Mechanotransduction mechanisms in growing spherically structured tissues

The shape changes that are observed as a tissue grows are in fact a combination of two physical processes, an increase in tissue volume and the consequent deformations due to mechanical interactions. To account for this within the framework of continuum elasticity theory we adopt the method of decomposition [42, 43, 45, 46]. This widely adopted approach explicitly separates the two physical processes into a growth phase and an elastic deformation, exploiting the separation of timescales between growth and elastic response. Specifically, we introduce an interim configuration S_g which is the configuration the tissue would adopt due to volume increase alone if all stresses were relaxed from every material element, see figure 1(b). The stresses are then calculated by considering how far from this target stress-free configuration S_g the tissue is currently observed to be (in configuration S_d). Throughout we assume volumetric isotropic growth so that there is no preferred direction of growth for the tissue.

Before considering complex structured tissues, we first consider as an instructional example a solid spherical tissue of unit radius that grows by 10%; in this case, if the growth is unconstrained, for example in a fluid medium, the tissue radius r₁ would consequently increase to r₁ $\approx$ 1.03. However, if we now consider the growth within a constraining gel we can expect that the observed radius would be ${r}_{1}\lt 1.03$ due to compression from the outside gel. Indeed in the limit of an infinitely stiff gel culture we see that r₁ = 1, where the unit radius now corresponds to a significant compression as compared with the target volume increase of 10%. Using the decomposition approach, the stress experienced within the tissue is calculated based on this difference from the target expanded state.

In this paper, we consider structured tissues such as epithelial acini, tumour spheroids or follicles, which share a common morphology of a central non-growing core region surrounded by an outer layer of proliferating cells. In each of these cases the proliferating tissue forms a growing spherical shell (so that by spherical symmetry we can see that all deformations must be a function of radial direction only). Consider volumetric growth, where the material expands uniformly and isotropically so that every microscopic element making up the material expands its linear dimensions by a factor λ. Thus λ = 1 corresponds to no growth and e.g. when λ = 2^1/3 ≈ 1.3 each tissue element has doubled in volume. Now it follows in spherical coordinates that the original configuration [r, θ, ϕ] maps to the intermediate configuration [λr, θ, ϕ], figure 1(b) (see e.g. [46, 56]). It is worth reiterating that this swollen configuration is not observed in reality but is only the target configuration for the tissue. Note that as the factor λ models growth, λ is assumed to be varying (indeed monotonically increasing) with time as the tissue grows. Importantly, however, λ is slowly varying with respect to the elastic response (the essential assumption of decomposition theory) so that we need only solve for the static mechanical equilibrium.

For an alternative visualisation of how volume increase drives a radial expansion in this interim configuration, consider a spherical shell of material at radial position r_A. As each volume element swells isotropically due to growth, the extra material must still, by symmetry, be accommodated into a spherical shell. Since the material cannot be compressed as this would violate the zero stress condition of the interim configuration, the spherical shell must increase in size, i.e. the radius must increase, and in fact scales with λ. This holds at all values of r_A, hence a consequence of isotropic swelling is that the central hole naturally increases in size (in the interim configuration) as the inner radius scales with λ. This enlargement under expansion is analogous to the well-known fact that a metal pipe increases its diameter under thermal expansion (something that must be taken into account when joining pipes with different coefficients of thermal expansion). Within tissues, the idea that swelling and growth of a tube can increase the target radius (i.e. reduce curvature) was demonstrated experimentally by Fung (see [57]) in his seminal tissue cutting experiments on e.g. arteries and hearts, where he observed that a stress-free configuration necessitates a larger target radius with reduced tissue curvature.

Assuming linear elasticity, the stress tensor σ_ij and strain tensor _ij satisfy the force balance equation

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot \sigma =0\ \mathrm{with}\,{\sigma }_{{ij}}=\displaystyle \frac{E}{1+\nu }\left({\epsilon }_{{ij}}+\displaystyle \frac{\nu }{1-2\nu }{\epsilon }_{{ll}}{\delta }_{{ij}}\right),\end{eqnarray} \tag{ 1 }$$

where the summation convention applies for repeated indices. Spherical symmetry implies that all displacements are in the radial direction so that ${\bf{u}}=u(r){{\bf{e}}}_{r}$ and that $[r,\theta ,\phi ]\to [r+u(r),\theta ,\phi ]$. The only non-zero strains are thus the radial component _rr = du/dr and circumferential components ${\epsilon }_{\theta \theta }={\epsilon }_{\phi \phi }=u/r$, which when substituted into (1) and integrated gives $u=\tfrac{A}{3}r+\tfrac{B}{{r}^{2}}$, where A, B are constants.

