Muscle fiber curvature has been hypothesized to play a role in maintaining mechanical equilibrium as well as in the asymmetry of deformation in the fiber cross-section.^1,2 Ultrasound studies have measured 2D curvature by examining limited portions of the muscle fibers under static and dynamic conditions.³ The focus of this work is (i) to study regional variations in 3D curvature in the medial gastrocnemius (MG) based on second order and third order polynomial fits of the fibers tracked from DTI data and (ii) to explore age based differences in fiber 3D curvature.

Five young (31.6 ± 7.0 years) and five senior (83.4 ± 3.2 yrs) women were scanned with IRB approval. Diffusion weighted images were acquired on a 3 Tesla GE scanner with a fat suppressed single shot EPI sequence, TE/TR/gradient directions/b-value/averages/resolution: 49 ms/4000 ms/32/400 s/mm²/4 averages/1.8 mm × 1.8 mm × 5 mm. DTI data was preprocessed for eddy current and susceptibility artifacts and denoised. Fiber tracking was performed using DTIStudio (https://www.dtistudio.org/) on the masked MG tensor volumes. Several automated checks were incorporated to ensure the integrity of the fibers including exclusion of fibers that tracked entirely along an aponeurosis or terminated far (> 4 voxels) from the superficial aponeurosis and trimming of fibers running parallel to an aponeuroses at either surface. The i^th point on the fiber tract is described by a position vector: $$$\vec{r_i}(x_i,y_i,z_i)$$$. The coordinates of the raw fiber are fitted by either 2nd or 3rd -order polynomial functions of point number. From the calculated coordinates of the fitted smooth fiber, curvature is calculated from the coordinates of the fitted fiber using the Frenet–Serret formula for each point:

$$k_i=\frac{\mid\dot{\vec{r_i}}\times\ddot{\vec{r_i}}\mid}{\mid \dot{\vec{r_i}}\mid^3}=\frac{\sqrt{(z''y'-y''z')^2+(x''z'-z''x')^2+(y''x'-x''y')^2}}{(x'+y'+z')^{3/2}}$$

where is the position vector of the ith point on the fitted fiber, and are the first and second differentials of the position vector. Curvature results are reported on a regional basis for fibers originating at the distal and at the center of the MG (distal 1/3 and next 1/3 of the MG muscle length). For each set of fibers, the distal and center regions were each divided into deep, middle, and superficial regions (1/3 each of the total length of the fiber from one aponeurosis surface to the other).

Figure 1 shows the fiber tracks, in 3D, tracked in the MG for an old and young subject. Figure 2 is the curvature values of fibers passing through one seed voxel using points calculated from 2^nd and 3^rd order fits of the fiber points for a senior subject. Figure 3 is a comparison of curvature from 2^nd and 3^rd order fits along the length of the MG (averaging along the fiber). Figure 4 lists the regional curvature values for the young and old subjects using 2^nd and 3^rd order fits. Significant regional difference in curvature was seen between the MG distal and central regions of the fiber (2^nd or 3^rd order fitting). There were no significant differences in the curvature values between young and old subjects.The 3D curvature extracted from DTI tracked fibers are in the same range as the 2D curvatures from ultrasound.³ The regional variation seen in ultrasound studies (small curvature in the middle regions of the MG and curvature increasing from distal to proximal) is also seen in the current study (Fig. 4). Ultrasound studies have also reported S-shape MG fibers where the curvature changed sign from deep to superficial aponeurosis passing through zero in the middle.³ It is not easy to extract the sign of the curvature in 3D but the 3D patterns show low values in the middle compared to deep and superficial regions (Fig. 4). An earlier study on 3D curvature of the Anterior Tibialis used a 2^nd order polynomial fit as the 3^rd order fit decreased the goodness of fit as well as they postulated that curvatures will not change sign (requiring higher order polynomials than 2).⁴ However, since S-shaped fibers in the MG have been shown by ultrasound, curvature extracted from 3rd order polynomials may be more accurate for MG fibers. It is feasible to extract 3D curvature from DTI tracked fibers of the MG using 3rd order polynomials and to detect regional changes in curvature. The absence of significant differences in curvature between young and old subjects requires confirmation from a larger study and possibly, curvature extraction methods less susceptible to noise.