# Strain and Strain Rate Tensor Mapping of Medial Gastrocnemius at Submaximal Isometric Contraction and Three Ankle Angles

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## Abstract

**:**

_{λ1)}and radial expansion strains (E

_{λ2}) and maximum shear strain (E

_{max}) based on paired t-test was also performed for each ankle angle. Results: Compressive strains/SRs were significantly lower at 25%MVC. Normalized strains/SR were significantly different between %MVC and ankle angles with lowest values for DF. Absolute values of E

_{λ2}and E

_{max}were significantly higher than E

_{λ1}for DF suggesting higher deformation asymmetry and higher shear strain, respectively. Conclusions: In addition to the known optimum muscle fiber length, the study identified two potential new causes of increased force generation at dorsiflexion ankle angle, higher fiber cross-section deformation asymmetry and higher shear strains.

## 1. Introduction

## 2. Materials and Methods

^{−1}3 direction velocity encoding). This resulted in 53 repetitions [([192 (phase encode lines) × 0.55 (partial FOV) × 2 (NEX)/4 (views per segment) = 53])] for the image acquisition. In total, twenty-two phases (using view-sharing) were collected within each contraction–relaxation cycle of ~3 s (isometric contraction). At each ankle position, diffusion tensor images (DTI) using 30 diffusion directions at b = 400 s/mm

^{2}with geometric parameters matched to the VE-PC images were acquired. The study protocol included at each ankle angle: large FOV image, VE-PC at 50%, followed by the 5-min DTI acquisition and then 25%MVC. The foot was then repositioned before repeating the imaging protocol. It should be noted that this order of imaging was implemented to minimize fatigue between the different dynamic acquisitions. The DTI data are not used in the current study but in a separate analysis to identify fascicles to derive fiber strains.

_{0}inhomogeneities and chemical shift and not from the velocity-encoding gradient. Velocity images extracted from the phase-corrected data are inherently noisy. As the calculation of the strain or SR tensor involves estimation of the spatial gradients of the displacement/velocity images that introduces additional noise into the image, the velocity images were first denoised using a 2D anisotropic diffusion filter [16]. The anisotropic diffusion filter reduces noise in homogenous regions while preserving edges, maintaining the effective resolution of the original velocity image. The filter was applied iteratively to reduce noise in homogenous regions, and was defined by the equation:

_{λ1}and E

_{λ2}are the normal principal strains while SR

_{λ1}and SR

_{λ2}, the normal principal strain rates (defined as perpendicular to the face of an element and represented by the diagonal terms of the E or SR tensor). It should be noted that during the compression part of the cycle, the eigenvector corresponding to E

_{λ1}and SR

_{λ1}is in a direction close to the muscle fiber direction while in the relaxation phase the eigenvectors are in a direction approximately orthogonal to the fiber direction. The reverse is true for E

_{λ2}and SR

_{λ2}. Two other strain and SR indices are calculated from the tensors: the out-of-plane strain denoted by E

_{out-plane}and SR

_{out-plane}and the maximum shear strain or shear strain rate denoted by E

_{max}and SR

_{max}, respectively. The out-of-plane strain and strain rate, which is in the fiber cross-section perpendicular to the imaging plane, was calculated from the sum of the principal eigenvalues at each voxel based on the assumption that muscle tissue is incompressible. A local contraction along the muscle will be accompanied by a local expansion in the plane perpendicular to the fiber. If considered in 3D, the sum of the three strain rates for an incompressible volume should be zero. However, only the 2D tensor is calculated here due to the constraints of single slice imaging, so the negative of the sum of the two eigenvalues (E or SR) yields the magnitude of the third eigenvalue.

_{λ1}for all strain rate indices and at the peak of E

_{λ1}for the strain indices.

_{λ2}~E

_{out-plane}and by the incompressibility of muscle tissue, these two strains will be equal to the half of E

_{λ1}. The maximum shear strain will be ~0.75|E

_{λ1}|. This is illustrated in Figure 2a.

