Skeletal Muscle Adaptations and Passive Muscle Stiffness in Cerebral Palsy a Literature Review

1. Introduction

Cerebral palsy (CP) results from an upper motor neuron lesion and has a significant effect on the musculoskeletal organisation. It develops during early on childhood and leads to musculus alterations including contracture, which is the chronic shortening of a muscle. Contracture results in muscle that cannot be stretched through its typical range of motility due to an increase in stiffness, and this has substantial effects on the ability of muscle to generate forcefulness and reduces daily performance. Typically, this is observed in the upper and lower limb flexor muscles. CP volition touch individuals differently, and the changes that tin occur will vary depending on the location of the musculus and illness severity (de Bruin et al., 2014; Handsfield et al., 2016; Lieber and Fridén, 2019). This variability increases the difficulty in quantifying the amount and types of changes that occur as a result of CP. However, despite the variability, alterations in the microstructural properties of skeletal muscle are commonly observed (Tisha et al., 2019), which will take a pregnant consequence on whole musculus behaviour, including force production and movement.

There are many structural differences comparison CP muscle to typically developed (TD) muscle, including changes in fat content (Ohata et al., 2009; D'Souza et al., 2020), extracellular matrix (ECM) stiffness (Lieber et al., 2003; Smith et al., 2011), amount of ECM (Lieber et al., 2003; Smith et al., 2011), fascicle length (Mohagheghi et al., 2008), fibre diameter (Mathewson et al., 2014a), fibre geometry (Barrett and Lichtwark, 2010), and sarcomere length (Lieber and Fridén, 2002; Smith et al., 2011; Mathewson et al., 2014a). Experimental studies take investigated CP muscle stiffness in vivo and accept found stiffer tissue compared to TD muscle using shear moving ridge elastography (Brandenburg et al., 2016; Lee et al., 2016) and through measuring joint movement (Barber et al., 2011; van der Krogt et al., 2016). However, these methods are unable to capture the underlying causes of this increased stiffness.

The exact microstructural changes that alter whole muscle stiffness have withal to be fully understood, as the extent of measured changes varies between studies (Lieber and Fridén, 2019). For instance, Smith et al. (2011) performed passive mechanical experiments on both muscle fibre bundles and single fibres extracted from CP and TD muscle. They found that CP musculus had longer in vivo sarcomere lengths and increased fibre bundle, but not fibre, stiffness, which suggests that the changes in muscle stiffness are due to alterations in the ECM. Another written report past Mathewson et al. (2014a), which used a similar experimental protocol equally Smith et al. (2011), also showed increased in vivo sarcomere lengths. Nonetheless, the authors demonstrated a difference in the unmarried fibre stiffness and not the fibre bundles, suggesting that at that place is not a meaning outcome from the ECM, and that any alterations to passive stiffness occur on the muscle fibre level. Smith et al. (2011) performed studies on the Gracilis and Semitendinosus muscles, whereas, Mathewson et al. (2014a) looked at the Gastrocnemii and the Soleus muscles. Mathewson et al. (2014a) mention that the difference in the results is possibly due to the locations of the muscles or different mechanical properties of ECM betwixt TD and CP muscle. Some other possible explanation for the differences between the two studies is the that the TD groups in the study past Mathewson et al. (2014a) had a much older average age (47.vii±15.3) compared to Smith et al. (2011) (15.8±1.8). Other studies have reported that CP muscle has a greater accumulation of fibrotic tissue, and potentially even results in an ECM with a larger volume fraction simply compromised stiffness (Lieber et al., 2003). Additionally, Booth et al. (2001) suggested that collagen plays a role in the increased muscle stiffness that is observed in CP. Nevertheless, it has been observed that fibrosis does not e'er alter the stiffness of muscle (Smith and Barton, 2014), and so there may be an upshot from the stiffness and structure of the ECM. In TD muscle the ECM has been shown to exist a major correspondent to passive whole muscle mechanics due its composition of potent collagen fibres (Gillies and Lieber, 2011; Meyer and Lieber, 2018). In particular, piece of work has shown that in mammalian muscle, the ECM is responsible for well-nigh one-half of load bearing in passive tension (Meyer and Lieber, 2018). Sarcomere length is a commonly observed amending in CP muscle, and has been said to accept a large effect on active muscle mechanics (Lieber and Fridén, 2002; Smith et al., 2011; Mathewson et al., 2014b). While many changes have been observed in CP muscle, the changes that are near common between studies are changes in ECM volume fraction, ECM stiffness, and sarcomere length. However, the private roles of the ECM properties and sarcomere length in passive whole musculus stiffness have still to be fully understood.

