The effect of intramuscular fat on skeletal muscle mechanics: implications for the elderly and obese

Abstract

Skeletal muscle accumulates intramuscular fat through age and obesity. Muscle quality, a measure of muscle strength per unit size, decreases in these conditions. It is not clear how fat influences this loss in performance. Changes to structural parameters (e.g. fibre pennation and connective tissue properties) affect the muscle quality. This study investigated the mechanisms that lead to deterioration in muscle performance due to changes in intramuscular fat, pennation and aponeurosis stiffness. A finite-element model of the human gastrocnemius was developed as a fibre-reinforced composite biomaterial containing contractile fibres within the base material. The base-material properties were modified to include intramuscular fat in five different ways. All these models with fat generated lower fibre stress and muscle quality than their lean counterparts. This effect is due to the higher stiffness of the tissue in the fatty models. The fibre deformations influence their interactions with the aponeuroses, and these change with fatty inclusions. Muscles with more compliant aponeuroses generated lower forces. The muscle quality was further reduced for muscles with lower pennation. This study shows that whole-muscle force is dependent on its base-material properties and changes to the base material due to fatty inclusions result in reductions to force and muscle quality.

1. Introduction

Skeletal muscle provides the forces that are necessary for the maintenance of body posture and for driving body movements for our activities of daily living. Muscle forces depend partly on the structural features [1] of the muscle that include fibre length, the pennation angle of the fibres relative to the line of action, the number of fibres and their physiological cross-sectional area (PCSA) [2]. Muscles forces additionally depend on the base-material properties of the muscle tissue, but much less is known about this. The structural and material properties of muscle vary between muscles and individuals [3,4] but can also change through our lifespan [5] and can be affected by disease (e.g. [6]). The purpose of this study was to investigate how the inclusion of fat within a muscle belly can affect its force output.

Intramuscular fat accumulates both in (intramyocellular) and out (extramyocellular) of the muscle fibres. Healthy muscle contains about 1.5% of intramyocellular fat and this can increase to over 5% in the obese [7]. The total intramuscular fat additionally contains extramyocellular components, and so the total intramuscular fat may exceed these values. Additionally in the obese, the muscles may remodel by hypertrophy to a larger size [8] and experience a transition to faster fibre types [7]. It has previously been shown that obesity can result in reductions in joint-specific torque (relative to lean or total body mass [9,10]), but we do not know the effect of intramuscular fat and its distribution on individual muscle mechanics, or the mechanisms that may cause deterioration of such performance.

Intramuscular fat can also increase as we age and can reach about 11% in the elderly [11]. Ageing also results in progressive muscle wasting called sarcopenia [12], which result in reductions in size, strength and a transition to slower fibre types [5]. Additionally, the lower levels of physical activity that accompany obesity in the elderly have been shown to accelerate muscle atrophy [13]. Connective tissue (tendon and aponeurosis) properties also change. Despite earlier experimental studies suggesting no effect [14] or an increase [15] in tendon stiffness with ageing, recent studies have reported a decrease in stiffness [16–18] of human tendons in the elderly. However, less is known about age-related changes to the aponeurosis stiffness.

Experimental measures of intramuscular fat have been achieved with a variety of imaging [11,19,20] and biochemical techniques [7,21]. However, in order to understand the mechanisms that may affect the fat-dependent loss of contractile performance, it is helpful to model the mechanical effect of fat inclusions within a muscle. Here, we test the effect of fat on skeletal muscle performance within a three-dimensional finite-element model that is based on the physics of continuum mechanics and represents the muscle as a composite biomaterial. A range of model variants were tested that represent the inclusion of intra- and extracellular fat. We additionally report on the influence of muscle structure and connective tissue properties on the deterioration of performance.

2. Material and methods

The muscle belly was modelled as fibre-reinforced composite biomaterial tissues with transversely isotropic properties using a three field (pressure, dilation and three coordinates of displacement) mathematical formulation [22]. The numerical solution for the system of nonlinear equations was obtained using a finite-element framework that was implemented in deal.II [23].

