Systematic Review of Skeletal Muscle Passive Mechanics Experimental Methodology

Abstract

Understanding passive skeletal muscle mechanics is critical in defining structure-function relationships in skeletal muscle and ultimately understanding pathologically impaired muscle. In this systematic review, we performed an exhaustive literature search using PRISMA guidelines to quantify passive muscle mechanical properties, summarized the methods used to create these data, and make recommendations to standardize future studies. We screened over 7,500 papers and found 80 papers that met the inclusion criteria. These papers reported passive muscle mechanics from single muscle fiber to whole muscle across 16 species and 54 distinct muscles. We found a wide range of methodological differences in sample selection, preparation, testing, and analysis. The systematic review revealed that passive muscle mechanics is species and scale dependent—specifically within mammals, the passive mechanics increases non-linearly with scale. However, a detailed understanding of passive mechanics is still unclear because the varied methodologies impede comparisons across studies, scales, species, and muscles. Therefore, we recommend the following: smaller scales may be maintained within storage solution prior to testing, when samples are tested statically use 2–3 minutes of relaxation time, stress normalization at the whole muscle level be to physiologic cross-sectional area, strain normalization be to sarcomere length when possible, and an exponential equation be used to fit the data. Additional studies using these recommendations will allow exploration of the multiscale relationship of passive force within and across species to provide the fundamental knowledge needed to improve our understanding of passive muscle mechanics.

1. Introduction

Skeletal muscle is one of many soft tissues where tissue passive biomechanical properties are strongly related to function (Fung, 1981). Mechanical testing of soft tissues such as skin, tendon, and ligament has been performed for decades since their passive mechanical role is their primary role. However, skeletal muscle passive mechanical properties are less well understood as these properties (often quantified as a “stiffness”) are often considered secondary to their primary role of active force generation (please see series of reviews in Herbert and Gandevia, 2019). However, passive muscle properties significantly impact normal and pathologic function (Blazevich, 2019; Herbert et al., 2019; Lieber and Friden, 2019). Therefore, quantifying muscle passive mechanical properties is of interest to understand normal muscle function and pathological adaptation to disease. Furthermore, as a soft tissue with multiple roles, muscles present unique measurement challenges compared to other tissues.

Skeletal muscle is a composite tissue composed typically of ~90% contractile muscle fibers and ~10% connective tissue supporting structures (Meyer and Lieber, 2012). Muscle connective tissues, typically referred to as endomysium, perimysium and epimysium, clearly have a primary role in transmitting force from fibers to tendon and resisting muscle elongation. However, these connective tissues also play more sophisticated roles in force transmission throughout the muscle, complicating understanding passive muscle mechanics. These roles include lateral force transmission among muscle fibers (Maas, 2019; Street, 1983), energy storage to improve locomotion efficiency (Alexander and Bennet-Clark, 1977; Morgan et al., 1978), buffering length changes to prevent injury (Garrett, 1990), aligning sarcomeres within and between muscle fibers (Morgan et al., 1982), and altering operating range by allowing sarcomere shortening (Lieber et al., 1992; Trestik and Lieber, 1993). Many of these properties are altered by exercise or disease; thus, quantifying such effects are an important goal of biomechanical analysis to understand their impact on function.

There are several muscle-specific challenges to quantifying its passive mechanical properties. First, because muscle is a complex multiscale composite, it is likely that its properties are also size dependent. Thus, the practice of normalizing properties to cross-sectional area across scales, as is typically done to quantify material properties, may not be sufficient. For example, muscle fiber and fiber bundle passive mechanical properties do not accurately predict whole muscle passive mechanics (Ward et al., 2020; Winters et al., 2011). Second, like most connective tissues, muscle passive mechanical properties are highly nonlinear. This means that if the goal is to define “stiffness,” it requires choosing a reference length or strain; this decision impacts the interpretation of whether there are similarities or differences among samples (Lieber and Binder-Markey, 2021). A more general description of passive properties can be made across strains but requires some equation of choice for curve fitting. While a multitude of equation forms may fit raw data equally well, their derivatives, which define stiffness (change in force divided by change in length), may be widely disparate. Finally, much of biomechanics is performed across species, and thus, true interspecies variation may be lost due to the confounding issues described above related to size, normalization, or analytic approaches.

We recently reviewed the structural properties of muscle relevant to passive mechanics and wished to define a “typical” skeletal muscle stiffness value to provide a generic recommendation for biomechanical modeling and tissue engineering (Lieber and Binder-Markey, 2021). Unfortunately, while we described much of the physiological bases for muscle passive mechanical properties, the actual quantitative definition of such properties proved much more challenging. The complexity of this comparison became obvious based on the factors described above, and thus, coming up with “typical” properties of skeletal muscle from the literature was impossible. In fact, it was not clear which species were most likely to be tested or at what scale most of the experiments were performed. We thus performed a systematic review of the English literature publications in which muscle passive mechanical properties were reported and examined the methods used to obtain these properties. The purpose of this review is to summarize this literature to quantify previously reported passive muscle mechanics by re-digitizing and compiling the original data, summarize the methods used to obtain these findings, and make recommendations on standardizing future studies. This set of recommendations will allow a better understanding of normal and pathologic skeletal muscle passive properties and their effect on function.

2. Methods

This systematic review follows the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines (Liberati et al., 2009; Moher et al., 2009).

2.1. Eligibility Criteria

Only manuscripts reporting original values for healthy mature skeletal muscle were used and only if they reported normalized mechanical properties as a tensile elastic stress (normalizing muscle force) and tissue strain or sarcomere strain to normalize muscle length. Scales ranged from single muscle fibers to whole muscles, including fiber (one isolated fiber), fiber bundle (~20 fibers), fascicle (~300 fibers with natural ECM divisions), tissue (biopsies), and whole muscle mechanics. The scales reported within this review are the self-reported scales stated within each manuscript. All species of animals and only peer-review journal manuscripts written in English were considered.

