Propagation of damage in brain tissue: coupling the mechanics of oedema and oxygen delivery

Abstract

Brain tissue swelling, or oedema, is a dangerous consequence of traumatic brain injury and stroke. In particular, a locally swollen region can cause the injury to propagate further through the brain: swelling causes mechanical compression of the vasculature in the surrounding tissue and so can cut off that tissue’s oxygen supply. We use a triphasic mathematical model to investigate this propagation, and couple tissue mechanics with oxygen delivery. Starting from a fully coupled, finite elasticity, model, we show that simplifications can be made that allow us to express the volume of the propagating region of damage analytically in terms of key parameters. Our results show that performing a craniectomy, to alleviate pressure in the brain and allow the tissue to swell outwards, reduces the propagation of damage; this finding agrees with experimental observations.

Appendices

Appendix 1: The coupled mechanics-oxygen damage model

In this Appendix, we state the governing equations at the $n$th iteration, for the coupled mechanics-oxygen damage model. At step $n=1$, we choose $R_{(1)}$ to be the initial radius of damage, and set $f_{(0)}(R)\equiv R$ everywhere (to represent that the tissue is initially undeformed).

• Step (a): Find the oxygen distribution $\mathcal {C}_{(n)}(R)$

We find the oxygen concentration profile at step $n$, $\mathcal {C}_{(n)}(R)$, by solving the steady diffusion-uptake equation for oxygen concentration (22). This equation can be rewritten, assuming spherical symmetry, as

$$\begin{aligned} \frac{\text {d}^2 \mathcal {C}_{(n)}}{\text {d} R^2}+\left( \frac{2f'_{(n-1)}}{f_{(n-1)}}-\frac{f_{(n-1)}''}{f_{(n-1)}'} \right) \frac{\text {d} \mathcal {C}_{(n)}}{\text {d} R}=f_{(n-1)}'^2\frac{\bar{V}_\mathrm{m}\mathcal {C}_{(n)}}{\mathcal {C}_{(n)}+\bar{k}_\mathrm{m}}, \end{aligned}$$

(28)

along with boundary conditions,

$$\begin{aligned}&\left. \frac{\text {d} \mathcal {C}_{(n)}}{\text {d} R}\right| _{0}=0, \left. \frac{\text {d} \mathcal {C}_{(n)}}{\text {d} R}\right| _{1}=0, \end{aligned}$$

(29)

$$\begin{aligned}&{\left\{ \begin{array}{ll} \left. {\mathcal {C}_{(n)}}\right| _{R_{\mathrm{cap}, i}}=1, &{} \text {if } {R_{\mathrm{cap}, i}}>R_{(n)},\\ \left[ {\mathcal {C}_{(n)}} \right] _{R_{\mathrm{cap}, i}}=0=\displaystyle \left[ \frac{1}{f'_{n{-}1}} {\frac{\text {d} \mathcal {C}_{(n)}}{\text {d} R}} \right] _{R_{\mathrm{cap}, i}},&{} \text {if } {R_{\mathrm{cap}, i}}\le R_{(n)},\\ \end{array}\right. } \end{aligned}$$

(30)

where ${R_{\mathrm{cap}, i}}$ is the position of the $i$th capillary (§2.1). The last boundary condition in (30) is a mathematical representation of the assumption that if the capillaries are closed (i.e., if they are located within the damaged region $0<R<R_{(n)}$). then they do not alter the flux or concentration of oxygen through them. However, if the capillaries are open ($R_{(n)}<R<1$) then they fix the concentration at that point to be 1. Note that the oxygen profile is solved on the deformed domain computed at the previous iteration (and hence $f_{(n-1)}$ appears in these governing equations).

Equation (28), along with boundary conditions (29)–(30), is solved using the multipoint boundary value solver ‘bvp5c’ in Matlab.