In brief, the gastrocnemius muscles were modelled as unipenate structures with superficial and deep aponeuroses. The structures were subdivided into many elements that had either muscle or aponeurosis properties. The muscle consisted of contractile elements that had active and passive force–length properties acting in parallel, and these elements were embedded within a ‘base material’ that limited the compressibility and provided an additional tissue stiffness. The net effect of contractile elements within a base material was to form a fibre-reinforced composite biomaterial that has transversely isotropic properties, and this was consistent with previous modelling representations of muscle [24–26]. Aponeuroses were represented with along fibre properties (AFPs) with specific force–length characteristics and were also embedded within a base material. The contractile elements in the muscle and fibres within the aponeuroses are collectively described by their AFPs; their orientation was specified in the initial geometry and could vary during contractions in the simulations. The effect of intramuscular fat was modelled by altering the base-material properties to include the effect of fat, and reducing the force generated by the contractile elements to reflect the reduced volume–density available for contractile elements within the muscle. A full mathematical description of the model for lean muscle has been reported previously [22], and a specific description of the geometries used in this study and the material representation of the intramuscular fat is described below.

2.1. Geometries, boundary conditions and muscle activations

The effect of fat inclusions was studied for the gastrocnemii. Despite being transversely isotropic, the model allowed asymmetric deformations of the contractile elements in their transverse directions due to stress distribution across the tissue; these asymmetries were consistent with recent imaging studies [27]. These ankle plantarflexor muscles were chosen because the plantarflexors have been shown to have greater loss of performance during walking in the elderly than the knee or hip extensors [28], and they are a major contributor to human balance and locomotion (e.g. [29]).

A simplified geometry of the human gastrocnemius (lateral or medial head) belly was used. Based on a recent study [30] using ultrasound imaging of young and elderly plantarflexor muscles, the initial fibre length (65 mm), initial belly width (55 mm) and length (273 mm) were kept constant for all the simulations (figure 1). The same study also showed that the initial pennation decreases with age and results in smaller PCSA of the muscle. However, muscle pennation may increase with obesity [8]. Here, we have chosen a parallelepiped geometry for the muscle [22] with a range of pennation angles representing sarcopenic (10°), healthy (15°) and obese (20°) states. The finite-element grid had 2772, 3696 and 4620 elements (muscle and aponeurosis combined) for geometries with 10°, 15° and 20° pennation, respectively. Each element had 27 integration (quadrature) points and fibre bundles passed through sets of integration points within the muscle tissue. Despite the change in the number of muscle tissue elements between different geometries, the number of muscle fibre bundles was the same and equal to 4158.

Figure 1.

Sample geometries of simplified human lateral gastrocnemius (LG) muscle with initial pennation of 10° (a) and 20° (b). Note that the change in cross-sectional area is only due to initial pennation because the fibre length and belly length are constant. Muscle tissue is shown in light grey and aponeuroses in dark grey. The belly and aponeuroses extended out of plane to a width of 55 mm.

The muscle belly was fixed at the muscle–tendon junctions before uniformly activating the muscle fibres. The activation was ramped up from zero towards a fully active muscle.

2.2. Material properties

The base and AFP of the fibre-reinforced composite muscle, and aponeuroses were as previously described [22]. Muscle fibres had active and passive force–length properties, and a maximum isometric stress of 200 kPa. Fat was assumed to be a nonlinear isotropic material. The neo-Hookean strain energy for the fat (W_fat) was adopted based on modelling work on human breast tissue [31] and is defined as

2.1

where I₁ is the first invariant of the Cauchy–Green deformation tensor. The adipose tissue (fat) had a larger stiffness than the isotropic muscle base material (figure 2) for the lean muscle (eqn 8 in [22]). However, fat stiffness was less than the passive response of the whole muscle tissue that includes both the base-material and fibre properties (figure 2). For the case of X% intracellular fat infiltration, the ischoric (volume perserving) component of the strain energy W_iso (eqn 7 in [22]) for fatty muscle can be written as

2.2

Figure 2.

Stress–stretch curves for fat (dotted), base muscle (dashed) and whole muscle (base + fibres; solid) materials used in simulations. The slope of the curves at each stretch is a representative of tissue stiffness where the combined muscle tissue has a greater stiffness than the fat or muscle base materials.

Here, W_muscle is the muscle along-fibre strain-energy and W_base,muscle is the base muscle strain-energy. Also, whenever assuming an X% loss of contractile elements of the fibres, the W_muscle component of isochoric strain energy was reduced by a factor of 1 − X/100.