_{λ2}. Then, deriving from the incompressibility of muscle tissue, E

_{out-plane}will be close to zero or very low values and E

_{λ2}will be equal to E

_{λ1}. The maximum shear strain will be ~E

_{λ1}. This is illustrated in Figure 2b.

_{λ2}greater than E

_{λ1}. In this case, the deformation in the fiber cross-section is such that the radial expansion in the in-plane direction exceeds that of the compressive strain in the fiber direction. From the incompressibility of the muscle tissue, this will lead to a compressive deformation in the out-plane direction. The maximum shear strain will be greater than E

_{λ1}. This is illustrated in Figure 2c.

_{λ1}, E

_{λ2}, E

_{out-plane}, E

_{max}) and the strain rate tensor (SR

_{λ1}, SR

_{λ2}, SR

_{out-plane}, SR

_{max}). Strain is unitless and the SR eigenvalues are in units of s

^{−1}. Normality of data was tested by using both, the Shapiro–Wilke test and visual inspection of Q-Q plots. Principal strains and strains rates as well the normalized strains and strain rates were normally distributed. Thus, changes between ankle angles, %MVC as well as potential interaction effects (ankle angle × %MVC), were assessed using two-way repeated measures ANOVAs and in case of significant ANOVA results for the factor ‘ankle angles’, Bonferroni-adjusted post hoc analyses were performed. Data are reported as mean ± SD for the variables since they were normally distributed. For all tests, the level of significance was set at α = 0.05. In addition to the above statistical tests, exploratory analysis using paired t-test was performed at each ankle angle between (i) absolute values of E

_{λ1}and E

_{λ2}using data from both force levels and (ii) absolute values of E

_{λ1}and E

_{max}using data from both force levels. The statistical analyses were carried out using SPSS for Mac OSX (SPSS 28.0.1.1, SPSS Inc., Chicago, IL, USA).

## 3. Results

#### 3.1. MVC at Different Ankle Angles

_{DF}= 289 ± 9 N, MVC

_{N}= 143 ± 14 N, MVC

_{PF}= 65 ± 10 N (average over all 6 subjects). The MVCs were significantly different between the three ankle angles: MVC

_{DF-N}(p = 0.0012), MVC

_{N-PF}(p = 0.0003), and MVC

_{DF-PF}(p = 0.0012), where the subscripts are the two ankle angles compared in paired t-tests.

#### 3.2. Strain and Strain Rate Maps and Temporal Plots of Deformation Indices

_{λ1}) and strain rate (SR

_{λ1}) maps through select frames of the dynamic cycle of 22 temporal frames for the three ankle angles at 50%MVC. The values of E

_{λ1}and SR

_{λ1}are superposed on the magnitude images of the VE-PC dataset using a colormap. Negative values of strain or strain rate (blue hue) are seen in the medial gastrocnemius and in the soleus (plantar flexor muscles) around frame 11 for the strain and around frames 8 and 16 for the strain rate maps. The temporal variation of strain and strain rate are shown in Figure 4a,b for one subject for a ROI placed in middle of the MG muscle for the three ankle angles and two %MVCs. Figure S2 shows for one subject, E

_{λ2}and SR

_{λ2}maps through select frames of the dynamic cycle of 22 temporal frames for the three ankle angles at 50%MVC. The values of E

_{λ2}and SR

_{λ2}are superposed on the magnitude images of the VE-PC dataset using a colormap. Positive values of strain or strain rate (red hue) are seen in the medial gastrocnemius and in the soleus (plantar flexor muscles) around frame 11 for the strain and around frames 8 and 16 for the strain rate maps.

