A model for one-dimensional morphoelasticity and its application to fibroblast-populated collagen lattices

Abstract

The mechanical behaviour of solid biological tissues has long been described using models based on classical continuum mechanics. However, the classical continuum theories of elasticity and viscoelasticity cannot easily capture the continual remodelling and associated structural changes in biological tissues. Furthermore, models drawn from plasticity theory are difficult to apply and interpret in this context, where there is no equivalent of a yield stress or flow rule. In this work, we describe a novel one-dimensional mathematical model of tissue remodelling based on the multiplicative decomposition of the deformation gradient. We express the mechanical effects of remodelling as an evolution equation for the effective strain, a measure of the difference between the current state and a hypothetical mechanically relaxed state of the tissue. This morphoelastic model combines the simplicity and interpretability of classical viscoelastic models with the versatility of plasticity theory. A novel feature of our model is that while most models describe growth as a continuous quantity, here we begin with discrete cells and develop a continuum representation of lattice remodelling based on an appropriate limit of the behaviour of discrete cells. To demonstrate the utility of our approach, we use this framework to capture qualitative aspects of the continual remodelling observed in fibroblast-populated collagen lattices, in particular its contraction and its subsequent sudden re-expansion when remodelling is interrupted.

Appendices

Appendix 1: effective strain and contraction in a 1-D morphoelastic body

1.1 The multiplicative decomposition of the deformation gradient

The idea that an observed deformation gradient tensor ${F}$ could be expressed in terms of elastic and plastic tensor fields through a multiplicative decomposition was first introduced by Bilby et al. (1957) and Kröner (1958, 1959). This was further developed by Stojanović et al. (1964, 1970) in the context of thermoelasticity and Lee (1969) for the description of metal plasticity at large deformations, while Rodriguez et al. (1994) and Cook (1995) were the first to utilise this in the context of biomechanics. The pioneering work of Rodriguez et al. (1994) and Cook (1995) was later extended and expanded by Hoger and coworkers (see especially Chen and Hoger (2000)), Goriely and coworkers (Ben Amar and Goriely 2005; Goriely and Ben Amar 2007; Goriely et al. 2008), Ambrosi and coworkers (Ambrosi and Mollica 2004; Ambrosi and Guana 2007; Ambrosi and Guillou 2007) and by Vandiver (2009). This biological work has developed alongside applications to thermoelasticity and plasticity, and achievements in these areas have informed each other. This is exemplified by the cross-disciplinary work of Lubarda (2001, 2004) and Rajagopal and coworkers (see, for example, Rajagopal and Srinivasa (2004)). For a comprehensive review of the history and applications of the multiplicative decomposition of deformation gradient, see Lubarda (2004).

In a recent comprehensive review of current work on modelling growth and remodelling, Ambrosi et al. (2011) describe a number of applications of the multiplicative decomposition of the deformation gradient, ranging from the remodelling of heart muscle to morphogenesis. The multiplicative decomposition of the deformation gradient is now being used in models of biomechanical phenomena ranging from tissue growth (Ben Amar and Goriely 2005) to the operation of the heart (Göktepe and Kuhl 2010; Rausch et al. 2011), although it is important to note that this approach will only be valid when the tissue behaves elastically on the timescale of remodelling (Jones and Chapman 2012) and that some authors have general reservations about the use of the multiplicative decomposition on theoretical grounds (Xiao et al. 2006). Indeed, Ambrosi et al. (2011) note that there are problems and ambiguities to be resolved when developing appropriate laws to describe the evolution of the growth part of the deformation gradient in response to remodelling. In the context of the present work, it is important to note that many of these difficulties are avoided as we restrict our analysis to the one-dimensional (1-D) Cartesian case, although it is still necessary to ensure that the constitutive relation is appropriate for the type of remodelling under consideration.

An accessible introduction to the use of the multiplicative decomposition in biological applications can be found in Goriely and Moulton (2011), which begins with an analysis of a 1-D growing material that is relevant to the research presented here. It is important to note that a 1-D body can never be residually stressed: it is impossible to encounter the situation in which the zero stress state cannot be achieved without introducing cuts. Moreover, ensuring observer independence of time derivatives is much simpler along a single dimension, as it excludes the possibility of a rotating observer. We now develop a simple mathematical framework for modelling the growth or contraction of a 1-D Cartesian morphoelastic body.

1.2 Strain evolution

The deformation gradient is given by the scalar function

$$\begin{aligned} F(X,\,t) = \frac{\partial x}{\partial X}, \end{aligned}$$

Following the 1-D version of Eq. (1) in Goriely and Moulton (2011), we express this as the product

$$\begin{aligned} F = \alpha \, \gamma , \end{aligned}$$

(29)

where the elastic stretch $\alpha$ is the local size ratio between the current state and the zero stress state, and the growth stretch $\gamma$ is the local size ratio between the zero stress state and the initial state (the growth stretch).