The purpose of this study was to determine which microstructural change occurring with CP has the largest contribution to whole muscle stiffness. In particular, whether the ECM, through changes in volume fraction or stiffness, or the sarcomere, through increases in length that result in an increased passive response from the titin, will have the greatest influence on passive whole musculus behaviour. During experimental studies, it is not possible to change a component of muscle, while keeping all other components constant; therefore, causation cannot be adamant. This is specially difficult in CP where many components of muscle are known to vary between individuals (Lieber and Fridén, 2019; Tisha et al., 2019). In this report, we endeavor non to decide the cause of changes to the microstructure, merely the relationship between changes to the microstucture and the overall muscle stiffness. Using a modelling approach, nosotros investigated the influence of the microstructural components on passive muscle stiffness. We utilised a three dimensional continuum model of skeletal musculus, adult in previous studies (Rahemi et al., 2014; Ross et al., 2018b; Wakeling et al., 2020; Konno et al., 2021), which tin can be modified to contain the effects of ECM and passive fibre properties on whole muscle mechanics. Here we practise not explicitly model the procedure of contracture, simply instead the resulting changes to the microstructure. Any mechanical response we detect in this written report volition too exist relevant to muscle without clear contracture, but with similar changes to the material properties. By modifying the material properties in the musculus, we investigated changes that occur with CP to empathize how each component contributes to whole muscle stiffness.

ii. Methods

2.1. Computational Model

In this study, nosotros utilised a continuum mechanical model of muscle as a fibre-reinforced composite biomaterial. The model uses a iii field formulation in terms of the velocity, u, pressure level, p, and dilation, J, and the balance of strain-energy potentials based on piece of work past Simo and Taylor (1991) and Weiss et al. (1996). In detail, nosotros want to minimise the full strain-energy

$\begin{array}{ll}{\mathrm{East}}_{\text{tot}}(\text{u},p,J)=U_{\text{int}}(\text{u},p,J)-W_{\text{ext}}(\text{u}),& (1)\end{array}$

where U _int is the internal strain-energy potential and West _ext is the total external work done on the muscle. We characterised the passive mechanical behaviour of muscle in terms of its stress-strain behaviour, which is the relationship betwixt the stress applied to the musculus and the strain experienced. Here we use the full Cauchy stress tensors, so the shear stress response is built into this tensor implicitly and has been investigated in Konno et al. (2021). The relationship betwixt the stress in the fabric, σ, and the strain-energy potential of a fabric, U, is given by the constitutive law

where B is a strain tensor measuring the deformation. To capture muscle, we separate the model into a 3 dimensional isotropic base material with one dimensional fibres running along the length of the muscle, making the composite textile anisotropic. The total stress response from the muscle (σ_muscle) is then additive contributions from the base textile (σ_base) and fibre (σ_fibre) components

$\begin{array}{ll}{\sigma }_{\text{muscle}}={\sigma }_{\text{fibre}}+{\sigma }_{\text{base}}.& (3)\end{array}$

Our model is a homogenized muscle material that included contributions from the fibres and base cloth at every element in the mesh. A more than precise conception of the model is described in the appendix of Wakeling et al. (2020) and in Konno et al. (2021). We used a finite element method to solve the continuum model that was implemented using an open source finite element library deal.2 (Arndt et al., 2017).

There are 2 main components of our muscle model: the fibre components and the iii base material. The base material encompasses effects from extracellular matrix and cellular components, including satellite cells and capillaries, while the fibre component runs along the length of the muscle and contains the passive effects from titin and active furnishings from the contractile elements. To investigate the role of CP on whole muscle stiffness, because the private effects from the ECM and cellular components is necessary. To do this, we let α be the volume fraction of the ECM, which includes furnishings from the collagen fibre matrix (Figure i). Meanwhile, 1 − α is the volume fraction of the cellular component, which includes effects from any other material in the muscle aside from the contractile units. In particular the cellular material includes the myofibres, satellite cells, and other cellular materials. We as well introduced a parameter s _ECM (Effigy 1), which is a stiffness factor multiplying the stiffness of ECM. A larger s _ECM corresponds to a stiffer ECM, which can occur equally a result of structural changes while the ECM book fraction remains the aforementioned (Gillies et al., 2016). Meanwhile, a smaller value of s _ECM results in an ECM with decreased stiffness, which can occur every bit a result of an ECM with compromised construction (Lieber et al., 2003). The total stress response from the base material is a homogenization of the ECM and cellular components given by

$\begin{array}{ll}{\sigma }_{\text{base}}(\text{B})=\alpha {\mathrm{south}}_{\text{ECM}}{\sigma }_{\text{ECM}}(\text{B})+(1-\alpha ){\sigma }_{\text{Cell}}(\text{B}).& (4)\end{array}$

σ_ECM(B) corresponds to the stress response from the ECM, while σ_CELL(B) is the stress response from the cellular component. To ensure the volume of the muscle remains about constant, the majority modulus for the cellular component is chosen to be 1 × x^vii Pa, while the bulk modulus of the ECM component was ready to ane × 10⁶ Pa (Konno et al., 2021). While the cellular component of the musculus contributes less to the overall stress response, information technology consists largely of h2o, so its majority modulus is gear up to be larger than the ECM component. The exact grade of the stress-strain response for the microstructural components are given in Konno et al. (2021) along with more detail on the homogenized base material.

