1. Dynamic simulation of mitral valve using the Immersed boundary method
Dr. Xiao Yu Luo
Department of Mathematics
University of Glasgow
Glasgow, G12 8QW UK

2. Acknowledgement Dr. Paul Watton, Mr. Min Yin
Dept. of Mathematics, University of Glasgow.
Professor Wheatley, Dr. G. Bernacca
Dept of Cardiac Surgery, University of Glasgow, Glasgow, UK
Professor Xiaodong Wang
Department of Mathematical Sciences, New Jersey Institute of Technology.

3. Anatomy of a mitral valve (MV)
MV: two leaflets, anterior leaflet (larger), and posterior leaflet (smaller).
Chordae run from valve leaflets to papillary muscles at the base of ventricle.
The human mitral valve (MV) is a complex anatomical structure consisting of two geometrically distinct flexible leaflets; a mitral annulus and
chordae tendinae that connect the valve leaflets to the papillary muscles (Berne & Levy, 1986) which are located in the walls of the ventricle.
The chordae act to reinforce the leaflet structure and prevent prolapse of the leaflets when the valve closes.
In addition, they also assist in maintaining the geometry and functionality of the ventricle.
Typical diseases of MV: Mitral stenosis is generally the result of rheumatic(ru:matic) heart disease.
MITRAL REGURGITATION (caused by MV prolapse, floppy MV, and other heart diseases caused in-coordination).
Both MV and AV are usually affected by disease due to high pressure. AV is studied extensively, but not MV.
Though over 50 MV designs are available, none is ideal with chordaes attached,
In most cases, AV is used in MV position.

4. (Mitral valve) MV: top view
The chordae reinforce the leaflet structure and prevent prolapse of the leaflets. They also assist in maintaining the geometry and functionality of the ventricle.

5. MV diseases
Typical diseases of MV: Mitral stenosis & mitral regurgitation.
MV needs to be repaired or replaced when damaged.
Two types of valve replacements: mechanical and bioprosthetic valves.

6. Mechanical mitral valves
Generation 1: (A) Starr-Edwards ball-and-cage.
2: (B) Medtronic-Hall tilting-disk, (C) Omnicarbon tilting-disk.
3: (D) St. Jude Medical bifleaflet, (E) Carbomedics bileaflet.
(F) ATS bileaflet. (G) ON-X bileaflet.
All require life long anticoagulant therapy, even the best designs have risks of hemorrhage related to anticoagulant therapy and thromboembolism
The first successful prosthetic mitral valve replacement was a device implanted by Nina Braunwald at the National Institute of Health in This was a homemade device with artificial chordae made of polyurethane. Two years later the first reliable device for replacement of the mitral valve was produced on a commercial basis. This was the Starr-Edwards ball-and-cage mitral valve that resulted from the collaboration of Albert Starr, a cardiac surgeon in Portland, and Lowell Edwards, a mechanical engineer in Southern California.7 This prosthesis was a great success and became the "gold standard" for many years, until the late 1960s, when second- and third-generation prosthetic valves began to appear. Although reliable hemodynamically, it was soon found that the Starr-Edwards valve had significant thromboembolic potential, particularly in the small ventricle, and aggressive anticoagulation was required to control thromboembolic events.8,9 The Silastic ball in the original prosthesis also had to be corrected because of inadequate durability.

7. Bioprosthetic mitral valves.
As the first, second, and third generations of prosthetic valves were developed, biologic or tissue replacement devices were developed concomitantly. The biologic valves showed a much lower frequency of thromboembolism and long-term anticoagulation seemed to be unnecessary.
The glutaraldehyde-fixed porcine aortic valve, principally developed by Hancock in the United States (1970) and Carpentier in Paris (1976), was the first commercially available bioprosthetic valve.29,30 These valves revolutionized mitral valve surgery by providing a biologic alternative that allowed long-term use without the need for lifelong warfarin anticoagulation. Both the Hancock and the Carpentier-Edwards valves became enormously popular in the 1970s and studies showed excellent 5-year durability (95%). But in the early 1980s structural valve dysfunction (SVD) became more apparent, with 15% to 20% of the prostheses failing within 10 years. The rate of deterioration seemed to accelerate in younger patients, with the valve gradually wearing down as a result of different biologically mediated dysfunctional processes.31–40
The first- and second-generation biologic valves were constructed from porcine aortic valves. Because of limited durability these valves have mostly been replaced by the third generation of biological valves, which still include porcine valves in addition to biomechanically engineered bovine pericardial valves. For this third generation of valves new technology has been incorporated aimed at improving valve longevity and hemodynamic function This has resulted in better mid-term results.41–48 These techniques include low-pressure or no-pressure fixation, antimineralization processes of the tissues, and low-profile, semiflexible stents that better define the biomechanical properties of the leaflets.
Generation 1: (A) Hancock II porcine heterograft,
2: (B) Carpentier-Edwards standard porcine heterograft, (C) Mosaic porcine
heterograft.
3: (D) Carpentier-Edwards pericardial bovine heterograft.
20% failing with 10 years. Rate of deterioation accelerates for young patients.