In the encapsulating gel (r > r₁) we denote the displacement by u_g(r), and find on applying the boundary condition ${u}_{g}(r)\to 0{\rm{}}\,{\rm{as}}\,{\rm{}}r\to \infty$ that

$$\begin{eqnarray*}&&{u}_{g}=\displaystyle \frac{{r}_{1}^{3}}{{r}^{2}}({\alpha }_{1}-1),\end{eqnarray*}$$

where ${\alpha }_{1}{r}_{1}={r}_{1}+{u}_{g}({r}_{1})$ is the current observed position of the interface between outer gel and growing tissue.

In the central compartment r < r₀, in the case where this is deformable, the deformation is denoted by u_c(r) and found (on applying the boundary condition u_c(0) = 0) to be

$$\begin{eqnarray*}&&{u}_{c}=({\alpha }_{0}-1)r,\end{eqnarray*}$$

where ${\alpha }_{0}{r}_{0}={r}_{0}+{u}_{c}({r}_{0})$ is the current observed radius of the inner compartment. Note that where the centre can be assumed to be infinitely stiff, as for example, would be the case for the ovarian follicle, u_c(r) = 0 and α₀ = 1, and it is shown in appendix B that this is the limiting solution of the full problem as the Young's modulus of the centre ${E}_{c}\to \infty$.

The only non-zero components of the stress tensor in the outside encapsulating gel are

$$\begin{eqnarray}&&{\sigma }_{{rr}}^{g}=-\displaystyle \frac{2{E}_{g}}{1+{\nu }_{g}}{\left(\displaystyle \frac{{r}_{1}}{r}\right)}^{3}({\alpha }_{1}-1),\ \ {\sigma }_{\phi \phi }^{g}={\sigma }_{\theta \theta }^{g}=\displaystyle \frac{{E}_{g}}{1+{\nu }_{g}}{\left(\displaystyle \frac{{r}_{1}}{r}\right)}^{3}({\alpha }_{1}-1).\end{eqnarray} \tag{ 2 }$$

In the centre we have the constant solution

$$\begin{eqnarray}&&{\sigma }_{\theta \theta }^{c}={\sigma }_{\phi \phi }^{c}={\sigma }_{{rr}}^{c}=\displaystyle \frac{{E}_{c}}{1-2{\nu }_{c}}({\alpha }_{0}-1).\end{eqnarray} \tag{ 3 }$$

The unknown constants α₀ and α₁ are to be determined by continuity of stress with the growing shell.

Here we shall introduce the subscript t to denote the solution in the growing tissue region. The radial displacement that determines the elastic response is in this case the displacement from the interim grown configuration $r\in [\lambda {r}_{0},\lambda {r}_{1}]$, due to the decomposition, and this displacement is denoted u_t(r). u_t(r) can be found similarly to section 2.2 as

$$\begin{eqnarray*}&&{u}_{t}(r)=\displaystyle \frac{{A}_{t}}{3}r+\displaystyle \frac{{B}_{t}}{{r}^{2}}.\end{eqnarray*}$$

The radial component of the stress in this case in the interim configuration is

$$\begin{eqnarray*}&&{\sigma }_{{rr}}^{t}=\displaystyle \frac{{E}_{t}}{(1+{\nu }_{t})(1-2{\nu }_{t})}\left[\displaystyle \frac{{A}_{t}}{3}(1+{\nu }_{t})-\displaystyle \frac{2{B}_{t}}{{r}^{3}}(1-2{\nu }_{t})\right].\end{eqnarray*}$$

To determine the unknown constants A_t, B_t, along with α₀, α₁, we apply continuity conditions at the inner and outer boundaries as detailed in appendix A for an undeformable central compartment and in appendix B for a deformable central compartment. Note that as the deformations in the growing phase are considered relative to the interim configuration, the assumption of small strain can hold at relatively large values of λ depending on the deformations induced by the constraining gel.

