
viscous blood flow across the AS is derived by taking into account
the aorta compliance. The derived relation of the new TPG is
expressed in terms of clinically available sur-rogate variables
(anatomical and hemodynamic data). The proposed relation includes
empirical constants which need to be empirically determined. We
used a numerical model including an anatomically 3D geometrical
model of the aortic root including the sinuses of Valsalva for
their iden-tification. The relation was evaluated using clinical
values of pressure drops for cases for which the modified Gorlin
equation is problematic (low flow, low gradient AS).
Keywords Pressure gradient relation Aortic stenosis Aortic valve
Pathologies Three-dimensional global model Fluidstructure
interaction
H. Mohammadi R. Mongrain (*) Mechanical Engineering Department,
McGill University, Montreal, QC H3A 0C3, Canadae-mail:
[email protected]
R. Cartier R. Mongrain Department of Cardiovascular Surgery,
Montreal Heart Institute, Montreal, QC H1T 1C8, Canada
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pC Convective component of the pressure gradient across
stenosis, dyn/cm2
pCo Pressure gradient component due to the vessel
distensibility, dyn/cm2
pL Pressure gradient component due to the local inertia,
dyn/cm2
pV Viscous component of the pressure gradient across stenosis,
dyn/cm2
ij Cauchy stress tensor, dyn/cm2Vi Material velocity vector,
cm/sVMJ Framework velocity, cm/sC Stiffness tensor of the vessel
wall material,
dyn/cm2 Strain tensor of the vessel wall material,
dimensionlessVf Vector containing the fluid unknownsVs Vector
containing the solid unknownsfs Common solidfluid interface
1 Introduction
In aortic stenosis (AS), calcified nodules on the valve
leaf-lets occur which lead to the thickening and stiffening of the
leaflets, restricting the natural motion of the valve [2931]. As a
consequence of the obstruction to the blood flow caused by the
stenosis, the hydraulic resistance increases. Therefore, high
systolic pressure is needed to maintain the necessary cardiac
output which may lead to left ventricu-lar hypertrophy which
eventually can result in heart failure. Because of the high rate of
mortality and morbidity due to the AS, assessing its stage and
severity is important for the clinician [22, 34, 36].
In the management of patients with AS, the first con-sideration
to perform corrective surgery is made largely on the presence or
absence of symptoms. In addition, the severity of the AS plays a
key role in determining which patients should undergo valve
replacement. Systolic trans-valvular pressure gradient (TPG) and
the effective orifice area (EOA) are commonly used to grade the
severity of the valvular dysfunction [36]. These parameters can be
assessed using either catheterization or Doppler echocar-diography
[3, 6]. However, the current models are prob-lematic under certain
conditions. Because of the role of compliance (windkessel effect),
it is hypothesized that the incorporation of compliance can improve
the pressure estimate.
Numerous experimental and analytical studies have been done to
relate the TPG across the stenosis to the blood flow rate and EOA.
In one of the first studies, based on fundamental hydraulics,
Gorlin [20] and his father developed a formula that can be used to
estimate the EOA of the stenotic valves and relate the pressure
difference and flow through the valve. The frictional effect of
the blood flow was not incorporated on the pres-sure loss. Young
and Tsai [39, 40], by doing an extensive series of model tests,
simulated the arterial stenosis and derived the empirical constants
of their proposed equa-tion. The obstructed geometry is more
diffuse than AS and to the best of the authors knowledge was not
trans-posed to AS. By considering the flow friction, Clark [11]
presented a detailed analysis of the instantaneous pressure
gradient across the aortic valve. In his study, for simplic-ity,
the contribution of the aortic wall distensibility was neglected.
Although the derived equation was validated with animal
experimentations, it was not translated to clinic. By using the
generalized Bernoulli equation for a rigid wall, several studies
investigated the pressure gradi-ent dependency on the blood flow
rate and the obstructed area [1, 5, 37]. However, the first
explicit model of the TPG was proposed by Garcia et al. [18]. They
developed an analytical model for the frictionless blood flow
across rigid aorta which has clinical potential. For simplicity,
none of these studies considered the effect of the aor-tic root
compliance on TPG. Again, the objective of this study is assessing
the aortic pressure drop for the transient viscous blood flow
across the AS, by taking into account the vessel wall
compliance.
