1. Electrical Wave Propagation in a Minimally Realistic Fiber Architecture Model of the Left Ventricle
Xianfeng Song, Department of Physics, Indiana University
Sima Setayeshgar, Department of Physics, Indiana University
March 17, 2006
Good morning! I am Xianfeng Song, and the topic of my talk is “Electrical wave propagation in a minimally realistic fiber Architecture model of the left ventricle”.

2. This Talk: Outline Goal Model Construction Results Conclusions
Here, I will discuss our work on constructing Minimally Realistic Fiber Architecture Model.
(The outline of this talk is as the followings: )
First, I will briefly discuss the goal we want to achieve by constructing such a model.
Next, I will talk about how we construct it.
Later, I will present some numerical results on our model comparing with the fully realistic model.
Finally, I will give our conclusion.
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

3. Minimally Realistic Model: Goal
Construct a minimally realistic model of the left ventricle for studying electrical wave propagation in the three dimensional anisotropic myocardium.
Adequately addresses the role of geometry and fiber architecture on electrical activity in the heart
Simpler and computationally more tractable than fully realistic models
More feasible to incorporate contraction into such a model
Easy to be parallelized and has a good scalability
Our goal is to construct a minimally realistic model of the left ventricle for studying electrical wave propagation in the three dimensional myocardium.
This minimally realistic model should adequately address the role of geometry and most importantly the fiber architecture on electrical activity in the heart
but be simpler and computationally more tractable than fully realistic models.
The model should also be more feasible to incorporate contraction.
And also the model should be easily parallelized and has a very good scalability.
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

4. Nested Cone Approximation
A simple nested cone geometry, represents the left ventricle which does not incorporate the valves.
fi=8
fe=16
Experiment shows that the real left ventricle has a nested layered geometry. So Here we adopted the similar nested cones geometry in our model to represent the left ventricle which does not incorporate the valves as the picture shows.
The cross sectional picture shows the construction of fiber surfaces. The red area is the cross section of the heart wall. The dotted line denotes the midwall. The nested cone was divided by the midwall into two halves. The fiber surface on the outer part have the same slope ae whereas the fiber surfaces on the inner part have the slope ai. The fiber surfaces on the outer part intersect the fiber surfaces on the inner part at the middle surface.
(A nested layered geometry for the left ventricle
A single macroscopic fiber bundle starting at the basal plane outside the midwall traverses down toward the apex on an outer surface, and at some point before reaching the apex, changes direction, traverses back along an inner surface reinserting at the basal plane inside the midwall)
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

5. Fiber construction Construction principles
Peskin Asymptotic Model (Derived by Peskin in 1996)
The fiber paths are approximate geodesics on the fiber surfaces.
Requiring the fibers to be circumferential where the double sheets meet at midwall
Euler-Lagrange equations (f: fiber trajectory):
Result
After the construction of the geometry and the fiber surfaces, we constructed our fiber on the fiber surface based on these two principles:
The fiber paths are approximate geodesics on the fiber surfaces when the heart wall is thin, which was derived by Peksin etc in 1996
We also require that the fibers to be circumferential where the double sheets meet at midwal. Thus we use Euler-Lagrange equation with the boundary condition to solve the fiber trajectory.
The result is shown in these two pictures: We let the fiber paths on the inner sheet be counter clockwise and the fiber paths on the out sheet be clockwise in order to make wave in the outer sheet can smoothly propagate along the fiber to the inner sheet.
(The shape of these two paths are a little bit different due to the different slope of the fiber surfaces.)
Fiber paths on the inner sheet
Fiber paths on the outer sheet
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

6. Parameters: a=0.1, m1=0.07,m2=0.3,k=8,e=0.01, Cm=1
Governing equations
Governing equation (a conventional parabolic partial differential equation)
Transmembrane current Im was described using a simplified excitable dynamics equations of the FitzHugh-Nagumo type (R. R. Aliev and A. V. Panfilov, 1996)
The governing equation we used is a conventional parabolic partial differential equation. The electro-physiological part in the governing equation is the simplified excitable dynamics equations of the FHN type with two variables, developed by A. V. Panfilov in The parameters we chose are in the unstable domain, which means the excitation will break into several scroll waves.
Parameters: a=0.1, m1=0.07,m2=0.3,k=8,e=0.01, Cm=1
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