#### 3.3. Region of Interest Values of Deformation Indices at Peak Contraction

_{λ1}and SR

_{λ1}) showed significantly lower absolute values at 25%MVC compared to 50%MVC while showing no significant changes between ankle angles. Radial strains and strain rates (E

_{λ2}, SR

_{λ2}, E

_{out-plane}, SR

_{out-plane}) in the fiber cross-section showed no significant changes with %MVC or with ankle angle though E

_{λ2}was consistently higher at 50%MVC. The absolute values of E

_{out-plane}and SR

_{out-plane}are much smaller than the strains and strain rates in the other two directions, indicating that deformation is the smallest in this direction. Further, while E

_{λ2}and SR

_{λ2}are clearly radial expansion strain and strain rates, respectively, in the fiber cross-section, E

_{out-plane}and SR

_{out-plane}are smaller in magnitude and exhibit negative signs indicating that these are small radial compressive strain and strain rates, respectively, in the out-of-plane direction. This indicates that the deformation is close to that shown in Figure 2, between Case 2 and Case 3. It should be noted that the largest absolute values of E

_{out-plane}and SR

_{out-plane}occur for the dorsiflexed ankle position. The maximum shear strain showed no significant changes with either %MVC (though larger absolute values at 50%MVC) or ankle angle. Maximum shear strain rate showed significant changes with %MVC (higher absolute values at 50%MVC) but no significant change with ankle angle.

_{λ1}normalized to force significantly changed with ankle angle, pairwise comparison revealed changes between (PF and DF), (PF and N), (N and DF) but showed no significant change with %MVC. SR

_{λ1}normalized to force significantly changed with %MVC and also with ankle angle, pairwise comparison revealed changes between PF and N. E

_{λ2}normalized to force showed significant change with %MVC as well as with ankle angles, pairwise comparison revealed changes between (PF and DF) and (N and DF). SR

_{λ2}normalized to force showed significant change with %MVC as well as with ankle angles, pairwise comparison revealed changes between (PF and N) and (PF and DF). Normalized E

_{out-plane}and SR

_{out-plane}showed no significant changes with either %MVC or with ankle angle. E

_{max}and SR

_{max}normalized to force significantly changed with %MVC and also with ankle angle, pairwise comparison revealed changes between (PF and DF), (N and DF).

_{λ1}, E

_{λ2}and E

_{max}for each ankle angle averaged over all subjects and both force levels. The mean of E

_{λ2}was greater than E

_{λ1}for all three ankle angles and paired t-tests comparing the absolute values of E

_{λ1}and E

_{λ2}yielded the following results: for PF ankle angle, the absolute values of E

_{λ1}and E

_{λ2}were significantly different (p = 0.04), for neutral ankle angle the difference was not significant while for the dorsiflexed ankle the difference was highly significant (p = 0.002) and the largest difference in means of the absolute values of E

_{λ1}and E

_{λ2}was seen in the DF ankle angle (~59% compared to ~26% for the other two ankle angles). The mean of E

_{max}was greater than E

_{λ1}for all three ankle angles and paired t-tests comparing the absolute values of E

_{λ1}and E

_{max}showed significant differences between the two values only for the dorsiflexed ankle angle (p = 0.004) and the size effect was also largest at the DF ankle angle (~27% compared to ~10%).

## 4. Discussion

_{λ1}, SR

_{λ1}and SR

_{max}(Table 1). Significant differences in E

_{λ1}and SR

_{λ1}with %MVC are anticipated as the higher force at the higher %MVC requires a larger contraction (strain). Surprisingly, there were no significant differences in the strain or SR indices between the different ankle angles despite a highly significant difference in force between the different ankle angles. This implies that similar strains (amount of contraction) at the different ankle angles were capable of producing significantly different forces. The deduced absolute values of E

_{out-plane}and SR

_{out-plane}are much smaller than the strains and strain rates in the orthogonal direction of the fiber cross-section indicating a strong anisotropy of deformation; this is true for all ankle angles. Anisotropy of fiber cross-section deformation has been reported in earlier studies for the neutral ankle angle [1,2,3,4,17]. The results from this study also show that anisotropy holds at plantarflexed and dorsiflexed ankle angles.