In order to investigate the function of the aortic valve, finite
element methods have been used for over 20 years. Due to the large
displacement that leaflets experience during the cardiac cycle,
numerical modeling of the aor-tic valve has proven to be a
challenging task, especially in a fluidstructure interaction (FSI)
study [28]. Numer-ous studies have been done on the numerical
modeling of aortic valve [8, 9, 12, 28, 33]. The objectives were to
reproduce the valve dynamics to study alteration to geo-metrical
and hemodynamic parameters and derive the stress patterns. A few
studies have been done for making these engineering results
meaningful for physicians in terms of diagnostic and prognostic
assessment. For this, it is required to map the engineering
variables into clinical indices based on anatomical and hemodynamic
data called surrogate variables (velocity, shear stress,
elasticity, pres-sure gradient, thickness, and dimensions). More
specifi-cally, anatomical data include: wall and valve
morpholo-gies, wall and leaflet thickening, wall thinning, dilation
of the aortic valve, and the hemodynamic surrogate variables
include: pressure drop, flow rate, back flow, leaflet stiffen-ing,
and dynamics. In this study, we develop a FSI model of the aortic
valve. Then, stenoses with different severi-ties are simulated.
Finally, the derived TPG is expressed in terms of the surrogate
variables, and the numerical model is used to extract the empirical
constants. Finally, the derived model is assessed using the results
from the literature.
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2 Methods
2.1 Analytical model of blood flow across AS
To derive an expression for the instantaneous TPG, a
two-dimensional model of the blood flow from the left ventricle
through the aortic root is used (Fig. 1). The cross section at any
position, x, is assumed to be circular. The blood is assumed to be
incompressible, and the vessel walls are lin-ear elastic.
Upstream of the stenosis, the flow accelerates due to the
obstruction presented by the stenosis which results in a jet with
its smallest diameter at the vena contracta (EOA). During this
convective acceleration, the pressure is con-verted to kinetic
energy. In this process, the pressure loss is minor. After passing
through the stenosis, the flow expands and fills the cross section
of the ascending aorta and decel-erates. This decelerating process
leads to recirculation and energy losses [27]. Applying Newtons
second law to an elemental disk of width dx as shown in Fig. 1
yields [11]:
where is the blood density, d the diameter, p the pressure, u
the velocity, and the viscous shear stress. For simplic-ity, the
velocity and pressure vary with position and time, while is assumed
to be dependent only on time. Integrat-ing Eq. (1) relates the
variables to the pressure difference as follows:
(1)p/x = u/t + uu/x + 4/d
(2)p =
(u/t)dx
pL
+
(uu/x)dx
pC
+ 4
(/
d)dx pV
The first term on the right side, pL, represents the pres-sure
loss due to local acceleration of blood particles. The second term,
pC, is the pressure loss because of the con-vective acceleration of
the blood flow, while the last term, pV, is due to the viscous
force. In this study, it is assumed that the contribution of the
inertial, frictional, and wall dis-tensibility terms in the TPG is
linear and will be analyzed separately. For this purpose,
initially, the effect of the local and convective inertia of the
blood flow on the pressure loss would be taken into account, and
the effect of the vis-cosity and compliance will be analyzed in the
subsequent sections.
2.1.1 The effect of fluid inertia
In order to derive the relationship between the pressure
gra-dient and flow, the model is split into two sections. Section
one is upstream of the stenosis, from location 1 in Fig. 1 to the
orifice area (location 2 in Fig. 1). Section two begins from
location 2 and ends at location 3, downstream of the stenosis where
the reattachment of the flow to the vessel wall occurs. In a first
approach, it is assumed that the veloc-ity profile is uniform, the
walls are rigid, and the fluid is inviscid. Using Eq. (2) for
section one yields:
If the wall is assumed to be indistensible, then the conti-nuity
yields Q = uuAu = u2EOA = udAd. So, for rigid walls, Eq. (3) can be
rewritten as:
In section two, from the orifice area to any point down-stream
of the stenosis, the flow can be disturbed and turbu-lent. For
analysis of this section, the control volume meth-ods are useful.
