Vadim Malis^{1}, Usha Sinha^{2}, Ryuta Kinugasa^{3}, and Sinha Shantanu^{4}

We quantified 3D strain tensor in the principle and muscle fiber basis along with two invariants (volumetric and octahedral shear strain) from multi-slice velocity encoded phase contrast images of the *in-vivo* human calf muscle under isometric contractions. Significant decreases in the medial gastrocnemius and soleus contractile strain eigenvalue and in the invariants with suspension may potentially arise from changes in muscle contractility and/or from extracellular remodeling. The significant reduction in shear strain may indicate a decrease in lateral transmission of force that may account for the disproportionate loss of force to loss of mass with atrophy.

Seven subjects (IRB approved) were scanned on a 1.5T GE scanner before and after a four-week period of muscle atrophy induced on the non-dominant leg using the ULLS model. Gated VE-PC images obtained during isometric contraction at 35% MVC (TE: 7.7ms, TR: 16.4ms, NEX: 2, FA: 20°, 7 contiguous slices, thickness 5mm / skip 0, sagittal-oblique orientation, FOV: 30 × 22.5cm, matrix: 256 × 192, 4 views/segment, 22 phases, 3D velocity encoding, VENC: 10 cm/s)^{5}. Lower leg was placed in a plaster cast with an embedded strain sensor and real-time visual feedback provided to the subject. Diffusion weighted images of the lower leg in relaxed state corresponding to first frame of VE-PC images and matching geometry were also acquired. Displacements were calculated by tracking voxels across the dynamic cycle in the phase images after phase correction and denoising. Strain tensor (E) was calculated by taking spatial gradient of the displacements (in x-,y-, and z-directions) obtained with respect to the first frame of the contraction-relaxation cycle. Principle basis eigenvalues (denoted by E_{fiber}, E_{in-plane}, E_{through-plane}) were obtained through eigenvalue decomposition, octahedral shear strain (E_{shear}) and volumetric strain (E_{vol}) were calculated using equation (1) and (2). In addition, components of the strain tensor in the diffusion basis (labeled f- muscle fiber, s- and t- secondary/tertiary diffusion eigenvector direction) were obtained by rotating SR tensor according to the equation (3) where R is the matrix of the diffusion eigenvectors in the voxel (obtained from the DTI data).

$$(1) \quad E_{shear} = \frac{2}{3}\sqrt{(E_{xx}-E_{yy})^2+(E_{xx}-E_{zz})^2+(E_{yy}-E_{zz})^2+6(E_{xy}^2-E_{xz})^2+E_{yz}^2}$$

$$(2) \quad E_{vol}=\frac{\delta V}{V}=E_{xx}+E_{yy}+E_{zz}$$

$$(3) \quad SR_{DTI} = R · SR · R^{T} $$

Quantitative analysis was performed for 3D regions of interest (28mm x 10mm x 15mm) placed inside the MG and SOL muscles. Position of each voxel inside ROI was tracked across the contraction-relaxation cycle. Differences in strain indices between pre- and post-ULLS groups extracted at the frame corresponding to max force were assessed using repeated measures two-way ANOVAs.

[1] De Boer MD, Maganaris CN, Seynnes OR, Rennie MJ & Narici MV. The Journal of Physiology2007; 583:1079–1091.

[2] Ramaswamy KS, Palmer ML, van der Meulen JH, Renoux A, Kostrominova TY, Michele DE, et al. J Physiol-London. 2011;589(5):1195-208

[3] Jensen ER, Morrow DA, Felmlee JP, Murthy NS, Kaufman KR.Physiol Meas. 2015;36:N135-46.

[4] Zhong X, Epstein FH, Spottiswoode BS, Helm PA, Blemker SS. J Biomech.2008;41(3):532-40.

[5]Malis V, Sinha U, Csapo R, Narici M, Sinha S. Magn Reson Med.2017;doi: 10.1002/mrm.26759.

Figure 1: Volumetric strain maps of the lower leg obtained at the peak of the force curve superimposed on the magnitude image pre- (a) and post- (b) suspension. Octahedral strain maps of the MG and SOL obtained at the peak of the force curve superimposed on the magnitude image pre- (c) and post- (d) suspension. The volumetric strain maps are heterogeneous and overwhelmingly E_{vol} is positive. In contrast, the shear strain maps are more uniform in the MG and SOL and clearly show the decrease with suspension. These images are one frame from the 22 temporal images obtained in the dynamic cycle.

Figure 2: Temporal plots of strain indices during isometric contraction for pre- (left column) post- (right column) suspension for an ROI placed in the MG. Top row: Strain Eigenvalues (along fiber (El_{1}), in-plane of muscle fibers (El_{2}) through-plane(El_{3}). Middle row: Octahedral shear strain. Bottom row: Volumetric strain. Similar plots were seen in ROIs placed in the soleus. The very small values of the through-plane can be seen in the eigenvalue plots.

Table 1: Pre- and post-suspension averages (over all subjects) of strain eigenvalues in the principal and muscle fiber basis and the two strain invariants for ROIs placed in the MG and in the SOL