8. Current designs offer no ideal substitutes
Use of aortic valve design, and put in reversed configuration:
Potential of changing the vortex structure in the flow inside the ventricle.
Have no chordae attached:
Potential of changing the ventricle wall mechanics, and causes prolapse.
Clinical observation: The durability of porcine valves is less with mitral bioprostheses than with aortic bioprostheses.
The durability of porcine valves is less with mitral bioprostheses than with aortic bioprostheses. The more rapid deterioration of mitral bioprostheses may be due to higher ventricular systolic pressures against the mitral cusps as compared with the diastolic pressures resisted by aortic bioprosthetic leaflets. Durability of bioprosthetic valves is directly proportional to age;108 deterioration occurs within months or a few years in children and young adults and only gradually over years in septuagenarians and octogenarians.33,40,42,43,92,93,103,109 Essentially all valves implanted into patients less than 60 years of age have to be replaced ultimately and valve failure is prohibitively rapid in children and in adults under 35 to 40 years of age; therefore, bioprosthesis are not advisable in these age groups

9. A New Bioprosthetic Mitral Valve
A new bioprosthesis (polyurethane) design developed by Dept. of Cardiac Surgery, University of Glasgow
Benefits:
durable
no need for anticoagulation therapy,
biostable (tested on sheep)
based on real MV geometry, similar mechanical property,
with chordae for the first time.
D.J. Wheatley (2002),
Mitral valve prosthesis
Pat. no. WO
Natural MV: reduce anticoagulation and optimize haemodynamics
Designed valve: D-shaped with a stiff frame incorporating two 7mm support posts. The valve frame was manufactured from PEEK polymer.
In this paper, we test a newly designed mitral prosthesis (Wheatley 2002) using an Immersed Boundary (IB) model (Watton et al. 2006).
It is a polyurethane bileaflet valve, with geometric and mechanical properties based on the native MV, which will combine the advantages of both
mechanical and bioprosthetic heart valves, i.e. long-term durability without the need for permanent anti-coagulation. The valve has a larger anterior
leaflet and incorporates chordae, which originate from the valve annulus and traverse each leaflet, exiting at the leaflet edges to attach to the papillary
muscle regions of the ventricle, see Fig. 1.

10. Evaluating the new MV design using simulations
A key question is what happens when inserted in the ventricle?
We test this design dynamically using the Immersed boundary (IB) method.

11. Immersed Boundary (IB) Method
Fluid: t
r: fibre point coordinates, T: tension
x: fluid coordinates, s s 
X(r,t)
Solid:
Interactions:
IBM: An immersed flexible structure contained in a viscous incompressible fluid with periodic boundary conditions, and a uniform computational grid.
s: Langrange coordinate attached to the material, X(s,t) the material position whose label is s at time t,
tao: fibre direction
F force density generated by the elasticity of the material.
Delta: 3D Dirac delta function.
The incompressible viscous Navier-Stokes equations are discretised on a fixed Eulerian lattice whilst the elastic fibre equations are discretised on a
moving Lagrangian array of points, which do not necessarily coincide with the fixed Eulerian mesh points of the fluid computation. The interaction
between the fibres and the fluid is handled by a smoothed approximation to the Dirac delta function used to interpolate the fluid velocity to the solid
and to apply the solid force to the fluid. The adoption of a regular Eulerian lattice and the periodic boundary conditions enables a Fast Fourier Transform
to be employed to solve the fluid equations.
where solid behaves like fibres immersed in the fluid. It imposes force f on the fluid, and is moved by the fluid.