Therefore, studies were excluded if they:

• Reported only impaired, young, or aged muscle mechanics

• Reported mechanics based on measurements of only:

  • Shear wave elasticity
  • Torque
  • Force
  • Percent of maximum isometric force
  • Stiffness (e.g., Young’smodulus, tangent stiffness)

• Reported data from only a single specimen

• Reported scales smaller than single muscle cell

• Performed experiments on cardiac or smooth muscle

• Tested muscle-tendon units rather than just muscle (thus affecting strain)

• Were reported only in books, conference proceedings, abstracts, or reviews

2.2. Sources and Search Strategy

A single reviewer performed web-based searches for relevant literature using medical subject headings (MeSH) and text words used within Pubmed, Web of Science, and Scopus. Within each database, relevant literature was found using the following search strategy: “(skeletal muscle OR muscle) AND passive AND (mechanics OR elasticity OR properties OR force OR stiffness OR stretching OR strain OR stress) NOT (smooth OR cardiac)” from the earliest date available (1949) through the search date of August 27^th, 2020. These broad search terms ensured all manuscripts with the potential of being included would be found and evaluated.

2.3. Study Selection

References were exported to Endnote (X9.3.3 Clarivate Analytics, Philadelphia, PA), where duplicates, conference proceedings, books, and book sections were removed. Two authors independently reviewed all titles and abstracts with any conflicting decisions of removal or inclusion discussed before final removal or inclusion. The 3rd author then arbitrated any split decisions.

2.4. Data Extraction

Two authors reviewed each manuscript and extracted the following data, if available:

• Species

• Size scale(s) (as defined in the manuscript)

• Muscle(s)

• Muscle preparation prior to testing

• Type of stretch (static or dynamic)

• Relaxation time

• Length measurement (strain or sarcomere length)

• Definition of slack length or 0% strain

• Units of stress

• Assumed shape for of cross-sectional area for calculating stress (e.g., circular, oval, or no assumed shape simply calculated from mass/volume and length)

• Whole muscle area (cross-sectional area (CSA) – orthogonal to the muscle’s long axis or physiological cross-sectional area (PCSA) – the theoretical cross sectional area of all of the muscle fibers in the muscle)

• Form of equation used to fit data (e.g., no fit, linear, quadratic, exponential, other)

The stress versus strain or sarcomere plots for every study were digitized using Engauge Digitizer (Mitchell et al., 2020) to enable quantitation of the stress-strain properties across species and scales. Data points beyond the yield point (if any) were excluded from the analysis.

2.5. Data Processing and Analysis

Digitized data were imported into MATLAB (vR2020a, The MathWorks Inc. Natick, MA) where stress was converted to kPa and, if sarcomere length was provided, species specific optimal sarcomere length (Burkholder and Lieber, 2001; Walker and Schrodt, 1974) was used to convert sarcomere length to strain. To determine how to best fit passive muscle mechanics data and compare across species and scales, each individual plot was fit three different ways using nonlinear least squares regression with the built-in MATLAB fit function:

• Linear: σ = aε + b

• Quadratic: σ = aε² + bε + c

• Exponential: $\sigma =\frac{a}{b}\left(e^{b\epsilon }-1\right)$

Values for r² across species and scales were compared to determine optimal equation form. The form that best fit across scales was then analyzed using ANOVA with the parameters of the curve as the dependent variable to determine independent and interaction effects of species, scale (fiber to muscle), stretch type (static or dynamic), and muscle region using SPSS (v26.0 IBM Corp, Armonk, NY). Because of the limited sample size for most muscles, muscles were categorized into regions: arm, leg, back, chest, etc. If an independent variable was significant, post-hoc t-test pairwise comparisons were performed with a Bonferroni correction for multiple comparisons.

3. Results

Of the over 7,500 individual papers screened for inclusion, we ultimately obtained 80 manuscripts that fit our inclusion criteria (Fig. 1 & Table 1).

Figure 1:

Flow chart of the screening and selecting process of the papers included within the review.

Table 1:

Summary table of studies included in this systematic review. ns – Not Specified within the manuscript, quad – Quadratic fit, exp – Exponential Fit

