On a three-dimensional constitutive model for history effects in skeletal muscles

Abstract

An exceptional property of skeletal muscles that distinguishes them from other soft tissues is their ability to contract by generating active forces, which in turn are initiated by an electrochemical trigger. Some of these so-called active material properties are generally characterised using isometric contraction experiments at various muscle lengths. In this context, experimental observations revealed that unlike the widespread assumption in muscle modelling, reaction forces indeed depend on so-called history effects, which can be classified into force enhancement and force depression. For the experimental settings of force enhancement, two subsequent isometric contractions are interrupted by an isokinetic extension. The isometric reaction force is increased after the isokinetic extension with respect to a reference measurement, while in the case of force depression, isokinetic shortening is responsible for forces below a certain isometric reference measurement. Most theoretical investigations of force enhancement and force depression use one-dimensional models to simulate the force response considering muscle deformation to be homogeneous. In contrast, the aim of the present study is to analyse history effects in skeletal muscle tissue using a three-dimensional geometry model of the whole muscle–tendon unit. Therefore, a purely phenomenological approach is presented. The model is implemented in the finite element framework to analyse history effects for the boundary value problem of the entire three-dimensional muscle–tendon geometry. The constitutive model shows good agreement with the experimental data. Furthermore, the simulations reveal information about the inhomogeneous stretch distributions within the muscle tissues.

Appendices

Appendix 1: Computation of $\varvec{w_a}$

The computation of the activation level $w_a$ involves a two-step approach: first, Eq. (20) is solved by using an approximation of $\mathrm LW$ as

$$\begin{aligned} \mathrm {LW}(y) \approx \dfrac{2\ln (1+Bz)-\ln (1+C\ln (1+Dz))+E}{1+(2\ln (1+Bz)+2A)^{-1}} \end{aligned}$$

(30)

with the substitution operation $z = \sqrt{{2e\,y}+2}$ of y, see Eq. (20)$_2$ and constants $A\approx 2.344$, $B\approx 0.8842$, $C\approx 0.9294$, $D\approx 0.5106$, and $E \approx -1.213$, see Winitzki (2003) for further information. The approximation of $w_a$ serves in the following step as an initial value to calculate the actual value of $w_a$ numerically. Reformulating Eq. (19) to the residual statement

$$\begin{aligned} r_{w_a} = P_{\mathrm{act}} - \dfrac{\mu }{4}e^{\displaystyle {\alpha ({{\hat{I}}}_p - 1)}}\left[ e^{\displaystyle {\alpha (w_a {{\hat{\lambda }}}^2)}}\left( {{\hat{I}}}^{\prime }_p +2 w_a {{\hat{\lambda }}} \right) -{{\hat{I}}}^{\prime }_p\right] \end{aligned}$$

(31)

and applying the Newton–Raphson iteration scheme solve the minimisation problem for the increment

$$\begin{aligned} d w_a = -\left[ \dfrac{\partial r_{w_a}}{\partial w_a}\right] ^{-1} r_{w_a}. \end{aligned}$$

(32)

Finally, the algorithmic update

$$\begin{aligned} w_a^{k+1} = w_a^k + d w_a \end{aligned}$$

(33)

is applied until the convergence criteria is reached. The second step of computing the accurate value of $w_a$ is necessary to predict the actual active stresses of Eq. (8).

Appendix 2: Incremental tangent moduli

The algorithmic treatment of the fibre-reinforced and nearly incompressible material in the finite element framework requires a consistent linearisation of the stresses defined in Eq. (27). Using the pullback operation

$$\begin{aligned} {\varvec{S}}{} = {\varvec{F}}{}^{-1} {\varvec{P}}{} \end{aligned}$$

(34)

to convert stresses in terms of the second Piola–Kirchhoff stress tensor, the relation between the incremental stresses $d {\varvec{S}}{}$ and the incremental strains $d {\varvec{C}}{}$ reads

$$\begin{aligned} d {\varvec{S}}{} = {\mathbb {C}} : \frac{1}{2} d {\varvec{C}}{} \end{aligned}$$