12. Generating Fibres N1 N2 N3 N4
The valve is represented by 4 node quadrilateral elements made of 2 fibres, each with three nodes.
Fibre 2
Each fibre is modelled as a Hookean spring:
Fibre 1
where s =fibre stiffness, E=Young’s modulus, A=cross-sectional area, l= fibre resting length, l=fibre stretch, T=tension.
A is the cross-sectional area of fibre. In general, the averaged area is used, i.e. Aave=he*Ae/L, where Ae is the planar corss-sectional area of the element. Note the stiffness of the element is provided by the stiffness of the fibre.

13. The tethering fibres where, u(r,t) is the velocity of the fibres.
Note if two nodal points on two different fibres have identical spatial positions at t=0, then this will be true for all successive times. Tethering fibres with a great stiffness, make use of this fact to enforce boundary conditions.
IBM: An immersed flexible structure contained in a viscous incompressible fluid with periodic boundary conditions, and a uniform computational grid.
s: Langrange coordinate attached to the material, X(s,t) the material position whose label is s at time t,
tao: fibre direction
F force density generated by the elasticity of the material.
Delta: 3D Dirac delta function.
The incompressible viscous Navier-Stokes equations are discretised on a fixed Eulerian lattice whilst the elastic fibre equations are discretised on a
moving Lagrangian array of points, which do not necessarily coincide with the fixed Eulerian mesh points of the fluid computation. The interaction
between the fibres and the fluid is handled by a smoothed approximation to the Dirac delta function used to interpolate the fluid velocity to the solid
and to apply the solid force to the fluid. The adoption of a regular Eulerian lattice and the periodic boundary conditions enables a Fast Fourier Transform
to be employed to solve the fluid equations.

14. Smooth approximation of Delta function
h: grid size
This is an even, smooth function, with has a continuous first derivative too. The implication for this is that in the true boundary when there is a velocity
Jump in normal derivative, this delta function smoothes such a jump out. An immersed interface method overcomes this difficulty (LeVeque & Li 1997).
Fluid: on a fixed Eulerian grid, Solid: network of fibres on a Lagrangian mesh.

15. Solving Fluid equations with FFT
The matrix equation of the discretized Navier-Stokes equations are
Kun+1=F(un)
where un is the solution vector at the nth time step.
This is linear in un+1, thus can be solved using the Fast Fourier Transform method (recall that the Eulerian grid is uniform).
In other words, we now need to have periodic boundary conditions.

16. ALGORITHM 1) Compute fibre force density, F
2) Distribute Fibre Force to Fluid Grid Points
3) Solve Navier-Stokes Equations (FFT)
4) Evolve fibres at the new local fluid velocity
The fluid (o) and fibre (x) grid points need not coincide. Grid x points are moving.
A smoothed approximation of Delta function is used.
Spatial discretization: finite difference
Temporal discretization: 2nd order accurate Runge-Kutta, midpoint rule?
If density is constant, the linear equations that needed to be solved on the Eulerian grid can be solved by FFT. To facilitate this, we typically formulate our problem on a period domain.

17. The IB valve modelling SOLIDWORKS (Valve Design)
GAMBIT (Mesh Software)
FIBRE GENERATOR
IB CODE
MATLAB (Graphics) a b c
Export: mesh files for (a) leaflets; (b) chordae; (c) fixed boundaries.

18. Chordae attached to the leaflets
CAP
14 chordae total
Anterior only (b) with posterior
CAP=Chordae Attachment Points

19. The IB Model
Cylindrical Tube: Length 16cm Diameter 5.6cm.
Experimentally determined periodic velocity profile prescribed.
Viscosity: 0.01g/m.s Density: 1g/cm3
Submerged within a fluid mesh: size 64x64x64

20. Validations – Static Aortic Valve
E=15MPa,
h = 0.125mm
Polyurethane
material
Valve housed in experimental rig and subject to an incremental steady pressure.
Displacement of centre of 3 leaflets measured and compared with IB results.
Good quantitative comparison for deformation obtained.