The acting forces on a fixed control volume with as boundary can be
expressed as [18, 19]:
where V is the fluid velocity vector, n is normal vector to the
surface and F are the body and surface forces acting on the control
volume. Neglecting the viscous forces and using Eq. (5) for the
dashed volume control shown in Fig. 1 gives:
(3)pu p2 = 2
1
u
tdx +
2
(u22 u2u
)
(4)pu p2 = Qt
x2x1
dxA+
2Q2(
1EOA2
1A2u
)
(5)
Vt
d +
V V .n d = F
(6)(p2 pd)Ad = Qt
x3x2
dl + Q(ud u2)
uA dAEOA
12 3
Volume
u
p dp p
u du
dx
+
+
Fig. 1 A schematic of blood flow from the left ventricle to the
aorta across stenotic aortic valve. Au cross-sectional area of
fluid upstream of the stenosis, EOA effective orifice area, Ad
cross-sectional area of the fluid downstream of the stenosis
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Using continuity equation simplifies Eq. (6) as follows:
where L23 is the distance from the location 2 to the location 3.
Summing Eqs. (4) and (7) gives:
where the first and second terms on the right side of equa-tion
correspond to pL and pC in Eq. (2), respectively.
Garcia et al. [18] used dimensional analysis and curve fitting
to replace the integral terms. In this study, a similar analysis is
done. By defining the parameter as:
and taking into account that the flow geometry and position of
location 3 and consequently L23 depends mainly on the ratio of EOA
and A. It is meaningful to express in terms of EOA, and Ad. A
dimensional analysis provides:
In order to determine the function g, it should be consid-ered
that according to Eq. (9), when EOA approaches zero (stenosis
becomes severe) tends toward +. In addi-tion, when the stenosis
approaches toward the non-stenotic case, location 3 tends toward
location 1 and consequently tends to zero. A simple function g
coherent with these two criteria is:
where is an empirical constant. Then, the net pressure drop
becomes:
the first term is the pressure loss due to local inertia, and
the second term represents the pressure loss caused by kinetic
terms in the sudden expansion from the orifice area to the aorta.
The introduced coefficient kc to this term is an empirical constant
which need to be evaluated.
(7)p2 pd = Qt
(L23Ad
)+ Q
2
Ad
(1
Ad 1
EOA
)
(8)
pu pd = Qt
L23
Ad+
x2x1
dxA
+ Q2
2
1
EOA 1
Ad
2+
1A2d 1
A2u
(9) =L23Ad+
x2x1
dxA
(10)
Ad = g(
AdEOA
)
(11)
Ad = (
AdEOA
1)
(12)
pu pd = 1AdQt
(Ad
EOA 1
)+ pkc Q
2
2
[(
1EOA
1Ad
)2+(
1A2d 1
A2u
)]
2.1.2 The effect of viscosity
The friction contribution in pressure loss is difficult to
evaluate. Clark suggested to use the equation of shear force
experienced by a flat plate oscillating in a viscous fluid for the
pulsatile flow across the valve which can be expressed as:
where is the dynamic viscosity, and is the heart fre-quency [11,
35]. Viscosity contribution to the pressure in the fluid flow, pV,
across any vessel with circular cross section is presented by the
last term of Eq. (2). Therefore, by substituting Eq. (13) in Eq.
(2) and using the continuity equation for a circular section,
pressure loss due to the vis-cosity can be expressed as:
where kv is an empirical constant and L13 is the distance from
location 1 to location 3.