• Step (b): Find the tissue deformation $f_{(n)}(R)$

We impose an FCD ${c^{(f)}_0}_{(n)}(R)$ wherever the oxygen concentration $\mathcal {C}_{(n)}$ falls below the critical value $\bar{\mathcal {C}}_\mathrm{crit}$,

$$\begin{aligned} {c^{(f)}_0}_{(n)}(R)= {\left\{ \begin{array}{ll} {\overline{c}^f_0}_I, &{} \text {where } \mathcal {C}_{(n)}(R)\le \bar{\mathcal {C}}_\mathrm{crit}, \\ 0, &{} \text {where } \mathcal {C}_{(n)}(R)> \bar{\mathcal {C}}_\mathrm{crit}, \\ \end{array}\right. } \end{aligned}$$

(31)

where ${\overline{c}^f_0}_I$ is a constant parameter representing the FCD of infarcted tissue.

If $\mathcal {C}_{(n)}> \bar{\mathcal {C}}_\mathrm{crit}$ everywhere then the tissue receives sufficient oxygen everywhere. Then there is no propagation (and hence we cease the iterations). If $\mathcal {C}_{(n)}< \bar{\mathcal {C}}_\mathrm{crit}$ everywhere then the entire tissue is ischaemic and we set the radius of ischaemic tissue, $R_{b (n)}=1$. Otherwise, let us define $R_{b (n)}$ as the value of $R$ such that $\mathcal {C}_{(n)}= \bar{\mathcal {C}}_\mathrm{crit}$ ¹.

Then, since the FCD is constant within each of the regions $0<R<R_{b (n)}$ and $R_{b (n)}<R<1$, the governing Eq. (23) can be solved with a constant FCD in each region. Boundary conditions must be applied at the interface between the damaged and healthy regions at $R=R_{b (n)}$ [Eq. (37)]. In a spherically symmetric geometry, the radial component of the governing equation $\nabla _\mathbf{X} \cdot \mathbf{S}_{e, n}=\mathbf{0}$ may be written,

$$\begin{aligned}&\frac{-{f_{(n)}^2c^{(f)}}^2}{R^2(J_{(n)}-\phi _0^s)\sqrt{{c^*}^2+{c^{(f)}}^2}}\nonumber \\&\quad \left[ \frac{f_{(n)}^2f_{(n)}''}{R^2} +\frac{2f_{(n)}f_{(n)}'^2}{R^2}-\frac{2f_{(n)}^2f_{(n)}'}{R^3}\right] \nonumber \\&\quad =\frac{\text {d} (S_{RR(n)})}{\text {d} R}+\frac{2(S_{RR(n)}-S_{\Theta \Theta (n)})}{R}, \end{aligned}$$

(32)

where $c^{(f)}$ is given by (7), $J_{(n)}=f_{(n)}^2f_{(n)}'/R^2$ is the volume change between the reference and deformed configurations, and $S_{RR(n)}$ and $S_{\Theta \Theta (n)}$ are the diagonal components of the effective Piola-Kirchhoff stress $\mathbf{S}_{e (n)}$. Note that in the outer region ($R_{b (n)}<R<1$), the FCD $c^{(f)}\equiv 0$ and thus the LHS of Eq. (32) is zero.

Several strain energy functions have been utilised in the literature to represent brain tissue. For example, an Ogden type hyperelastic model was applied to brain tissue by García and Smith (2009) to model infusion tests, whilst the Fung model was used by Elkin et al. (2010) and Lang et al. (2014) to model the equilibrium behaviour of brain tissue slices in solution baths of differing concentrations. Lang et al. (2014) investigated the behaviour of several of these strain energy functions and showed that, for moderate homogeneous swelling (bulk volume changes of up to 50 %), the neo-Hookean, Fung and Ogden models give quantitatively similar behaviour for parameters relevant to the brain.