Effigy 1. Comparing of typically developed (TD) and cerebral palsy (CP) muscle. CP results in contracture, which is the chronic shortening of muscle, decreasing muscle length relative to TD muscle (not investigated in this study). Longer sarcomere lengths relative to the rest of the muscle have also been observed compared to TD muscle (Smith et al., 2011; Mathewson et al., 2014b). At that place is an increase in passive forces due to the increased stretch in the titin proteins. The longer sarcomeres lengths in muscle affected by CP reduces the regions of overlap of the actin and myosin filaments, which results in decreased contractile forces. Additionally, there is an increase in extracellular matrix (ECM) volume fraction, and a possible increase or subtract in ECM stiffness. Any combination of changes to the sarcomere length or ECM properties could occur with CP.

The other alteration typically observed in CP is an increase in in vivo sarcomere length (Smith et al., 2011; Mathewson et al., 2014b), which alters the passive musculus stiffness past stretching the titin protein. Experimentally, the sarcomere lengths for muscles in the lower extremities were measured with ninety degrees hip and knee flexion (Smith et al., 2011; Mathewson et al., 2014a), and with the talocrural joint in full dorsiflexion for the lower leg muscles (Mathewson et al., 2014a). Our model does not have a joint angle, so we define the in vivo length of the sarcomeres to be the length of the sarcomeres when the whole muscle is at its resting length. This alter in length also decreases the contractile forcefulness produced when the musculus is active by reducing the number of attached actin-myosin crossbridges (Figure 1). We modelled this using a dimensionless parameter, c _sarco, which corresponds to a shift in the passive force-length curve of the sarcomeres

$\begin{array}{ll}{\sigma }_{\text{fibre}}={\sigma }_{\text{fibre}}({\overline{\lambda }}_{\text{tot}}+c_{\text{sarco}}),& (\mathrm{five})\end{array}$

where ${\overline{\lambda }}_{\mathrm{\text{tot}}}$ is the total boilerplate stretch of the fibres over the muscle volume. This parameter acts as an additive contribution to the intrinsic stress-stretch human relationship, resulting in larger stresses in the fibres at a given stretch (Figure 2). It is important to note that, while the fibre component of the muscle depends on c _sarco, the intrinsic stress response from the base textile, σ_base(B), just depends on the deformation and stretch of the muscle, and non c _sarco. At a value of c _sarco = 0.0, the behaviour of the sarcomere is the same as that of TD muscle. Increasing values of c _sarco results in longer lengths of the sarcomeres given past

$\begin{array}{ll}{\mathrm{fifty}}_{\text{sarco}}=l_0({\overline{\lambda }}_{\text{tot}}+c_{\text{sarco}}),& (6)\end{array}$

where fifty _sarco is the new length of the sarcomere and we assume l ₀ = 2.2μm is the optimal length of a sarcomere (Burkholder and Lieber, 2001). This volition vary depending on the value of c _sarco. The fibres in the model are based off the one dimensional Loma type model (Hill, 1938; Zajac, 1989) and are described in Wakeling et al. (2020). It is possible the behaviour of the fibre will vary in more than a shift of its force-length bend, such equally a different force-length relationship, which may be due to factors in the myofibres other than the sarcomeres. Still, equally this is not observed often and information are limited, nosotros focus on the effects from the sarcomere length.

Figure ii. The influence of c _sarco on the intrinsic fibre stress-stretch relationship. Hither we plot the stress from the fibre component of our model against the stretch in the fibres. The curve respective to typically developed musculus (c _sarco = 0) was obtained based on curves and data in Ross et al. (2018a). This was washed through trigonometric polynomial and 2nd-gild piecewise polynomial fits to experimental data from Winters et al. (2011) (denoted by the dots).

ii.2. Whole Muscle Experiments

To investigate the passive effects of α, due south _ECM, and c _sarco on skeletal musculus, we synthetic a rectangular block of muscle with dimensions iii cm × 1 cm × 1 cm. These dimensions, while non the aforementioned as muscle affected by CP, sufficiently capture the behaviour of muscle on a macroscopic scale. Using a block geometry reduces the demand to consider the additional effects from architecture, aponeurosis, and pennation bending, which affects muscle behaviour (Wakeling et al., 2020), and instead immune us to isolate the furnishings due to CP independent of a specific architecture. Additionally, this geometry has been previously validated to capture the full general qualitative behaviour of muscle when compared to the mechanics within a MRI derived whole muscle geometry (Wakeling et al., 2020) and has the benefit of computational simplicity. To compare the passive behaviour of the model to experimental data, we performed stress-strain tests. This involved constraining one end face of the model from movement in all directions, while a normal stress was applied to the opposite face stretching the muscle. In addition to the stress-strain tests, we investigated the stiffness of the muscle to compare with experimental studies (east.g., Smith et al. (2011); Mathewson et al. (2014b)). α, s _ECM, and c _sarco each have an individual contribution to the overall stiffness of muscle. To investigate the stiffness in the model, the modulus (in Pa) of the muscle fabric was calculated during the stress-strain experiments using the slope of the tangent line to the overall stress-stretch relationship. This was washed by performing a nonlinear to the lowest degree-squares fit of a cubic polynomial to the overall stress-strain data in the longitudinal management. This method for calculating the modulus is just representative of the stiffness at the given stretch value, since the stress-stretch curves are nonlinear; all the same, we exercise this to compare with experimental studies.

