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American Institute of Aeronautics and Astronautics Paper 2010-4733

Numerical Study of Pulsatile Flow through Models of Vascular and Aortic Valve Stenoses and Assessment of

Gorlin Equation

Ramesh K. Agarwal*, Robert Rifkin**, E. Okpara***, T. Foote+, and J. Daiber+

Mechanical, Aerospace, and Structural Engineering Department Washington University in Saint Louis, MO 63130

Abstract

For determining the blockage (stenosed valve area) in the aortic valve, Gorlin equation has been used in clinical practice for past fifty years [1]. It has been derived using the Bernoulli equation across the stenosed valve and making the approximation that the velocity of the fluid behind the stenosis is much greater than the velocity upstream of the stenosis (it is a good assumption for a valve with severe stenosis). Based on many clinical studies, it is well documented that the Gorlin equation has large error in predicting the stenosed valve area under mild stenosis and at low flow rates [2]. In last fifty years, a large number of theoretical/numerical studies have been reported in the literature to improve upon the predictions of the Gorlin equation; however none of them has found acceptance in clinical practice. The goal of this paper is to study the pulsatile flow in models of aortic valve with actual waveform of the heart using the commercial CFD software FLUENT. Computations for steady and pulsatile Newtonian flow are performed for four axisymmetric models with valve areas of 0.5 cm2, 1.0 cm2, 1.5 cm2 and 2.0 cm2 at flow rates of 5.0 l/min, 7.5 l/min, 10.0 l/min, 12.5 l/min, 15.0 l/min, 17.5 l/min, and 20 l/min. Thus a total of 28 cases are computed to assess the range of validity of the Gorlin equation. The flow is turbulent in all cases downstream of the stenosis; thus a modified k-epsilon turbulence model is employed in the computations. Using the calculated pressure drop across the stenosis, Gorlin equation was used to determine the stenosed area of the valve. For all the 28 cases, the error in valve area computed from Gorlin equation varied from 15 to 100%. The Gorlin equation was modified so that it retains its basic features but better fits the computational data. The stenosed valve areas computed with modified Gorlin equation give results within 3 to 5% error when compared to the exact valve areas used in FLUENT computations. Clinical data using 35 patients covering the whole range of flow rates and severity of stenosis of the aortic valve was obtained by Dr. Rfikin of the Washington University School of Medicine. The valve area from clinical data was compared with that obtained from original Gorlin equation and modified Gorlin equations for various flow rates and pressure drops. For high flow rates, the original Gorlin equation predicts the valve area which is significantly different from clinical valve area; however the modified Gorlin equation is in very good agreement with the clinical results for all physiologically relevant flow rates and stenoses from mild to severe. Therefore we recommend the use of modified Gorlin equation by the physicians.

1. Introduction

In the United States, heart and other circulatory system diseases are among the leading causes of death in

the adult population. Aortic valve stenosis is a disease that occurs when there is narrowing of the aortic valve due to formation of plaque. Since aortic valve stenosis occurs inside the cardiovascular system, it

* William Palm Professor of Engineering, Fellow AIAA

** Professor, Washington University School of Medicine

*** Doctoral Student, + Undergraduate Student

40th Fluid Dynamics Conference and Exhibit28 June - 1 July 2010, Chicago, Illinois

AIAA 2010-4733

Copyright © 2010 by R. Agarwal, R. Rifkin, E. Okpara, T. foote, J. Daiber. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


is not easy to diagnose it since it can not be inspected visually. Using invasive pressure drop

measurements from catheters upstream and downstream of the aortic valve, an estimate of the area of the

stenosed valve can be obtained by performing simple fluid dynamics analysis. Gorlin [1] employed the

Bernoulli equation upstream and downstream of the stenosed valve to determine the valve area using the

pressure values upstream and downstream of the valve measured by the catheter and the flow rate. Gorlin

equation has been used in clinical practice for past fifty years for determining the blockage (stenosed valve area) in the aortic valve. In applying the Bernoulli equation across the stenosed valve, it assumes

that the velocity of the fluid downstream of the stenosis is much greater than the velocity upstream of the

stenosis (it is a good assumption for a valve with severe stenosis). Based on many clinical studies, it is

well documented that the Gorlin equation has large error in predicting the stenosed valve area under mild

stenosis and at low flow rates [2]. In last fifty years, a large number of theoretical/numerical studies have

been reported in the literature to improve upon the predictions of the Gorlin equation; however none of

them has found acceptance in clinical practice.

