Parametric Modeling of the Mouse Left Ventricular Myocardial Fiber Structure

Abstract

Magnetic resonance diffusion tensor imaging (DTI) has greatly facilitated detailed quantifications of myocardial structures. However, structural patterns, such as the distinctive transmural rotation of the fibers, remain incompletely described. To investigate the validity and practicality of pattern-based analysis, 3D DTI was performed on 13 fixed mouse hearts and fiber angles in the left ventricle were transformed and fitted to parametric expressions constructed from elementary functions of the prolate spheroidal spatial variables. It was found that, on average, the myocardial fiber helix angle could be represented to 6.5° accuracy by the equivalence of a product of 10th-order polynomials of the radial and longitudinal variables, and 17th-order Fourier series of the circumferential variable. Similarly, the fiber imbrication angle could be described by 10th-order polynomials and 24th-order Fourier series, to 5.6° accuracy. The representations, while relatively concise, did not adversely affect the information commonly derived from DTI datasets including the whole-ventricle mean fiber helix angle transmural span and atlases constructed for the group. The unique ability of parametric models for predicting the 3D myocardial fiber structure from finite number of 2D slices was also demonstrated. These findings strongly support the principle of parametric modeling for characterizing myocardial structures in the mouse and beyond.

Appendix

Representations of the Standardized Prolate Spheroidal LV Geometry

Based on the general equation of an upright prolate spheroidal surface with longitudinal radius a and transverse radius b in Cartesian coordinates,

$$\frac{{x^{2} + y^{2} }}{{b^{2} }} + \frac{{z^{2} }}{{a^{2} }} = 1 ,$$

(8)

as depicted in Fig. 5, the geometry of the LV was approximated by a constant-thickness volume consisting of concentric prolate spheroidal shells specified by,

Figure 5

Approximation of the LV by an approximate prolate hemispheroidal volume in Cartesian coordinates. Dimensions in the anatomical space (left), including the areas of the LV in the equatorial short-axis plane and its distance to the apex, were used to compute the transverse and longitudinal axis lengths of the prolate hemispheroid (right) as specified by Eq. (9). The area A ₁ included the ventricular cavity.

$$\frac{{x^{2} + y^{2} }}{{\left( {b_{1} - \tau w} \right)^{2} }} + \frac{{z^{2} }}{{\left( {a_{1} - \tau w} \right)^{2} }} = 1 ,$$

(9)

where τ ∈ [0, 1] was the transmural distance variable, a ₁ and b ₁ were the outer longitudinal and transverse radii, a ₂ and b ₂ were the inner radii, and $w = a_{1} - a_{2} = b_{1} - b_{2}$ was the wall thickness of the volume.

The Cartesian coordinates of a prolate spheroid could be alternatively represented using the spherical radial μ, circumferential ψ and azimuthal ν coordinate variables and focal distance ρ according to :3

$$x = \rho \sinh \left( \mu \right)\sin \left( \nu \right)\cos \left( \psi \right)$$

(10)

$$y = \rho \sinh \left( \mu \right)\sin \left( \nu \right)\sin \left( \psi \right)$$

(11)

$$z = \rho \cosh \left( \mu \right)\cos \left( \nu \right)$$

(12)

Using trigonometric manipulation, the above 3 equations were combined and reduced to

$$\frac{{x^{2} + y^{2} }}{{(\rho \sinh \left( \mu \right))^{2} }} + \frac{{z^{2} }}{{(\rho \cosh \left( \mu \right))^{2} }} = \cos^{2} \left( \nu \right) + \sin^{2} \left( \nu \right) = 1 .$$

(13)

By comparing Eq. (13) to Eq. (9), the spherical coordinates of all points in the above prolate spheroidal LV volume were found by first determining $\mu$ and ρ according to

$$\mu = \tanh^{ - 1} \left( {\frac{{b_{1} - \tau w}}{{a_{1} - \tau w}}} \right) , \;{\text{and}}$$

(14)

$$\rho = \sqrt {\left( {a_{1} - \tau w} \right)^{2} - \left( {b_{1} - \tau w} \right)^{2} } = \frac{{a_{1} - \tau w}}{\cosh (\mu )}$$

(15)

followed by ψ and ν through

$$\psi = \tan^{ - 1} \frac{y}{x} , \;{\text{and}}$$

(16)

$$\nu = \cos^{ - 1} \left( {\frac{z}{\rho \cosh (\mu )}} \right) .$$

(17)

In the spherical coordinate system, the local tangential basis vectors, $\hat{c}$, $\hat{l}$ and $\hat{n}$ in the circumferential, longitudinal and normal (or radial) directions, respectively, were then computed according to

$$\hat{c} = \left[ {\begin{array}{*{20}c} { - \sin (\psi )} \\ {\cos (\psi )} \\ 0 \\ \end{array} } \right] ,$$

(18)

$$\hat{l} = \frac{ - 1}{{\sqrt {\sinh^{2} \left( \mu \right) + \sin^{2} \left( \nu \right) } }} \left[ {\begin{array}{*{20}c} {\sinh \left( \mu \right)\cos (\nu )\cos (\psi )} \\ {\sinh \left( \mu \right)\cos (\nu )\sin (\psi )} \\ { - \cosh \left( \mu \right)\sin (\nu )} \\ \end{array} } \right] ,$$

(19)

$$\hat{n} = \hat{l} \times \hat{c} .$$

(20)

Mapping of the Standardized Geometry to Specimen-Specific Anatomy

The standardized prolate spheroidal volume was mapped to the specimen-specific anatomy by first determining the size-related parameters in Eq. (9). As also shown in Fig. 5, since the entire transverse axis radii lay in the equatorial plane, b ₁ and b ₂ were respectively estimated from the areas A ₁ of the LV (with filled cavity) and A ₂ of its cavity in the mid-ventricular cardiac short-axis slice, according to,

$$b_{1} = \sqrt {{\raise0.7ex\hbox{${A_{1} }$} \!\mathord{\left/ {\vphantom {{A_{1} } \pi }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\pi $}}} ,\;{\text{and}} ,$$

(21)

$$b_{2} = \sqrt {{\raise0.7ex\hbox{${A_{2} }$} \!\mathord{\left/ {\vphantom {{A_{2} } \pi }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\pi $}}} .$$

(22)

The wall thickness w was computed from $w = b_{1} - b_{2}$. The distance from the equatorial slice to the cardiac apex was taken to be the outer longitudinal axis length a ₁.

Since the prolate spheroidal shape was only an approximation of the actual LV anatomy, a one-to-one and invertible mapping between the two was obtained by registering the former to the latter via diffeomorphic demons.39 For computing the local myocardial fiber orientation quantities (e.g., the helix angle) while ensuring orthogonality of the reference axes, the rotational component of the diffeomorphic transformation was used to map2 the tangential basis vectors, $\hat{c}$, $\hat{l}$ and $\hat{n}$, for each point in the prolate spheroidal volume onto the anatomical space.

Finally, to account for size variability among hearts and facilitate numerical modeling as polynomials, the radial coordinate variable (μ, originally spanning [$\tanh^{ - 1} \left( {b_{1} /a_{1} } \right)$,$\tanh^{ - 1} \left( {b_{2} /a_{2} } \right)$]) and azimuthal variable (ν, spanning [0, π]) were both normalized via linear transformation to span the interval [−1, 1].