_{λ1}and E

_{λ2}for the three ankles showed that the dorsiflexed ankle position had the largest and most significant difference (|E

_{λ2}| > |E

_{λ1}|); the deformation is similar to that shown in Figure 2c. On the other hand, while |E

_{λ2}| > |E

_{λ1}| for both plantarflexor and neutral ankle angles, the differences were smaller and tentatively, the deformation patterns in these two ankle angles may be between the schematics shown in Figure 2b,c. One explanation for the PF and N to have smaller in-plane deformations (smaller E

_{λ2}) compared to the dorsiflexed position may be related to the initial (at rest) fiber radial size. The plantarflexed position has the largest fiber cross-section of the three ankle angles and the larger initial radial size may provide a constraint to further radial expansion. Azizi et. al. showed with a physical model that constraints to radial expansion limits the contractility and thus, the force generated [19]. Thus, the constraints to radial expansion in the PF (arising from the larger radius) may also be a contributor to force reduction in this ankle angle position. Further, computational modeling studies have predicted that when there is a strongly anisotropic constraint the force output may increase by a factor of two [8]. This latter computational model showed that maximum force output was obtained by introducing anisotropy of passive material stiffness along the fiber cross-sectional axes such that there was very little deformation along one axis (the through-plane axis) during a muscle length change. In this anisotropic model, the stiffness in one direction was reinforced such that it was stiffer by a factor of 4 compared to the orthogonal direction that resulted in a near doubling in force compared to an isotropically stiff material. The authors postulated that the structural muscle proteins called costameres were a potential candidate for introducing such an anisotropy in the passive material properties [8]. Highly asymmetric deformation in the fiber cross-section seen in DF may be facilitated since in this ankle position, the fiber is longest and consequently, the fiber cross-section area is the smallest allowing larger radial expansions. A strongly anisotropic constraint, as is seen in DF, provides another potential mechanism of higher force in DF from the highly asymmetric deformation at this ankle angle. It should be noted that the strain in the fiber cross-section (E

_{λ2}) is also highly likely to be determined by the extracellular matrix (e.g., a stiffer ECM will offer a greater constraint to deformation).

_{λ1}and E

_{max}at each ankle angle also showed that E

_{max}was greater than E

_{λ1}in all three ankle angles but was significantly so only for the dorsiflexed ankle angle. In terms of the deformation pattern, this also indicates that while PF and N ankle angles are potentially between the schematics shown for asymmetric to highly asymmetric (Figure 2b,c), the dorsiflexion case may potentially correspond to highly asymmetric. Prior MR studies found a significant positive correlation of force in a cohort of young and old subjects or force loss due to unloading to the absolute value of the max shear strain (E

_{max}) [7,20]. Thus, another potential reason for the higher force generated may arise for higher absolute values of E

_{max}in the dorsiflexed position.

_{out-plane}or SR

_{out-plane}. Further, care was taken to ensure that the acquired oblique sagittal slice captured the MG fibers in the imaging plane. (ii) The cohort size is small but the repeated measures design provides higher statistical power and statistically significant differences were seen between the normalized strain and strain rate values between the different ankle angles. This is a proof-of-concept paper where a new technique is established (2D strain tensor analysis to track changes of compressive, radial expansive and shear strains for different fiber architecture). The technique will be expanded in a future study to include a larger number of subjects and applied to studying differences with age and in disease conditions such as muscular dystrophy. The proposed MRI-based strain technique can also be adapted for elastography and compared to other techniques such as elastosonography [24].

## 5. Conclusions

## Supplementary Materials

_{λ2}) and positive strain rate (SR

_{λ2}). Table S1: Strain rate indices for different ankle angles and %MVC. Table S2: Strain rate indices normalized to force for different ankle angles and %MVC.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**

**Left top**panel: Subject setup with dominant leg in the foot pedal device, centered in a cardiac flex coil (labeled RF coil), with visual feedback projected onto the screen for the subject to follow. Pressure against the carbon-fiber plate in the foot pedal was detected by the transducer and converted into voltage and then converted into measurements of force.

**Left bottom**panel: The foot pedal output was processed to generate a trigger to synchronize with the MR acquisition and also displayed to the subject on a screen.