For the fluid flow across a distensible wall, the flow rate, Q,
varies with distance because of the transient storage of fluid
associated with the distensible boundary [11, 17]. The variation of
the flow rate with distance due to the compli-ant walls will
influence all terms in the right-hand side of Eq. (2). The main
influence of compliance on the pressure drop is related to
additional convective pressure loss due to the post-stenotic
dilatation of the aorta resulting from flow disturbances in this
region [14]. For simplicity, since the convective pressure term has
the highest contribution on the pressure loss [11], in this study,
only the changes in this term due to the wall distensibility are
analyzed. This effect can be calculated from the last term in Eq.
(12) compared to the same conditions for the non-distensible case.
Since the area of the ventricle outflow tract is of the same
caliber as the aorta, the last parenthesis can be neglected. For a
dis-tensible wall with fixed EOA, flow disturbance downstream of
the stenosis causes a change in the cross-sectional area by an
amount dA.Then, for the compliant vessel, the con-vective pressure
loss becomes:
For a linearly elastic vessel wall, by expanding Eq. (15) and
neglecting the higher-order effect of the wall compli-ance, the
compliance pressure loss can be expressed as (Appendix):
(13) =
2u
(14)pV = kv L13EOA3/ 2 Q
(15)pC = Q2
2A2d
(Ad + dA
EOA 1
)2
(16)pCo =kpQ2
Eh
(Ad
EOA
)(1
EOA 1
Ad
)
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where pCo corresponds to the contribution of the vessel
compliance to the convective pressure loss. In this equation, kp is
an empirical constant which has pressure dimension. Additional
details for Eq. (16) are provided in Appendix.
2.1.3 Global pressure drop across the aortic valve
The instantaneous global TPG can be then expressed as:
substituting Eqs. (12), (14), and (16), Eq. (17) becomes:
2.2 Numerical model of the global pressure drop
The finite element software LS-DYNA (Livermore Soft-ware
Technology Corporation, Livermore, CA, USA) is used to perform FSI
simulations to assess the proposed relation for computing the
global pressure model geometry.
2.2.1 Aortic root model geometry
By combination of imaging modalities (MRI, CT scan) and
physiological data, an anatomically 3D geometrical model of the
aortic root including the sinuses of Valsalva was developed. The
averages of the various dimensions reported in the previous studies
are used for anatomical sites [28, 33]. This geometrical model is
composed of four
(17)pu pd = pV +pC +pL +pCo
(18)
pu pd = kv L13EOA3/ 2 Q+ kc Q
2
2
[(
1EOA
1Ad
)2+(
1A2d 1
A2u
)]
+ 1Ad
Qt
(Ad
EOA 1
)
+ kpQ2
Eh
(AdEOA
)( 1EOA
1Ad
)
distinct parts. These are the ventricle outflow tract, leaflets,
and aortic wall in the solid domain and an encasing fluid domain.
This domain is additionally subdivided into the ventricular inlet,
aortic outlet, middle reservoir including the interfaces with the
corresponding solid structures, right coronary outlet, and left
coronary outlet. An exploded view of the full assembly is shown in
Fig. 2.
A model of the aortic root was previously reported [28, 33]. The
explicit finite element method by means of Arbitrary
LagrangianEulerian (ALE) algorithms was used to model the
interaction between the structural and fluid parts. The ALE solver
was originally designed for modeling high-speed dynamic problems
involving a compressible fluid for simu-lating explosions, shock
waves, impacts etc. [21]. The advan-tage of this solver for
modeling heart valves was its capabil-ity to support large
deformation rate of the leaflets. Using this solver, the blood was
modeled as slightly compressible (using Newtonian fluid with a
linear polynomial equation of state with bulk modulus of 2.5 104
kPa to avoid instability in the ALE solver [25]). In this study,
the incompressible flow solver (ICFD), recently added to LSDYNA, is
fully coupled with the solid mechanics solver. This coupling
permits robust strong FSI analysis for an incompressible fluid
[26].
