University of British ColumbiaMECH 410L - Mechanics of Biofluids

Term Project Report

Heart Valves:A Fluid Dynamics Perspective

Author:Colin RussellEngineering PhysicsMechanical OptionStudent Number 42289025crussell@interchange.ubc.ca

Instructor:Dr. Dana GrecovAssistant Professor

Mechanical Engineeringdgrecov@mech.ubc.ca

April 25, 2008

Abstract

Heart valves, which serve to regulate blood flow direction within the circulatory sys-

tem’s central pump, are a prime example of the principles of biofluid mechanics. In

particular, the design of prosthetic heart valves is a complex and interesting problem.

This report gives an introduction to heart valve anatomy and pathology, reviews

some typical prosthetic valve designs, investigates some mathematical models of me-

chanical valve hydrodynamics, and discusses a flow loop which could be used to

measure valve hydrodynamics.

The report concludes that the prosthetic heart valve industry is destined to con-

tinue to inspire future innovations similar in importance to the milestones of the last

half century, as heart valves are a vital aspect of cardiovascular health and valve

diseases are increasingly common in modern society.

This document was created with LATEX.

ii

Contents

1 Introduction 1

2 Prosthetic Heart Valve Review 3

2.1 Mechanical Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Porcine Tissue Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Bovine Pericardial Tissue Valve . . . . . . . . . . . . . . . . . . . . . 4

2.4 Summary of Advantages and Disadvantages . . . . . . . . . . . . . . 7

3 Model of Mechanical Valve Flow Field 8

3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Simplified Model of Valve Hydrodynamics 10

4.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 Simplified Governing Equations . . . . . . . . . . . . . . . . . . . . . 10

4.3 Empirical Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Experimental Flow Loop for Measurement of Valve Hydrodynamics 13

6 Conclusion 15

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List of Figures

Figure 1 Diagram of the heart [Figure taken from Wikipedia (2008a)] . 1

Figure 2 Illustration of the heart and its valves [Illustrations taken from

Netter (1975)] . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Figure 2(a) Heart valves during diastole . . . . . . . . . . . . . . . 1

Figure 2(b) Heart valves during systole . . . . . . . . . . . . . . . 1

Figure 3 St. Jude Medical bileaflet mechanical heart valve [Figure taken

from Akay (2006)] . . . . . . . . . . . . . . . . . . . . . . . . . 3

Figure 4 Hancock II bioprosthetic heart valve [Image taken from CTSNet

(2008)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Figure 5 Mitroflow aortic pericardial heart valve . . . . . . . . . . . . . 4

Figure 6 Artificial heart valve from bovine pericardium [Figure and cap-

tion text taken from Morsi et al. (2004)] . . . . . . . . . . . . 5

Figure 7 Mitroflow valve construction . . . . . . . . . . . . . . . . . . . 5

Figure 8 Suture ring of the Mitroflow valve . . . . . . . . . . . . . . . . 6

Figure 9 Benefits of Mitroflow low profile valve design . . . . . . . . . . 6

Figure 9(a) Interference with sinotubular junction avoided . . . . . 6

Figure 9(b) Supra-annular or intra-annular seating options . . . . . 6

Figure 10 Stereo photogrammetry flow loop schematic [Figure taken from

Gao et al. (2000)] . . . . . . . . . . . . . . . . . . . . . . . . . 13

Figure 11 A three-dimensional PIV system set up to study the mixing of

an air jet with cross duct flow. [Figure and caption text taken

from Cengel and Cimbala (2006)] . . . . . . . . . . . . . . . . 14

Figure 12 Diagram of proposed flow loop . . . . . . . . . . . . . . . . . . 14

Figure 13 FEA used in the design of the Mitroflow valve [Image taken

from Mitroflow website (www.mitroflow.com)] . . . . . . . . . 15

List of Tables

Table 1 Valve comparison . . . . . . . . . . . . . . . . . . . . . . . . . 7

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1 Introduction

The human heart’s four valves are a vital part of the circulatory system, responsible

for maintaining unidirectional blood flow while subject to high intracardial pressure

gradients. As shown in Figure 1, the heart contains two atrio-ventricular (AV) valves,

the mitral or left AV valve, and the tricuspid or right AV valve. Exiting the left and

Figure 1: Diagram of the heart [Figure taken from Wikipedia (2008a)]

right ventricles, respectively, are the aortic and pulmonary valves. All of the valves

are in fact tricuspid, except for the bicuspid mitral valve, as can be seen in Figure 2.

