Changes in strain tensor resulting from atrophy induced by Unilateral Limb Suspension of the calf muscle.
Vadim Malis1, Usha Sinha2, Ryuta Kinugasa3, and Sinha Shantanu4

1Physics, UC San Diego, San Diego, CA, United States, 2Physics, San Diego State University, San Diego, CA, United States, 3Kanagawa University, Yokohama, Japan, 4Radiology, UC San Diego, San Diego, CA, United States


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.


Chronic muscle atrophy induced by limb suspension (Unilateral limb suspension (ULLS)) is characterized by a loss of muscle mass and force1. However, several studies also indicate extracellular remodeling and its functional impact on lateral transmission of force as potential contributors to the loss of muscle function with age2. Here, we extract the full 3D strain tensor in the principle and muscle fiber (fiber aligned strains) basis and two strain invariants (volumetric strain and octahedral shear strain) from multi-slice dynamic imaging. Volumetric strain is clinically important as it relates to intramuscular pressure3 while shear strain is relevant as shear in the endomysium has been postulated as the most likely mechanism of lateral force transmission4. This study explores ULLS induced changes in 3D strain tensor in the medial gastrocnemius (MG) and soleus (SOL) from multi-slice velocity encoded phase contrast (VE-PC) images acquired under isometric contraction.


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 Efiber, Ein-plane, Ethrough-plane) were obtained through eigenvalue decomposition, octahedral shear strain (Eshear) and volumetric strain (Evol) 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.


Fig. 1 shows the maps of volumetric and shear strain in one slice pre- and post-suspension (maximum strain frame of the dynamic strain maps). Fig. 2 shows the temporal plots of the principal strains, volumetric strain and octahedral shear strain averaged over all subjects (pre- and post-suspension) in the MG. Table 1 lists the MG and SOL strains in the principle and muscle fiber basis and the two invariants; significant changes with suspension are seen in Efiber, Ein-plane, and in Eshear (trend to significance in Evol) while Ethrough-plane is very small in pre- and post-suspension. The strain indices in the muscle fiber basis were smaller than in the principle basis and did not show any significant differences with suspension.


The strain indices were reduced post-suspension for MG and SOL and may be related to decrease in inherent contractility as well as an increase in the stiffness of the extracellular matrix. The through-plane strain is very small and confirms earlier studies that reported asymmetric deformation in the fiber cross-section5. Force generated in the muscle fibers is transmitted longitudinally through the myotedinous junction and laterally through myofascial pathways (through shear of the extracellular matrix). It is to be noted that Eshear is larger than Evol for both MG and SOL possibly indicating shear as a dominant mechanism of force transfer. Further, a significant reduction in Eshear with suspension in both the MG and SOL may be a consequence of extracellular matrix remodeling and may reflect a decrease in lateral transmission of force with suspension.


3D strain tensor analysis along with invariant indices derived from it have the potential to reveal changes in contractility and in the extracellular matrix and may enable the non-invasive determination of lateral transmission of force.


This work was supported by the National Institute of Arthritis and Musculoskeletal and Skin Diseases Grant R01AG056999-01


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[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 Evol 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 (El1), in-plane of muscle fibers (El2) through-plane(El3). 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

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)