Study	Species	Scale	Muscle(s)	Muscle Prep	Type Of Stretch	Relaxation Time (Seconds)	Length	0% Strain	Units Of Stress	Assumed Shape for Norming Stress	Whole Muscle Norming Stress	How Are Data Fit
Ahamed et al. (2016)	caterpillar	whole muscle	other	ns	dynamic		strain	ns	MPa	oval	CSA	no fit
Alipour-Haghighi and Titze (1985)	dog	tissue	Vocal muscles	fresh	static	100	strain	nominal force	kPa	ns		no fit
Alipour and Titze (1999)	dog	tissue	Cricothyroid	fresh	dynamic		strain	nominal force	kPA	ns		no fit
Alipour et al. (2005)	dog	tissue	Interarytenoid, Posterior Cricoarytenoid, Lateral Cricoarytenoid	fresh	dynamic		strain	nominal force	kPa	mass and length		no fit
Bol et al. (2019)	pig	fiber	biceps femoris (BF)	fresh	dynamic		strain	nominal force	kPa	area measured		data: no fit model: exp
Brooks et al. (1995)	mouse	whole muscle	Extensor Digitorium Longus (EDL)	fresh	static	180	strain	optimal fiber length	kN/m2	mass and length	CSA	no fit
Brown et al. (1996)	cat	whole muscle	Sartorius anterior, semitendinosus (ST), caudofemoralis, BF, tenuissimus	fresh	static	3	strain	nominal force, also max length	N/cm2	mass and length	PCSA	no fit
Brown et al. (2012)	rat	fiber bundles	rectus abdominis (RA), external oblique (EO), internal oblique (IO), and transverse abdominis (TrA)	maintained in storage sol	static	180	strain	nominal force	kPa	circle		quad
Brynnel et al. (2018)	mouse	fiber tissue whole muscle	Diaphragm (Dia), Soleus (Sol), EDL	fresh	static	20	sarcomere		mN/mm2	oval	PCSA	exp
Burnett et al. (2020)	human	bundles	Coccygeus, Iliococcygeus, Pubovisceralis	maintained in storage sol	static	180	strain	nominal force	kPa	circle		quad
Calvo et al. (2010)	rat	whole muscle	Tibialis Anterior (TA)	fresh	dynamic		strain	ns	MPa	mass and length	CSA	data: no fit model: exp
Danos and Azizi (2015)	frog	bundles	cruralis, Plantaris	fresh	ns	ns	sarcomere		N/mmm2	mass and length		exp
de Bruin et al. (2014)	human	fiber bundles	Flexor Carpi Ulnaris (FCU)	maintained in storage sol	static	240	sarcomere		mN/mm2	oval		no fit
Dorfmann et al. (2007)	caterpillar	whole muscle	ventral interior lateral	fresh	dynamic		strain	ns	MPa	oval	CSA	no fit
Farkas et al. (1985)	Dog	tissue	Dia Intercostals (IC)	fresh	static	ns	strain	optimal fiber length	kg/cm2	mass and length		no fit
Feit et al. (1989)	chicken	bundles	Pectoralis	maintained in storage sol	dynamic		strain	visible slack	MN/m2	oval		no fit
Gosselin et al. (1994)	rat	tissue	Dia	fresh	dynamic		strain	optimal fiber length	N/cm2	mass and length		no fit
Granzier et al. (1991)	fish	bundles	Levator operculi anterior, Hyohyoideus	fresh	static	10	strain	ns	N/cm2	oval		exp
Granzier and Wang (1993)	waterbug	fiber	Dorsal-Longitudinal Muscle	fresh	static	260	sarcomere		kN/m2	oval		no fit
Hashemi et al. (2020)	sheep	whole muscle		fresh	dynamic		strain	ns	MPa	ns	CSA	data: no fit model: exp
Henkin et al. (2004)	fly	fiber	Indirect flight muscles	fresh	static	ns	strain	visible slack	kN/m2	circle		no fit
Hernandez et al. (2011)	rabbit	tissue	RA, EO, IO, TrA	fresh	dynamic		strain	ns	MPa	mass and length		data: no fit model: exp
Horowits (1992)	rabbit	fiber	Psoas, Sol	fresh	static	300	sarcomere		kg/cm2	circle		no fit
Irving et al. (2011)	rabbit	fiber	Psoas	maintained in storage sol	static	1200	sarcomere		mN/mm2	ns		no fit
Jalal and Zidi (2018)	pig	tissue	BF	fresh & maintained in storage sol	static	300	strain	nominal force	kPa	ns		no fit
Kelly et al. (1993)	rat	tissue	Dia, IC	fresh	dynamic		strain	nominal force	kPa	mass and length		no fit
Kovanen et al. (1984)	rat	whole muscle	Sol, RF	fresh	dynamic		strain	visible slack	N/mm2	mass and length	CSA	no fit
Lassche et al. (2020)	human	fiber	VL, TA	maintained in storage sol	static	90	sarcomere		mN/mm2	mass and length		no fit
Lieber et al. (2003)	human	bundles	adductor pollicis, pronator teres	maintained in storage sol	static	120	sarcomere		kPa	circle		quad
Lim et al. (2019)	human	fiber	VL	maintained in storage sol	static	60	sarcomere		uN/mm2	oval		no fit
Magid and Law (1985)	frog	whole muscle	ST	ns	static	ns	sarcomere		N/m2	ns	ns	exp
Marcucci et al. (2019)	human	fiber bundles	VL	maintained in storage sol	static	600	sarcomere		kPa	circle		data: no fit model: exp
Maruyama et al. (1977)	frog	fiber	sartorius	maintained in storage sol	ns	ns	sarcomere		kg/cm2	ns		no fit
Mathewson et al. (2014)	human	fiber, bundles	Gastrocnemius (GM), Sol	maintained in storage sol	static	180	sarcomere		kPa	circle		no fit
Mead et al. (2014)	dog	tissue	Dia	fresh	dynamic		strain	optimal fiber length	g/cm2	ns		no fit
Meyer et al. (2011)	mouse	fiber	EDL	maintained in storage sol	static	180	sarcomere		kPa	circle		no fit
Meyer and Lieber (2018)	mouse, frog	fiber bundles	ST, EDL	maintained in storage sol	static	180	sarcomere		kPa	ns		exp
Mohammadkha h et al. (2016)	chicken	tissue	Pectoralis	fresh	dynamic		strain	visible slack	kPa	measured		no fit