The bulk modulus, κ, for the volumetric part of the strain energy W_vol (eqn 5 in [22]) was chosen to be κ_fat = 0.25 × 10⁶ for fat. This was based on the fat compressibility properties used in modelling the human heel pad [32]. κ_fat had a smaller value than the muscle tissue (κ_muscle = 1.0 × 10⁶) indicating that it is more compressible, nonetheless in the most compressible case (20% fat) the total volume change of the tissue was less than 0.5% for 20% activation. In this study, the volumetric part of the strain energy for X% intracellular fat accumulation had the form of

2.3

where J is the determinant of the deformation gradient tensor and represents dilation [22]. The deep and superficial aponeuroses were assumed to have the same material properties in these simulations and had a stiffness level that was either compliant, normal or stiff. Similar to our previous work [22], the compliant, normal and stiff aponeurosis were designed to have 10, 5 and 2% strains when loaded to a force equivalent to the maximum isometric force of the muscle.

2.3. Distributions (model variants) and intensities of intramuscular fat accumulation

The properties of the transversely isotropic muscle tissue were changed in six model variants. Lean muscle (M1–M2) had no fat in the muscle base material or between the fibres. For the other four model variants, an X% accumulation was introduced into muscle, where the effects of fat were simulated differently. Variations of the model are: (M1) lean muscle (no fat) with 100% AFPs in muscle fibres; (M2) lean muscle with X% reduction in AFPs; (M3) muscle with X% fat in the base muscle material and 100% AFPs; (M4) muscle with X% of fat in the base muscle material and X% reduction in AFPs; (M5) muscle with a random and sparse distribution of X% pure fat (W_iso = W_fat, κ = κ_fat) at the integration points dispersed within the lean muscle tissue; and (M6) muscle with a random and sparse distribution of X/2% pure fat points, X/2% of fat in the base muscle material and an X/2% reduction in AFPs. M1 represented a control condition for lean muscle. M2 was a lean muscle with a loss of AFPs, M3 and M4 were models with intracellular fat, M5 represented extracellular fat and M6 contained a combination of intracellular and extracellular fat. As the volume of intramuscular fat increased, models M2 and M4–M6 had a reduced number of AFPs to reflect the reduced volume available for contractile elements. The different variations of the model are summarized in table 1. The sparse distribution of fat in the M5 and M6 models were chosen such that fat was not contained in adjacent integration points, and we assumed that the sparse distribution has a negligible effect on the fibre orientations in the belly. The change in the contractile capacity of elderly muscles measured by different methods such as the interpolated twitch technique (e.g. [10]) was the rationale to change the AFP.

Table 1.

The model variants for X% fat infiltration in the muscle. Fatty variants (M3–M6) represent possible intramyocellular (IMC) and extramyocellular (EMC) fat distributions.

model	base muscle properties (%)	fat properties (%)	contractile elements (%)	fat distribution
M1	100	—	100	—
M2	100	—	100 − X	—
M3	100 − X	X	100	IMC
M4	100 − X	X	100 − X	IMC
M5	100 (muscle points)	100 (for X fatty points)	100 (muscle points)	EMC
M6	100 − X/2 (muscle points)	100 (for X/2 fatty points)	100 − X/2 (muscle points)	IMC and EMC

Three levels of fat were used for the models M3–M6, having 2, 10 or 20%. The 2% fat represents a healthy condition, with higher levels reflecting the incereased intramuscular fat in the elderly and obese. As an example, figure 3 shows a muscle with 15° pennation which is a M5 variant at a 20% fat level.

Figure 3.

A muscle belly geometry with 15° pennation angle and 20% sparse fat distribution (M5 variant). The dots show the positions of the integration points with aponeuroses (grey), muscle (red) and fat (yellow) properties.

2.4. Calculated parameters and analysis method

The resultant force (F) at the muscle–tendon junction, mean pennation angle relative to aponeuroses and mean fibre length were calculated to assess structural changes and performance of the muscle in the simulated scenarios.

To accommodate the effect of different initial pennation on the force that muscles with different PCSA can develop, the force (F) was normalized by the PCSA of the muscle to give the specific force. Here, the PCSA is defined as

2.4

where β, V_muscle, l_fibre, l_belly and w are the initial values for pennation, muscle tissue volume, fibre length (65 mm), muscle belly length (273 mm) and muscle width (55 mm), respectively.