The model was meshed in ANSYS Mechanical. The solid components
were discretized into 10,892 shell ele-ments. The fluid medium, on
the other hand, consisted of 25,984 unstructured triangular
elements. In order to ana-lyze the model in LS-DYNA, the input deck
including the whole geometry, boundary conditions, loads, and
material properties were prepared.
2.2.2 FSI governing equations
The momentum equations for both the solid and the
incom-pressible fluid domains to be solved are [17]:
(19)DViDt
= ijxj
+ fi
Fig. 2 Exploded view of the various components of the aortic
root model
Ventricular inlet
Aortic outlet
Middle reservoir fluid
Right coronary outlet
Left coronary outlet
Leaflets
Ventricle outflow tract
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where is the density, ij is the Cauchy stress tensor, Vi denotes
the material velocity vector, fi is the specific body force. For
the incompressible part of the domain, mass conservation is
[17]:
for the fluid domain, acceleration vector may be expressed in a
moving framework different than the particle displace-ment as
[15]:
where DFVi/Dt and VMJ represent the framework accelera-tion and
velocity, respectively. The fluid and solid domains are different
only in constitutive equations. The constitutive equation for the
Newtonian fluid can be expressed in terms of the rate of
deformation dij, and pressure p as [17]:
while is the dynamic viscosity of the fluid. On the other hand,
for an elastic solid, the constitutive equations as a function of
the strain are [17]:
where C and are the stiffness and strain tensors, respec-tively,
and Ui corresponds to the displacement vector.
In order to implement the interaction between the solid and
fluid domains, a strongly coupled scheme is adopted. In this
approach, the system of equations is split into the solid unknowns
(the velocity or displacement) and the fluid unknowns (velocity and
pressure) and they are solved sepa-rately. The boundary conditions
at the interface are [23]:
where Vf and Vs are the vectors containing the fluid and solid
unknowns, respectively, and fs is the common solidfluid interface
[13, 15, 16]. Because a fully Lagrangian frame of reference is used
to model the interface, Eq. (24) imposes the consistency conditions
which guaranties that the fluid and solid meshes are tightly
coupled along the interface. Equation (25) guaranties the balance
of stresses along the interface.
2.2.3 Boundary conditions and material properties
Constrains are applied on the aortic annulus, ascending aorta,
and coronary ostia in order to avoid any rigid body motion,
twisting, rotation, and translation in the solid domain.
(20)Vixi
= 0
(21)DViDt
= DFViDt
+ (Vj VMJ)Vixj
(22)ij = 2dij pij with dij = 12(Vixj
+ Vjxi
) 1
3Vlxl
ij
(23)ij = Cklij kl with ij =12
(Uixj
+ Ujxi
+ Uixj
Ujxi
)
(24)(Vf Vs)T = 0 on fs
(25)f + s = 0 on fs
The axial deformation of the inlet is constrained, while its
circumferential motion and consequently the root expan-sion are
unconstrained. Rotational and translational motion on the outlet,
located in the ascending aorta, are also con-strained. Finally,
element deformation, on the coronary ostia periphery, is fully
constrained.
The aortic root and leaflets are modeled as linear elas-tic
material with a Youngs modulus of 3.34 and 4.00 MPa, respectively,
and a Poissons ratio of 0.45 [24, 38]. This is in good agreement
with a study by our group that has shown that cardiac tissue in the
physiological regime can essentially be considered linear [10].
For the fluid domain, four sets of boundary conditions including
the time-dependent pressure difference between the left ventricle
and the ascending aorta on the ventricular inlet, zero gauge
pressure on the aortic outlet, physiological pulse wave for
coronary flow are imposed [32]. The blood is modeled as a Newtonian
fluid with dynamic viscosity of 3.5 mPa s and density of 1,060
kg/m3.
2.3 Aortic stenosis models
Equation (18) includes empirical parameters which need to be
evaluated. This can be done with experimental data or simulated
data. We used the 3D FSI numerical model to generate data for their
determination. To achieve this goal, as illustrated in Fig. 3, by
constraining the motion of leaf-lets tip, five stenosis models with
different severity were created. In this study, the percentage of
the reduction in the area occupied by blood from the left ventricle
out tract to the orifice is used as an index to evaluate the
stenosis severity.