Therefore, for simplicity we consider a neo-Hookean strain energy (see Ogden 1984 for example). The components $S_{RR}$ and $S_{\Theta \Theta }$ required for the stress balance (32) may be written as functions of the radial deformation $f(R)$:

$$\begin{aligned} S_{RR(n)}&=\frac{1}{f_{(n)}'}\left[ \bar{\mu }_s(f_{(n)}'^2-1)+\bar{\lambda }_s J_{(n)}(J_{(n)}-1) \right] , \end{aligned}$$

(33)

$$\begin{aligned} S_{\Theta \Theta (n)}&=\frac{R}{f_{(n)}}\left[ \bar{\mu }_s\left( \frac{f_{(n)}^2}{R^2}-1\right) +\bar{\lambda }_s J_{(n)}(J_{(n)}-1) \right] . \end{aligned}$$

(34)

The Cauchy stress $\varvec{\sigma }_{(n)}$ is defined as,

$$\begin{aligned} \varvec{\sigma }_{(n)}={\varvec{\sigma }_{e_{(n)}}}-\left[ \sqrt{{\bar{c}^*}{}^2+\left( {\frac{\phi _0^w{{c^{(f)}_0}_{(n)}(R)}}{ J_{(n)}-1+\phi _0^w}}\right) ^2}-\bar{c}^*\right] , \end{aligned}$$

(35)

where the effective Cauchy stress ${\varvec{\sigma }_{e, n}}$ is related to the effective Piola-Kirchhoff stress $\mathbf{S}_{e (n)}$ by ${\varvec{\sigma }_{e (n)}}=(1/{J_{(n)}})\mathbf{F}_{(n)}\cdot {\mathbf{S}_{e (n)}}$. Note from Eq. (35) that in the absence of FCD ($c^{(f)}_0=0$), the Cauchy and effective Cauchy stresses are equal.

The boundary conditions for $f_{(n)}$ consist of no displacement at the origin,

$$\begin{aligned} \left. f_{(n)} \right| _0=0, \end{aligned}$$

(36)

along with continuity of displacement and radial stress at the interface between the damaged and non-damaged region $R=R_{b (n)}$,

$$\begin{aligned} \left[ f_{(n)} \right] _{R_{b (n)}}&=0, \quad \left[ \hat{\mathbf{e}}_r\cdot {\varvec{\sigma }_{(n)}}\cdot \hat{\mathbf{e}}_r\right] _{R_{b (n)}}=0. \end{aligned}$$

(37)

Note that if $R_{b (n)}=1$ (i.e., the entire tissue has insufficient oxygen, so that $c^{(f)}_0(R)={\bar{c}_0^f}{}_I$ everywhere), then this boundary condition (37) is unnecessary. At the outer boundary $R=1$, we impose either no displacement or no stress,

$$\begin{aligned} \left. f_{(n)} \right| _{1}&=1, \quad \text {or} \quad \left. \hat{\mathbf{e}}_r\cdot {\varvec{\sigma }_{(n)}}\cdot \hat{\mathbf{e}}_r\right| _{1}&=0. \end{aligned}$$

(38)

The governing Eq. (32), along with boundary conditions (36)–(38), is solved using a multipoint boundary value solver, ‘bvp5c’, in Matlab.

• Step (c): Find the new damage radius, $R_{(n+1)}$

We now consider the maximum compressive stress ${\sigma _{\mathrm {max}}}_{(n)}$ $(R):= \max (-(\hat{\mathbf{e}}_r\cdot {\varvec{\sigma }_{(n)}}\cdot \hat{\mathbf{e}}_r), -(\hat{\mathbf{e}}_\theta \cdot {\varvec{\sigma }_{(n)}}\cdot \hat{\mathbf{e}}_\theta ))$. We shall show in §1 and §4.2 that ${\sigma _{\mathrm {max}}}_{(n)}(R)$ is a monotonic decreasing function over $R_{(n)}<R<1$; thus, there are three possible cases:

• (i) ${\sigma _{\mathrm {max}}}_{(n)}<\bar{\sigma }_\mathrm{crit}$ for all $R_{(n)}<R<1$: the damage does not propagate further, and $R_{(n+1)}=R_{(n)}$.