21. Validations -Static Mitral Valve
ANSYS IB
P=120mmHg.
Leaflets:
E = 4.29MPa,
H = 1.32mm,
Chordae:
E = 47MPa,
A = 0.4mm2.
Colormaps of surface displacements (mm)
Peak stress =1.4MPa predicted by both IB and Ansys
IB has no problem with the thin wall, but with contact problem.
Model based on Kunzelman’s model of native mitral valve.
Deformation and peak stress predicted by IB and ANSYS in agreement.

22. IB versus ANSYS
IB is much better with thin structures. Ansys has “locking” phenomenon, hence a thicker shell has to be used (increasing thickness by 3-4 times, while reducing Young’s modulus by the same scale).
IB is fast
IB is designed for dynamic modelling. Ansys is not.
Current IB has no bending.
Current IB is not so good with contact problems.
This ensures that the stretching stiffness of the shell remain unchanged. For IB, the both are equivalent, for ANSYS, this results in a slightly higher bending stiffness.
(numerical) Locking: as t^3Ab+tAm=F (Ab+Am/t^2)=F/t^3, where t is the thickness. As t->0 (rho=3), tAm->0, the only soln. is zero (when the exact soln is not). I.e. the structure is increasingly stiffen. Thus called locking.
IB: 64*64*64, solid twice as dense
Ansys: 12,429 elements (4-node shell) MV, 2964 elements for AV.
Static simulations: both Ansys and IB took about 18 CPU hours (MV), but Ansys only did ½ of the domain. The dynamic simulations take about a week
On titania (Sheffield grid, which is now dead).

23. Validations: Dynamic MV with fixed CAP
Flow Rate (ml/s)
Time (s)
Experimentally determined flow profile prescribed.
Polyurethane leaflets: E = 5MPa, h = 0.125mm
Chordae: E = 30MPa, Area = 0.4mm2
NOTE: Chordae are in leaflet surface in computational model but not visually represented.
Computation model of valve behaves too flexibly.
Unrealistic crimping of leaflets occurs.
IB unable to model bending effects.

24. Validations: Pressure Gradient, fixed CAP
Pressure Gradients
Exp. IB
mmHg
Given Flow Profiles
mmHg
The higher and delayed peak pressure is caused by the “sticking effect” of IB: when two fibre nodes are too close together, they share a very similar flow field (due to the uniform
grid used), therefore it is not easier to move them apart since the alpha distribution smear things out.
Predicted pressure gradients consistent with experimental ones. However, note a higher (and delayed) peak pressure in IB.

25. Physiological Chordae Attachemnt Points (CAP) motions
Analyse Human MRI data with CMRTOOLS - software package for analysing Cardiovascular Magnetic Resonance (CMR) images (Imperial College
Determine dynamic geometry of ventricle and papillary muscle axes.
Intersection of data enables papillary muscle regions of ventricle to be identified

26. Track Mitral Annulus DIASTOLE SYSTOLE
Software package Slice-o-Matic used to analyse MRI data.
Tracks motion of Mitral Annulus through 2 planes (32 Time Slices of Data)
Note relative displacement of mitral annulus plane to apex of ventricle is obtained.

27. Determining Motion of Mitral Apparatus
CMRTOOLS - Determine geometry of ventricle and papillary positions.
Slice-O-Matic - Track 4 points on mitral annulus.
Fortran code - Convert CMRTOOLS data to Matlab surface data.
Matlab scripts - 1) Transform Slice-O-Matic and CMRTOOLS data to common coordinate system.
2) Tag papillary attachment regions on ventricle.
3) Determine relative motion of mitral apparatus.

28. Relative Motion of CAP to Annulus
Relative motion to a fixed annulus
CAP=Chordae Attachemnt Point, which will be the attachment point to the ventricle during surgery.
The relative motion of CAP to mitral annulus is determined. Only perpendicular direction is considered. The averaged distance-time curve from the 4 points are used for the two CAP positions.
Each CAP is positioned 40mm from MV annulus, with cross-sectional area of 0.4mm^2, and E=30MPa. Annulus is spatially fixed.