(a) Heart valves during diastole (b) Heart valves during systoleFigure 2: Illustration of the heart and its valves [Illustrations taken from Netter (1975)]

Heart valves can suffer two main diseases, stenosis and incompetence. Stenosis

is caused by calcification of the valve leaflets which makes them stiffer and results

in a smaller effective orifice area when opening under pressure, while incompetence

is the incomplete sealing of the closed valve leaflets which may be due to rheumatic

1

disease or bacterial endocarditis and results in large amounts of backward blood flow,

or regurgitation (Webster, 2006).

Valvular pathology is most common in the left heart due to the higher pressures

present, and many heart valve prostheses have thus been developed over the years

to replace failing aortic and mitral valves. In fact, the industry has grown to such a

degree that more than 275,000 prosthetic valves are implanted each year around the

world (Morsi et al., 2004).

This report gives a review of available prosthetic heart valves, followed by mathe-

matical modeling of the flow field around a mechanical valve and a further simplified

hydrodynamic model. Finally an experimental flow loop to allow measurement of

valve hydrodynamics is proposed, and some general conclusions regarding prosthetic

heart valves are drawn.

2

2 Prosthetic Heart Valve Review

Many prosthetic valves have been designed over the last four decades, with most

designs falling into either the mechanical or ‘natural’ (ie. tissue-derived) categories

(Webster, 2006). In the interests of brevity, three representative valve designs will be

reviewed, highlighting the advantages and disadvantages of each.

2.1 Mechanical Valve

The main advantage of a mechanical valve is its durability; a well designed valve will

not wear out over time or become calcified as may its tissue counterparts. However,

anticoagulants must be taken by the patient on a regular basis to prevent throm-

boembolisms, or blood clots.

One of the most popular modern mechanical valve designs is the St. Jude Medical

bileaflet valve (St. Jude Medical Inc., St. Paul, MN), pictured in Figure 3.

Figure 3: St. Jude Medical bileaflet mechanical heart valve [Figure taken from Akay(2006)]

This bileaflet valve was a major innovation of the late 1970’s, using two hinged

semicircular leaflets to limit flow disturbance (Webster, 2006). The two leaflets open

to 85◦, and result in a lower closing force than previous tilting disc or ball valve

designs. Like most modern mechanical valves, the housing and leaflets are hardened

with a pyrolytic carbon coating to greatly enhance durability and biocompatibility.

3

2.2 Porcine Tissue Valve

Porcine tissue valves are made from harvested pig valves, with one common design

being the Hancock II Aortic and Mitral Bioprostheses (Medtronic Inc., Minneapo-

lis, MN). The Hancock II valve, shown in Figure 4, improved on its predecessor by

replacing the stiffer right coronary cusp of the porcine aortic valve with a similarly

sized but less stiff cusp from another valve (Webster, 2006).

Figure 4: Hancock II bioprosthetic heart valve [Image taken from CTSNet (2008)]

The valve is treated with T6 tissue treatment solution, a chemical agent used to

retard calcification of the valve (Thiene, 1986). A flexible polypropylene stent and

silicone rubber foam fiber suture ring with polyester cloth cover are attached to the

tissue to complete the valve (Webster, 2006).

2.3 Bovine Pericardial Tissue Valve

One example of a bovine pericardial tissue valve is the Mitroflow Aortic Pericardial

Heart Valve, produced by the local Mitroflow Division of the Sorin Group Canada

Inc. in Burnaby, BC. The stented Mitroflow valve, seen in Figure 5, is available in

19, 21, 23, 25, 27, and 29 millimetre sizes, and only for aortic valve replacement 1.

Figure 5: Mitroflow aortic pericardial heart valve

1All images in this section are taken from the Mitroflow website (www.mitroflow.com), unlessotherwise noted.

4

Like other bovine pericardial valves, the leaflets for the Mitroflow valve are cut

from the cow’s pericardium sac which encases the heart, and are fit together to mimic

a natural heart valve. Since the leaflets can thus be created to the specifications of

the human aortic valve, the hemodynamics of bovine pericardial valves are typically

better than those of porcine valves (Webster, 2006).