• (ii) ${\sigma _{\mathrm {max}}}_{(n)}=\bar{\sigma }_\mathrm{crit}$ for some $R_{(n)}<R<1$: we define $R_{(n+1)}$ such that $\left. {\sigma _{\mathrm {max}}}_{(n)}\right| _{R_{(n+1)}}=\bar{\sigma }_\mathrm{crit}$.

• (iii) ${\sigma _{\mathrm {max}}}_{(n)}>\bar{\sigma }_\mathrm{crit}$ for all $R_{(n)}<R<1$: the entire tissue is damaged; we set $R_{(n+1)}=1$.

We repeat steps (a) to (c) for step $n+1$ and iterate. A flow chart summary of this process is shown in Fig. 2.

Appendix 2: The mechanics-only damage model

Initially, we impose some initial radius of damage, $R_{(1)}$, and in subsequent steps, we take the value of $R_{(n)}$ from the previous step. At step $n$, we impose an FCD,

$$\begin{aligned} {c^{(f)}_0}_{(n)}(R)={\left\{ \begin{array}{ll} {\overline{c}^f_0}_I,&{} R<R_{(n)},\\ 0, &{} R\ge R_{(n)},\\ \end{array}\right. } \end{aligned}$$

(39)

and solve Eq. (32) in each of the regions $0<R<R_{(n)}$ and $R_{(n)}<R<1$, along with the boundary conditions (36) at $R=0$, (38) at $R=1$, and at $R_{(n)}$:

$$\begin{aligned} \left[ f_{(n)} \right] _{R_{(n)}}&=0, \quad \left[ \hat{\mathbf{e}}_r\cdot {\varvec{\sigma }_{(n)}}\cdot \hat{\mathbf{e}}_r\right] _{R_{(n)}}=0, \end{aligned}$$

(40)

where the Cauchy stress $\varvec{\sigma }_{(n)}$ is given by (35). We then define the maximum compressive stress as ${\sigma _{\mathrm {max}}}_{(n)}(R):= \max (-(\hat{\mathbf{e}}_r\cdot {\varvec{\sigma }_{(n)}}\cdot \hat{\mathbf{e}}_r), -(\hat{\mathbf{e}}_\theta \cdot {\varvec{\sigma }_{(n)}}\cdot \hat{\mathbf{e}}_\theta ))$, and find $R_{(n+1)}$ such that $\left. {\sigma _{\mathrm {max}}}_{(n)}\right| _{R_{(n+1)}}=\bar{\sigma }_\mathrm{crit}$ as described in step (c) of the ‘mechanics-oxygen’ model (§2.5). A flow chart of the stages shown in Fig. 4.

Appendix 3: Analytical solutions for infinitesimal deformations

In this Appendix, we consider the mechanics-only damage model described in §3.3, with the finite deformation governing equations replaced by governing equations valid only for infinitesimally small deformations. In the infinitesimal framework, deformations are assumed to be small and thus the Eulerian ($r$) and Lagrangian ($R$) frames are the same to leading order.

1.1 Inner and outer solutions

At the $n$th iteration we have a particular damage radius $R_{(n)}$, such that ${{\bar{c}_0^f}{}}={{\bar{c}_0^f}{}_I}$ for $0\le r\le R_{(n)}$ (the inner region), and ${{\bar{c}_0^f}{}}=0$ for $R_{(n)}<r\le 1$ (the outer region). Let us consider each of these regions separately. We define $u_{I (n)}(r)$, $u_{O (n)}(r)$ as the radial displacement in the inner and outer regions, respectively. In each region, the linear strain tensor is defined $\mathbf{e}_{i (n)}=\text {diag}[{\text{ d }}u_{i (n)}/{\text{ d }}r, u_{i (n)}/r, u_{i (n)}/r]$, and the dilation as,

$$\begin{aligned} e_{i (n)}&=\text {tr}(\mathbf{e}_{i (n)})=\frac{1}{r^2}\frac{\partial }{\partial r}\left( r^2 u_{i (n)}\right) ,&(i&=I, O), \end{aligned}$$