7. Numerical Implementation
Working in spherical coordinates, with the boundaries of the computational domain described by two nested cones, is equivalent to computing in a box.
Standard finite difference scheme is used to treat the spatial derivatives, along with explicit Euler time-stepping
We discretized the model in the spherical coordinates along the theta, phi, r directions. Working on this coordinate, with the boundaries of the computational domain described by two nested cones, is equivalent to computing in a box. The standard finite difference scheme is used to treat the spatial derivatives along with explicit Euler time-stepping.
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

8. Diffusion Tensor Local Coordinate Lab Coordinate
Transformation matrix R
In order to implement the model in spherical coordinate, we have to calculate the diffusion tensor in this coordinate.
Here the first picture shows the fiber in its local coordinate. The diffusion tensor is only a simple diagnoal matrix with three components: D//, the diffusion constant along the fiber direction, two perpendicular components which means the diffusion constants on these two perpendicular directions. We use D//=1 and Dperp=0.5 in our simulation.
As we already constructed the fiber paths, we are able to calculate the transformation matrix from the local fiber coordinate to the lab coordinate. Using the transformation matrix, we are able to calculate the diffusion tensor in the lab coordinate using this formula.
The result is a full matrix which is more complicated than in local coordinate.
Local Coordinate
Lab Coordinate
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

9. Parallelize the code
The communication can be minimized when parallelized along the theta direction
Computational results show the model has a very good scalability
CPUs
Speed up 2
1.40 4
3.65 8
7.80 16
 15.50 32
 29.20
Our numerical implementation can be easily parallelized. The best choice is to divide the computational domain along theta direction as the picture shows. The communication between different computational nodes only occurs on the boundary.
The plot here shows the speed up ratio for different CPU numbers. The green curve shows the ideal speed up, which shows our speed up ratios are very close to the ideal ones thus proves our model has a very good scalability.
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

10. Finding the filament Finding all tips
Choose an unmarked tip as current tip
Add current tip into a new filament,
marked as the head of this filament
set reversed=0
Add current tip into
current filament
Find the closest unmarked tip
Is the
distance smaller than a certain
threshold?
Mark the current tip
Yes
Set the closest tip as current tip
Set reversed=1
The good physical quantities we used to quantitively describe the system are the filament number and the filament length. Thus, we developed our filament finding algorithm based on the 2D tip finding algorithm.
This flow chart shows our algorithm.
At the first step, we find all tips on all fi surfaces, which are approxately the fiber surfaces.
Then we randomly choose a tip as the current tip and find the nearest tip.
If the nearest tip is within a certain threshold distance, we consider these two tips are in the same filament and using the new tip as current tip to repeat the same loop.
If the nearest tip is not within a certain threshold, we reverse the search direction and repeat the same loop here.
We only reverse the search direction once before considering the current filament is complete.
After that, we repeat the whole process until we consume all tips.
Here, we redefined the distance between two tips in the algorithm, makes the distance calculated between two tip on non-adjacent surfaces to be very big, which inhibit the connections between two tips on non-adjacent surfaces. No
Is revered=0?
Set the head of current
filament as current tip
Definition: Distance between two tips
If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity
Otherwise, the distance is the distance along the fiber surface
Yes No
Are there any unmarked tips?
Yes No
End

11. Finding the filament
1. Find all tips
2. Connect the closest tip
3. Continue search the closest tip
The good physical quantities we used to quantitively describe the system are the filament number and the filament length. Thus, we developed our filament finding algorithm based on the 2D tip finding algorithm.
This flow chart shows our algorithm.
At the first step, we find all tips on all fi surfaces, which are approxately the fiber surfaces.
Then we randomly choose a tip as the current tip and find the nearest tip.
If the nearest tip is within a certain threshold distance, we consider these two tips are in the same filament and using the new tip as current tip to repeat the same loop.
If the nearest tip is not within a certain threshold, we reverse the search direction and repeat the same loop here.
We only reverse the search direction once before considering the current filament is complete.
After that, we repeat the whole process until we consume all tips.
Here, we redefined the distance between two tips in the algorithm, makes the distance calculated between two tip on non-adjacent surfaces to be very big, which inhibit the connections between two tips on non-adjacent surfaces.
4. The closest tip is too far
5. Reverse the search direction
6. Complete the filament
Distance between two tips (Our definition): If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
7. Start a new filament and repeat