2.5. Data analyses

We ran 10 iterations of the M5 randomized distribution for a particular combination of the other three factors, namely 15° pennation, 10% fat and normal aponeuroses stiffness. At 20% activity level, the coefficients of variation (standard deviation/mean) for the force, specific force, fibre stress, final fibre length and pennation were 0.2%, 0.2%, 0.2%, 0.01% and 0.03%, respectively. As the values for the coefficients of variation were small for the randomized variants of the model (M5–M6), we used the result of only one instance of each combination of randomized model variants, fat level, pennation and aponeuroses stiffness.

The effects of fat level (X), model variant (M1–M6), initial pennation (β) and aponeurosis stiffness (k) (factors) on the force, specific force, fibre stress, final fibre length and final pennation (response variables) were compared by their least-square means (adjusted means) of the deterministic muscle model responses (JMP v. 11.0, SAS Institute Inc., Cary, NC, USA).

2.6. Fat clump simulation

A further model was simulated that included 10% fat as a concentrated clump inside the muscle belly. The clump of fat in the muscle belly was a tube extending for 16 elements along the length of the belly and had a symmetric and polygonal cross-sectional area (figure 4a–c). Fibre orientations at integration points up to two elements from the fatty clump were changed so that the fibres curve smoothly around the fatty clump. This simulation was run with a 15° pennation muscle belly and normal aponeuroses stiffness. It was similar to M5 apart from the fat being clumped into the centre of the muscle belly and minor deviations to the neighbouring muscle fibre directions. This simulation allows for a comparison of extreme distributions between clump and completely sparse (M5) distributions of extracellular fat.

Figure 4.

The clump fat simulation. The integration points for a 15° muscle geometry (a) with cutting planes corresponding to transverse (b) and longitudinal (c) sections of the muscle. The muscle points are shown in red, fat points are in yellow and aponeurosis points are shown in grey. The deformed shape of the muscle belly at 20% activity (d) is coloured with a contour showing the magnitude of the displacement of the integration points. Comparison of the muscle belly force between the clumped-fat simulation, the lean variants M1–M2 and variant M5 that had a sparse distribution of extracellular fat for simulations with the same initial geometry and connective tissue properties, and X = 10 (e).

3. Results

The simulations were for an isometrically contracting muscle belly. As the activation level increased, the fibres shortened and expanded in their transverse direction, rotating to greater pennation angles and causing the aponeurosis to stretch as it became loaded. Simulations were intended to continue until the muscle became fully active and that happened in many of the simulations. Unfortunately, across the full range of input parameters (pennation, fat percentile, model variants and aponeuroses stiffness), there were simulations that were not computationally stable up to full activation (figure 5). Therefore, in order to compare the simulation results at a common level of activity, the highest common activity of 20% was selected (figure 6). The fibre stresses at 20% activation reached up to 17% of the maximum isometric stress of 200 kPa, but were reduced in cases of low initial pennation, reduced aponeurosis stiffness and increased fat accumulation.

Figure 5.

Force–activation plots for the different variants M1–M6. Lines show variant M1 (black circles), M2 (red diamonds), M3 (blue squares), M4 (green triangles), M5 (purple inverted triangles) and M6 (orange stars) at 2% (a), 10% (b) and 20% (c) fat levels.

Figure 6.

Main effects of the fat level, model variant, pennation and aponeurosis stiffness on the final pennation, muscle fibre length, stress and force. Points show the least-squares means, with their standard errors.

The increased initial pennation of the muscle was a major factor for greater muscle force. The geometries with higher pennation angle, and therefore larger PCSA, developed higher forces at each level of activity. For instance, the mean force for 10° pennated muscle geometry was 41% less than the 15° pennated geometry. When the effect of increased PCSA was removed, by calculating the specific force (force/PSCA) of the whole muscle and the stress of the muscle fibres, it was seen that changes in specific force and fibre stress showed similar patterns to the changes in muscle force (figure 6). Therefore, despite normalizing the force by the PCSA, the effects of pennation change still persisted on the specific force and fibre stress. The extent of fibre rotation and shortening as well as the muscle belly force depended on the aponeurosis stiffness. A stiffer aponeurosis resulted in smaller rotation and shortening of the fibres and an increase in the force and stress (figure 6).