(41)

We use the infinitesimal stress tensor for the effective Cauchy stress, so that ${\varvec{\sigma }_e}_{i (n)}=\bar{\lambda }_s e_{i (n)}+2\bar{\mu }_s \mathbf{e}_{i (n)}$. In infinitesimal deformation elasticity, the osmotic term in the Cauchy stress tensor (12) may be linearised. Then,

$$\begin{aligned}&\varvec{\sigma }_{i (n)}={\varvec{\sigma }_e}_{i (n)}\nonumber \\&\quad -\left[ \sqrt{{{\bar{c}_0^f}{}_i}^2+{\bar{c}^*{}}^2}-\bar{c}^*-\left( \frac{{{\bar{c}_0^f}{}_i}^2}{\phi _0^w \sqrt{{{\bar{c}_0^f}{}_i}^2+{\bar{c}^*}{}^2}} \right) e_{i (n)}\right] \mathbf{I}, \end{aligned}$$

(42)

where ${{\bar{c}_0^f}{}_I}$ is constant in the inner region, and ${{\bar{c}_0^f}{}_O}=0$ in the outer region.

In each region, the stress satisfies $\nabla \cdot {\varvec{\sigma }}_{i (n)}=0$. Substituting in the Cauchy stress, (42), we find that

$$\begin{aligned} 0=\frac{\partial }{\partial r}\left( \frac{1}{r^2}\frac{\partial }{\partial r}\left( r^2 u_{i (n)}\right) \right) , \end{aligned}$$

(43)

in both the inner and outer regions. The general solution of Eq. (43) is,

$$\begin{aligned} u_{i (n)}&=a_{i (n)}r+\frac{b_{i (n)}}{r^2}, \end{aligned}$$

(44)

where $a_{i (n)}$ and $b_{i (n)}$ are constants that must be determined from the boundary conditions.

At the origin, we have a no-displacement boundary condition,

$$\begin{aligned} \left. u_{I (n)}\right| _{0}&=0, \end{aligned}$$

(45)

which immediately gives $b_{I (n)}=0$. At the boundary between the outer and inner region, both the radial component of stress and the displacement must be continuous. Thus,

$$\begin{aligned} \left. u_{I (n)}\right| _{R_{(n)}}&=\left. u_{O (n)}\right| _{R_{(n)} },&\hat{\mathbf{e}}_r\cdot \left. {\varvec{\sigma }}_{i (n)}\right| _{R_{(n)}}\cdot \hat{\mathbf{e}}_r&=\hat{\mathbf{e}}_r\cdot \left. {\varvec{\sigma }}_{O (n)}\right| _{R_{(n)}}\cdot \hat{\mathbf{e}}_r. \end{aligned}$$

(46)

At the outer edge, we apply either no-displacement or no-stress,

$$\begin{aligned} \left. u_{O (n)}\right| _{1}&=0,&\text {or}&\hat{\mathbf{e}}_r\cdot \left. {\varvec{\sigma }}_{O (n)}\right| _{1}\cdot \hat{\mathbf{e}}_r&=0. \end{aligned}$$

(47)

We may now substitute the general expression for the displacement (44) into the boundary conditions (45)–(47) to solve for the displacement. Once $u_{i (n)}$ is determined, it is a simple matter to determine $\varvec{\sigma }_{i (n)}$ and hence the region of $r$ for which $\sigma _{\mathrm{max}}>\bar{\sigma }_{\mathrm{crit}}$. We consider the cases of no-displacement and no-stress boundary conditions separately.