Moore et al. (2006)	human	fiber	Dia	maintained in storage sol	static	60	sarcomere		kN/m2	circle		quad
Moss and Halpern (1977)	frog	whole muscle	ST	fresh	static	ns	sarcomere		N/m2	mass and length	CSA	no fit
Muñiz et al. (2001)	rat	whole muscle	Sol, Plantaris	fresh	dynamic		strain	optimal fiber length	N/m2	mass and length	CSA	sigmoidal
Mutungi et al. (2003)	rat	fiber	Sol, EDL	fresh	static	5	sarcomere		kN/m2	circle		ns
Nagle et al. (2014)	human	tissue	Levator ani	fresh	dynamic		strain	in situ length	MPa	mass and length		power law
Noonan et al. (2020a)	rat	fiber bundles	EDL, Sol, Vastus intermedius	maintained in storage sol	static	120	sarcomere		kPa	circle		logistic
Noonan et al. (2020b)	human	fiber	VL	maintained in storage sol	static	120	sarcomere		kPa	circle		logistic
Olson and Marsh (1993)	scallop	bundles	adductor	fresh	ns	ns	strain	ns	N/cm2	mass and length		no fit
Olsson et al. (2006)	human	fiber	VL	maintained in storage sol	static	180	sarcomere		mN/mm2	circle		no fit
Ottenheijm et al. (2012)	mouse	bundles	Sol, Tibialis Cranial	maintained in storage sol	static	30	sarcomere		mN/mm2	circle		no fit
Ottenheijm et al. (2006)	human	fiber	Quadriceps Femoris	maintained in storage sol	dynamic		sarcomere		mN/mm2	oval		no fit
Ottenheijm et al. (2006)	human	fiber	Dia	maintained in storage sol	static	60	strain	optimal fiber length	mN/mm2	ns		no fit
Pavan et al. (2020)	human	fiber bundles	VL	maintained in storage sol	static	variable	sarcomere		kPa	circle		3rd order polynomial
Perlman and Alipour-Haghighi (1988)	dog	tissue	vocal	fresh	static	ns	strain	in situ length	kPa	ns		no fit
Persad et al. (2021)	rabbit	fiber bundles	Psoas, EDL, GM, Longissimus dorsi (LD), Dia	maintained in storage sol	static	120	sarcomere		mN/mm2	circle		quad
Prado et al. (2005)	mouse	fiber	TA	fresh	static	30	strain	nominal force	kPa	circle	CSA	data: no fit model: exp
Rehorn et al. (2014)	rat	tissue	Dia	fresh	ns	ns	strain	optimal fiber length	N/cm2	mass and length		quad
Reid et al. (1987)	chicken	whole muscle	Flexor digitorum profundus	maintained in storage sol	dynamic		strain	ns	kPa	mass and length	CSA	no fit
Reyna et al. (2020)	mouse	fiber	EDL	maintained in storage sol	static	120	strain	nominal force	Pa	circle		quad
Shah et al. (2012)	human	fiber bundles	Gracilis, ST	maintained in storage sol	static	180	sarcomere		kPa	circle		linear & quad
Smith et al. (2011)	fish	fiber	axial	fresh	dynamic		sarcomere		kN/m2	ns		no fit
Spierts et al. (1997)	dog	tissue	Dia	fresh	dynamic		strain	visible slack	g/cm2	area measured		no fit
Strumpf et al. (1993)	pig	tissue	LD	fresh	dynamic		strain	nominal force	kPa	measured		no fit
Takaza et al. (2013)	rabbit	bundles	hamstrings	maintained in storage sol	dynamic		strain	ns	kPa	circle		no fit
Tamura et al. (2016)	rabbit	bundles	hamstrings	maintained in storage sol	static	300	strain	nominal force	kPa	oval		exp
Tamura et al. (2019)	rat	whole muscle	EDL, sol	fresh	static	80	strain	nominal force	N/mm2	mass and length	PCSA	no fit
Toscano et al. (2010)	rat, rabbit	fiber	Sol	maintained in storage sol	static	180	sarcomere		kN/m2	circle		quad
Toursel et al. (2002)	mouse	bundles	Sol, EDL	maintained in storage sol	static	90	sarcomere		mN/mm2	oval		no fit
van der Pijl et al. (2020)	rabbit	whole muscle	TA	fresh & fresh maintained	dynamic		strain	in situ length	kPa	ns	PCSA	no fit
Van Ee et al. (1998)	rabbit	whole muscle	TA	fresh & fresh maintained	dynamic		strain	in situ length	kPa	ns	PCSA	no fit
Van Ee et al. (2000)	rat	fiber	Dia, Sol	maintained in storage sol	static	60	sarcomere		mN/mm2	oval		no fit
van Hees et al. (2010)	rat	fiber	Dia, Sol	maintained in storage sol	static	60	sarcomere		mN/mm2	oval		no fit
van Hees et al. (2012)	mouse	whole muscle	TA	fresh	static	300	strain	ns	mPa	ns	ns	no fit
Virgilio et al. (2020)	rabbit	fiber bundles fascicle whole muscle	TA, EDL, ED2	fresh & maintained in storage sol	static	180	strain	nominal force	kPa	circle	PCSA	linear & quad
Ward et al. (2020)	rabbit	tissue	TA	fresh	static	300	strain	ns	kPa	measured		no fit
Wheatley et al. (2016a)	rabbit	tissue	TA	fresh	static	300	strain	ns	kPa	ns		no fit
Wheatley et al. (2016b)	rabbit	whole muscle	TA	fresh	static	300	strain	ns	kPa	mass and length	PCSA	data: no fit model: exp
Wheatley et al. (2018)	rabbit	whole muscle	TA	fresh	static	180	strain	ns	kPa	mass and length	PCSA	data: no fit model: exp
Wheatley et al. (2017)	cow	tissue	genioglossus	fresh	dynamic		strain	ns	kPa	ns		data: no fit model:quad
Yousefi et al. (2018)	mouse, rat, rabbit	bundles	Multifidus (Mul), Erector Spina (ES)	maintained in storage sol	static	120	sarcomere		kPa	circle		logistic
Zwambag et al. (2019)	rat	fiber bundles	Mul, lumbar ES, thoracic ES	maintained in storage sol	static	120	sarcomere		kPa	circle		quad
Zwambag et al. (2018)	human	Whole muscle	gracilis	fresh	static	ns	sarcomere		kPa	mass and length	PCSA	exp