(35)

with

$$\begin{aligned} {\mathbb {C}} = 2\dfrac{\partial {\varvec{S}}{}}{\partial {\varvec{C}}{}} + 2\dfrac{\partial {\varvec{S}}{}}{\partial w_a}\dfrac{\partial w_a}{\partial {\varvec{C}}{}}, \end{aligned}$$

(36)

to be the so-called algorithmic tangent. While the partial derivations $\partial {{\varvec{S}}{}}/\partial {{\varvec{C}}{}}$ and $\partial {{\varvec{S}}{}}/\partial {w_a}$ are directly available by straightforward calculation, $\partial w_a/\partial {\varvec{C}}{}$ is accessible after the derivation of Eq. (31) with respect to the right Cauchy–Green strain tensor. Using the chain rule the derivation

$$\begin{aligned} \frac{dr_{w_a}}{d{\varvec{C}}{}} = \left( \dfrac{\partial r_{w_a}}{\partial {{\hat{\lambda }}}} + \dfrac{\partial r_{w_a}}{\partial w_a}\dfrac{\partial w_a}{\partial {{\hat{\lambda }}}} \right) \dfrac{\partial \hat{\lambda }}{\partial {\varvec{C}}{}} = {\varvec{0}}{} \end{aligned}$$

(37)

can be reformulated to

$$\begin{aligned} \dfrac{\partial w_a}{\partial {\varvec{C}}{}} = \dfrac{\partial w_a}{\partial {{\hat{\lambda }}}} \dfrac{\partial \hat{\lambda }}{\partial {\varvec{C}}{}} = -\left[ \dfrac{\partial r_{w_a}}{\partial w_a}\right] ^{-1} \dfrac{\partial r_{w_a}}{\partial {{\hat{\lambda }}}} \dfrac{\partial {{\hat{\lambda }}}}{\partial {\varvec{C}}{}}. \end{aligned}$$

(38)

For a better overview of the computation steps, see Table 3.

Table 3 Local Newton update of the internal variable $w_a$

Appendix 3: One-dimensional analyses

Following the parameter identification scheme as illustrated in Sect. 2.4, Fig. 7 illustrates the comparison between experiment and simulation for (a) the passive force–stretch and (b) the total, i.e. active and passive, force–stretch relation. In both cases, the modelling concept is able to adequately reproduce the experimental data.

Fig. 7

Comparison of MT-1D with experimentally recorded data (rabbit soleus muscle) for a the passive force–stretch and b the total force–stretch relation. Note that the stretch in b is defined as relation between the deformed and undeformed muscle length

Based on experimental investigations of Siebert et al. (2015), history effects are presented in Fig. 8 in terms of FE (a) and FD (b) effects during and after muscle stretching and muscle shortening, respectively. In both cases, the model is able to reproduce the experimental data. In part, there are some deviations at the maximum (a) and minimum forces (b) at the end of muscle stretch and shortening, respectively. In comparison with, for example, the gastrocnemius and plantaris muscles, which feature significant fatigue effects leading to non-stable plateaus (Siebert et al. 2015), the soleus muscles present significantly less fatigue effects. However, as the proposed modelling approach does not include any formalism to describe fatigue effects, those effects cannot be captured, resulting in idealised horizontal plateaus; see Fig. 8.

Fig. 8

Comparison of MT-1D with experiments (Siebert et al. 2015), see also Figure 1, featuring a FE and b FD characteristics of rabbit soleus muscle

The MT-1D model approach is able to track passive (Fig. 7a) and total force–stretch (Fig. 7b) and force–time relations (Fig. 8) in a convincing way. However, based on its one-dimensional representation, only homogeneous deformations can be realised, and with the simplification of the muscle geometry, it is rather difficult to link the model with the physiological behaviour of a real muscle–tendon unit.