1.2 No-displacement outer boundary (intact skull)

We will first consider the no-displacement outer boundary condition ($\left. u_{O (n)}\right| _{1}=0$). Using Eq. (44) and boundary conditions (45)–(47), we find that the solutions for the inner and outer displacement may be written,

$$\begin{aligned} u_{I (n)}&=A_{(n)}\left( \frac{1}{R_{(n)}^3}-1\right) r,&u_{O (n)}&=A_{(n)}\left( \frac{1}{r^2}-r\right) , \end{aligned}$$

(48)

where the value of $A_{(n)}$ depends on $R_{(n)}$ according to,

$$\begin{aligned} A_{(n)}:=A(R_{(n)})&=\frac{\alpha _1R_{(n)}^3}{\alpha _2-3R_{(n)}^3\alpha _3}, \end{aligned}$$

(49)

where the constants $\alpha _1$, $\alpha _2$, $\alpha _3$ are given by,

$$\begin{aligned} \alpha _1&= \sqrt{{{\bar{c}_0^f}{}_I}^2+{\bar{c}^*{}}^2}-\bar{c}^*,&\alpha _2&= 3\bar{\kappa }_s+4\bar{\mu }_s+ 3\alpha _3, \end{aligned}$$

(50)

$$\begin{aligned} \alpha _3&=\frac{{{\bar{c}_0^f}{}_I}^2}{\phi _0^w \sqrt{{{\bar{c}_0^f}{}_I}^2+{\bar{c}^*}{}^2}}, \end{aligned}$$

(51)

and $\bar{\kappa }_s=\bar{\lambda }_s+2/3\bar{\mu }_s$ is the dimensionless bulk modulus. Using the general form of the Cauchy stress tensor (42), we can also write the radial component of the Cauchy stress in each of the inner and outer regions,

$$\begin{aligned} \sigma _{I (n)}&=3A_{(n)}\left( \bar{\kappa }_s+ \alpha _3 \right) \left( \frac{1}{R_{(n)}^3}-1\right) -\alpha _1,\end{aligned}$$

(52)

$$\begin{aligned} \sigma _{O (n)}&=-A_{(n)}\left[ 3\bar{\kappa }_s+\frac{4\bar{\mu }_s}{r^3}\right] . \end{aligned}$$

(53)

The maximal compressive stress in the tissue is,

$$\begin{aligned} \sigma _{\mathrm{max} (n)}(r)=&{\left\{ \begin{array}{ll}\displaystyle \alpha _1-3A_{(n)}\left( \bar{\kappa }_s+ \alpha _3 \right) \left( \frac{1}{R_{(n)}^3}-1\right) ,&{} r\le R_{(n)},\\ \displaystyle A_{(n)}\left( 3\bar{\kappa }_s+\frac{4\bar{\mu }_s}{r^3}\right) , &{} r>R_{(n)}. \end{array}\right. } \end{aligned}$$

(54)

Equation (54) shows that at every step of the iteration, the maximum compressive Cauchy stress is constant within the inner region ($0<r<R_{(n)}$) and monotonic decreasing in the outer region ($R_{(n)}<r<1$). This trend agrees well with the numerical results obtained in §4.2 (see Fig. 8) for the finite elasticity case.

For some radius of damage $R_{(n)}$, we now consider the damage radius following one iteration, $R_{(n+1)}$. Following the procedure described in §3.3, there are three cases to consider. Firstly, if the maximal compressive stress in the tissue $\sigma _{\mathrm{max} (n)}(r)$ is below the critical stress $\bar{\sigma }_\mathrm{crit}$ everywhere, then, damage does not propagate and $R_{(n+1)}=R_{(n)}$ $\forall n>0$. Secondly, if there is some value of $R_{(n+1)}$ such that $R_{(n)}<R_{(n+1)}<1$ and $\sigma _{\mathrm{max} (n)}(R_{(n+1)})=\bar{\sigma }_\mathrm{crit}$, then the damage will propagate up to that value. Thirdly, if the compressive stress $\bar{\sigma }_\mathrm{crit}$ exceeds the critical stress everywhere, then the entire tissue is damaged and hence $R_{(n+1)}=1$.