3.1. Types of muscles tested

From these 80 reports, 173 data sets were digitized from 16 species and 54 distinct muscles, across the range of scales from muscle fiber, fiber bundle, muscle fascicle, muscle tissue, and whole muscle. From the 173 data sets, the most common species reported were rat (n=40), rabbit (n=38), and human (n=28) (Sup Table 1). The most frequently tested muscles were the extensor digitorum longus (n=18), soleus (n=16), diaphragm (n=15), and tibialis anterior (n=13). Fibers (n=57) and bundles (n=53) were the most frequently tested size scales, with fascicles being the least frequently tested size (n=3) (Sup Table 1 and 2).

3.2. Journals publishing

To understand how studies within this review were being disseminated and audiences most likely to review these papers, we tracked the journals in which these papers appeared. 44 different journals were present, and only 8 of those journals had 3 or more publications. The top journals publishing this work were the Journal of Biomechanics (8 studies), Journals of Applied Physiology (8 studies), and Mechanical Behavior of Biomedical Materials (6 studies) (Sup Table 3).

3.3. Year paper published

54% (43/80) of the papers within this review were published within the past 10 years, 73% (58/80) in the past 20 years and 27% (22/80) of the papers were published before 2000. There was an increase in papers published on the topic in the mid-1990s (Fig. 2).

Figure 2:

Histogram of the number of papers published within 5-year bins within the review.

3.4. Sample Storage

Sample preparation and storage may affect passive force measurements. 71 of the 173 samples specified that they tested the samples fresh while 90 samples were specified that they were maintained in a storage solution prior to testing, 2 samples were freshly chemically skinned prior to testing. The preparations of the remaining 10 samples were not specified (Sup Table 4). All whole muscle samples were tested fresh (tested within hours of collection and never stored nor fixed), whereas the majority of fibers, bundles, and fascicles were stored prior to testing.

3.5. Testing Procedure

Another factor affecting passive stress is the testing protocol used—static or dynamic. Dynamic testing would include viscous properties in addition to the elastic properties. Of the 80 studies, 51 specified that they tested under static conditions, 25 studies reported dynamic testing, and 4 studies did not report their testing protocol. Within static tests, relaxation times between static data points ranged widely from 3 to 600 seconds, with the two most common times being 120 and 180 seconds (Fig. 3).

Figure 3:

Histogram of relaxation time frequency indicated within the studies.

3.6. Stress and Strain Normalization

To compare across studies accurately, stress and strain normalization should be performed consistently. However, we found a wide range of processes and assumptions made during the normalization process that affects the values obtained. For example, stress was reported in 12 different units, the most common being kPa (Sup Table 5). Units can easily be interconverted and then compared. However, the assumed shape (e.g., circular, oval, or no assumed shape calculated simply from mass/volume and length) of the sample’s cross-sectional area (CSA) can also affect reported stress, which cannot be corrected. Across the 80 studies reviewed, CSA shape assumption was not reported in 17 studies. For those that reported CSA shape, there were 4 categories of assumed shapes: 1) circle (24/80), 2) no specific shape for mass was divided by length and density (21/80), 3) oval (13/80), and 4) experimentally measured area (5/80) (Sup Table 6). Of the 21 studies testing whole muscle mechanics, 10 normalized to the physiological cross-sectional area (PCSA – the theoretical sum of all the fibers cross-sectional area; Lieber and Fridén (2000)), 9 studies normalized to estimated CSA (orthogonal to the long axis of the muscle), and 2 did not specify their normalization assumptions. (Sup Table 7).

To report length change, 34 of the 80 studies measured sarcomere length, with the remaining 46 reporting sample strain. However, stress versus strain relationship depends on the length at which 0% strain is defined. We found studies used 4 broad categories to determine the 0% strain point. They were either 1) nominal resting force, 2) optimal fiber length, 3) visible slack, or 4) in situ length (Table 2). Surprisingly, many studies reporting strain (15/46) did not explicitly report their definition of 0% strain.

Table 2:

0% Strain Length Definition

0% Strain Length Definition	Number of Studies
Nominal force	15
Optimal fiber length	7
Visible slack	5
in situ length	4
Nominal force, also max length	1
Not specified	15
Grand Total	26

3.7. Data Curve Fitting

Finally, the way in which the data were analyzed affects values obtained because each equation form has its own innate properties. Across the 80 studies, only 24 studies reported that the data were fit to a specific form of an equation. Across these 24 studies, 7 different equation forms were assumed—quadratic, exponential, logistic, linear, 3^rd order polynomial, sigmoidal, or a power law (Table 3), while the majority did not even fit the data within the study (55/81). Interestingly though, in 9 other studies (separate from those above) that modeled the elastic properties of muscle, 8 assumed an exponential function, and 1 assumed a quadratic function.

Table 3:

The equation of the form used to fit the data within each study

Equation Type	Number of Studies
Quadratic	9
Exponential	7
Logistic	3
Linear (fiber) & Quadratic (larger scales)	2
3rd Order Polynomial	1
Sigmoidal	1
Power Law	1
Not specified	1

3.8. Data Analysis across forms, scales, and species

3.8.1. Equation Form

To provide a uniform assessment of equation form, we digitized all included 173 raw data sets (Fig. 4). Each data set was fit to 3 separate equations—linear, quadratic, and exponential. Overall, the linear form had the worst fits across all scales (Table 4; overall r²=0.854). Quadratic and exponential forms fit comparably better (Table 4; r²=0.970 and r²=0.946, respectively). The quadratic form fit better at smaller scales, fiber and bundle, while the exponential fit was better at the whole muscle scale.