29. Physiological Boundary Conditions
Experimental Flow Rate
Flow prescribed at the tube entrance and exit.
Pressure gradient (calculated).
Relative CAP motion is prescribed perpendicular to the mitral annulus plane.
ml/s
Valve opens
Pressure Gradient
mmHg
Onset of systole
Minimum distance between the annulus and CAP of the ventricle wall occurs during systole.
An increase of 10mm occurs as the ventricle enlarges and the mitral annulus moves upwards.
For the native MV, the papillary muscles contract in systole and relax in diastole, thus helping to maintain the distance between leaflet edges and papillary regions of the ventricle.
The valve design should not restrain the natural motion of the ventricle – otherwise this may lead undesirable remodelling response of the ventricle. mm
Physiological CAP motion
Time (s)

30. Pressure Gradients Opening pressure gradient reduces with CAP motion.
The pull of the chordae assists the opening of the valve.
Opening pressure gradient reduces with CAP motion.

31. Leaflet stretches with CAP motion
Anterior: circumferential
Anterior: longitudinal
Posterior: circumferential
Posterior : longitudinal
ANTERIOR: small difference, POSTERIOR: stretches increase.

32. Leaflet stretches with CAP motion
Anterior: circumferential
Anterior: longitudinal
Posterior: circumferential
Posterior : longitudinal
ANTERIOR: small difference, POSTERIOR: stretches increase.

33. Leaflet stretches with CAP motion
Anterior: circumferential
Anterior: longitudinal
Posterior: circumferential
Posterior : longitudinal
ANTERIOR: small difference, POSTERIOR: stretches increase.

34. Chordae Stretches No Stretch Stretch
ANTERIOR CHORDAE
Stretch
POSTERIOR CHORDAE
Stretch
Anterior leaflet can accommodate the CAP motion without its chordae becoming taut
CAP motion results in high stretches of the posterior chordae (lower right).
NOTE: Stretches <1, denote chordae relaxed.

35. Up to 20% strain increase due to papillary motion
Anterior leaflet can accommodate CAP motion.
Posterior leaflet subject to significantly higher stretches detrimental effect on the long-term durability of the valve.

36. Improved Flow Dynamics of Mitral Valve
Aortic: central jet and double vortices.
New design yields improved flow characteristics.
This may conserve energy of flow when it is ejected from ventricle, and aid closure of aortic valve.
Mitral: flow asymmetry, larger vortex

37. Ongoing work To add ventricle model to model detailed flow and vortex.

38. Ongoing work Improvements for IB code:
Developing a better solid model with bending stiffness.
Use adaptive mesh to get rid of sticking effects when opening.
Better boundary conditions.

39. Summary
Dynamic analysis for a mitral valve with chordae attachment points (CAP) moving with ventricle is carried out with a IB code. Results show that:
CAP motion assists the opening of the valve (lower opening pressure gradients).
Current design: posterior is over-stretched.
Recommendation for design improvements:
Modify geometry to allow a greater movement of the posterior leaflet.
Modify stiffness of posterior chordae: low modulus external chordae that can accommodate high stretches, and high modulus internal leaflet chordae, which resist deformation
Design a stiffer posterior leaflet which does not require chordae.
A mitral prosthesis must function without the mechanical role of the papillary muscles. Consequently, the valve design may need to differ from that of the native valve.

40. Thank you

41. Modifications to IB Code
Enable ARBITRARY VALVE GEOMETRIES to be modelled easily using a Fibre Generator Fortran Code.
Static loading of valves added to perform static analysis (validations).
Variable Young’s modulus for valve materials.
Motion of papillary muscle included.
Set up modelled, I.e. valve housing and tubing.

42. Locking in FEM Where thickness parameter is the unknown solution,
is the scaled bending energy,
is the scale membrane energy,
is the scaled external virtual work.
If r =3, bending-dominated
r =1, membrane-dominated
ill-posed membrane problem

43. Fibre Representation of Prosthetic Mitral Valve
Fibres generated automatically from an FEM mesh.
Stiffness dependent on valve material and element sizes.

44. Smooth approximation of Delta function
h: grid size
This is an even, smooth function, with has a continuous first derivative too. The implication for this is that in the true boundary when there is a velocity
Jump in normal derivative, this delta function smoothes such a jump out. An immersed interface method overcomes this difficulty (LeVeque & Li 1997).

45. Leaflet stretches with CAP motion
Anterior: circumferential
Anterior: longitudinal
Posterior: circumferential
Posterior : longitudinal
ANTERIOR: small difference, POSTERIOR: stretches increase.