TD muscle has been observed to have a value for α between 0.02 and 0.10 (Binder-Markey et al., 2020), while larger volume fractions (α≈0.6) have been observed for fibrotic tissue (Lieber et al., 2003; Smith and Barton, 2014). Experimental studies have only constitute an increase in in vivo sarcomere lengths (Lieber and Fridén, 2002; Smith et al., 2011; Mathewson et al., 2014a), so nosotros varied c _sarco from 0 to 0.75. This corresponded to a 0–75 % increment in the sarcomeres relative to the sarcomeres in typically developed musculus, which has been observed in the literature (Lieber and Fridén, 2002; Smith et al., 2011; Mathewson et al., 2014b). Experiments are often performed on severe cases of CP, and so larger sarcomere lengths have been reported (c _sarco > 0.75); notwithstanding, in this written report, we considered less severely stretched sarcomeres to stand for less severe cases of CP. It is possible that muscle altered past CP does not always have such a substantial increase in sarcomere length, since measurements are typically taken from children with severe CP undergoing surgery (Lieber and Fridén, 2019). The last parameter that was manipulated in the model is s _ECM. While the stiffness of muscle can vary depending on the blazon of muscle, the properties of the base material, including the effects from the ECM, represent and have been validated for TD musculus (Konno et al., 2021), so we prepare southward _ECM = i for TD muscle (this corresponds to a value of 150 in Konno et al., 2021). During our stress-strain tests, we and so considered the possibility of the ECM component of musculus being stiffer (s _ECM = 1.33) and less stiff (s _ECM = 0.66). Note that a s _ECM value of 1.33 corresponds to a stiffness of 133 % compared to TD muscle, while a value 0.66 corresponds to a stiffness of 66 % relative to TD muscle. Data for the changes in stiffness of the ECM are not available experimentally, so these values of south _ECM were chosen to investigate the effects of altering this component. In summary, to investigate the effects of CP, our parameter ranges were α = 0.02 to 0.6, c _sarco = 0.0 to 0.75, and s _ECM = 0.66 to i.33, while TD musculus had parameters α = 0.05, c _sarco = 0.0, and s _ECM = 1.0.

iii. Results

3.ane. Effects of c _sarco

For TD musculus (c _sarco = 0.0, south _ECM = i.0, and α = 0.05), we observed typical overall stress-stretch behaviour for passive skeletal muscle: as the stress increased, the muscle stretch and sarcomere lengths besides increased (Figure 3). The muscle block in its resting and stretched states is shown in Figures 3C,D, respectively. The shift in the intrinsic passive sarcomere stress-stretch relationship, c _sarco, affected the musculus behaviour in both the stress-length for the sarcomeres and overall stress-stretch for the whole muscle relationships of the model (Figure three). At in vivo lengths, we constitute that there is no longer zero stress for c _sarco > 0.0 (Figure 3A). This indicates that larger forces are required to stretch muscles with increased sarcomere lengths, too as to concord it at the resting length of the muscle. We also see that there is a nonlinear relationship between the consequence of c _sarco on the fibre component (Figure 2) and the overall muscle response which is influenced by the base material (Figure three). Additionally, optimal length of the sarcomeres no longer occurred at the aforementioned resting length of the whole muscle. For the same range of stress values (0–3 × ten^v Pa), we saw a larger range of whole muscle stretches with larger c _sarco (Figure 3). This is probable due to effects from the base fabric and, therefore, the ECM, which acts to deform the muscle back to optimal length. At stretch values less than 1.0, the base material works to extend the musculus to optimal length, while the sarcomeres are still working to shorten the muscle for c _sarco > 0.0.

Figure 3. Plots of whole musculus stress in the along fibre management confronting sarcomere length (A) and whole muscle stretch (B). Plots are from the computational model during passive lengthening with ECM volume fraction, α, of 0.05 and ECM stiffness gene, s _ECM, of i.0. Each curve represents a shift in the sarcomere stretch by a factor of c _sarco. * represents in vivo sarcomere length for respective c _sarco. (C,D) show the mesh at resting length and at a deformed land after the stress has been applied to the model.

In addition to the behaviour in the forth-fibre management of the muscle, c _sarco also affects the behaviour transverse to the musculus fibres (Figure 4). Nosotros observed a like modify in concavity of the stress-stretch curves in both the stress-stretch relationships in the longitudinal (Figure 3B) and transverse (Figure 4) directions. This demonstrates similar effects from the muscle ECM component in both directions. For stretch values in the y direction less than 0.85, the influence of the sarcomeres on the stress-stretch relationship decreased, and there was larger influence from the ECM. For smaller normal stresses in the longitudinal directions, there were larger effects from sarcomere length relative to larger stresses (Figure iv). The sarcomeres, acting merely in the forth-fibre direction, altered three dimensional deformation, which could impact muscle strength product (Wakeling et al., 2020).