In Goriely and Moulton (2011), the constitutive relation for 1-D growth is assumed to depend on the rate of growth $g(x,\,t)$:

$$\begin{aligned} \frac{\partial \gamma }{\partial t} = g(x,\,t), \end{aligned}$$

(30)

While (30) is useful at small deformations (i.e. when $F \approx 1$), it leads to inconsistencies if the current state is significantly different from the initial state, as in the case of FPCL contraction. In order to obtain an equivalent of (30) for large deformations, we first note that $g(x,\,t)$ should satisfy

$$\begin{aligned} \frac{\mathrm {d} }{\mathrm {d} t} \int _{X_A}^{X_B} \gamma (X,\,t) \, \mathrm {d}X = \int _{x(X_A,\,t)}^{x(X_B,\,t)} g(x,\,t) \, \mathrm {d}x. \end{aligned}$$

(31)

That is, $g(x,\,t)$ should be defined with reference to the current configuration, but it should measure the rate of change of the zero stress state of any collection of material particles. It follows from (31) that $g(x,\,t)$ is related to the material derivative of $\gamma (X,\,t)$:

$$\begin{aligned} \frac{\mathrm {D} \gamma }{\mathrm {D} t} = F \, g(x,\,t), \end{aligned}$$

(32)

Note that this reduces to (30) when $F \equiv 1$.

Now, we expect that the stress at any point in the body will be related to the difference between the zero stress state and the current state. A plausible constitutive law that relates the stress, $\sigma$, to the elastic stretch, $\alpha$ is

$$\begin{aligned} \sigma = E \, \left( 1 - \alpha ^{-1}\right) , \end{aligned}$$

(33)

where E is the Young’s modulus. This is analogous to Hooke’s law for a linear elastic material, but uses an Eulerian rather than a pseudo-Lagrangian measure of strain, since

$$\begin{aligned} e^{E}\equiv & {} 1 - \alpha ^{-1} = \lim _{\Delta x \rightarrow 0} \frac{\Delta x - \Delta z}{\Delta x}, \text { while }\\ e^{L}\equiv & {} \alpha - 1 = \lim _{\Delta x \rightarrow 0} \frac{\Delta x - \Delta z}{\Delta z}, \end{aligned}$$

where $\Delta x$ and $\Delta z$ relate to the changes in the current and zero stress states, respectively. In cases where the current state is close to the zero stress state, and hence, $\alpha \approx 1$, we see that $e^{E}\approx e^{L}$ and (33) is equivalent to other plausible constitutive laws, such as those in Goriely and Moulton (2011):

$$\begin{aligned} \sigma= & {} E \, (\alpha - 1), \end{aligned}$$

(34)

$$\begin{aligned} \sigma= & {} \frac{E}{3} \, \left( \alpha ^2 - \alpha ^{-1}\right) , \end{aligned}$$

(35)

Experimental observations indicate that most of the change in size of a contracting FPCL is due to the permanent rearrangement of fibres by fibroblasts (Guidry and Grinnell 1985). Hence, it is appropriate to use (33) and assume a linear relationship between stress and strain, rather than the nonlinear model (35). Moreover, (33) has an interesting advantage over (34) and (35), namely that the evolution of Eulerian strain in response to growth can neatly be expressed as an advection equation with a source term that is independent of $e^E$.

In order to see this, we substitute (29) into (32) to obtain

$$\begin{aligned} F \, \frac{\mathrm {D} }{\mathrm {D} t} \, \alpha ^{-1} + \alpha ^{-1} \, \frac{\mathrm {D} F}{\mathrm {D} t} = F \, g(x,\,t). \end{aligned}$$

(36)

Since $F^{-1} \, D F/D t = \partial v/\partial x$ where v is the velocity and where $\partial v/\partial x$ is the velocity gradient, it follows that $\alpha ^{-1}$ satisfies the equation

$$\begin{aligned} \frac{\partial }{\partial t} \alpha ^{-1} + \frac{\partial }{\partial x} \left( v \, \alpha ^{-1}\right) = g(x,\,t), \end{aligned}$$

and using $e^{E} \equiv 1 - \alpha ^{-1}$ we thus obtain the mechanical model for a morphoelastic solid with small effective strain

$$\begin{aligned} \frac{\partial e^E}{\partial t} + \frac{\partial }{\partial x} \left( e^E \, v \right) = \frac{\partial v}{\partial x} - g(x,\,t), \end{aligned}$$

(37)

Note that thus far we have made the assumption that the relation between stress and strain is purely elastic

$$\begin{aligned} \sigma = E \, e^E. \end{aligned}$$

However, as discussed in Sect. 3.1, this can easily be extended to viscoelastic bodies through the use of a Kelvin–Voigt viscoelastic constitutive law. In Sect. 3 of the main text, we use this formulation together with (37) to derive a set of governing equations for the contraction of a 1-D morphoelastic body.