3 Results
For all models, the heart rate was fixed at 74 bpm. A sec-tion
view of the blood velocity vector at 0.12 s into the car-diac cycle
during which the blood velocity in the left ven-tricle is maximum,
for healthy and stenosed models with severity of 79 %, is presented
in Fig. 4.
All of the six models, including the healthy model, were
simulated with four distinct cardiac outputs of 3, 4, 5, and 6
L/min. Then, by measuring TPG, blood flow rate, rate of change of
the blood flow rate, EOA, all corresponding terms in Eq. (18) were
evaluated. Therefore, a system of twenty-four linear equations were
generated which can be expressed in matrix form as:
where M is a 24 4 matrix whose rows are the calculated terms on
the right-hand side of Eq. (18). K is the matrix of the empirical
constants with dimension of 4 1, and p
(26)MK = p
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is a 24 1 matrix of the measured TPG corresponding to each
model. Since M is a non-square matrix, a MoorePen-rose
pseudo-inverse approach is used to invert the resulting
over-determined system of linear equations. The approach that was
proposed by Moore (1920) and Penrose (1955) aims to compute the
least squares solution to a system of linear equations which lacks
a unique solution [4]. Accord-ing to this method, the generalize
inverse of matrix M in Eq. (26) is defined as (MT )1MT, where MT
denote the transpose of M. Hence, the empirical constants can be
cal-culated using the following equation.
The calculated values for kv, kc, , and kp are presented in
Table 1.
The overall square root error of the approach was 0.0965.
Therefore, the derived global transvalvular equa-tion can be
rewritten as:
The calculated pressure gradient across the valve is visualized
for two simulated stenotic models with EOA of 0.862 and 1.14 cm2 in
Fig. 5 showing a good agreement with the results of the Eq.
(28).
(27)K = (MT )1MTp
(28)
pu pd =20.05 L13EOA3/ 2 Q+ 0.79Q
2
2
[(
1EOA
1Ad
)2+(
1A2d 1
A2u
)]
+ 6.89 1Ad
Qt
6(
AdEOA
1)
82162Q2
Eh
(Ad
EOA
)(1
EOA 1
Ad
)
The temporal average of pressure drop calculated from Eq. (28)
is compared with results of studies done by Gor-lin, Garcia et al.,
and Clark for a fixed flow of 5 L/min in Fig. 6a and for a fixed
EOA of 0.85 cm2 in Fig. 6b. Also, the impact of stenosis severity
and blood acceleration for the global pressure drop is shown in
Fig. 6c. The relative contribution of wall compliance to global
pressure gradi-ent for as a function of EOA for different cardiac
output is presented in Fig. 6d.
4 Discussion
A global relation of TPG that takes into account geometri-cal
and hemodynamic parameters including the vessel wall compliance was
derived. The proposed relation includes empirical parameters. A
numerical model incorporating an anatomical 3D geometrical model of
the aortic root with the sinuses of Valsalva was used for their
identification. The ICFD from LSDYNA is fully coupled with the
solid mechanics solver which permits robust FSI analysis. The
contribution of the solid elements on the interface is added to the
fluid elements when the pressure Laplace equation is built. This
procedure greatly improves the convergence of the FSI coupling [23,
26].