Figure 4. Plot of the normal stress applied in the along-fibre direction (10) confronting the stretch in the muscle transverse to the fibres. Given the symmetry in the muscle geometry transverse to the fibres (y), the stress-stretch response shown is the aforementioned in the z direction. Each line represents a shift in the sarcomere stretch by a factor of c _sarco.

3.2. Effects of α and s _ECM

The ECM backdrop also had a substantial effect on the overall stress-stretch relationships (Figure 5). Given the range of possible values for the ECM volume fraction, α, (0.02–0.six), it had a larger consequence on the muscle stiffness compared to the ECM stiffness parameter, s _ECM. As southward _ECM was increased, with fixed α = 0.05 or 0.6, the overall stress-stretch human relationship became more linear and covered a smaller range of stretch values (Figure 5). Increases in both α and due south _ECM reduced the effect from the sarcomere length on the stress-stretch relationship (Figure 5). Due to the lack of available data for the stiffness of the ECM, the s _ECM was only varied betwixt 0.66 and 1.33, which is a relatively modest range compared to the volume fraction of the ECM. Similar effects were observed from changing s _ECM and α, but α had more effect on the overall stress-stretch behaviour. This is expected given the change in limerick of the base material; withal, due to the express information for s _ECM, it is possible that real muscle has a larger range of values than investigated in this study, but to ensure that we remain within realistic physiological ranges nosotros chose this limited range. Our results showed that southward _ECM had very little consequence on the overall stress-stress relationship for smaller values of α and it has the most effect when α is big. The main effects from the south _ECM are in altering the influence of c _sarco on the overall muscle stress-stretch relationship (Effigy v). Varying southward _ECM over a larger range of values would likely but influence the upshot of the sarcomere length on the overall stress-stretch response.

Figure 5. Stress-stretch plot during passive lengthening of the muscle model for various values of ECM book fraction, α, and stiffness, s _ECM. The traction was linearly increased on the +x face up of the muscle to 3 × 10^v Pa, while the −ten face was constrained in all directions. Private lines on each plot stand for a shift in sarcomere stretch by c _sarco. Typically adult (TD) muscle has values of α = 0.05, southward _ECM = 1.0, and c _sarco = 0.0, while cognitive palsy muscle could take a combination of α > 0.1, s _ECM = 0.66 or 1.33, and c _sarco > 0.

three.3. Musculus Stiffness

As c _sarco was increased upwards to a value of 0.v, the modulus of the muscle in the x direction increased at optimal length of the muscle (Figures 6A,B). However, after a value of 0.v, the muscle modulus decreased, and this was observed when looking at the variations in c _sarco with abiding α and s _ECM (Figures 6A,B). The modulus at optimal length was dominated past α. For changes in c _sarco and s _ECM, the change in modulus (at most four × 10⁵ Pa) was less than the possible variation in modulus with changes in α (upwardly to 15 × 10⁵ Pa for highly fibrotic tissue). While holding c _sarco constant, we found a linear increase in the modulus when increasing α and s _ECM (Figures 6C,D).

Figure 6. Plot of whole muscle modulus vs. c _sarco (A,B), α (C), and s _ECM (D) at optimal length (${\stackrel{-}{\lambda }}_{\mathrm{\text{tot}}}=\mathrm{one}.0$). Where α is the ECM book fraction, s _ECM is the ECM stiffness factor, and c _sarco is the shift in the sarcomere stretch. In (A,D) α is held abiding at 0.05, and in (B,C) due south _ECM is held constant at 1.0.

While belongings c _sarco constant, there was a larger consequence from volume fraction of the ECM, α, than the stiffness of the ECM, due south _ECM, on the overall muscle stiffness (Figures 7A,B). However, as α was increased, in that location was a greater effect of s _ECM. The nonlinear behaviour, which showed increasing muscle modulus with increasing α and s _ECM was more than pronounced at larger stretches (Figures 7B,D). At a stretch of 1.20 in the x direction, the stiffness appeared to be more nonlinear when moving along the lines of constant s _ECM and when moving along the lines of constant α for α > 0.2 (Figures 7C,D). When property s _ECM abiding, there was a larger effect α on the modulus of the muscle compared to c _sarco, peculiarly at larger stretch values (Figures 7C,D). As the stretch increased at that place was an increase in modulus from the ECM parameters; withal, there was a decrease in the furnishings of c _sarco (Figures 7E,F). The reduced influence of c _sarco was due to more pronounced behaviour from the base fabric at larger stretches (Figures five, vii).

Figure seven. Surface plot of the whole muscle modulus at an average musculus stretch, ${\overline{\lambda }}_{\mathrm{\text{tot}}}$, of i.0 (A,C,E) and 1.two (B,D,F). The ECM stiffness factor, south _ECM, was varied between values of 0.66 and 1.33; ECM volume fraction, α, from 0.02 to 0.6; and shift in sarcomere stretch, c _sarco, from 0.0 to 0.75. Modulus values were extracted from passive lengthening simulations with c _sarco = 0.0 in (A,B); southward _ECM = ane.0 in (C,D); and α = 0.05 in (E,F).