Appendix 2: derivation of the spatial and temporal transformations between coordinate systems

In one spatial dimension, we use X to represent the Lagrangian coordinate and x to represent the Eulerian coordinate. At any given time t, there will be a one-to-one mapping from the initial configuration to the current configuration. Thus, we can always write $X = X(x,\,t)$ and $x = x(X,\,t)$; moreover, the fact that particles are not permitted to move through each other implies that ${\partial x}/{\partial X} > 0$.

Now, the Eulerian displacement gradient is the spatial derivative of $u(x,\,t) =x-X(x,\,t)$, i.e.

$$\begin{aligned} w{(x,t)}=1-\frac{\partial X}{\partial x}. \end{aligned}$$

(38)

Similarly, the Lagrangian displacement gradient is

$$\begin{aligned} W(X,\,T)=\frac{\partial }{\partial X}\left( x-X\right) =\frac{1}{1-w}-1. \end{aligned}$$

We thus have the relation

$$\begin{aligned} 1+W=\frac{1}{1-w}. \end{aligned}$$

(39)

Using the chain rule, the Eulerian spatial derivative is

$$\begin{aligned} \frac{\partial }{\partial x}\equiv \frac{\partial X}{\partial x}\frac{\partial }{\partial X}+\frac{\partial T}{\partial x}\frac{\partial }{\partial T}. \end{aligned}$$

The derivative $\partial T/\partial x$ is equal to zero, and using (38) we obtain the expression

$$\begin{aligned} X=x-\int _{0}^{x}\,w(\xi ,t) \mathrm {d}\xi . \end{aligned}$$

(40)

Furthermore, using (38) and (39), we obtain the following transformation for the spatial derivative

$$\begin{aligned} \frac{\partial }{\partial x}\equiv \frac{1}{1+W}\frac{\partial }{\partial X}. \end{aligned}$$

(41)

We similarly use the chain rule to obtain the following expression for the Eulerian temporal derivative

$$\begin{aligned} \frac{\partial }{\partial t}\equiv \frac{\partial X}{\partial t}\frac{\partial }{\partial X}+\frac{\partial T}{\partial t}\frac{\partial }{\partial T}. \end{aligned}$$

The derivative $\partial T/\partial t$ is equal to one, and so using (40) we have

$$\begin{aligned} \frac{\partial }{\partial t}\equiv \frac{\partial }{\partial t}\left( \int ^{x}_{0} 1 - w(\xi ,t)\,\mathrm {d}\xi \right) \frac{\partial }{\partial X}+\frac{\partial }{\partial T}. \end{aligned}$$

Using (15), we have

$$\begin{aligned} \frac{\partial }{\partial t} \equiv \int ^{x}_{0} \frac{\partial }{\partial \xi }\left( w(\xi ,t)\,v(\xi ,t)-v(\xi ,t)\right) \,\mathrm {d}\xi \frac{\partial }{\partial X}+\frac{\partial }{\partial T}, \end{aligned}$$

which, on using (39) and the fact that $v(0,\,t)=0$, yields

$$\begin{aligned} \frac{\partial }{\partial t}\equiv \frac{\partial }{\partial T}-\frac{v}{1+W}\frac{\partial }{\partial X}. \end{aligned}$$

Now, from (10), we have $v\,(1-w)=\partial {u}/\partial {t}$. Since $u\equiv U$, using the above expression in conjunction with (39), this yields

$$\begin{aligned} v = (1+W)\,\left( V-\frac{v}{1+W}\,W \right) , \end{aligned}$$

where we have used the definitions of V and W. This implies that $v\equiv V$, and so we obtain the following transformation for the temporal derivative

$$\begin{aligned} \frac{\partial }{\partial t}\equiv \frac{\partial }{\partial T}-\frac{V}{1+W}\frac{\partial }{\partial X}. \end{aligned}$$

(42)

Now, from (15) we have $\partial {w}/\partial {t} + \partial {(w\,v)}/\partial {x} = \partial {v}/\partial {x}$. Using (39), (41) and (42), this reduces to the following relation between the Lagrangian displacement gradient and velocity:

$$\begin{aligned} \frac{\partial W}{\partial T}=\frac{\partial V}{\partial X}. \end{aligned}$$

(43)

We use this expression in Sect. 3.2 of the main text to derive our morphoelastic model of FPCL contraction.