Figure 4:

Digitized stress strain curves grouped by scale across the 172 samples, colored by species groups of invertebrates (red), fish (green), amphibians (magenta), birds (black), small mammals (dark blue), and large animals (light blue).

Table 4:

Average fit, r², for all the digitized samples using each of the 3 equation forms.

	fiber	bundle	fascicle	tissue	muscle	overall
Linear	0.914	0.872	0.927	0.797	0.761	0.854
Quadratic	0.985	0.986	0.999	0.947	0.932	0.970
Exponential	0.928	0.934	0.996	0.893	0.981	0.946

A particular attraction of the exponential form of curve fitting is that it provides an interpretation of sample parameters independent of strain. By taking the natural-log of the exponential, the “b” value becomes the linear natural-log stiffness of the sample. Therefore, a specific strain does not need to be chosen to compare the material properties across samples. However, it should be appreciated that with the exponential form of the equation, the passive forces are dictated to start at 0% strain, where the “a” value is the initial stiffness at zero strain. The quadratic equation allows passive force to start at strain values other than the 0% strain point. Unfortunately, there is no simple transformation for the quadratic form that allows such an easy comparison across samples. Since both forms have comparable fits across size scales, we chose to use the exponential form within our analysis. Average species and size species curve fit parameters, using the exponential form, are summarized in supplemental table 10.

3.8.2. Effect of Species, Scale, and Muscle

To determine the effects of species, scale, muscle region (categorized as arm, leg, back, chest, etc.), and type of stretch (dynamic vs. static) on the overall muscle stiffness, “b” in the exponential, ANOVA was performed with each main and interaction effects. We found significant main effects of species, size, and type of stretch and significant interaction effects of species with scale and type of stretch, and type of stretch with scale and muscle region (p<0.002) (Sup table 8).

Post-hoc analysis among species revealed the flies, water bugs, and chickens were significantly different from other species (Sup Table 9). However, these samples sizes are very small, so implications and interpretation are limited.

Post-hoc analysis across scales revealed that whole muscle was significantly different from all other scales, fascicles and bundles were significantly different from all other scales except each other, and fiber and tissue were significantly different from all other scales, except each other (Sup Table 10).

Post-hoc analysis of the type of stretch demonstrates a significant difference between dynamic and static stretches (p < 0.001).

3.8.3. Stiffness Scaling

The “b” values from the exponential fit were compared across scales to understand how stiffness changes across scale. This was performed by calculating the ratio of b values between each scale. This analysis was restricted to species with larger samples sizes (n>10), which included frog, human, mouse, rabbit, and rat (Table 5). However, it did not include comparisons to the tissue scale due to a lack of data. Overall, sample stiffness increased with scale: fiber to muscle, the nominal increase was four-fold.

Table 5:

Ratio of b stiffness between scales

	Bundle: Fiber	Fascicle: Bundle	Muscle: Bundle	Muscle: Fascicle	Muscle: Fiber
Frog	2.54		0.35		0.89
Human	0.92	0.32	2.27	7.17	2.10
Mouse	1.68		1.70		2.84
Rabbit	1.97	0.50	5.79	11.54	11.40
Rat	0.68		4.01		2.74
Mean	1.56	0.41	2.82	9.36	4.00

4. DISCUSSION

Understanding passive skeletal muscle mechanics is critical to define structure-function relationships in skeletal muscle and ultimately to understand pathological conditions, such as fibrosis, that make muscles stiffer. However, as reported in this review, while there are many studies of passive muscle mechanics, they are performed across a variety of species, muscles, and scales using variable methodology. Therefore, differentiating the basis for observed differences among studies is quite challenging, resulting in a muddled and incomplete understanding of this topic. This systematic review quantified the passive mechanical properties of muscle and systematically examined current methodological variations among studies to gain insights into the sources of variations across the passive muscle mechanics literature. As a result, we have created a set of recommendations to standardize the collection of passive mechanics data in future experiments. Should these recommendations be followed, studies can be easily compared, leading to a greater understanding of passive muscle mechanics.

Probably the most important factor that impedes comparisons among studies is the methodological approaches used to perform and analyze the experiment. Factors that vary among studies include the choice of sample, sample storage, testing procedure, data normalization, and data analysis. These are considered sequentially.

4.1.1. Sample Choice Variations

Within this study, 171 data sets were examined from 16 species, 53 distinct muscles, and across a range of scales from muscle fiber to whole muscle. This large number of studies gives the impression that there are a large number of studies on the topic, yet when broken down by species, muscle, and scale, only a few studies remain within each classification. For example, only a single study examined a single species (rabbit) and compared 3 muscles (TA, EDL, & ED2) across scales from single fiber to whole muscle (Ward et al., 2020). Within this study, muscle scaling from fiber to whole muscle was muscle specific, with each muscle having unique passive mechanical properties. Studies performed across species demonstrate that muscle mechanics at a single scale are species specific (Meyer and Lieber, 2018; Toursel et al., 2002; Zwambag et al., 2019). Therefore, generalizing passive muscle mechanics across species, scale, and muscles, as is often done in biomechanical musculoskeletal modeling, is likely inaccurate for each species and muscle may have unique passive mechanical properties that also scale differently.