The calculated values for the empirical constants of Eq. (18)
are listed in Table 1. Therefore, the pressure drop across a
compliant wall is expressed by Eq. (28). The first term on the
right-hand side of this equation corresponds to frictional loss,
the second term takes into account the pressure loss due to
convective acceleration, the third term is responsi-ble for local
inertia of the blood flow, and the last term is the pressure loss
due to the dilation of the compliant vessel. As
Fig. 3 Schematic of the cre-ated models in their maximum opening
state: a healthy model, b stenosis with severity of 61 %, c
stenosis with severity of 72 %, d stenosis with severity of 79 %, e
stenosis with severity of 84 %, f stenosis with severity of 92
%
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it was discussed previously, Gorlin developed a formula that can
be used to estimate the EOA of the stenotic valves and relate the
pressure difference and flow through the valve:
(29)p = 12Q2 1.28
EOA2
This formula was derived with the assumptions of a rigid
circular conduit, non-viscous, and steady flow, while val-vular
orifices are compliant and the flow is viscous and pul-satile. It
is reported that the error for the calculated area by this formula
increases in the following conditions: (1) low flow rate and (2)
small area [2, 7]. As presented in Fig. 6a, the pressure drop
calculated from our study for stenoses of different severities,
through which a fixed amount of flow (5 L/min) is passing, is
higher than the Gorlin result, while for a fixed size stenosis
(Fig. 6b), our results yield higher values of pressure gradients
for lower flow rate.
Based on a theoretical model, Clark [11] proposed the following
equation for TPG by neglecting the effect of the compliance:
where cd is the discharge coefficient to include the fric-tional
effects. His suggested range for discharge coefficient was 0.81.
His model includes an integral term for the local acceleration
which needs to be expanded for clinical appli-cation. The temporal
average of pressure drop plotted in terms of EOA for a constant
flow in Fig. 6a and for a con-stant EOA as a function of Q in Fig.
6b. Hence, the Clark equation is almost superimposed to the Gorlin
equation in Fig. 6a, while there is a small difference for high
flow for a fixed EOA (Fig. 6b).
Garcia et al. used a theoretical model and derived a simi-lar
equation for TPG in which only the convective and local inertial
terms were considered.
(30)
p = Q2
2c2d
[(1
EOA2 1
A2u
)+ 2
(1 Ad
/EOA
)A2d
]
+ Qt
AdAu
dxA
(31)
pu pd = Q2
2
[(1
EOA 1
Ad
)2]
+ 6.28 1Ad
Qt
1
EOA 1
Ad
Fig. 4 Blood velocity vector across leaflets at 0.12 s into the
cardiac cycle in a healthy model b stenosis with severity of 79
%
Table 1 Values of derived empirical constants
Parameter kv kc kp
Calculated value 20.05 0.79 6.89 82,162 dyn/cm2
Fig. 5 Visualized pressure gradient across the valve for
stenotic models with a EOA of 0.862 cm2 and b EOA of 1.14 cm2
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They used a dimensional analysis to derive the inertial pressure
loss 6.28 1Ad
Qt
(Ad
EOA 1)0.5
. We have used a similar dimensional analysis but introducing
only one parameter for simplicity. Hence, the value for our
param-eter , 6.89 is comparable with their value of 6.28. As it is
shown if Fig. 6a, b, pressure drop calculated from their equation
results in lower values compared to our results. The reason is that
they did not consider the frictional effect. Therefore, it
underestimated the pressure drop
In Fig. 6c, the net pressure drop calculated from Eq. (30) is
presented as a function of EOA for a normal flow rate of 5 L/min
for three selected values of flow acceleration to illustrate the
effect of local term. This term is generally neglected in most
studies, while it is clear from Fig. 6c that as the stenosis
becomes severe, the contribution of the local acceleration in
global pressure drop becomes higher.
The last term, which is the contribution of the aorta compliance
to the pressure loss, is the main contribution of this study. As it
was discussed before, some assumptions have been used for its
derivation. The calculated value for parameter (kp) is 82,162
dyn/cm2. The contribution of this term to the total pressure
gradient for the three selected val-ues of flow is plotted as a
function of orifice area in Fig. 5d. So, as the flow rate
increases, because of more dilation of wall, this term becomes
higher, and for a normal cardiac output of 5 L/min, about 10 % of
the pressure is stored in the wall deformation for the AS with
severity of 84 %.