Figure 6 shows the general steps involved in the preparation of a bovine peri-

cardium valve, including intermediate fixation of the tissue in glutaraldehyde, and

the generic leaflet shape cut from the pericardium sheets. Figure 7 shows how the

Mitroflow valve is similarly constructed.

Figure 6: Artificial heart valve from bovine pericardium. Diagram showing the steps in-volved in creating a heart valve device. Bovine pericardium is collected (a) thencrosslinked in glutaraldehyde (b), fashioned into three leaflets (c), mounted ona stent (d), and fitted with a sewing ring. While opening, blood flow (arrow)pushes leaflets toward the outside (e) and creates an almost circular orifice (in-sert). Upon closure, leaflets are pushed toward the center of the valve (f) andcomplete closure is ensured by central coaptation of leaflets (insert). [Figureand caption text taken from Akay (2006)]

Figure 7: Mitroflow valve construction

5

The suture ring, which includes a radiopaque tungsten-impregnated insert to aid

in imaging, is highlighted in Figure 8.

Figure 8: Suture ring of the Mitroflow valve

The unique design and low profile of the Mitroflow valve allow it to be seated either

intra- or supra-annularly, while avoiding interference with the nearby sinotubular

junction as much as possible (see Figure 9).

(a) Interference with sinotubular junctionavoided

(b) Supra-annular or intra-annular seatingoptions

Figure 9: Benefits of Mitroflow low profile valve design

6

2.4 Summary of Advantages and Disadvantages

A comparison of some of the advantages and disadvantages of the three valve designs

is presented in the following table.

Valve name Advantages DisadvantagesSt. Jude • Does not require replacement • Requires anticoagulant ther-

apy• Unnatural hemodynamics (al-though better than previousvalve designs)

Hancock II • Better hemodynamics thanmechanical valves

• Stiffer than natural valves• Requires replacement after 10-15 years

Mitroflow • Better hemodynamics thanporcine valves• Minimal interference withsinotubular junction

• Stiffer than natural valves• Requires replacement after 10-15 years• Can only replace aortic valve

Table 1: Valve comparison

In general there is no single valve design which is superior to all others, but

some designs may be favorable for certain patients. For example, mechanical valves

are usually used for younger patients, to avoid future replacement which would be

necessary for less durable tissue valves.

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3 Model of Mechanical Valve Flow Field

3.1 Governing Equations

A mathematical model of the flow field around the St. Jude Medical bileaflet valve

must include the governing equations of conservation of mass, or the continuity equa-

tion, and conservation of momentum, or the Navier-Stokes equations. The differ-

ential form of the continuity equation and Navier-Stokes equations are given below

(Wikipedia, 2008b).∂ρ

∂t+∇ · (ρv) = 0 (1)

Inertia︷ ︸︸ ︷ρ( ∂v

∂t︸︷︷︸Unsteady

acceleration

+ v · ∇v︸ ︷︷ ︸Convectiveacceleration

)= −∇p︸ ︷︷ ︸

Pressuregradient

+ ∇ · T︸ ︷︷ ︸Viscous forces

+ f︸︷︷︸Otherforces

(2)

Here f includes the force of gravity on the blood as well as forces arising from the

rotation of the non-inertial reference frame; the human body can rarely be viewed as

an inertial reference frame except when at rest, adding further complexity to to an

already under-defined problem.

Another difficult term in the general Navier-Stokes equations is the deviatoric

stress tensor, T, defined below (Wikipedia, 2008c):

Tij = σij −σkk3δij (3)T11 T12 T13

T21 T22 T23

T31 T32 T33

=

σ11 σ12 σ13

σ21 σ22 σ23

σ31 σ32 σ33

−h 0 0

0 h 0

0 0 h

(4)

=

σ11 − h σ12 σ13

σ21 σ22 − h σ23

σ31 σ32 σ33 − h

. (5)

Where the mean hydrostatic stress h = σkk

3= σ11+σ22+σ33

3and thus,

T =

σ11 − σ11+σ22+σ33

3σ12 σ13

σ21 σ22 − σ11+σ22+σ33

3σ23

σ31 σ32 σ33 − σ11+σ22+σ33

3

. (6)

Here σ11, σ22, and σ33 are normal stresses, and σ12, σ13, σ21, σ23, σ31, and σ32 are

8

shear stresses.

