Muscle passive stiffness provides insight into the biomechanical function, physiology, and health of our musculoskeletal system; thus, its accurate measurement is important. Most studies do not differentiate between bundles of muscle fibers with different sizes. Thus, these studies assume that the elastic modulus (i.e. passive stiffness/CSA) of these muscle specimens is independent of size. The same assumption is made for single fibers. The results of our current study demonstrated that the size matters and in general, larger sizes were associated with lower elastic moduli in fibers and bundles of fibers. Therefore, future studies should consider maintaining consistent bundle sizes for measurement of passive properties especially when comparing different groups against each other.
This same finding was observed for single fibers recently by Noonan et al.20 where larger fibers in vastus lateralis of 10 healthy volunteers manifested smaller elastic moduli. They demonstrated that considering a constant thickness and higher elastic modulus for the basement membrane, relative to the modulus of the contractile area of the fiber, results in larger elastic modulus for smaller fibers. The effect is due to the CSA of the basement membrane being proportional to the fiber diameter while the contractile area of a fiber is proportional to its diameter squared. At the fiber bundle level however, no study has examined the effect of size on elastic modulus.
The lower elastic modulus of larger bundles in the current study may arise from their extracellular matrix (ECM) content. It is well established that bundles of fibers have a larger elastic modulus than single fibers due to the high stiffness of ECM10,19,21,27,28. Assuming that the ECM and fibers within a bundle are both homogenous, the elastic modulus of a bundle can be calculated following the rule of mixture for composites as:
$${E}_{Bundle}={f}_{ECM}{E}_{ECM}+left(1-{f}_{ECM}right){E}_{Fiber}$$
where ({E}_{Bundle}, {E}_{ECM},) and ({E}_{Fiber}) are the elastic moduli of the bundle, the extracellular matrix, and fibers, respectively; and ({f}_{ECM}) denotes the percentage of the extracellular matrix within the bundle. For example, the elastic modulus of a bundle containing 5% ECM (i.e. ({f}_{ECM}=0.05)) with a fiber elastic modulus of 20 kPa and ECM elastic modulus of 1 MPa28 is calculated as 69 kPa. The predicted bundle elastic modulus would be 79 kPa or 59 kPa had the ECM percentage been changed to 6% or 4%, respectively.
To further explore this idea, we performed an immunostaining analysis to quantify ECM content in bundles of different sizes from multifidus of one rat in G1 as an example. While the exact relationship between ECM elastic modulus and its constituents is not yet clear, many studies report collagen I as the major contributor to elastic modulus of ECM15,29. Therefore, we measured and contrasted collagen I content in bundles of different sizes. A portion of the collected biopsy was separated and immediately snap frozen in isopentane cooled by liquid nitrogen. The biopsy was sectioned, placed on slides, and immune-stained for collagen I. Using Image J software, bundles of different sizes were arbitrarily defined, segmented and their collagen I area fraction (i.e. the ratio of collagen I area over the entire bundle cross sectional area) was measured. As no collagen I exists inside muscle fibers, the number of fibers within each bundle can easily be counted. However, as the exact border of bundles cannot be determined from the images, two segmentations (types A and B) were performed for each bundle: the borders of one segmentation (type A) passed internal to the edge of boundary fibers of the bundle (Fig. 5A,D), while borders of the other segmentation (type B) passed externally, including the boundary edge of the fibers immediately adjacent but external to the bundle (Fig. 5B,E). The collagen I deposition was measured for these two segmentations and their average was considered for the selected bundle.
Segmentation of fiber bundles from immunohistochemistry images. Two segmentations were performed for each bundle: the borders of one segmentation (type A) passed internal to the edge of boundary fibers of the bundle (A,D), while borders of the other segmentation (type B) passed externally, including the boundary edge of the fibers immediately adjacent but external to the bundle (B,E). Schematic representations for a simulated bundle of 6 fibers, are shown for segmentation type A (A), segmentation type B (B), and how a real bundle may present (C). The collagen I deposition was measured for the two segmentations (D,E) and their average was considered for the selected bundle.
Following this approach, collagen I deposition of larger bundles was measured to be smaller for the studied biopsy (Fig. 6). The three bundles studied had 8, 16, and 25 fibers but their collagen I content was 5.3%, 3.8%, and 3.4% of the area, respectively. Assuming an elastic modulus of ~ 20 kPa for muscle fibers and ~ 1 MPa for the ECM28 and using the rule of mixture, the corresponding elastic moduli of these three bundles are estimated as 72, 57, and 53 kPa, respectively.
Inverse correlation between collagen I deposition and size of three bundles of different sizes from multifidus of one rat in G1. The measured percentage of collagen I deposition for bundles A, B, and C were 5.3%, 3.8%, and 3.4%, respectively.
Another factor contributing to the elastic modulus of a bundle could be the distribution of fiber sizes within that bundle. As our results suggest, larger fibers have a smaller modulus. Therefore, for two muscle bundles with same amount of ECM, the one having more fibers (i.e. consisting of smaller fibers) will have a higher elastic modulus (Fig. 7).
Effect of fiber size on bundle elastic modulus. In all images, black color represents the contractile elements of muscle fibers, blue color represents basement membranes of muscle fibers, and red color represents the ECM. Given that larger fibers have smaller elastic moduli (A,B), for two muscle bundles with same ECM content (C,D), the one having more fibers (i.e. consisting of smaller fibers) will have a higher elastic modulus.
While both factors, i.e. collagen I deposition and fiber size distribution, could potentially explain the results of our study, further investigation is required to find out their relative effects. Although we reasonably used the average of type A and B segmentations for our sample measurement of collagen I percentage, the actual bundle that was mechanically tested might have a different collagen I content (Fig. 5C) that could be closer to either segmentation A or B. Notably, for the three bundles in Fig. 6, segmentation type A would result in larger percentages for larger bundles, while type B segmentation and the average method showed an opposite trend. Therefore, it is quite important to know how much ECM remains on an actual extracted bundle, for example by measuring the volumetric collagen deposition of a bundle using a three dimensional imaging technique or hydroxyproline assay29.
It is noteworthy that if the larger fibers/bundles manifest shorter slack sarcomere lengths, because of how we used a set force threshold to determine this length, then 30% strain would occur at shorter absolute sarcomere lengths which could influence the calculated moduli. To explore whether the smaller elastic modulus of larger bundles was due to differences in their slack sarcomere lengths we fitted a linear regression model to identify any relationship between the slack sarcomere lengths and CSAs of the tested bundles. No significant linear correlation was found for any of the groups except for G1 longissimus fibers (P < 0.05). This confirmed that the observed difference in elastic moduli of the tested bundles with varying sizes was not an artifact of the slack sarcomere length measurement method.
The linear regression approach in the current study revealed a statistically significant effect of CSA on the elastic modulus of all fibers and fiber bundles (except for G3 multifidus fibers and G2 longissimus bundles). That the ({R}^{2}) values for this association were relatively low (all ({R}^{2}<0.30)) is not surprising. The low values for the ({R}^{2}) could stem from the biological variations in the subjects or from the variations in sites within biopsies from where fibers and fiber bundles were extracted; thus suggesting possible role of other factors that were not studied here. What is noteworthy is the found dependence of elastic modulus on CSA despite being already normalized by the CSA, which means that the size of the fibers and fiber bundles needs to be considered.
For the second objective of this study, the intriguing observation was that muscle fibers and fiber bundles were not cylindrical (all P < 0.0001). In contrast to the shape of a single muscle fiber, the shape of a muscle fiber bundle could be somewhat controlled by the person extracting the bundle from the biopsy tissue. However, during extraction to maintain the integrity of a bundle, it may be necessary to avoid separating extra fibers that can result in loss of bundle integrity. This will result in bundle shapes that may not be cylindrical. Many studies only measure fiber and fiber bundle diameter along a single axis typically from the top view14,15,18,19,20,21. The cross section of fibers and fiber bundles is then assumed to possess a cylindrical shape. However, measurement of diameters from two orthogonal axes (top and side view) in the current study revealed that such an assumption was not valid in rodents or humans, especially at the bundle level. The ratio of major axis over the minor axis of a cylindrical sample should be equal to 1, whereas the median for ratio of major axis over the minor axis of the samples tested in this study ranged between 1.15 and 1.29 for fibers and 1.27 and 1.44 for fiber bundles. The implication of this finding is that the measurement of elastic modulus values could be off by a factor of 1.15 to 1.29 for fibers and 1.27 to 1.44 for fiber bundles if the diameter along only one axis is measured. Therefore, it is recommended to measure the fiber and bundle diameters from both top and side views.
Muscle is an organized material with several structural levels. In a recent study, Ward et al.15 measured the elastic modulus of rabbit muscles at multiple levels, i.e. single fibers, fiber bundles (~ 20 fibers), fascicles (~ 300 fibers), and whole muscles. They found that elastic modulus increases nonlinearly with these size scales as does the collagen content. The results of the current study demonstrated that larger bundles were associated with lower elastic moduli. These results are not in conflict, but rather they are complementary. Our results suggest that for fiber bundles of less than ~ 50 fibers larger sizes will be associated with smaller elastic moduli. However, beyond a certain size (e.g. ~ 300 fibers), bundles will transition to true fascicles, including the presence of perimysium and higher amounts of collagenous tissue, thereby leading to larger elastic moduli compared to bundles. In conclusion, the findings of our study suggest that similar size of bundles should be tested when comparing for differences between groups.