5 Conclusion
In this study, we have derived a theoretical model of the
transient viscous blood flow across the AS tak-ing into account the
aorta compliance. Then, by using a numerical model including an
anatomical 3D model of the aortic root including the sinuses of
Valsalva, the derived relation of the new TPG is expressed in terms
of clinically available surrogate variables (anatomical and
hemodynamic data). The results showed that the proposed relation
provides physiologically compatible results even for cases for
which current models fail (low flow). The model reveals that for a
normal cardiac output of 5 L/min, about 10 % of the pressure drop
is used to deform the wall for a severe AS, while this is neglected
in the models. This generalized model can be used to estimate the
effective valve orifice area for determining the severity of the
stenosis in cases where the tissue still has compliance.
Acknowledgments We are thankful to the support of McGill
Engineering Doctoral Award (MEDA), Natural Sciences and
Engi-neering Research Council of Canada (NSERC) and Montreal Heart
Institute (MHI). We would also like to thank Mr. Facundo Del Pin, a
scientist at Livermore Software Technology Corporation for
developing the ICFD solver. This work was made possible by the
facilities of the Shared Hierarchical Academic Research Comput-ing
Network (SHARCNET: www.sharcnet.ca) and Compute/Calcul Canada.
Fig. 6 a Temporal mean of pressure drop calculated from Eq.
(28), Gorlin, Garcia et al., and Clark study results for cardiac
output of 5 L/min. b Pressure drop predicted from our result,
Gorlin, Gar-cia et al., and Clark study as a function of flow for a
fixed EOA of
0.85 cm2. c Global pressure drop for selected values of flow
accel-eration. d Percentage of pressure loss due to vessel wall
compli-ance to total pressure gradient, for all cases, heart rate
is 74 bpm and Au = Ad = 4.91 cm2
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Med Biol Eng Comput
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Appendix
For a distensible wall with fixed EOA, the flow disturbance
downstream of the stenosis causes a change in the cross-sectional
area by an amount dA.Then, for the compliant vessel, the convective
pressure loss becomes:
If the vessel is modeled with a thin-walled cylinder obeying
Hookes law (assuming physiological deforma-tion), since the
longitudinal stress is much smaller that the circumferential one,
then
where e is the circumferential strain, a the radius of the
vessel, a0 the initial radius, E the Youngs modulus of the wall
material, and h the wall thickness. Then, the cross-sec-tional
variation in Eq. (32) can be expressed as:
Hence, by expanding Eq. (32), neglecting the small higher-order
terms and using Eqs. (33) and (34), the effect of the wall
compliance in pressure loss can be expressed as:
The first term in the right-hand side (pRigid) is pressure loss
caused by convective inertia in the rigid vessel, while the second
term (pCo) corresponds to the contribution of the ves-sel
compliance to convective pressure loss as the following:
where dp is the pressure variation in the aorta. This term can
be scaled as a fraction of the pulse pressure by intro-ducing an
empirical constant kp which has the dimension of the pressure.
Hence, by including the effect of all constant coefficients of Eq.
(36) in kp, it can be simplified as:
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(32)pC =Q22A2d
(Ad + dA
EOA 1
)2
(33)e = daa= adp
Eh
(34)dA = 2Ad
Ad
dpEh
(35)
pC = pRigid +pCo = Q2
2
[(
1EOA
1AAO
)2+ 4
Addp
Eh
(1
EOA
)(1
EOA 1
Ad
)]
(36)pCo = 2Q2
AddpEh
(1
EOA
)(1
EOA 1
Ad
)
(37)pCo = kpQ2
Eh
(Ad
EOA
)(1
EOA 1
Ad
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Derivation ofa simplified relation forassessing aortic root
pressure drop incorporating wall complianceAbstract 1 Introduction2
Methods2.1 Analytical model ofblood flow acrossAS2.1.1 The effect
offluid inertia2.1.2 The effect ofviscosity2.1.3 Global pressure
drop acrossthe aortic valve
2.2 Numerical model ofthe global pressure drop2.2.1 Aortic root
model geometry2.2.2 FSI governing equations2.2.3 Boundary
conditions andmaterial properties
2.3 Aortic stenosis models
3 Results4 Discussion5 ConclusionAcknowledgments References