There was an effect of the model variant (fat and muscle distribution) used in the simulations on the muscle force and stress of the muscle fibres. However, there was no effect of the model variants on the final pennation angle of the muscle fibres. A reduction in the along-fibres components showed a decrease in the belly force: for example, the M2 variant with 15° pennation muscle, normal aponeurosis stiffness and a 10% reduction in AFPs (X = 10%) showed an 11.3% decline in force compared with the lean M1 variant (figure 4e) at 20% activity. The longer fibre lengths for the fatty models would predispose them to greater forces due to their force–length properties [22]; however, the force and stress were reduced due to the intramuscular fat despite this effect (figures 4–5). For example, a 15° pennation muscle with normal aponeuroses stiffness showed an average of 25% and 45% decrease in force for 10% and 20% of fat accumulation, respectively. Despite the substantial effect of fat on muscle force, specific force, fibre stress and final fibre length, there was no effect of the percentage of fat on the final fibre pennation.

The simulation with 10% fat clumped in the centre of the muscle belly showed a lower force (60.1 N) compared with the lean variants M1–M2 (80.4 N and 72.2 N, respectively); however, the force from the clumped-fat simulation was greater than for the M5 variant (48.8 N) that had 10% extracellular fat distributed across the muscle belly (figure 4d).

4. Discussion

Fat accumulation in skeletal muscles is a common phenomenon in ageing and obese populations. Studying the effect of fat infiltration on the mechanical performance of human skeletal muscles is an experimental challenge as muscle forces cannot be measured directly. In addition, it is impossible to experimentally manipulate factors such as connective tissue stiffness, fibre pennation and the percentage and distribution of fat that affect muscle performance in the elderly and obese populations. In this work, we used a model to uncouple the effect of such factors on muscle belly force output. This study focused on the human plantarflexor gastrocnemii muscle group as a major contributor to human balance and locomotion.

The purpose of this study was to evaluate the general effects of muscle geometry and composition on the contractile force from the whole muscle. In order to test these ideas, abstract geometries were created, and these allow the fundamental concepts to be tested without being clouded by subject-specific idiosyncracies that would occur between individuals. It should be remembered that the actual force generated by muscle within the body would additionally be influenced by subject-specific geometries, activation patterns, muscle fibre properties and interactions with adjacent tissues.

Skeletal muscle models have previously been used to study the effects of ageing and obesity on human locomotion. Thelen in 2003 [33] introduced a framework for comparing young and elderly dorsi- and plantarflexors performance during isometric and isokinetic contractions. He showed that elderly muscle with 30% decrease in maximum isometric strength, 20% decline in maximum contraction velocity and an increased deactivation rate of 20% compared with young models had about 40% or more decline in ankle torque and power. In another study [34], a decline in maximum contraction velocity and maximum isometric force, an increase in muscle stiffness and altered shape of force–length curve were predicted when mechanical properties of the elderly muscle was estimated using an inverse dynamics optimization technique. In the case of the obese population, a recent modelling study [35] estimated an increase in gastrocnemii force and a decrease in vasti muscle group force in altered gait patterns of obese people. Despite the similarities of our results, such as increased muscle tissue stiffness due to fat accumulation and increased gastrocnemii muscle force in obese muscle with larger PCSA, the previous modelling studies addressing muscle performance in ageing and obesity used point-to-point muscle models that had no base-material representation, and this limits the study of muscle structural parameters and the effect of fat accumulation. In previous three-dimensional finite-element modelling frameworks for active skeletal muscle [24–26], the heterogenetic effects of fat accumulation have not been considered. However, Hodgson et al. [36] used a finite-element model to show that increases to the stiffness of the base material resulted in decreased muscle force. This model was essentially two-dimensional and unable to account for the transverse bulging of the fibres that is known to occur [27], and the base material was not modelled using known fat properties; however, they parallel our modelling results.

The simulations tested three different muscle geometries that varied by their pennation. The geometries with higher initial pennation had more muscle fibres acting in parallel, and thus they had greater PCSAs. It is known that muscle force increases with PCSA [1,37], and indeed the models with higher PCSA generated greater force (figure 6). The force was normalized by the PCSA to result in the specific force, and this is similar to the term ‘muscle quality’ that is the force a muscle can produce per unit size [38,39]. The specific force showed changes to the simulation parameters mirrored the changes in absolute force (figure 6), although at a lower magnitude, and indeed these patterns were also reflected at the level of fibre stress. Thus, the inclusion of intramuscular fat and changes to aponeurosis stiffness changed the muscle quality independent of the effect of muscle size or PCSA. Thus, intramuscular fat and aponeurosis stiffness are important factors that affect the contractile performance of muscle. The fatty models (M3–M6) generated specific forces that were lower than for the lean models (M1–M2). However, the fatty models generated these forces at longer fibre lengths (figure 6). The fibre lengths for the fatty models were closer to their optimum length of 65 mm (figure 1; [22]). Due to the force–length properties of the contractile elements [40], these longer fibre lengths would predispose the fatty models to generate higher forces, but this was not the case. Thus, the fatty models generated lower specific forces despite, and not because of, their longer fibre lengths.

