#
A Mathematical Spline-Based Model of Cardiac Left Ventricle Anatomy and Morphology^{ †}

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## 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 ${\varphi}_{i}$, $i=0,1,\dots n-1,$ where measurements are taken;
- coordinates ${\rho}_{i,j}^{epi}$, ${z}_{i,j}^{epi}$, $i=0,1,\dots n-1,$ $j=0,1,\dots {n}_{i}^{epi}-1,$ of marked points on the epicardium;
- coordinates ${\rho}_{i,j}^{endo}$, ${z}_{i,j}^{endo}$, $i=0,1,\dots n-1,$ $j=0,1,\dots {n}_{i}^{endo}-1,$ 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 $[0,1]$. We utilize the expression for $\psi (\gamma ,z)$ from Formula (1) and substitute it into (5):$$\rho ={\rho}^{epi}(\psi (\gamma ,z),\varphi )(1-\gamma )+{\rho}^{endo}(\psi (\gamma ,z),\varphi )\gamma .$$We solve this equation with respect to γ. Let the root be γ. Hence, the point’s special coordinate $\psi =arcsin\left(\frac{Z-z}{Z-h\gamma}\right).$
- Differentiate (numerically or analytically) the function $\rho (\gamma ,\psi ,\varphi )$ with respect to all arguments and obtain three partial derivatives ${\rho}_{\gamma},$ ${\rho}_{\psi}$ and ${\rho}_{\varphi}.$ We can find one derivative analytically:$${\rho}_{\gamma}={\rho}^{endo}(\psi ,\varphi )-{\rho}^{epi}(\psi ,\varphi ).$$
- The LV model point is an image of a point on the sector $\mathrm{P}\le 1,$ $\mathrm{\Phi}\in [\pi {\gamma}_{0},\pi {\gamma}_{1}]$. This point, the preimage, has polar coordinates $\mathrm{P}=1-\frac{2\psi}{\pi}$, $\mathrm{\Phi}=\pi \gamma ({\gamma}_{1}-{\gamma}_{0})+\pi {\gamma}_{0}$ and Cartesian coordinates $X=\mathrm{P}cos\mathrm{\Phi}$, $Y=\mathrm{P}sin\mathrm{\Phi}$.
- The LV point $\overrightarrow{r}$, parameterized by Φ, has Cartesian coordinates:$$x\left(\mathrm{\Phi}\right)=\rho (\gamma \left(\mathrm{\Phi}\right),\psi \left(\mathrm{P}\left(\mathrm{\Phi}\right)\right),\varphi (\gamma \left(\mathrm{\Phi}\right),{\varphi}_{min},{\varphi}_{max}))cos\varphi (\gamma \left(\mathrm{\Phi}\right),{\varphi}_{min},{\varphi}_{max}),$$$$y\left(\mathrm{\Phi}\right)=\rho (\gamma \left(\mathrm{\Phi}\right),\psi \left(\mathrm{P}\left(\mathrm{\Phi}\right)\right),\varphi (\gamma \left(\mathrm{\Phi}\right),{\varphi}_{min},{\varphi}_{max}))sin\varphi (\gamma \left(\mathrm{\Phi}\right),{\varphi}_{min},{\varphi}_{max}),$$$$z\left(\mathrm{\Phi}\right)=\rho \left(\gamma \right(\mathrm{\Phi}),\psi (\mathrm{P}\left(\mathrm{\Phi}\right)\left)\right),$$
- The non-normalized vector $\overrightarrow{w}=(wx,wy,wz)=\frac{d\overrightarrow{r}}{d\mathrm{\Phi}}$ of the fiber direction has components:$$\begin{array}{cc}\hfill wx=& \frac{sin\mathrm{\Phi}}{(\pi -2\psi )({\gamma}_{1}-{\gamma}_{0})}\xb7\left(y{\varphi}_{max}-({\rho}_{\gamma}+{\rho}_{\varphi}{\varphi}_{max})\xb7cos\varphi \right)-cos\varphi \xb7{\rho}_{\psi}\xb7\frac{\pi}{2}\xb7cos\mathrm{\Phi},\hfill \\ \hfill wy=& \frac{sin\mathrm{\Phi}}{(2\psi -\pi )({\gamma}_{1}-{\gamma}_{0})}\xb7\left(x{\varphi}_{max}+({\rho}_{\gamma}+{\rho}_{\varphi}{\varphi}_{max})\xb7sin\varphi \right)-sin\varphi \xb7{\rho}_{\psi}\xb7\frac{\pi}{2}\xb7cos\mathrm{\Phi},\hfill \\ \hfill wz=& \frac{hsin\mathrm{\Phi}sin\psi}{(2\psi -\pi )({\gamma}_{1}-{\gamma}_{0})}+\left(Z-h\gamma \right)cos\psi \xb7\frac{\pi}{2}\xb7cos\mathrm{\Phi}.\hfill \end{array}$$

## 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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**Figure 1.**Vertical (meridional) sections of the LV free wall (on the

**left**) and the IVS (on the

**right**) of a human heart. The points represent DT-MRI data; the solid red line represents the model epicardium; the dashed blue line represents the model endocardium. The circles show the interpolation (marked) points given by a user. The horizontal axis is ρ; the vertical axis is z. On the right panel, we see a papillar muscle in the RV cavity (vertical one, $\rho =25,\dots ,45$ mm) and an RV free wall (inclined).