**Right top**panel: The force curve averaged over ~53 contractions (required to acquire the MR images) is shown along with upper and lower boundaries of the force curve.

**Right lower**panel: Plot of the force curves for one VE-PC acquisition and the green vertical lines are the triggers. The foot positioning for neutral (N) ankle angle is shown here, dorsiflexed (DF) and plantarflexed (PF) positions were obtained by adjusting the ankle angle of the foot.

**Figure 2.**Schematic of different deformation patterns in the fiber cross-section, the non-deformed state is shown as a solid line and the deformed state with a dashed line. (

**a**): Symmetric deformation in fiber cross-section leads to |E

_{λ2}|~|E

_{out-plane}|~0.5|E

_{λ1}| and |E

_{max}|~0.75|E

_{λ1}|. (

**b**): Asymmetric deformation in fiber cross-section with little to no deformation in the out-plane direction leads to |E

_{λ2}|~|E

_{λ1}|, E

_{out-plane}~0 and |E

_{max}|~|E

_{λ1}|. (

**c**): Highly asymmetric deformation in fiber cross-section with |E

_{λ2}| > |E

_{λ1}| will lead to E

_{out-plane}being negative (compressive strain in the fiber cross-section) and |E

_{max}| > |E

_{λ1}|.

**Figure 3.**Maps of the negative strain (E

_{λ1}) (

**top**panel) and negative strain rate (SR

_{λ1}) (

**bottom**panel) projected on magnitude images at the corresponding temporal frame. Images shown here were acquired at 50%MVC, at each foot position PF, N, DF (order of rows

**top**to

**bottom**). Overlay allows for better identification of the underlying muscle, aponeuroses, and fascicles. The color maps are color-coded according to the legend with the figures. Select frames (from the acquired 22 dynamic frames) where peak strains and strain rates occur are shown here, the frame number is indicated on the top row. The peak of the strain occurs around frame 10–11 (arrow points to MG) where the blue shade corresponding to compressive strains in the MG and in the soleus can be seen. The peak of the strain rate occurs in the contraction (~frame 7–8, arrow to MG) and relaxation (~frame 16) phases and this is visualized as blue shades in the MG and in the soleus around these frames. The regions of interest in the MG (seen as red boxes in the first frame of each row) used to extract the deformation indices are shown in the first frame.

**Figure 4.**(

**a**) Temporal plots for strain indices for one subject for an ROI placed in the middle of the MG muscle for the three ankle angles (PF, N, DF). Organized in column by foot angle position, and in row order of: E

_{λ1}, E

_{λ2}, E

_{λ3 (or out-plane)}, and E

_{max}. (

**b**) Temporal plots for strain rate indices for one subject for an ROI placed in the middle of the MG muscle for the three ankle angles (PF, N, DF). Organized in column by foot angle position, and in row order of: SR

_{λ1}, SR

_{λ2}, SR

_{λ3 (or out-plane)}, and SR

_{max}.

Ankle Position | %MVC | Peak Force (N) | E_{λ1} * | E_{λ2} | E_{out-plane} | E_{max} |
---|---|---|---|---|---|---|

Plantarflexion | 50 | 32 ± 6.959 | −0.141 ± 0.009 | 0.168 ± 0.03 | −0.036 ± 0.031 | −0.151 ± 0.019 |

25 | 16.72 ± 3.694 | −0.102 ± 0.022 | 0.137 ± 0.031 | −0.046 ± 0.019 | −0.117 ± 0.026 | |

Neutral | 50 | 74.24 ± 9.655 | −0.185 ± 0.015 | 0.208 ± 0.025 | −0.013 ± 0.033 | −0.19 ± 0.014 |

25 | 37.68 ± 4.883 | −0.134 ± 0.024 | 0.202 ± 0.037 | −0.068 ± 0.052 | −0.165 ± 0.022 | |