3.2 Boundary Conditions

Along with the governing equations outlined above, a solution for the valve hydrody-

namics requires appropriate boundary conditions to be specified.

Boundary conditions at the surrounding vessel walls are that both the normal

component (no transfer through wall) and tangential component (no-slip condition)

of the fluid velocity are zero at the interface. A time-varying physiological pressure

corresponding to aortic pressure can be specified as an outlet condition, and a similar

inlet condition would correspond to the left ventricular pressure.

Certainly, the solution of a realistic model of valve hydrodynamics is a very difficult

problem. Even approximate solutions using Computational Fluid Dynamics will most

likely require some simplifying assumptions to be made in order to make the problem

more tractable.

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4 Simplified Model of Valve Hydrodynamics

4.1 Assumptions

Several simplifying assumptions are commonly used to facilitate the study of blood

flow. In the subsequent discussion the following assumptions will be made:

1. Blood is a Newtonian fluid

2. Blood is incompressible (ρ = constant)

3. Flow is axisymmetric and and irrotational (uθ = 0 and ∂∂θ

= 0

4. Flow is fully developed and steady ( ∂∂t

= 0)

5. The body is an inertial reference frame

6. Effect of gravity on flow is negligible

4.2 Simplified Governing Equations

Under the assumption of a Newtonian fluid, ∇·T can be simplified to µ∇2v. Now in

cylindrical coordinates, which fit the geometry near the valve well, the Navier-Stokes

equations become

ρ

(∂ur∂t

+ ur∂ur∂r

+uθr

∂ur∂θ

+ uz∂ur∂z− u2

θ

r

)=

−∂p∂r

+ µ

[1

r

∂

∂r

(r∂ur∂r

)+

1

r2

∂2ur∂θ2

+∂2ur∂z2

− urr2− 2

r2

∂uθ∂θ

]+ ρgr (7)

ρ

(∂uθ∂t

+ ur∂uθ∂r

+uθr

∂uθ∂θ

+ uz∂uθ∂z

+uruθr

)=

−1

r

∂p

∂θ+ µ

[1

r

∂

∂r

(r∂uθ∂r

)+

1

r2

∂2uθ∂θ2

+∂2uθ∂z2

+2

r2

∂ur∂θ− uθr2

]+ ρgθ (8)

ρ

(∂uz∂t

+ ur∂uz∂r

+uθr

∂uz∂θ

+ uz∂uz∂z

)=

−∂p∂z

+ µ

[1

r

∂

∂r

(r∂uz∂r

)+

1

r2

∂2uz∂θ2

+∂2uz∂z2

]+ ρgz, (9)

while the continuity equation becomes

1

r

∂

∂r(rur) +

1

r

∂uθ∂θ

+∂uz∂z

= 0. (10)

10

Applying assumptions 3, 4 and 6 cancels many of the terms in the preceding

equations, leaving the following relations:

ρ

(ur∂ur∂r

+ uz∂ur∂z

)= −∂p

∂r+ µ

[1

r

∂

∂r

(r∂ur∂r

)+∂2ur∂z2

− urr2

](11)

ρ

(ur∂uz∂r

+ uz∂uz∂z

)= −∂p

∂z+ µ

[1

r

∂

∂r

(r∂uz∂r

)+∂2uz∂z2

](12)

1

r

∂

∂r(rur) +

∂uz∂z

= 0. (13)

While this is a significant simplification of the Navier-Stokes and continuity equa-

tions, a simple solution is still difficult to achieve, therefore a standard empirical

equation will be used to consider some numerical values for valve hydrodynamics.

4.3 Empirical Equation

The standard empirical equation used in the study of heart valves is the Gorlin

equation, given below (Waite and Fine, 2007):

EOA =CO

K · TE ·HR ·√

∆P(14)

Where,

EOA = Effective Orifice Area of the valve [cm2]

CO = cardiac output [cm3/min]

K = Gorlin constant [ cms√mmHg

]

TE = Ejection time, or systolic time [s/beat]

HR = heart rate [bpm]

∆P = mean pressure gradient over ejection period [mmHg]

The Gorlin constant is derived empirically for various configurations and flow rates

and is generally assigned a numerical value of 44.3 for the aortic valve. Hartrumpf

et al. (2003) studied the influence of tilt and rotation on the hemodynamic per-

formance of the St. Jude Medical bileaflet aortic valve, and made use of the Gorlin

equation in their calculations. Measured pressure gradients from the Hartrumpf et al.