4.1.2. Sample Storage Variations

Preparation and storage affect passive muscle mechanics. All of the whole muscle samples considered here were tested fresh; the majority of fiber, bundle, fascicle, and tissue samples were maintained within a storage solution prior to testing (88/142) (Table 1). At the whole muscle and tissue level, the data indicate that freezing the sample initially stiffens the sample post-thaw then with successive stretches, the sample becomes more compliant, affecting the passive mechanics of the muscle (Jalal and Zidi, 2018; Van Ee et al., 1998, 2000; Wheatley et al., 2016b). Though none of the studies included in this review froze fiber, bundle, and fascicle samples, most were maintained at sub-zero temperatures in a storage solution, as is standard procedure when storing muscle samples. This process does not freeze the sample and preserves the active mechanics and passive mechanics of fibers for an extended period of time (Einarsson et al., 2008; Moss, 1979; Wood et al., 1975). However, it is unknown how maintaining muscle bundle and fascicle samples within storage solution affects the passive mechanics due to potential alterations in the connective tissue structures. Therefore, further studies into the effect of sub-zero storage within a storage solution on bundle and fascicle passive mechanics should be further explored.

4.1.3. Testing Procedure Variations

Due to the inherent viscoelastic properties of muscle, variations in testing procedure, specifically whether a muscle is tested in a static versus dynamic state, will affect results. Testing in the dynamic state will include additional viscous properties, increasing the muscle’s apparent stiffness. In contrast, static testing does not include these viscous properties (given the sample has sufficiently relaxed and reached an equilibrium state). However, within the studies included in this review that tested in a static state, we found a large range of relaxation times. The most common times were 120 and 180 seconds, or 2 to 3 minutes of relaxation. Two studies of relaxation time show that after 2 to 3 minutes, samples reach ~10% of equilibrium (Fridén and Lieber, 2003; Lieber et al., 2003) and the viscous properties dissipated, revealing the muscle’s elastic properties. However, the exact relaxation time likely varies with scale and strain (Fung, 1993). Though, after stress-relaxation, elastic properties can be compared independent of muscle viscous properties.

4.1.4. Data Normalization Variations

Normalization of raw data to both stress and strain dramatically affects reported values. Our inclusion criteria were very strict in that data must be reported in stress versus strain or sarcomere length to allow post-hoc comparison. However, many studies were excluded because they either reported raw force or percent of maximum force to enable the comparison of elastic modulus or stress and strain across studies. Within the 80 studies included in this review, a wide array of units were used (12 different units, Sup table 5); however, it is simple to interconvert units to a common unit for comparison.

The studies we surveyed used multiple methods to estimate the CSA. Most studies (37/80) estimate CSA at the fiber, bundle, fascicle, or tissue scales as a shape, a circle or oval, from diameter measurements. Other methods used to calculate CSA at these scales were either experimentally measured CSA or CSA calculated from sample volume and length (Sup Table 6). Additionally, many studies (17/80) did not explicitly report how CSA was determined. Also, because muscle is isovolumetric, the strain at which CSA is measured affects area. However, the strain at the time of measurement is generally not specified within studies, thus difficult to correct, but the effect of strain on CSA is likely negligible. However, at the whole muscle level, the definition of area can significantly affect the calculated stress. Half of the studies measuring whole muscle properties did not correct for architectural properties of the muscle and used the anatomical CSA, the area orthogonal to the long axis, of the muscle rather than the PCSA, which calculates the total fiber of cross sectional area. This step is critical to understand scaling to whole muscle mechanics (Lieber and Fridén, 2000).

In addition to variations in stress normalization, there were variations in strain calculations originating from a variety of definitions of 0% strain. The 0% strain point determines the relationship between passive forces and the samples strain (Fig. 5 of Lieber and Binder-Markey (2021)). As is a reoccurring theme, a third of the studies (15/46) did not report the definition of 0% strain. Of the studies that reported 0% strain definition, most defined this point as the length at which a small or nominal force was produced (14/46) (Table 1). In addition to studies reporting in muscle strain, we chose to include studies that measure length change in sarcomere length, for the sarcomere’s length is an intrinsic property of muscle and provides an accurate measurement of length deformation that can be converted to strain using species-specific optimal sarcomere length (Burkholder and Lieber, 2001; Walker and Schrodt, 1974). This normalization allows consistent measurement across studies and muscles because a sarcomere’s optimal length is an intrinsic property of the muscle allowing for a consistent reference point. But, this assumption plots the passive stress versus the muscle’s active strain. Additional information on the interaction between active and passive properties may be observable when the passive stress is plotted versus active strain. This observation might demonstrate that 0% passive strain, or the start of passive forces, is independent of the active properties and likely does not align with optimal active sarcomere length, as is often assumed. However, the sarcomere length at which passive forces start will likely be species and muscle specific and may be associated with either the active sarcomere operating range (Loren et al., 1996; Ward et al., 2009a; Ward et al., 2009b) or maximum muscle length (Brown et al., 1996) rather than directly linked with the active mechanical properties.

Figure 5:

Box and whisker plots with individual data points from the fit exponential parameter for stiffness “b” from each digitized data series across species by scale. Please note the substantial differences in y-axis scales among the graphs.

4.1.5. Data Analysis Variations

Because passive mechanics vary over a range of strains, the decision of how to analyze the data dramatically affects the interpretation of results within and between studies. One solution often utilized is to pick a single strain or strain range at which to compare. However, if each study chooses its own strain to compare, comparisons across studies become increasingly difficult due to the nonlinear nature of the mechanics (Lieber and Binder-Markey, 2021; Ward et al., 2020). In addition, the “correct” or “typical” range is nearly impossible to define across studies or among muscles (Ward et al., 2020). However, even when studies report the stress versus strain curves, it becomes difficult to compare data without curve fitting. Within this review, a majority (55/80) of studies present the data without a fit and within the studies that fit the data 7 different assumed curve fits were used, with quadratic and exponential being the most common (Table 3).