Dorsiflexion | 50 | 141.2 ± 9.246 | −0.143 ± 0.019 | 0.209 ± 0.008 | −0.083 ± 0.023 | −0.173 ± 0.006 |

25 | 73.82 ± 4.26 | −0.095 ± 0.027 | 0.169 ± 0.018 | −0.092 ± 0.025 | −0.129 ± 0.019 |

*****Significant difference between 25% and 50%MVC.

Ankle Position | %MVC | Peak Force (N) | E_{λ1} ^{o,!,ẟ} | E_{λ2} ^{o,!,}* | E_{out-plane} | E_{max} ^{o,!,}* |
---|---|---|---|---|---|---|

PF | 50 | 32 ± 6.96 | −0.0053 ± 0.001 | 0.0061 ± 0.0012 | −0.001 ± 0.0016 | −0.0056 ± 0.001 |

25 | 16.72 ± 3.69 | −0.0069 ± 0.0007 | 0.01 ± 0.0012 | −0.0041 ± 0.0009 | −0.0083 ± 0.0009 | |

N | 50 | 74.24 ± 9.65 | −0.0027 ± 0.0004 | 0.0032 ± 0.0004 | −0.0003 ± 0.0004 | −0.0028 ± 0.0003 |

25 | 37.68 ± 4.88 | −0.0043 ± 0.0011 | 0.0058 ± 0.0007 | −0.0012 ± 0.0018 | −0.005 ± 0.0006 | |

DF | 50 | 141.2 ± 9.25 | −0.001 ± 0.0002 | 0.0015 ± 0.0001 | −0.0006 ± 0.0002 | −0.0012 ± 0.0001 |

25 | 73.82 ± 4.26 | −0.0015 ± 0.0005 | 0.0025 ± 0.0003 | −0.0013 ± 0.0004 | −0.0019 ± 0.0003 |

^{!}Significant difference between PF and DF;

^{ẟ}Significant difference between PF and N;

^{o}Significant difference between N and DF.

**Table 3.**Comparison of absolute values (std dev) of E

_{λ1}to E

_{λ2}and E

_{λ1}to E

_{max}for PF, N, DF.

Ankle Position | Abs (E_{λ1}) | E_{λ2} | Abs (E_{max}) | %Diff (E_{λ1}, E_{λ2}) | %Diff (E_{λ1}, E_{max}) |
---|---|---|---|---|---|

Plantarflexion * | 0.121(0.044) | 0.152(0.073) | 0.134(0.056) | 25.50% | 10.10% |

Neutral | 0.160(0.054) | 0.205(0.073) | 0.178(0.044) | 28.60% | 11.40% |

Dorsiflexion *^{,}** | 0.119(0.060) | 0.189(0.039) | 0.151(0.040) | 59.40% | 27.30% |

_{λ1}and E

_{λ2}; ** Significant difference between E

_{λ1}and E

_{max}.

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**MDPI and ACS Style**

Hernandez, R.; Sinha, U.; Malis, V.; Cunnane, B.; Smitaman, E.; Sinha, S.
Strain and Strain Rate Tensor Mapping of Medial Gastrocnemius at Submaximal Isometric Contraction and Three Ankle Angles. *Tomography* **2023**, *9*, 840-856.
https://doi.org/10.3390/tomography9020068

**AMA Style**

Hernandez R, Sinha U, Malis V, Cunnane B, Smitaman E, Sinha S.
Strain and Strain Rate Tensor Mapping of Medial Gastrocnemius at Submaximal Isometric Contraction and Three Ankle Angles. *Tomography*. 2023; 9(2):840-856.
https://doi.org/10.3390/tomography9020068

**Chicago/Turabian Style**

Hernandez, Ryan, Usha Sinha, Vadim Malis, Brandon Cunnane, Edward Smitaman, and Shantanu Sinha.
2023. "Strain and Strain Rate Tensor Mapping of Medial Gastrocnemius at Submaximal Isometric Contraction and Three Ankle Angles" *Tomography* 9, no. 2: 840-856.
https://doi.org/10.3390/tomography9020068