4. Word

In CP, alterations occur on the microstructural level that can influence whole muscle stiffness and reduce function. In particular, alterations to ECM backdrop and sarcomere length can occur; however, their relative contributions to muscle stiffness in CP is unknown. Isolating private effects on passive musculus stiffness is difficult to do in experimental studies as there is large variability between subjects and private muscles (Calvo et al., 2010; Takaza et al., 2013; Mohammadkhah et al., 2016). Therefore, to determine whether the ECM backdrop or sarcomere lengths have more effect on the passive muscle behaviour, nosotros used a three dimensional continuum model (Rahemi et al., 2014; Ross et al., 2018b; Wakeling et al., 2020; Konno et al., 2021). This model does not actually develop articulation contractures; still, it allows us to isolate the effects from individual microscopic components, and investigate the relative contributions to whole muscle function.

4.one. Physiological Changes to Musculus During Cerebral Palsy

The ECM is composed of a highly structured arrangement of collagen fibres and plays a substantial role in skeletal muscle mechanics (Gillies and Lieber, 2011). In this study, we investigated the effects of changes to ECM volume fraction and stiffness on the whole muscle stiffness. Changes in ECM volume fraction have been observed in previous studies, particularly as a consequence of fibrosis (eg. Smith et al. (2011)). This corresponds to an increase in ECM material, while the contributions from the cellular components in muscle, such equally the contractile fibres and other cells, decreases. In addition, fibrosis creates a physical barrier that can touch on muscle regeneration (Chen and Li, 2009), which will reduce the ability for musculus to grow and add sarcomeres in the musculus fibres, further decreasing the compliance of the musculus. Additionally, it is possible that while the volume fraction stays constant, changes in the structure or limerick of collagen types varies. However, studies accept institute that the ratio of collagen isoforms is the same in both TD and CP muscle (Smith et al., 2019), and so it is unlikely that a deviation in collagen isoforms in muscle accounts for the increase in whole muscle stiffness with CP. It is possible that at that place are alterations in ECM structure, such as the organisation of collagen fibres, that occur with CP (Lieber et al., 2003), and this could increment or decrease ECM stiffness depending on the change. Therefore, both furnishings were considered in the model to investigate the relative contributions to stiffness on a whole muscle level.

Information technology has likewise been well documented that increases in the sarcomere length occur with CP (Lieber and Fridén, 2019; Tisha et al., 2019). Stiffness changes have been reported on the fibre level by looking at the stress-strain human relationship for TD and CP musculus fibres (Mathewson et al., 2014a). It is possible that these effects are non just a result of increased sarcomere lengths, but due to dissimilar titin isoforms, every bit they could issue in a unlike stress-strain relationship for the individual sarcomeres (Prado et al., 2005). However, Smith et al. (2011) found that there is no modify in the composition of titin isoforms between TD and CP muscle. Therefore, changes in stiffness due to the sarcomeres are non likely due to changes in titin isoforms. While information technology is possible that an increased stretch of the titin is responsible for the increased passive stiffness of the fibres, this could likewise be a event of changes to other mechanical properties in the myofibres. More investigation is required to confirm the master cause of the increased stiffness at a fibre level.

4.2. Microstructural Contributions to Whole Musculus Behaviour

It has been demonstrated experimentally that the ECM has a pregnant contribution to muscle passive stiffness (Gillies and Lieber, 2011), and that fibrosis has been observed in CP (Lieber et al., 2003). We found that the book fraction of the ECM had a larger influence on whole muscle stiffness compared to ECM stiffness and sarcomere length. The contribution from the ECM increased as stretch increased (Figure 7), demonstrating a nonlinear relationship betwixt the ECM volume fraction and muscle stretch. At larger stretch values, the ECM contributes more to the whole muscle stiffness; these nonlinear effects imply that fibrosis will substantially reduce the ability of a muscle to deform at larger stretch values. The ECM is composed of crimped collagen fibres, which likely do not contribute equally much to the stress initially (Gillies and Lieber, 2011), and this is reflected in a smaller outcome from the ECM volume fraction at optimal length. Currently, experimental data for the variation in stiffness of the ECM due to structural changes are not available; even so, in the ranges tested in this written report, the stiffness of the ECM did non alter whole muscle stiffness equally much as the book fraction. At larger volume fractions of the ECM at that place was a more than substantial contribution from the ECM stiffness (Figure 7B), and since larger volume fractions are typically seen in CP, this could play a larger part.