When curve fitting, the decision of equation form to be fit determines the inherent properties of the curve and comparing curve fits across different equation forms becomes difficult. Since most studies compare “stiffness” or “modulus,” the derivative of the stress-strain function will have a defined form even when curve fits of the raw stress-strain data are nearly the same. For example, though quadratic and exponential curve fits have similar fits (Table 4), the stiffness of a quadratic fit will be a linear function, whereas the stiffness of an exponential fit will remain an exponential function. Additionally, the strain at which passive forces begin is also an inherent property of the curve. The exponential equation dictates forces begin at 0% strain, whereas the quadratic equation allows passive force to start at strain values other than 0% strain. The ability to start at values other than 0% strain may explain why the quadratic form has slightly better fits at the fiber and bundle scales where strain is often normalized to optimal sarcomere length. In comparison, the exponential form had better fits at the tissue and whole muscle scales, where slack length is often used to define 0% strain.

4.2. Analysis of compiled data

We found that species, size, and type of stretch had a significant effect on muscle stiffness, whereas muscle region did not have a significant effect. Though species and scale were significant factors, there was wide variability across and within species and scale (Figs. 4 and 5). Interestingly, comparisons among species demonstrated significant differences between the water bug and fly and every other species, suggesting a potential divide in muscle properties between insect muscles and all others. These stiffness differences in insect muscles parallel differences observed within insect muscle connective tissue networks (Pringle, 1967; Tregear, 2011). Additionally, chicken muscles were significantly different from many other species, indicating potential differences in flight muscles from mammalian muscles. However, these comparisons overall should be interpreted with caution due to the low sample size within many of the species. Additionally, comparisons among scales found significant differences between most of the scales except bundles and fascicles and tissue and fiber. Fascicles and bundles likely have similar connective tissue structures since the fascicle may represent a large bundle, but this should be further explored. Finally, comparisons revealed significant differences between muscles tested statically or dynamically. This is not unexpected, for the dynamic test includes viscous properties in addition to elastic properties.

Overall, the data analyzed demonstrate that passive mechanical properties increase non-linearly with scale (Figs. 4 and 5). Using only species with larger numbers of studies (n>10; frogs, human, mouse, rabbit, and rat), we analyzed how stiffness changes across scales by calculating the ratios between each scale (Table 5). Overall, sample stiffness increased with scale (Fig. 5). However, it was interesting to note that in frogs, whole muscle was actually less stiff than either the bundle or fiber. In all other species, whole muscle was 2–10 times stiffer compared to fibers and bundles. Though there is a large amount of variability across these data, in mammals, muscle stiffness increases non-linearly with size scale (fiber to whole muscle), but the magnitude of increase is species specific.

However, towards the initial goal of wishing to define a “typical” skeletal muscle stiffness value to provide a generic recommendation for biomechanical modeling and tissue engineering: this review further clarifies that a detailed understanding of passive mechanics and this “typical” value remains unclear. Due to methodological variations summarized within this review, the substantial variations in observed stiffnesses across studies impedes comparisons and conclusions based on scale, species, and/or muscle. Therefore, we suggest the following standardized approach for testing and reporting muscle passive mechanical properties to limit these variations and advance our understanding and field forward.

4.3. Recommendations:

• Testing Procedure: To fully characterize the passive viscoelastic properties, samples should be tested both statically and dynamically. When testing statically, allow 2–3 minutes of stress relaxation prior to recording passive force. However, the strain rate at which to test is outside the scope of this review.

• Stress Normalization: An assumption of a circular or oval cross-sectional shape is likely sufficient from the fiber to the fascicle. At the whole muscle level, PCSA should be the standard to account for muscle architecture.

• Strain Normalization: Sarcomere length should be the gold standard. Since sarcomere length is an intrinsic property of muscle, using sarcomere length will give insight into the passive mechanics across species and muscle, and insights into how active and passive mechanics relate to each other. Normalization to optimal sarcomere length will likely be most useful in species with well-established optimal sarcomere lengths. However, among invertebrates, optimal sarcomere lengths can vary by more than an order of magnitude (Walcott and Dewey, 1980) thus should be used with caution. At the whole muscle level, sarcomere length may be difficult to measure but would still be the preferred length measurement to accurately compare across scales. Advances in technology may make measurement of sarcomere lengths in whole muscle feasible, see Young et al. (2017) and Llewellyn et al. (2008). However, until these technologies are widely available, we propose that whole muscle excised at slack length or a true resting length may be the best definition of 0% strain. However, further studies on how to best define 0% strain are needed to allow for improved comparisons across scales and species.

• Equation Form of Curve Fitting: We recommend using the exponential form of:

  $$\sigma =\frac{a}{b}\left(e^{b\epsilon }-1\right)$$

This form provides an intuitive singular parameter, “b” that allows interpretation of a muscle’s nonlinear stiffness independent of strain for comparisons among studies, species, scales, and muscles and allows the derivative, stiffness, to be nonlinear. However, this exponential form dictates that passive forces start at 0% strain. This is appropriate if strain is calculated from a slack length. However, if strain is measured from optimal sarcomere length, an intrinsic active property, the passive 0% strain may not align with active zero strain, and therefore determination that these two points do not differ significantly needs to be determined prior to use of the exponential equation. However, within this analysis, any differences between the two 0% strain definitions did not substantially affect the fits of the exponential curves. This further emphasizes the importance of standardization and declaration of reference strain.