Alternately, using X-ray contrast angiography, images of severe stenosis can be obtained, but these

images provide little or no information about flow properties such as pressure, velocity and wall shear

stress. Doppler ultrasound techniques can also be used to measure velocities and waveforms in stenotic

valves and to determine the estimates of the valve area, however this method also has limitations in providing accurate predictions of the valve area because of several simplifying assumption involved. In

recent years, the application of Phase Contrast Magnetic Resonance Imaging (PC-MRI) has become

popular in the measurement of velocity field in vascular flows, and has shown some promise as a tool for

diagnosing the disease [3].

Surgeries on stenotic valves are often advised based on the percent area reduction of the valve due

to stenosis, but all the techniques currently available for determining the valve area are not very accurate

and have some limitations in predicting the area. Surgical procedures on aortic valves with stenoses are

expensive and carry high mortality rates, thus reliable techniques to determine the necessity of such

procedures are of great value. Furthermore it is now known that several flow related phenomenon play a

critical role in the progression of the vascular disease. These are oscillating and low wall shear stress, high blood pressure, and flow phenomenon such as recirculating flow regions and turbulence that may

occur after the onset of stenosis.

In a recent paper, Okpara and Agarwal [4] presented the results of simulations of steady and

sinusoidal pulsatile flow in axisymmetric and 3D concentric phantoms and their comparison with

experimental data. In another recent paper, Thompson, Pinzón and Agarwal [5] extended the work in [4]

by modeling a stenosis using the physiological waveform of the heart; they validated their computational

results against the experimental data of Peterson and Plesniak’s [6]. The goal of this paper is to extend

the work reported in [5] by computing the pulsatile flows with actual physiological waveform of the

heart in models of valves with varying degrees of stenoses (mild to severe) for different flow rates.

Computations for steady and pulsatile Newtonian flow are performed for four axisymmetric models with

valve areas of 0.5 cm2, 1.0 cm2, 1.5 cm2 and 2.0 cm2 at flow rates of 5.0 l/min, 7.5 l/min, 10.0 l/min, 12.5 l/min, 15.0 l/min, 17.5 l/min, and 20 l/min. Thus a total of 28 cases are computed. These cases are used

to assess the range of validity of the Gorlin equation. The flow is turbulent in all cases downstream of the

stenosis; thus a modified k-epsilon turbulence model is employed in the computations. Using the

calculated pressure drop across the stenosis, the original Gorlin equation is used to determine the

stenosed area of the valve. For all the 28 cases, the error in valve area computed from Gorlin equation

varied from 15 to 100%. The Gorlin equation is then modified so that it retains its basic features but

better fits the computational data. Clinical data using 35 patients covering the whole range of flow rates

and severity of stenosis of the aortic valve has been obtained by Dr. Rfikin of the Washington University

School of Medicine. The valve area from clinical data is compared with that obtained from the original

Gorlin equation and modified Gorlin equations for various flow rates and pressure drops. For high flow

rates, the original Gorlin equation predicts the valve area which is significantly different from clinical valve area; however the modified Gorlin equation is in very good agreement with the clinical results for

all physiologically relevant flow rates and stenoses from mild to severe. Therefore we recommend the

use of modified Gorlin equation by the physicians


2. CFD Solver FLUENT

FLUENT is a numerical flow solver in a software package of the same name sold by Fluent Inc. which is

utilized in this paper to model pulsatile flow fields through stenosed aortic valves. While the software has broader capability, it is employed here to solve the governing equations for incompressible,

Newtonian fluid using a finite-volume method. It can solve both steady and unsteady flows using

turbulence models of zero, one, two or five equations. The grids can be either structured or unstructured.

The version of FLUENT used in this paper is 6.3.26 with Cortex version 3.7.3. GAMBIT is the pre-

processing software included in the Fluent Inc. package that is used to build the geometry and the mesh

on which FLUENT operates.