Focusing first on the mechanical stress that the growing tissue and central compartment experience as growth progresses, we introduce the parameter

$$\begin{eqnarray}&&{\lambda }_{s}=\displaystyle \frac{{E}_{t}(1+{\nu }_{g})}{{E}_{g}(1+{\nu }_{t})},\end{eqnarray} \tag{ 4 }$$

which, given that the Poisson's ratio of cells and matrix are in most circumstances likely to be very similar (typically in the range ν = 0.45 to ν = 0.49 [58]), quantifies the relative stiffnesses of the constraining gel and growing tissue. λ_s < 1 thus in practice translates to the circumstance of the outside constraining gel being stiffer than the growing tissue, while λ_s > 1 implies that the growing tissue is stiffer than the outside gel. We will consider these two cases separately as we will show that the stress experienced within the tissue differs markedly in each of these two regimes.

Consider first the case where the tissue is softer than the gel (${\lambda }_{s}\lt 1$) and in which the centre is rigid. We plot in figures 2(a), (b) typical profiles for the stresses σ_rr and ${\sigma }_{\theta \theta }={\sigma }_{\phi \phi }$, parameter values given in the figure caption (Young's modulus measured in $\mathrm{kPa}$). We see in this case that the radial stress is uniformly compressive and at a maximum at the outer radius of the tissue, and the circumferential stresses are also compressive but with a maximum at the inner radius. The stress increases in magnitude as tissue grows, i.e. as λ increases. In particular, on the growth initiation of the surrounding tissue, the inner compartment, which may be considered to be e.g. the central oocyte of an ovarian follicle, experiences an immediate compressive radial stress, which increases over time as the tissue grows.

Zoom In Zoom Out Reset image size

In stark contrast, if the tissue is stiffer than the surrounding gel (λ_s > 1), then initially at the inner surface there is a positive radial stress, indicating that the central compartment initially experiences a pulling force, while the circumferential stresses remain compressive, see figures 2(c), (d). At the outer surface r = r₁, it can be seen from figure 2(c) that the radial stress is compressive, and that within the growing region there is a radius at which the stress is zero. As the tissue continues to grow, i.e. as λ increases, it can be seen that the compressive stress at the outer surface increases, and the radius at which σ_rr = 0 decreases.

We plot in figure 3 the radial and circumferential stresses at the inner radius in the regime λ_s > 1. Significantly the radial stress at this inner boundary is initially positive, so that the inner compartment is initially pulled outwards. This pulling force increases with growth until a maximum is reached, after this the radial stress decreases to zero before becoming negative so that the inner compartment now experiences a compressive force towards its centre.

Zoom In Zoom Out Reset image size

To find the critical value of growth at which such a qualitative switch could occur, we set ${\sigma }_{{rr}}^{t}(\lambda {r}_{0})=0$ in the solution from appendix A to obtain the following quadratic equation for the critical values of λ

$$\begin{eqnarray}&&(1+{\nu }_{t})\left(({\alpha }_{1}-\lambda )-{\left(\displaystyle \frac{{r}_{0}}{{r}_{1}}\right)}^{3}(1-\lambda )\right)=2(1-2{\nu }_{t})(1-{\alpha }_{1}),\end{eqnarray} \tag{ 5 }$$

with α₁(λ) given by (A.1). This has roots λ = 1, corresponding to the initial conditions of no growth and zero stress, and λ = λ_s. Thus we see that the point at which the radial stress at the inner surface switches from a tensile to a compressive force is λ = λ_s, thus giving an a posteriori justification for the significance of the parameter λ_s. This then can be termed a switching point of the system.

For a structured tissue with a deformable central compartment, corresponding to the solution presented in appendix B, similar dynamics are observed, see ESI figures S1, S2 available online at stacks.iop.org/NJP/20/043041/mmedia. Significantly, the parameter λ_s plays the same role as before. For λ_s < 1 the inner compartment experiences uniform compression, if λ_s > 1 then the inner compartment initially experiences tension before switching to compression once the growth parameter passes a critical threshold. To show this, we set ${\sigma }_{{rr}}^{t}(\lambda {r}_{0})=0$ using the solution from appendix B to obtain a cubic equation corresponding to (5) for the critical values of λ

