A Mathematical Spline-Based Model of Cardiac Left Ventricle Anatomy and Morphology †
Abstract
:1. Introduction
1.1. Examination of Individual Hearts and Empirical Models
1.2. Theoretical Models of Cardiac Anatomy
1.3. Relation to Other Anatomical Ventricular Models
2. Construction of the LV Model
- meridians , where measurements are taken;
- coordinates , , of marked points on the epicardium;
- coordinates , , of marked points on the endocardium.
2.1. Spiral Surfaces
2.2. Filling a Spiral Surface with Fibers
2.3. Fitting the LV Form
3. Methods for Model and Experimental Data Comparison
4. Results of Comparing the Model with DT-MRI Data
4.1. Comparison of the Model with Human Heart Data along Straight Pinning Lines
4.2. 3D Verification
5. Constructing a Model Based on Sonography Data
6. Numerical Method for Solving Reaction-Diffusion Systems on the Model
The Method to Rarefy the Computation Mesh
7. An Example of the Spline-Based Model’s Practical Usage in Electrophysiological Simulations
8. Discussion
8.1. Advantages and Disadvantages of the Proposed Model
- It has relatively few parameters for shape, fibers and sheets.
- The LV apex is smooth thanks to the special choice of function z. This is a useful feature for integration methods sensitive to the smoothness of the boundary.
- It uses only simple one-variable splines, and no two- or three-parametric splines are required.
- The fitting of shape and fiber directions are independent tasks in this model.
- It is flexible enough to fit not only normal, but pathological LVs.
- It yields not only fibers, but also sheets.
- Similar to our previous models, its base is flat, which is not the case in real mammal anatomy.
- The fibers are not geodesic lines in the model sheets. It is difficult to determine if this is a disadvantage of the fibers, the sheets or both.
- Model fibers end at the base. However, this demerit can be amended by combining this model with the toroids-based one described in [62].
- It is unclear whether the model is generalizable for both ventricles.
8.2. The Present Model Compared with Other Qualitative Models
8.3. Comparison with Experimental Data and Quantitative Models
8.4. Further Development and Usage of the Model
Acknowledgments
Conflicts of Interest
Abbreviations
CT | computed tomography |
DT-MRI | diffusion tensor magnetic resonance imaging |
DTI | diffusion tensor imaging |
IVS | interventricular septum |
LV | left ventricle |
RV | right ventricle |
SS | spiral surface |
Appendix A. Calculation of the Fiber Direction in Points of the LV Model
- Use Formula (5) to find the special coordinates γ and ψ of the point numerically. This problem can be reduced to solving one algebraic equation with one unknown quantity γ on the segment . We utilize the expression for from Formula (1) and substitute it into (5):We solve this equation with respect to γ. Let the root be γ. Hence, the point’s special coordinate
- Differentiate (numerically or analytically) the function with respect to all arguments and obtain three partial derivatives and We can find one derivative analytically:
- The LV model point is an image of a point on the sector . This point, the preimage, has polar coordinates , and Cartesian coordinates , .
- The LV point , parameterized by Φ, has Cartesian coordinates:
- The non-normalized vector of the fiber direction has components:
Appendix B. Formulae for the Laplacian in the Special Coordinates
Appendix C. The No-Flux Boundary Conditions in the Special Coordinates
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True Fiber | Canine | Human | ||
---|---|---|---|---|
Angle | Raw Data | Averaged | Raw Data | Averaged |
mean | ||||
median | ||||
st. dev. |
Helix | Canine | Human | ||
---|---|---|---|---|
Angle | Raw Data | Averaged | Raw Data | Averaged |
mean | ||||
median | ||||
st. dev. |
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Pravdin, S. A Mathematical Spline-Based Model of Cardiac Left Ventricle Anatomy and Morphology. Computation 2016, 4, 42. https://doi.org/10.3390/computation4040042
Pravdin S. A Mathematical Spline-Based Model of Cardiac Left Ventricle Anatomy and Morphology. Computation. 2016; 4(4):42. https://doi.org/10.3390/computation4040042
Chicago/Turabian StylePravdin, Sergei. 2016. "A Mathematical Spline-Based Model of Cardiac Left Ventricle Anatomy and Morphology" Computation 4, no. 4: 42. https://doi.org/10.3390/computation4040042