The parameters used by FLUENT in this paper are chosen to best represent the physiological blood

flow from the heart through the aortic valve, with simplifications appropriate to the accuracy of the

model. We use a blood mimicking density of 1030 kg/m3 and viscosity of 0.00255 kg/ms in the

calculations assuming the blood to be a Newtonian fluid. Inlet velocity profiles are specified with appropriate User Defined Functions (UDFs). The flow is assumed to be laminar and fully developed

upstream of the valve. An extrapolation boundary condition is applied at the outlet, which assumes zero

normal gradients for all flow variables except pressure. The second-order upwind solver for solution of

the momentum equations is employed. Pressure-velocity coupling in incompressible flow is treated using

the SIMPLE scheme, and the pressure is computed using the standard Poisson solver. Reynolds numbers

of the flow as well as viscous effects at the valve are sufficient to require a turbulent model; the two

equation k-epsilon model is used. Each computational case is initialized using the inlet velocity profile at

the first time step of the UDF being used. They are then run in steady flow mode until the flow

converges, and then the unsteady flow simulation is started without reinitializing. This minimizes any

errors from starting the flow. Each unsteady case is run through six cycles of 1600 time steps with

monitors recording data for each time step.

3. Heart Waveform Inlet Functions

The physiological waveform of the heart is generated in velocity inlet function UDFs for FLUENT in

order to accurately represent the blood flow through the aortic valve. A set of data quantifying the blood

flow rate through an artery over the course of a heartbeat cycle is derived by digitizing a figure in

Peterson’s [6] thesis. Using a third order spline over 11 discrete sections of the data, a curve fit is formed

and used to create a new set of flow rate data with a constant time interval. The data and spline are

shown in Figure 1.

Fig. 1: Digitized data with spline fit of physiological waveform from Peterson [6].


The new set of flow data is then scaled from the 0.0064 meter radius artery used by Peterson to the

assumed aortic artery radius of 0.01 meters. A period of 0.906173 seconds is derived. At this stage, ten

sets of this data are created with each scaled to a desired flow rate. These range from 5 to 27.5 liters per

minute at 2.5 liters/min intervals. The remaining steps of the UDFs’ creation are carried out for each set.

The data sets only contain the flow rates, so the Womersley solution is used to create an inlet velocity

profile. It is given as:

tin

n

nc

neFQ

QREAL

R

r

u

u,0

1

111

1

2

1 , (1)

where

nn

n

n

n

n

iJi

iJ

iJ

R

riJ

F

230

23

231

230

230

,02

1

1

In equation (1), nQ are the Fourier coefficients determined by a fast Fourier transform of the column

vector containing the flow rate data. Eighth order polynomial fits were then determined for the real and

imaginary components of each of the 11 terms in the summation in equation (1). To generate the inlet

profile, the 22 polynomials, as powers of the radial distance r, become coefficients, F, for the sinusoidal

terms inside the summation in equation (2). Thus it is rational and FLUENT can evaluate it across the

radius of the artery and through time. The form of the equation in the UDF appears as:

11

1

611

2

sincos101,n

imagrealc ntFntFrR

rutru

nn (2)

The centerline velocity used is the average centerline velocity, and is assumed to be twice the average

velocity as for a parabolic flow. The powers of ten and the radius are from scaling down the polynomials

to avoid floating point errors.

4. CFD Validation with Experimental Data for a Model of Vascular Stenosis

The computed velocity (using FLUENT) and the theoretical velocity (given by equation (2)) at various

radial locations z (z = 0 being the centerline) of the tube at different time are compared in Figure 2;

excellent agreement between the two is obtained. It should be noted that the graphs in Figure 2 also

match perfectly with the experimental data obtained by Peterson [6]. The validated Womersley inlet

profile (Figure 2) was then used as a user defined function at the inlet of a tube with a symmetric

stenosis. The uniform inlet region was 5D long, where D is the diameter of the tube. The stenosis was 2D

long and the outlet tube was of uniform diameter 23D long. The equation for the stenosis geometry in

both the experiment of Peterson [6] and computation is as follows:

L

xxs

DxS o

o2

cos112

(3)

Equation (3) describes an axisymmetric model of 75% area reduction stenotic occlusion. Computations were performed using the FLUENT with inlet profile given in Figure 2. The centerline

velocities were computed at various points downstream of the stenosis and are shown below in Figure 3.

It should be noted that the computed values in Figure 3 are in excellent agreement with the experimental

data. The three-dimensional plots of velocity profiles at select postions downstream of the stenosis

specifically at a distance 2D and 4D from the beginning of the stenosis region are shown in Figure 4.

Figure 4 shows the corresponding experimental velocity plots at a distance 2D and 4D downstream from

the beginning of the stenosis [6]. Figures 4 and 5 are in reasonably good agreement; there are some small


discrepancies in the reattachment region of the flow. These calculations were performed using the full

Reynolds stress model (RSM) of turbulence.

Figure 2: Comparison of computed and theoretical Womersley solution for velocity at

various radial locations.

0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t/T

uC

L/u

c

-4D

2D

3D

4D

6D

8D

Figure 3: Computed values of the centerline velocity at various distances downstream of the

beginning of the stenosis given by equation (2).

(a) (b)

Figure 4: 3D plots of computed velocity profiles at (a) 2D and (b) 4D distance downstream

from the beginning of the stenosis given by equation (3)


Figure 5: 3D plots of experimental velocity profiles at (a) 2D and (b) 4D distance downstream

from the beginning of the stenosis given by equation (3)

For this axisymmetric case (Figure 6), pressure plots upstream and downstream of the stenosis are shown in Figures 6 and 7. Figures show the expected pressure drop downstream of the stenoses. The peak value

of the pressure drop is an important measure of the area reduction of the occlusion and is used by the

physician in assessing the severity of the stenosis.

Figure 6: Axisymmetric stenosis (Equation 3) with specific locations at which pressure

was computed

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t/T

Pre

ssu

re [

Pa]

Figure 7: Average pressure at location L1 upstream of the

stenosis

-0.180

-0.160

-0.140

-0.120

-0.100

-0.080

-0.060

-0.040

-0.020

0.000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t/T

Pre

ssu

re [

Pa]

Figure 8: Average pressure at location L2

downstream of the stenosis

The results presented in this section show that overall the computations and the experiments are in

acceptable agreement given the uncertainty in the computations due to turbulence modeling as well as the

uncertainty in the measurements. The lessons learnt from this CFD validation for flow in a model of

vascular stenosis were then applied to calculate the flow in models of stenosed aortic valves.

L1 L2 L3


5. Computation of Flow in Models of Stenosed Aortic Valve

The models of stenosed aortic valves are concentric and are represented as axisymmetric walls extending

from the outer artery wall to constrict the valve flow area to the desired dimension as shown in Figure 9. The four stenosed valve areas considered are 0.5, 1.0, 1.5, and 2.0 square centimeters. The outer wall of

the artery is taken to be 0.01 meters in radius and 0.24 meters long. The location of the stenosed valve is

0.08 meters from the inlet of the artery. Axisymmetric flow meshes in tubes shown in Figure 9 are built

in GAMBIT. The meshes are 34 cells across the radius and 300 cells along the axis giving a total of

10200 cells.

Fig. 9: Cross-Sections of the four valves tested (left); Axisymmetric representation of aorta and valve

employed in the simulations (right).

Computations for pulsatile Newtonian flow (with actual waveform of the heart), using the density

and viscosity of the blood, were performed for four axisymmetric models (with a circular cross-section)

shown in Figure 9 with valve areas of 0.5 cm2, 1.0 cm2, 1.5 cm2 and 2.0 cm2. Calculations were

performed for each of these four models at flow rates of 5.0 l/min, 7.5 l/min, 10.0 l/min, 12.5 l/min, 15.0

l/min, 17.5 l/min, and 20 l/min. Thus a total of 28 cases were computed to assess the range of validity of the Gorlin’s equation. The flow was turbulent in all cases downstream of the stenosis; thus a k-epsilon

turbulence model was employed in the computations. Using the calculated pressure drop across the

stenosis, Gorlin equation was used to determine the stenosed area of the valve. As shown in Table 1 for

all the 28 cases, the error in area computed from Gorlin equation varied from 15 to 100% depending

upon the degree of stenosis and flow rate. The original Gorlin equation can be expressed as [1]:

where Q = flow rate through the valve (ml/sec), Av = area of aortic valve (cm2) and ∆Pd = pressure difference across the valve (dynes/cm2). The Gorlin equation was modified so that it retains its basic

features but better fits the computational data. The modified Gorlin equation is as follows:

The stenosed valve area computed with equation (5) gives results within 3 to 5% error when compared to

the exact valve areas used in FLUENT computations as shown in Table 2. We now compare the valve

area computed by equation (4) and (5) with the clinical valve area in the Figure 10 below. For high flow

rates, the original Gorlin equation predicts the valve area which is significantly different from clinical

valve area; however the modified Gorlin equation is in good agreement with the clinical results for all

physiologically relevant flow rates and stenoses from mild to severe. Therefore we recommend the use of equation (4) by the physicians.

dP

QvA

4.50 (4)

(5) 8.0)