The contribution of the sarcomere length to whole muscle stiffness varied depending on the ECM properties. There was minimal effect of the sarcomeres on the passive stiffness in the fibrotic tissue (Figure 6), which corresponds to book fractions of ECM greater than 10%, and this larger outcome from the ECM has been observed during experiments (Smith et al., 2011). Furthermore, the sarcomere effects are mitigated at larger stretches as the ECM begins to dominate the muscle stiffness. At whole musculus stretches well-nigh 1.0, we found a larger effect of sarcomere length (Figure 6), which agrees with the results from Mathewson et al. (2014a) for fibre bundles. The work by Mathewson et al. (2014a) indicates that there was no increase in the stiffness of the ECM during CP in the muscles investigated. They found that there was no difference in the stiffness of fibre bundles at larger stretches, which could be explained by the dominating behaviour of the ECM. This agrees with our findings that demonstrate that if a muscle is operating nearly optimal length, so there might exist a noticeable effect of sarcomere length. However, if the muscle has a larger range of movement, then the ECM would probable have a larger contribution to muscle passive stiffness. Information technology is likely that the lengthening of the sarcomeres during CP has a larger effect on the active backdrop of the muscle (which we have non evaluated in this study) compared to the passive properties as noted by Lieber and Fridén (2019).

Using this model we can obtain a deeper understanding of the three-dimensional furnishings that occur in muscle altered by CP. Equally shown in previous modelling (Ryan et al., 2020; Wakeling et al., 2020) and experimental studies (Randhawa and Wakeling, 2018), the power of a muscle to deform both in the along and transverse fibre directions can modify muscle office. Additionally, our results agree with experimental testify that the whole muscle response is not the same as the response from individual fibres (Ward et al., 2020). In our model, the three-dimensional behaviour is captured in part by the base material, which works to render the muscle to its original country. At longer muscle lengths, the base material will work in the aforementioned direction every bit the sarcomeres, which are trying to shorten the muscle. Meanwhile, when the whole muscle stretch is less than one, the ECM will be working to render the musculus dorsum to the original musculus length. We have observed in the model that the stiffness of the muscle decreases after sarcomere lengths greater than 3.3 μm (Effigy six), and this is due to the iii dimensional behaviour of the model we are using. In a ane dimensional model, there are no effects from the book conserving nature of the base material or other furnishings transverse to the fibres. This is a nearly incompressible and nonlinear model, and so the effects from the volumetric component of the model contribute more with larger shifts in the sarcomere force-length curve. While these furnishings have been observed based on our assumptions for the model (come across Wakeling et al., 2020; Konno et al., 2021), which are typical of many finite chemical element models (Blemker and Delp, 2005; Sharafi and Blemker, 2011; Spyrou et al., 2017), these effects take not been reported experimentally. Experimentally, the decrease in muscle stiffness may non be as substantial equally the changes observed in this study; notwithstanding information technology is probable a like trend would announced. Another of import event of the three-dimensional behaviour is that changes occurring strictly in the forth-fibre direction, such equally changes in the sarcomere length, affect the stretch transverse to the fibres (Figure 4). In particular, the bulging and stretching in the transverse management is decreased by increased in vivo sarcomere lengths, which increases the passive stiffness of the muscle fibres. Given this reduced movement in the transverse management, it is likely that there would be a decreased contractile forcefulness produced given the significant outcome of three dimensional deformation (Ryan et al., 2020). This demonstrates that to accurately capture all of the effects from CP, investigating 3 dimensional behaviour is required to completely understand the mechanical behaviour of the muscle.

4.3. Model Parameters

Experimental studies are cardinal to agreement the mechanical changes that occur with CP; however, many of the procedures are invasive and unable to determine the exact role each change due to CP plays in altering musculus stiffness (Smith et al., 2011; Lieber and Fridén, 2019; Tisha et al., 2019). Additionally, at that place is contradicting data as to whether fibres, ECM, or both have a substantial contribution to passive stiffness (Smith et al., 2011; Mathewson et al., 2014a), which likely depends on the severity of the disease (Tisha et al., 2019). At that place are less invasive procedures that accept been developed to investigate the human relationship between musculus stiffness and CP (Lee et al., 2016); however, they are withal unable to isolate the part of individual factors. For case, experimental studies have establish that stiffness of CP musculus is twice equally high as TD muscle (von Walden et al., 2017); however, they were not able to determine which microstructural changes led to this increase in stiffness. While this model cannot directly determine which microstructural changes will cause this experimental increase in stiffness, it tin provide insight into how diverse changes on the microscopic level could lead to these effects on muscle stiffness. We have observed that there is approximately double the increase in stiffness when the volume fraction of ECM in our model increases from 5 to twenty%. Some other possible way to achieve this increase in muscle stiffness is through increasing the stiffness of the ECM, or some combination of the 2. The possible changes that cause increased stiffness can exist investigated through our modelling arroyo and can indicate which factors may have the most bear upon on musculus behaviour. It is difficult to perform experimental tests on whole muscles affected past CP, although tests take been done on mice with spasticity or fibrosis (Ziv et al., 1984; Smith and Barton, 2014), equally muscle can only be dissected during surgery making it difficult to obtain data for an accurate comparison to similar TD muscle tissue. Muscle is a three-dimensional material, so applying a continuum model to CP muscle allows usa to understand the underlying muscle mechanics. In item, developing an understanding of the complete behaviour of muscle will give insight into the role each microstructural alteration that occurs in CP volition play in whole musculus mechanical behaviour.