A reduction in the number of contractile components within the muscle caused a decrease in the muscle force, and this can be seen in the change from M1 to M2 (figure 6). However, not only does the muscle force depend on the contractile components but also on the nature of the interaction between the contractile components and the base material within the muscle. Fat has stiffer material properties than muscle [31,32], and thus the introduction of fat into the muscle resulted in a stiffer base material, and this increase in stiffness would act to resist muscle fibre shortening and transverse bulging. The fatty models all showed lower specific force than the lean models, even for equivalent levels of contractile components (figure 6), and this is due to the increase in stiffness of the base material due to the inclusion of fat. These results support experimental findings that report a loss of muscle quality in both the elderly [38,39] and obese [41], and show that one of the causes for such a loss in contractile performance is the increase in muscle belly tissue stiffness as the concentration of intramuscular fat increases. While the source of the increased tissue stiffness was modelled due to the increase in adiposity of the tissue, connective tissue stiffness may also increase within sarcopenic muscle [20], and this would likely additionally contribute to a decrease in muscle quality.

The simulation with a single clump of fat (figure 4) within the muscle belly showed a reduced force compared with the lean muscle, but the reduction was not as pronounced as when the same amount of fat was dispersed throughout the muscle, as in model M5. The clump of fat acted to separate the medial and lateral aspects of the muscle within the middle of the muscle belly: if the muscle was totally divided into two halves, then it would be expected that the force from each half would be half the value for the lean muscle, and that the two halves combined would then be the same as for the lean condition, but this was not the case. This result shows that the distribution of fat through the muscle will alter the muscle force. The actual distribution of fat would likely be somewhere between the extremes that we have tested here: between a fine but uniform distribution to a single clump containing the entire amount of intramuscular fat. There are currently few data to inform the exact three-dimensional nature of fat distribution within skeletal muscle, and for this reason we did not focus on testing a range of possible intermediate fat distributions. It will be important to experimentally quantify fat distributions in different populations if we are to fully understand the impact of intramuscular fat on the contractile mechanics of muscle.

The simulations show that muscles with more compliant aponeuroses generate lower forces (figure 6). For the more compliant aponeuroses, the aponeurosis would stretch more allowing the fibres to rotate to greater pennation angles and shorten to shorter lengths, and both these effects would result in a reduction in the force that would be developed for each specific muscle geometry (figure 6; [40]). It is not totally clear if there are general changes to the aponeurosis stiffness as we age, although the consensus would suggest that aponeurosis stiffness is reduced in the elderly [16–18]. Thus, this effect of increased aponeurosis compliance causing reductions in muscle force may be a contributing factor to the reduction in muscle forces that occur in the elderly.

In conclusion, a mathematical modelling framework was used to simulate the effect of intramuscular fat on muscle force, to predict its effect for the elderly and obese. Both the concentration of intramuscular fat, and the stiffness of the aponeusoses were shown to have an important effect on muscle fibre stress and whole muscle force. The effect is partly due to the increased stiffness of the base-material properties that affect the extent of fibre shortening, lateral expansion of the fibres, and thus their interaction with the aponeuroses. The simulations in this study (M1–M6) were for muscle with uniform distributions of activity and intramuscular fat. It should be remembered that muscle force additionally depends on regional variations in muscle activity [22,42], fat distribution (figure 4) and fibre-type composition [43,44], and that muscle contribution to joint torque also depends on its moment arm that can vary with ageing and obesity [45,46]. Nonetheless, the results from this study show that the inclusion of intramuscular fat and the base-material properties of the muscle tissue have an important effect on muscle force.

Authors' contributions

H.R. developed the mathematical framework, designed and ran the simulations, analysed the data and drafted the manuscript. N.N. contributed to model development and design of the study and data analysis. J.W. conceived the study, designed the study, contributed to simulation set-ups, coordinated the study and helped draft the manuscript.

Competing interests

We have no competing interests.

Funding

We gratefully acknowledge funding from Natural Sciences and Engineering Research Council of Canada (N.N. and J.W.) and the Canada Research Chairs Program (N.N.).