35.08.0( 5.02

dv

P

QA


Table 1: % Error between the exact valve area used in FLUENT and the area calculated

using the original Gorlin equation for three different flow rates Q (5, 12.5 and 20 liters/min)

and four different valve areas (0.5, 1.0, 1.5 and 2 cm2)

Table 2: % Error between the exact valve area used in FLUENT and the area calculated

using the modified Gorlin equation for three different flow rates Q (5, 12.5 and 20

liters/min) and four different valve areas (0.5, 1.0, 1.5 and 2 cm2)

FLUENT FLUENT FLUENT GORLINQ (L/min) A (cm 2̂) Pd (Pascal) A (cm 2̂) % Error

5 0.5 3850 0.59 17.005 1 800 1.29 29.005 1.5 225 2.42 62.005 2 81 4.04 102.00

12.5 0.5 23660 0.59 18.0012.5 1 5000 1.29 29.0012.5 1.5 1546 2.31 54.0012.5 2 527 3.96 98.0020 0.5 60330 0.59 18.0020 1 12500 1.30 30.0020 1.5 4025 2.29 53.0020 2 1350 3.96 98.00

FLUENT FLUENT FLUENT MODIFIED GORLINQ (L/min) A (cm 2̂) Pd (Pascal) A (cm 2̂) % Error

5 0.5 3850 0.53 6.345 1 800 0.96 -3.985 1.5 225 1.43 -4.815 2 81 2.08 4.04

12.5 0.5 23660 0.53 6.4212.5 1 5000 0.96 -4.0312.5 1.5 1546 1.42 -5.1312.5 2 527 2.07 3.3020 0.5 60330 0.53 6.6220 1 12500 0.96 -3.7520 1.5 4025 1.43 -4.8520 2 1360 2.07 3.43


Flow Rate vs Valve Area Av

0

1

2

3

4

5

6

0 100 200 300 400 500

Flow Rate (ml/sec)

Av(c

m^

2)

Clinical Av

Av Gorlin

Altered Gorlin

Figure 10: Comparison of Stenosed valve area obtained from the clinical data (blue) with the original

Gorlin equation (4) ( purple) and the modified Gorlin equation (5) ( yellow)

6. Conclusions A modified Gorlin equation has been developed by performing computations in axisymmetric models of

stenosed aortic valve. Twenty eight computations were performed covering all physiologically possible

flow rates (from high to low) and stenosed valve areas (from mild to severe). The modified Gorlin

equation agrees with the clinical data within 3 to 5% of the value of the stenosed valve area while the

original Gorlin equation gives error ranging from 10 to 80% against the clinical data. Therefore the use

of modified Gorlin equation is recommended for clinical practice.

References

1. Gorlin, R. and Gorlin, S.G., “Hydraulic formula for calculation of the area of the stenotic mitral valve,

other cardiac valves, and central circulatory shunts I,” Am. Heart J., Vol. 41, 1951, pp. 1-29.

2. Baumgartner, H., Hung, J., Bermejo, J., Chambers, J.B, Evangelista, A., Griffin, B.P, Iung, B., Otto,

C.M., Pellikka, P.A., and Quinones, M., “ Echocardiographic Assessment of Valve Stenosis: EAE/ASE

Recommendations for Clinical Practice,” J. of Am. Soc. of Echocardiography, Vol. 22, 2009, pp. 1-23.

3. Moghaddam, A.N., Behrens, G., Fatouraee, N., Agarwal, R.K., Choi, E.T., and Amini, A.A., “Factors

Affecting the Accuracy of Pressure Measurements in Vascular Stenoses from Phase-Contrast MRI,”

MRM journal, Vol. 52, No. 2, 2004, pp. 300-309.

4. Okpara, E. and Agarwal, R. K., “Numerical Simulation of Steady and Pulsatile Flow Through Models of Vascular and Aortic Valve Stenoses,” AIAA Paper 2007-4342, AIAA Fluid Dynamics Conference,

Miami, FL, 25-28 June 2007.

5. Thompson, J. D., Pinzón, C. and Agarwal, R. K., “Numerical Simulations of Pulsatile Flow through

Models of Vascular Stenosis with Physiological Waveforms of the Heart,” AIAA Paper 2008-3953,

AIAA Fluid Dynamics Conference, Seattle, WA.

6. Peterson, S., “On the Effect of Perturbations on Idealized Flow in Model Stenotic Arteries,” Ph.D.

Thesis (supervisor: M. Plesniak), Purdue University, December 2006.

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