While the model has the ability to investigate behaviour of musculus that is difficult to examine experimentally, it relies on accurate experimental data for its intrinsic backdrop. Unfortunately, mechanical data for the furnishings of stiffness of the ECM are not available, then the value for the ECM stiffness parameter was chosen to vary past 33% from healthy muscle. Information technology is possible that changes in the structure of the ECM would modify by more than this value; however, these values were chosen to probe the behaviour of the ECM stiffness. Given the derivation of the whole muscle stress in the model (Equation 4) it is likely that the volume fraction of the ECM would still have the largest contribution to whole musculus stiffness. Both the ECM volume fraction and stiffness multiply the ECM stress response, so they have similar contributions to whole muscle behaviour for small variations in their values. However, only the ECM volume fraction decreases the contributions from the cellular components. This attribute of the model is realistic, since it is not likely changes in the structure of the ECM volition decrease the contribution of the fibres to whole muscle stiffness.

4.4. Limitations and Future Directions

A limitation of this model is the lack of current experimental certainty on changes that occur with CP. Many changes to individual components take been observed in CP afflicted muscle; however, the extent to which microstructural changes occur are varied (Tisha et al., 2019). Therefore, the effectiveness of the model in providing a comparison to CP muscle will depend on the specific muscle. Additionally, there are very piddling data available for the changes in stiffness of the ECM, and so it is possible that this could vary more than than investigated in this study. This would effect in a large influence of the stiffness of the ECM component. Work by Brashear et al. Brashear et al. (2021) plant that stiffness and orientation of the ECM component may have more result than the amount of the ECM, so this lack of experimental data for ECM stiffness is a limitation of our model. Additionally, information technology is possible that a focalized accumulation of collagen de Bruin et al. (2014); Von Walden et al. (2018) could occur in contrast to the even distribution investigated in this written report. This would likely influence the mechanical response of muscle; however, this was not investigated in this study. The response of the base of operations material likely changes in response to compression every bit opposed to tension. Currently, the available data for the ECM is express to tension, and and then a different response for compression was not implemented in the model. However, we await the effect on the output from our model would be minimal.

In the model, we have causeless for simplicity that with changes in the volume fraction of the ECM, there is no effect on the corporeality of force produced by the fibres. Any reduction in contribution from the musculus fibres is assumed to exist included in the subtract in cellular component contribution to the base material response. In addition to changes in the microstucture, information technology is possible that changes occur to the geometry of the musculus in CP. The results of this written report but demonstrate the furnishings of the changes to the fabric properties, and the effect of changes to the geometry could exist investigated in time to come piece of work. Nosotros take not investigated the active behaviour of musculus in this study, although it is key in muscle function. In CP, the contractile force produced has been seen to decrease (Stackhouse et al., 2005), so using this model to investigate the influence of ECM and sarcomere properties on agile force would exist valuable and would requite additional insight into how the structural alterations that occur with CP individually touch on musculus contraction. In this model, the properties of our TD muscle may not be representative of all muscles every bit the material properties vary both across and within studies (Calvo et al., 2010; Takaza et al., 2013; Mohammadkhah et al., 2016). So, while the qualitative passive behaviour is captured in this model, the exact values could vary between muscles. However, nosotros would expect the full general trends observed during this study to hold.

five. Determination

The purpose of this written report was to make up one's mind the furnishings of the microstructural changes that are normally observed during experimental studies of CP musculus, including variation in ECM volume fraction, stiffness, and sarcomere length, on whole muscle stiffness. To practice this, a 3 dimensional computational model of skeletal musculus was used, and overall stress-stretch relationships and musculus stiffness were calculated by measuring the passive stress of the whole musculus. We found that the book fraction of the ECM had a larger issue on overall muscle stiffness compared to the ECM stiffness and sarcomere length, and that the furnishings of the sarcomere length were mitigated at larger ECM volume fractions. Investigating these effects provides a causal relationship between changes to microstructural properties and the overall response of the muscle. Experimental research is currently unable to vary independent components of muscle, and then this piece of work tin can be used to help directly future experimental research. In this report, we were able to decide the crucial role that the microstructural alterations observed in CP have on whole skeletal muscle behaviour.

Data Availability Argument

The original contributions presented in the study are included in the commodity/supplementary material, further inquiries can exist directed to the corresponding author.

Author Contributions

RK carried out experimental design, performed simulations, analyses of results, and drafted and edited the manuscript. NN, JW, and SR carried out experimental pattern, analyses of results, and editing of the manuscript. All authors gave final approval for publication and agree to exist held accountable for the work performed therein.

Funding

We would like to acknowledge funding from Natural Sciences and Technology Research Quango of Canada for Discovery Grants to NN and JW.

Conflict of Interest

The authors declare that the research was conducted in the absence of whatsoever commercial or fiscal relationships that could be construed as a potential conflict of interest.

Publisher'due south Annotation

All claims expressed in this commodity are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Whatsoever product that may be evaluated in this article, or claim that may be made by its manufacturer, is non guaranteed or endorsed by the publisher.

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