(2003) study are thus appropriate to combine with approximate physiologic values of

the remaining Gorlin equation variables in order to arrive at an approximate EOA

for the bileaflet valve.

For a valve diameter of 23 mm, the measured mean pressure gradient was 2.3

mmHg. Assuming a cardiac output of 4 L/min, a systolic period of 0.35 s, and a

11

heart rate of 70 bpm, the resulting effective orifice area is then

EOA =4000

44.3 · 0.35 · 70 ·√

2.3= 2.43 cm2. (15)

The contraction coefficient, CC , can then be calculated by comparing the effective

orifice area to the geometric orifice area (GOA) (Garcia and Kadem, 2006):

CC23mm =EOA

GOA=EOA

πr2=

2.43

π(

2.32

)2 =2.43

4.15= 0.58. (16)

Similarly for the 21 mm and 25 mm valves, with corresponding mean pressure

gradients measured as 4.2 and 1.0 mmHg, respectively,

CC21mm =1.80

3.46= 0.52, (17)

and

CC25mm =3.69

4.91= 0.75. (18)

This demonstrates that the smaller bileaflet valves suffer from progressively

smaller contraction coefficients, further emphasizing the benefit of larger valves.

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5 Experimental Flow Loop for Measurement of

Valve Hydrodynamics

To empirically determine the hydrodynamics of an artificial valve, a flow loop may be

devised which makes use of artificial blood and a variety of equipment and measuring

devices to simulate and quantify valve performance. What follows is a description of

some possible components of such a flow loop.

Experimental flow loops can be very complex, as with the flow loop shown in

Figure 10 which Gao et al. (2000) used to monitor heart valve leaflet motion via dual

camera stereo photogrammetry. However, a simpler flow loop will be proposed here

for the measurement of valve hydrodynamics using standard devices.

Figure 10: Stereo photogrammetry flow loop schematic [Figure taken from Gao et al.(2000)]

Provided the artificial blood is sufficiently transparent, as well as the piping near

the valve, a Particle Image Velocimeter (PIV) would be the ideal device for measuring

local velocity profiles around the valve. A stereo PIV system, as shown in Figure 11,

13

provides a snapshot of the three-dimensional velocity field for a cross-section of the

flow by measuring the scattering of a laser sheet off of particles in the flow (Cengel

and Cimbala, 2006).

Figure 11: A three-dimensional PIV system set up to study the mixing of an air jet withcross duct flow. [Figure and caption text taken from Cengel and Cimbala(2006)]

A simple manometer may be used to measure the pressure difference across the

valve, and a flow meter could be placed upstream or downstream from the valve

to quantify the bulk flow. Finally, a pump to drive the artificial blood flow would

complete the flow loop, depicted in Figure 12.

PIV

Heart Valve

Manometer

Pump

Flow Meter

Figure 12: Diagram of proposed flow loop

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6 Conclusion

The prosthetic heart valve industry has undergone many changes throughout its his-

tory, and innovations continue to this day. Recent advances include percutaneous

valve replacement, which allows valve implantation via catheter rather than the tra-

ditional open heart surgery, and novel methods of annuloplasty, or heart valve repair,

which can spare candidate patients from the drawbacks of valve replacement alto-

gether. Modern design tools such as Finite Element Analysis (FEA) are also changing

the field, and in fact FEA was used by Mitroflow to assist with valve research and

development (see Figure 13).

Figure 13: FEA used in the design of the Mitroflow valve [Image taken from Mitroflowwebsite (www.mitroflow.com)]

Due to the importance of heart valves to the function of the circulatory system, and

the rising prevalence of various valve diseases plaguing modern society, research into

the understanding of valve pathology and the design of new and improved prosthetic

valves or annuloplasty methods will undoubtedly continue for years to come.

15

References

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Cengel, Y. and J. Cimbala. Fluid mechanics: fundamentals and applications, chap. 8.

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CTSNet. Hancock II aortic and mitral bioprostheses — CTSNet, 2008. [Online;

accessed 20-April-2008].

URL http://www.ctsnet.org/medtronic/product/598

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leaflet motion monitored by dual camera stereo photogrammetry. Journal of Biome-

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Garcia, D. and L. Kadem. What do you mean by aortic valve area: geometric orifice

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