**Figure 2.**Horizontal chords on the sector $\mathrm{P}\le 1,$ $\mathrm{\Phi}\in [\pi {\gamma}_{0},\pi {\gamma}_{1}]$. Here, ${\mathrm{\Phi}}_{0}$ and ${\mathrm{\Phi}}_{1}$ are polar angles of the right and left chord ends, respectively. Parameter ${\gamma}_{0}=0.1$, and ${\gamma}_{1}=0.95$.

**Figure 3.**Images of the sector chords in $(\gamma ,\psi )$ coordinates. The horizontal axis is γ, and the vertical axis is ψ. Parameter ${\gamma}_{0}=0.1$, and ${\gamma}_{1}=0.95$.

**Figure 4.**Epicardial and endocardial surfaces of the model that was constructed based on DT-MRI data. (

**left**) The epicardium is opaque; (

**right**) The epicardium is semi-transparent. Color shows z-coordinates.

**Figure 5.**A spiral surface (SS) with fibers on it. Two views are shown. The left of the figure depicts a view from the top and side of the SS, while the right of the image provides a side view with the mid-myocardial part on the left and the epicardial layer on the right. Color shows z-coordinates.

**Figure 6.**The definition of the local coordinate system is shown on the (

**left**) of the image. $Oxyz$ is the global Cartesian coordinate system. The blue axis is normal, u, to the epicardium. The red axis is parallel tangent, v. The dark green axis is a meridian tangent, w. The colorful surface is the epicardium; the color depends on altitude z. The curve inside is a model fiber, and it does not lie on the epicardium. The normal axis intersects the curve at a point. On the (

**right**), the definitions of the true fiber angle α and the helix angle ${\alpha}_{1}$ are shown. The thick, dashed line is a tangent to a myofiber segment constructed at the origin of the coordinates. The dashed-and-dotted lines are projections of the myofiber tangent. (Copied from [53] (Figure 9) with a deletion of angle ${\alpha}_{2}$.)

**Figure 7.**Fiber angles in the model and in the experimental data. The basal area ($\psi =10\xb0$) of the LV free wall is shown. (

**A**) A horizontal LV section. The points are myocardial points from a human heart DT-MRI scan. (

**B**) A meridional LV section. The solid (dashed) curve is the model epicardium (endocardium), and the points are myocardial points from the DT-MRI scan. (

**C**,

**D**) The angles α and ${\alpha}_{1}$, respectively. The X-axis displays the position of points in the wall depth; 0 corresponds to the endocardium, and 1 corresponds to the epicardium. The points show the experimental data, while the solid curves show our model data, and the dashed curves display results yielded by the model from [46].

**Figure 8.**Fiber angles in the model and experimental data. The LV free wall in the middle area ($\psi =25\xb0$) is shown. The conventional signs are the same as in Figure 7.

**Figure 9.**Fiber angles in the model and experimental data. The LV free wall in the apical area ($\psi =60\xb0$) is shown. The conventional signs are the same as in Figure 7.

**Figure 10.**Fiber angles in the model and experimental data. The IVS in the basal area ($\psi =5\xb0$) is shown. The conventional signs are the same as in Figure 7.

**Figure 11.**Fiber angles in the model and experimental data. The IVS in the middle area ($\psi =30\xb0$) is shown. The conventional signs are the same as in Figure 7.

**Figure 12.**Fiber angles in the model and experimental data. The IVS in the apical area ($\psi =65\xb0$) is shown. The conventional signs are the same as in Figure 7.

**Figure 13.**Histogram of angles between fibers in experiment and model data. The human heart dataset is used. X-axis is the angle; Y-axis is the number of points in the segments; the segment $[0,{90}^{\circ}]$ was divided into 100 segments of equal length.

**Figure 14.**Ultrasound imaging as a base for model construction. (

**left**) A sonographic two-chamber view of a patient’s heart and two measurements: the first for the power-function-based model (white lines) and the second for the spline-based model (blue lines show the endocardium; red lines show the epicardium). The patient suffered from an old myocardial infarction, and a scar is present. The marked points are shown as crosses. This view gives data for two model meridians. The model axis and base are shown as grey lines. The image has been rotated so that the model axis $Oz$ is vertical. (

**right**) A spline-based model of this patient’s LV. The view is semitransparent to show both the epicardial and endocardial surfaces. The surface mesh was made using Ani3D-AFT software [57].

**Figure 15.**Flowchart displaying the steps for the construction and use of the spline-based LV model.

True Fiber | Canine | Human | ||
---|---|---|---|---|

Angle | Raw Data | Averaged | Raw Data | Averaged |

mean | $12.3\xb0$ | $10.9\xb0$ | $12.5\xb0$ | $10.1\xb0$ |

median | $9.2\xb0$ | $8.7\xb0$ | $8.8\xb0$ | $7.8\xb0$ |

st. dev. | $10.8\xb0$ | $9.1\xb0$ | $12.2\xb0$ | $9.2\xb0$ |

Helix | Canine | Human | ||
---|---|---|---|---|

Angle | Raw Data | Averaged | Raw Data | Averaged |

mean | $15.2\xb0$ | $12.3\xb0$ | $18.9\xb0$ | $13.3\xb0$ |

median | $11.0\xb0$ | $9.8\xb0$ | $12.1\xb0$ | $10.2\xb0$ |

st. dev. | $16.5\xb0$ | $9.8\xb0$ | $22.6\xb0$ | $11.5\xb0$ |

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pravdin, S.
A Mathematical Spline-Based Model of Cardiac Left Ventricle Anatomy and Morphology. *Computation* **2016**, *4*, 42.
https://doi.org/10.3390/computation4040042

**AMA Style**

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 Style**

Pravdin, 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