These three cases depend on the size of $R_{(n)}$, relative to two length scales $r_a$ and $r_b$,

$$\begin{aligned} r_{a}&=\left( {\frac{\alpha _2\bar{\sigma }_\mathrm{crit}-4\alpha _1\bar{\mu }_s}{3(\alpha _1\bar{\kappa }_s+\alpha _3\bar{\sigma }_\mathrm{crit})} }\right) ^{1/3}, \end{aligned}$$

(55)

$$\begin{aligned} r_{b}&=\left( {\frac{\alpha _2\bar{\sigma }_\mathrm{crit}}{4\alpha _1\bar{\mu }_s+3(\alpha _1\bar{\kappa }_s+\alpha _3\bar{\sigma }_\mathrm{crit})}}\right) ^{1/3}, \end{aligned}$$

(56)

the values of $R_{(n)}$ such that $\sigma _{\mathrm{max} (n)}(R_{(n)})=\bar{\sigma }_\mathrm{crit}$ and $\sigma _{\mathrm{max} (n)}(1)=\bar{\sigma }_\mathrm{crit}$ respectively.

These three cases can be summarised as,

(57)

where the middle line in Eq. (57) is obtained by finding the value of $R_{(n+1)}$ such that ${\sigma }_{\mathrm{max} (n)}(R_{(n+1)})=\bar{\sigma }_\mathrm{crit}$ in Eq. (54).

Ultimately given some initial radius of damage, $R_{(1)}$, we are interested in finding $R_{(\infty )}$, the final radius of damage. We thus consider fixed points $R^*$ of the first-return map (57), by seeking solutions to (57) such that $R^*=R_{(n)}=R_{(n+1)}\forall n\ge 1$. It is trivial to observe that $R^*$ can take all values less than or equal to $r_a$, and also $1$. However, $R^*$ cannot take any values between $r^a$ and 1.

We now consider the final radius of damage, $R_{(\infty )}$, for a given initial radius of damage $R_{(1)}$. If $R_{(1)}\le r_a$, then $R_{(1)}$ is a fixed point of the map; damage does not propagate and trivially $R_{(\infty )}=R_{(1)}$. Rearranging the expression (57), we find that if $r_{a}<R_{(n)}\le r_b$ then $R_{(n+1)}>R_{(n)}$. Thus if $r_{a}<R_{(1)}\le r_b$ then the damage radius $R_{(n)}$ will increase at each iteration until $R_{(n)}>r_b$, following which $R_{(n+1)}$ reaches a fixed point so that $R_{(\infty )}=1$; propagation is halted as the entire tissue is damaged. Finally, if $r_b<R_{(1)}\le 1$ then $R_{(\infty )}=1$. These cases are summarised in Eq. (25).

1.3 No-stress outer boundary (craniectomy)

We now consider the no-stress outer boundary condition ($\hat{\mathbf{e}}_r\cdot \left. {\varvec{\sigma }}_{O (n)}\right| _{1}\cdot \hat{\mathbf{e}}_r=0$). Using Eq. (44) and boundary conditions (45)–(47) we find that the solutions for the inner and outer displacement may be written,

$$\begin{aligned} u_{I (n)}&=B_{(n)}\left( \frac{4\bar{\mu }_s}{3\bar{\kappa }_s}+\frac{1}{R_{(n)}^3}\right) r,&u_{O (n)}&=B_{(n)}\left( \frac{4\bar{\mu }_s}{3\bar{\kappa }_s}r+\frac{1}{r^2}\right) , \end{aligned}$$

(58)

where $B_{(n)}$ is a function of $R_{(n)}$ defined by,

$$\begin{aligned} B_{(n)}:=B(R_{(n)})&=\frac{\bar{\kappa }_s \alpha _1R_{(n)}^3}{\bar{\kappa }_s \alpha _2+ 4\bar{\mu }_s \alpha _3R_{(n)}^3}, \end{aligned}$$

(59)

the constants $\alpha _1$, $\alpha _2$, $\alpha _3$ are given by Eq. (51).