12. Result - Simulation Filament initially time=2
FHN Model:
Filament initially time=2
The movie here shows the result of our simulation. The color denotes the slow variable in the electro-physiological model. We can clearly see the wave breaks as it spreads. This pictures on the right show the filament found by our algorithm at initial condition, which looks like a straight line. The second picture shows the filaments after the wave broken into several scroll waves.
The nice filaments found by our algorithm also shows the stability of our filament finding algorithm.
Animation: heart size: (wave length approximately 40) r1=87.5 r2=210 D//=1, Dp1=Dp2=0.5
Mesh points: 168*42*672 (dr=0.7)
time:30 (parfile.42.1)
Simulation: 7 hours for one CPU and 15 minutes for 32 CPUs
Color denotes the u variable in FHN model. The movie shows the spread of excitation in the cone shaped model from time=0-30.
The filament after breaking up time=999
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

13. Filament number and Filament length vs Heart size
Result - Convergence
The results of filament length agree within error bar for three different mesh sizes
The results of filament number agree within error bar between dr=0.7 and dr=0.5. The result for dr=0.5 is slightly off, which could be due to the filament finding algorithm
The computation time for dr=0.7 for one wave period in normal heart size is approximately 3 hours of cpu time using our electro-physiological model
In order to show the convergence, we use four different mesh sizes to simulate the model and compare the filament number and filament length for different mesh sizes.
Here we show the results for 3 different mesh sizes: dr=1.1, dr=0.7 and dr=0.5. For these 3 different mesh sizes, as we can see from this plot, the results of filament lengths are all consistent wthin the error. However the result of filament number for mesh size dr=1.1 are slightly off the result for mesh size dr=0.5, and the result for dr=0.7 agree with dr=0.5 within the error bar.
The slightly off on the results of filament number between mesh size 0.5 and mesh size 1.1 could be explained by the defects of filament finding algorithm. For mesh size 1.1, the mesh could be too rough for filament finding algorithm to precisely determine the connections among tips, however the filament length are less affected by the connections among tips agrees within error bar.
The mesh size for dr=0.7 is computational acceptable. Thus, we showed our model is convergent by reducing the mesh size and can reach the convergent condition very easily.
Filament number and Filament length vs Heart size
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore

14. Result - Filaments
(After we got the information on the filaments for different heart size, we can explore the relation between them. )
Here we explored the relation between filaments and the heart size.
On the left, the blue curve shows the logarithm of filament length versus the logarithm of heart size and the red curve shows the logarithm of filament number versus the logarithm of the heart size. Which clearly shows that they have linear relationship in loglog plots. This is consistent with the simulation on the fully realistic model using the same electro-physiological model by A. V. Panfilov (though the scale factors are different. The difference could due to the different geometry.)
The right figure shows the average filament length/average heart thickness vs the heart size, which shows the average filament length/average heart thickness tends to be a constant. This is also consistent with the result from A. V. Panfilov for the fully realistic model.
Filament number: y=2.787x-9.992
a_err=1.48 b_err=0.30
Filament length: y=3.739x-11.07
a_err=1.35 b_err=0.27
Panflov: Filament number a=2.21 +/
Filament length a=2.98 +/
!! The xlabel is wrong!!
Both filament length
Scaling of ventricular turbulence. The log of the total length and the log of the number of filaments both have linear relationship with log of heart size, but with different scale factor.
The average filament length/avearge heart thickness versus the heart size. It clearly show that the this average tends to be a constant

15. Conclusion
We constructed a minimally realistic model of the left ventricle for studying electrical wave propagation in the three dimensional myocardium and developed a stable filament finding algorithm based on this model
The model can adequately address the role of geometry and fiber architecture on electrical activity in the heart, which qualitatively agree with fully realistic model
The model is more computational tractable and easily to show the convergence
The model adopts simple difference scheme, which makes it more feasible to incorporate contraction into such a model
The model can be easily parallelized, and has a good scalability
In conclusion, we constructed a minimally realistic model of the left ventricle for studying electrical wave propagation in the 3D myocardium and developed a stable filament finding algorithm based on this model.
Our model can adequately address the role of geometry and fiber architecture on electrical activity in the heart which can be shown by comparing the result with that of the fully realistic model.
The model is using very simple finite difference scheme and equivalently computes in a box, which makes our model more computational tractable. We showed our model can easily achieve the convergence. Also, the simple finite difference scheme makes it more feasible to incorporate contraction into such a model.
Finally the model can be easily parallelized and has a very good scalability.
Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore