How to estimate the sarcomere size based on oblique sections of skeletal muscle

Abstract Ultrastructural analysis of muscular biopsy is based on images of longitudinal sections of the fibers. Sometimes, due to experimental limitations, the resulting sections are instead oblique, and no accurate morphological information can be extracted with standard analysis methods. Thus, the biopsy is performed again, but this is too invasive and time‐consuming. In this study, we focused our attention on the sarcomere's shape and we investigated which is the structural information that can be obtained from oblique sections. A routine was written in MATLAB to allow the visualization of how a sarcomere's section appears in ultrastructural images obtained by Transmission Electron Microscopy (TEM) at different secant angles. The routine was used also to analyze the intersection between a cylinder and a plane to show how the Z‐bands and M‐line lengths vary at different secant angles. Moreover, we explored how to calculate sarcomere's radius and length as well as the secant angle from ultrastructural images, based only on geometrical considerations (Pythagorean theorem and trigonometric functions). The equations to calculate these parameters starting from ultrastructural image measurements were found. Noteworthy, to obtain the real sarcomere length in quasi‐longitudinal sections, a small correction in the standard procedure is needed and highlighted in the text. In conclusion, even non‐longitudinal sections of skeletal muscles can be used to extrapolate morphological information of sarcomeres, which are important parameters for diagnostic purposes.

They form the basic units of the contractile apparatus (Squire, 1997).
In TEM images, the sarcomeric architecture appears with several differently electron-dense regions, each area resulting composed of many filaments connected with the peripheral areas, the so-called Z-disks, or within the central region, the M-band, which divides into two specular halves the sarcomere (Pinotsis, Abrusci, Djinovic -Carugo, & Wilmanns, 2009). In particular, the different electron densities of the sarcomere in TEM images, due to the presence of thick and thin filaments of myosin and actin respectively, reveal the division of sarcomeres into different regions. Beyond the Z-disk or Z-line, limiting each sarcomere, there is the I-band, surrounding the Z-line, composed of thin filaments. Following the I-band there is the A-band containing both thick and thin filaments. Within the A-band, there is the H-zone composed of thick filaments and within the Hzone, there is the M-line that is in the middle of the sarcomere (Hu, Ackermann, & Kontrogianni-Konstantopoulos, 2015;Lange, Ehler, & Gautel, 2006).
In a previous study, the parameters characterizing the sarcomere and myofibril arrangement were defined and a semi-automatic routine to measure the parameters on ultrastructural (TEM) images was presented (Cisterna, Malatesta, Zancanaro, & Boschi, 2021). The routine was tested on skeletal muscle samples of trisomic (Down syndrome) and euploid mouse models, showing differences in sarcomere shape/arrangement in health and disease, confirming and expanding the results obtained by Cisterna, Sobolev, Costanzo, Malatesta, and Zancanaro (2020). Apart from Down syndrome, sarcomere size can be affected in different other disease. For instance, sarcomere length could be increased in diabetes, altering ventricular function (Isola et al., 2021).
Generally, morphological pattern in ultrastructural images is evaluated on sections obtained by cutting the myofibers with a plane parallel to the fibers' axis (longitudinal sections). In histological practice, the sections are not always longitudinal and redoing the sampling might be invasive and time-consuming. Here, which are the morphological information obtainable from virtual sections of muscle fibers, was investigated. The study was focused on the sarcomere's radius and length, and the angle formed by the secant plane with the sarcomere's axis. A routine simulating how a sarcomere appears when it is cut at different angles was written in MATLAB. Hypothesizing that the sarcomere's shape can be considered as a cylinder and using the Pythagorean theorem and trigonometrical equations, has been found that it is possible to obtain the cylinder's size (the sarcomere dimensions) starting from the shape and dimensions of the cylindric sections (i.e., the intersection of a cylinder's surface with a plane) for many different secant angles (oblique sections). Finally, using the same routine, how the Z-band and M-line lengths change, varying the secant angles, is described. This study refers principally to the skeletal muscles; in fact, cardiomyocytes have a general shape that cannot be related to a cylinder.

| Hypothesis and geometrical parameters
The sarcomere is considered here as a right circular cylinder ( Figure 1) and many sarcomeres are aligned along the same axis to form a straight myofibril. The geometrical parameters of the sarcomere are: L = cylinder's length (i.e., sarcomere's length), r = cylinder's radius (half size of Z-line length), α = secant angle (the angle between the plane and the cylinder's axis).
The intersection of the cylinder with a plane (secant plane), the cylindric section, can have a different shape (circular, ellipse or parallelogram, or others). The bases of the cylinder are the Z-disks; the intersection of the cylinder with the plane passing in the center of the cylinder and parallel to the bases is the M-line. Both Z-disks and the M-line are generally well visible in ultrastructural images due to the contrast with the paler surrounding areas and it aids the measurements.

| MATLAB routine
A routine was written in MATLAB (R2018b version, Mathworks) to graphically visualize the sarcomere's sections at different secant angles. Using a Cartesian coordinate system (x, y, z), the sarcomere is represented by a right cylinder, centered on the origin, oriented along the z-axis, with r = 1 and L = 8, from z = −4.0 to z = +4.0. We assumed here that the ratio L/r is equal to 8 as reported in the literature (Cisterna et al., 2021). The cylinder is subdivided into different regions with dimensions similar to the sarcomere's bands and colored with different gray shades similar to the ones observed in TEM images (Cisterna et al., 2021).

| Longitudinal sections
In longitudinal sections (α = 0°), the sarcomere's length and the Z-disks size can be measured directly on the images. Ideally, all the sarcomeres composing the same myofibril are visible in TEM images.

| Almost longitudinal sections
In almost longitudinal section (α ~ 0°), a number n of aligned sarcomere belonging to the same myofibril are visible in ultrastructural images. Being DE = 2r, it follows that r = DE/2. Indicating AC = d and using the Pythagorean theorem, L can be determined as: It is worth noting that L is not equal to AC/n or d/n, or, in fact, the distance between two consecutive Z-bands measured in the ultrastructural images (the apparent sarcomere's length) is not equal to the real sarcomere length (being the hypotenuse and a cathetus of a right triangle, respectively).
More precisely, the real and the apparent lengths are related by the following equation: Clearly, the greater the number of visible aligned sarcomeres in the ultrastructural section, the lower the difference between the apparent and the real lengths.
Using the trigonometric functions, α can be calculated as follow Knowing L from Equation 1, α can be obtained also from:

| Oblique sections with two visible Z-lines
In oblique sections, if α is low, many sarcomeres are intersected by the plane and the situation is the same as described in the previous but h is also related to d max and α by the following equation (Figure 3b): Thus, α can be evaluated as: Moreover, in the right triangle CFG (Figure 3b), the following relationship is valid: which leads to the final result: (4) = tan −1 CB AB = tan −1 2r nL or, alternatively, looking at the right triangle CEF (Figure 5b), = sin −1 r + h d , being more useful, because d is more easily measurable than d max .

| Almost transversal section
In almost transversal section, the secant plane could not intersect the cylinder's bases and the cylindric intersection is an ellipse. In the ultrastructural images, no Z-lines are visible ( Figure 6). It is no longer To do this, it is sufficient to measure MN = 2R and CF ( Figure 6).
Also, in this case r = MN/2.
In the right triangle, CEG ( Figure 6b) the following relationship is valid: Then, α is:

| Transversal section
In perfect transversal section (α = 90°) the cylindric section is a circle and the radius is the only obtainable parameter. It is sufficient to measure the diameter and halve the obtained value.

| Z-lines and M-line length measurement
The routine written in MATLAB shows the cylindric section and the appearance of the sarcomere cut by different planes. Figure 7a shows In this work, only through geometrical considerations, we showed that it is possible to calculate the length and the radius of the sarcomeres as well as the inclination of the secant plane starting from ultrastructural images of skeletal muscle also in the case of oblique sections. Moreover, in this work, a digital model of a sarcomere useful to simulate how it appears in ultrastructural images at different secant angles was presented.
TA B L E 1 Varying the secant angle α from 0° to 90° (1st column), i.e., from longitudinal to transversal section, the visible sarcomeres' features in ultrastructural images (2nd column), the shape of the cylindric section (3rd column), the sarcomeres' size (r, L) and secant angle (α, 4th, 5th and 6th columns, respectively) obtainable from the images are shown. In the 7th column the measures necessary to obtain the sarcomeres' morphological parameters are indicated. The hypothesis of an almost cylindrical shape on which the mathematical evaluations are based, is particularly true for skeletal muscle, in spite of cardiomyocytes having a shape that cannot be related to a cylinder. The skeletal muscle sarcomere features obtainable at different cut angles are explained in the text and the equations to obtain the results are reported. In addition, we showed that, even in not longitudinal perfectly sections, the length of the sarcomere is not the apparent distance between the two Z-bands that confine it, but that distance must be corrected for the cosine of the secant angle (Equation 2).

Sarcomere's visible features
In this study, we did not consider the case of longitudinal sections in which the cutting plane, parallel to the myofibril's axis, it is quite far from it. In this condition, the sarcomeres appear with a radius lower compared to those of the adjacent myofibrils, so the measurements of those sarcomeres are generally excluded from the analysis.
In addition, it is important to mention that although sarcomere behavior in contraction it is considered to be uniform, the occurrence of sarcomere length nonuniformities is well known in the literature (Burkholder & Lieber, 2001;de Souza Leite & Rassier, 2020;Edman & Reggiani, 1984;Edman & Reggiani, 1987;Julian & Morgan, 1979;Ralston et al., 2008). Moreover, the size of the different types of myofibrils can be increased or reduced by the fixative reagents (Davidowitz, Rubinson, Jacoby, & Philips, 1996). In longitudinal sections, a high number of sarcomeres are visible along the same myofibril, and to increase the number of measurements, the analysis can be performed on the same myofibril and in the adjacent ones.
Instead, in the case of oblique sections only one sarcomere (the cut one) is visible for each myofibril, so measurements should be made also on many adjacent myofibrils. Thus, a large number of measurements is necessary to reduce the effects of nonuniformities and to achieve statistical significance.
The proposed method is based on the manual extraction of the measurements. A completely automated software able to extract the same information on a large number of sarcomeres could be based on a fit between the real data and the shape of the theoretical cylindric section or on the Hough transform (Duda & Hart, 1972;Hart, 2009). The high contrast of Z-bands and of the M-line with respect to the surrounding areas can help the automated analysis.
Besides the length, the approach here presented might also be useful to calculate the radius of the sarcomere and the myofibrils diameter. This specific parameter has to be considered when comparing different muscles since the sarcomere size could vary among different muscles (Burkholder & Lieber, 2001;Schönbauer et al., 2011) and further the improper regulation of the sarcomere diameter could be a sign of myopathies (González-Morales et al., 2019; Katti et al., 2022). For these reasons, it could be interesting to compare not only the mean sarcomere size but also the sarcomere size distribution of the data among different pathological conditions.
The proposed equations allow for comparing sarcomere lengths from different samples and species as well as in physiological and pathological conditions. Furthermore, in the case of the anisotropic effects of the fixation procedure on the sarcomere size (e.g., greater reduction in sarcomere length compared to the reduction in the radial direction) the proposed equations still work.
In conclusion, this work opens the possibility to consider extracting all possible information from ultrastructural images even in the case of not perfectly longitudinal sections. Next step could be the application of the presented equation on real TEM images. This approach could be considered a useful tool for skeletal muscle morphological analysis.

FU N D I N G I N FO R M ATI O N
Author FB declares that no funding was received for conducting this study.

CO N FLI C T O F I NTE R E S T S TATE M E NT
Author FB declares that he has no conflict of interest.

DATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.