Using the general form of the Cauchy stress tensor (42), we can also write the maximum compressive Cauchy stress in each of the inner and outer regions,

$$\begin{aligned} \sigma _{\mathrm{max} (n)}(r)=&{\left\{ \begin{array}{ll}\displaystyle \alpha _1-\left( 3\bar{\kappa }_s+ \alpha _3 \right) B_{(n)}\left( \frac{4\bar{\mu }_s}{3\bar{\kappa }_s}+\frac{1}{R_{(n)}^3}\right) , &{} r\le R_{(n)},\\ \displaystyle 4\bar{\mu }_sB_{(n)}\left( \frac{1}{r^3}-1\right) , &{} r>R_{(n)}. \end{array}\right. } \end{aligned}$$

(60)

Again, we observe that $\sigma _{\mathrm{max} (n)}$ is constant for $r\le R_{(n)}$ and monotonic decreasing for $r>R_{(n)}$, consistent with the stress profiles shown in Fig. 8 for the finite elasticity case.

We now consider the relationship between the initial radius of damage, $R_{(1)}$, and the radius of damage after a single iteration, $R_{(2)}$. We find that there is a value $r_c$,

$$\begin{aligned} r_c=\left( {1-\frac{\bar{\sigma }_\mathrm{crit}(\bar{\kappa }_s\alpha _2+4\bar{\mu }_s\alpha _3)}{ 4\bar{\mu }_s\left( {\bar{\kappa }_s} \alpha _1+\bar{\sigma }_\mathrm{crit}\alpha _3\right) }}\right) ^{1/3}, \end{aligned}$$

(61)

such that $\sigma _{\mathrm{max} (n)}(R_{(1)})\ge \bar{\sigma }_\mathrm{crit}$ if $R_{(1)}\le r_c$ and $\sigma _{max (n)}(R_{(1)})$ $< \bar{\sigma }_\mathrm{crit}$ if $R_{(1)}> r_c$. Note that $r_c<1$. We find an expression for $R_{(2)}$,

$$\begin{aligned} R_{(n+1)}= {\left\{ \begin{array}{ll} \displaystyle \left( {\frac{4\bar{\mu }_sB_1}{\bar{\sigma }_\mathrm{crit}+4\bar{\mu }_sB_1}}\right) ^{1/3},&{} R_{(n)}\le r_c,\\ R_{(n)}, &{} R_{(n)}>r_c, \end{array}\right. } \end{aligned}$$

(62)

where the uppermost line in Eq. (62) is obtained by setting ${\sigma }_{\mathrm{max} (1)}(R_{(n+1)})=\bar{\sigma }_\mathrm{crit}$ in Eq. (60), and $r_c$ is the value of $R_{(1)}$ such that $\sigma _{\mathrm{max} (1)}(R_{(n)})=\sigma _{\mathrm {crit}}$. We observe that if $\bar{\sigma }_\mathrm{crit}>4\bar{\mu }_s\alpha _1/\alpha _2$ then there are no positive values for $r_c$. In this case no propagation can occur, since then $R_{(n+1)}=R_{(n)}$ for all positive $R_{(n)}$.

Finally, we consider the fixed points $R^*$ of the map (62), in order to identify the relationship between the initial damage radius $R_{(1)}$ and the final damage radius $R_{(\infty )}$. By seeking solutions of (62) for which $R^*=R_{(n+1)}=R_{(n)}\forall n \ge 1$, we find that there are no fixed points that take values less than $r_c$ (since $R_{(n+1)}>R_{(n)}$ if $R_{(n)}<r_c$), but observe that all radial values greater than $r_c$ are fixed points. Furthermore, rearrangement of Eq. (62) gives that $R_{(n+1)}\le r_c$ if $R_{(n)}\le r_c$, meaning that $R_{(\infty )}=r_c$ if $R_{(1)}\le r_c$. These results are summarised in Eq. (27).