$$\begin{eqnarray}&&(1+{\nu }_{t})\left(({\alpha }_{1}-\lambda )-{\left(\displaystyle \frac{{r}_{0}}{{r}_{1}}\right)}^{3}({\alpha }_{0}-\lambda )\right)=2(1-2{\nu }_{t})({\alpha }_{0}-{\alpha }_{1}),\end{eqnarray} \tag{ 6 }$$

with α₀ and α₁ given now by (B.1) and (B.2). The solutions of (6) are λ = λ_s, λ = 1 (i.e. the initial conditions of zero stress, no growth) and λ = 0, which is unphysical. That λ_s plays the same role as before is initially surprising, however, it is clear that the switching condition for which the radial stress at the inner surface changes sign must be the same, for at this point the inner surface is transmitting zero stress and so the properties of the central region can have no influence on the mechanical equilibrium achieved.

We thus see, in both cases considered, that when the material surrounding the growing structured tissue is softer than the growing tissue, a new and interesting mechanism for growth induced mechanotransduction is introduced. In this case, the central compartment will initially experience tension as the tissue grows, before switching to compression when the growth is such that $\lambda =({E}_{t}(1+{\nu }_{g}))/({E}_{g}(1+{\nu }_{t}))$. A switching mechanism based on such significant qualitative change has the potential to be particularly robust. Biological tissues vary greatly in their measured Young's modulus (ranging anything from kPa to MPa) for different tissues [44, 59–61], so the Young's modulus is relatively unconstrained, however note that it is only the relative stiffness of gel and tissue that determines the switching point. Equally, λ_s is independent of the shell thickness. The shell thickness and the respective Young's moduli and Possion's ratios, however, do determine the magnitude of the stress in each case. Experimentally changing the material properties of the external constraining gel would change λ_s and could potentially take this parameter through the critical threshold of λ_s = 1 generating significant change in the system.

Considering now how growth could generate qualitative changes in cellular strain, we see that the circumferential strains ${\epsilon }_{\theta \theta }^{t}={\epsilon }_{\phi \phi }^{t}={u}_{t}/r$ within the growing tissues are always negative as a constraining gel will always induce negative displacements relative to the grown configuration. Considering the radial component of strain (in the case of a rigid central compartment) we find that

$$\begin{eqnarray}{\epsilon }_{{rr}}^{t} & = & \gamma (\lambda )\left((1+{\nu }_{t}){\left(\displaystyle \frac{\lambda {r}_{0}}{r}\right)}^{3}-(1-2{\nu }_{t}){\left(\displaystyle \frac{{r}_{0}}{{r}_{1}}\right)}^{3}\right.-\displaystyle \frac{{E}_{g}}{{E}_{t}}\displaystyle \frac{(1+{\nu }_{t})(1-2{\nu }_{t})\lambda }{1+{\nu }_{g}}\left(1-{\left(\displaystyle \frac{{r}_{0}}{{r}_{1}}\right)}^{3}\right))\end{eqnarray} \tag{ 7 }$$

within the growing region $r\in [\lambda {r}_{0},\lambda {r}_{1}]$, where

$$\begin{eqnarray*}&&\gamma (\lambda )=\displaystyle \frac{2(\lambda -1)}{\widehat{B}{\lambda }^{2}+\lambda (\widehat{A}+1+{\nu }_{t})}{\left(1-{\left(\displaystyle \frac{{r}_{0}}{{r}_{1}}\right)}^{3}\right)}^{-1}\end{eqnarray*}$$

and we note that γ(λ) > 0 everywhere. Plots of this radial strain are shown in figure 4(a).

Zoom In Zoom Out Reset image size

In contrast to the circumferential strains, this radial strain can be positive and this can occur even when the radial component of stress is negative. To see this, consider first the case of a thin shell where r₁ = r₀ + δ, δ ≪ 1, which gives ${\epsilon }_{{rr}}^{t}=2{\nu }_{t}(\lambda -1)/\lambda (1-{\nu }_{t})+O(\delta )\gt 0$ for all values of the material properties. This counter-intuitive result can be understood in the context of the compressive circumferential strains. Given that the material is not infinitely compressible, these necessitate an expansion (or 'squeezing out' of the material) in the radial direction for these thin shells, even in the presence of compressive radial forces.

As for stress, there is the possibility for strain to also exhibit sign switching under the action of growth, introducing an additional potential mechanotransductive mechanism linked to tissue growth in these structured tissues. To see this, consider again ${\epsilon }_{{rr}}^{t}(r)$ as given by (7); this is a decreasing monotonic function of r (within the growing region $r\in [\lambda {r}_{0},\lambda {r}_{1}]$) with a zero at $r=\eta {(\lambda )}^{-1/3}\lambda {r}_{0}$ where

$$\begin{eqnarray*}&&\eta (\lambda )=\displaystyle \frac{1-2{\nu }_{t}}{1+{\nu }_{t}}{\left(\displaystyle \frac{{r}_{0}}{{r}_{1}}\right)}^{3}+\displaystyle \frac{{E}_{g}}{{E}_{t}}\displaystyle \frac{(1+{\nu }_{t})(1-2{\nu }_{t})\lambda }{1+{\nu }_{g}}\left(1-{\left(\displaystyle \frac{{r}_{0}}{{r}_{1}}\right)}^{3}\right).\end{eqnarray*}$$

For $r\lt \eta {(\lambda )}^{-1/3}\lambda {r}_{0}$ the radial strain will be positive (indicating stretch) and for $r\gt \eta {(\lambda )}^{-1/3}\lambda {r}_{0}$ the radial strain will be negative. The condition that positive radial strains exist within the growing region is thus that $\eta (\lambda )\lt 1$, i.e. that the radius at which zero strain is felt lies in r > λ r₀. However, as η(λ) is an increasing function, we see that eventually, at some value for the growth λ, these positive strains will be lost and that there will be a qualitative switch from positive to negative strains as the encapsulating gel begins to dominate. We note that this switching mechanism is controlled not just by the relative stiffnesses of the tissue and gel, but also by the thickness (r₀/r₁) of the proliferating shell.

Any such change in strain can underpin a stretch-activated mechanism of switching as it will induce molecular conformation changes at this point, see figure 4(b). Interestingly, in contrast to the stress-based mechanism this qualitative strain switching can occur even if the tissue is softer than the constraining gel, see figure 4(b). This behaviour may thus be more likely in vivo as the surrounding stiffness of tissues is typically softer than that of in vitro gels [59].

When the outer gel is softer than the growing tissue (λ_s > 1), an additional consequence of the positive stress on the inner boundary is that the inner compartment can increase in size. This occurs even in the absence of material growth in this region, at least in the case that E_c is finite. To see this, we plot in figure 5(a) α₀ as a function of the growth parameter λ. Where α₀ > 1, this corresponds to an expansion of the inner compartment due to the shell expansion pulling it outwards. In this case, it might appear that there has been growth initiation in this central compartment, but in fact the inner region has expanded merely as a result of the stretch exerted by the outside tissue. This has particular significance in considerations of growth initiation, for example, in ovarian follicles, where an increase in oocyte size cannot alone be taken as a marker of growth initiation. This highlights the need for a full consideration of mechanics in interpreting the experimental data and the need for additional biologic readouts in interpreting growth data.

Zoom In Zoom Out Reset image size

It is interesting to also compare this with the case where the central region is empty (so that we apply a zero stress boundary condition at the inner boundary), as is the case for an epithelial cyst. We plot this expansion factor α₀ in figure 5(b) for different values of tissue stiffness E_t with ${r}_{1}=\tfrac{10}{9}{r}_{0}$. For all values of λ_s the model predicts that even without additional mechanical forces the central lumen will close (α₀ → 0) when

$$\begin{eqnarray*}&&\lambda =\displaystyle \frac{{E}_{t}}{{E}_{g}}\displaystyle \frac{1+{\nu }_{g}}{1+{\nu }_{t}}+3\displaystyle \frac{1-{\nu }_{t}}{1+{\nu }_{t}}{\left(1-{\left(\displaystyle \frac{{r}_{0}}{{r}_{1}}\right)}^{3}\right)}^{-1},\end{eqnarray*}$$

although this may in practice be a point beyond the validity of the linear model, see figure 5(b). Variations in shell thickness have a weak influence on the expansion factor α₀ as compared with variations in the constraining gel stiffness in this initial phase of growth. For example, for a growth factor of λ = 1.2, unless the outer gel is relatively soft, once ${r}_{1}\gt \tfrac{5}{4}{r}_{0}$ changes in shell thickness do not have a discernable affect on α₀ for both the hollow centre and deformable centre cases, see ESI figures S3(a), (b).