Using Experimental Data and Information Criteria to Guide Model Selection for Reaction–Diffusion Problems in Mathematical Biology

Abstract

Reaction–diffusion models describing the movement, reproduction and death of individuals within a population are key mathematical modelling tools with widespread applications in mathematical biology. A diverse range of such continuum models have been applied in various biological contexts by choosing different flux and source terms in the reaction–diffusion framework. For example, to describe the collective spreading of cell populations, the flux term may be chosen to reflect various movement mechanisms, such as random motion (diffusion), adhesion, haptotaxis, chemokinesis and chemotaxis. The choice of flux terms in specific applications, such as wound healing, is usually made heuristically, and rarely it is tested quantitatively against detailed cell density data. More generally, in mathematical biology, the questions of model validation and model selection have not received the same attention as the questions of model development and model analysis. Many studies do not consider model validation or model selection, and those that do often base the selection of the model on residual error criteria after model calibration is performed using nonlinear regression techniques. In this work, we present a model selection case study, in the context of cell invasion, with a very detailed experimental data set. Using Bayesian analysis and information criteria, we demonstrate that model selection and model validation should account for both residual errors and model complexity. These considerations are often overlooked in the mathematical biology literature. The results we present here provide a straightforward methodology that can be used to guide model selection across a range of applications. Furthermore, the case study we present provides a clear example where neglecting the role of model complexity can give rise to misleading outcomes.

Appendices

Appendix A: Analysis of Wave Front Concavity

The Generalised Porous Fisher model (Eq. (4) in the main text) with $\lambda = 0$, in one dimension, is

$$\begin{aligned} \frac{\partial {C}}{\partial {t}} = \frac{\partial }{\partial x} \left[ D_0 \left( \frac{C}{K}\right) ^r\frac{\partial {C}}{\partial {x}}\right] , \quad -\infty< x < \infty , \end{aligned}$$

(16)

where $D_0$ is the free diffusivity and K the cell carrying capacity density. For the initial condition $C(x,0) = C_0\delta (x)$, Eq. (16) has an exact solution,

$$\begin{aligned} C(x,t) = {\left\{ \begin{array}{ll} \dfrac{K}{h(t)}\left[ 1 - \left( \dfrac{x}{d_0 h(t)}\right) ^2\right] ^{1/r}, &{} \quad |x| \le d_0 h(t), \\ 0, &{}\quad |x| > d_0 h(t), \end{array}\right. } \end{aligned}$$

where $d_0 = C_0 \varGamma (1/r + 3/2)/(K\sqrt{\pi }\varGamma (1/r + 1))$, $t_0 = d_0^2 r /(2D_0(r+2))$, $h(t) = (t/t_0)^{1/(r + 2)}$ and $\varGamma (x)$ is the Gamma function. This solution, often called the source solution for the porous media equation, has compact support, $x \in [-d_0 h(t),d_0h(t)]$. Here, $|x| = d_0 h(t)$ are the contact points. This solution is very different to the source solution for the linear diffusion equation, $r=0$, which is a Gaussian function without compact support (Barenblatt 2003; Crank 1975).

Without loss of generality, we now only consider the positive real line $x \ge 0$. The cell density is always decreasing as we approach the contact point, that is, $\partial C /\partial x < 0$ for $0< x < d_0 h(t)$. Specifically, we have

$$\begin{aligned} \frac{\partial C}{\partial x} = \frac{-2 K x}{d_0^2 h(t)^3 r}\left[ 1 - \left( \frac{x}{d_0 h(t)}\right) ^2\right] ^{1/r -1}. \end{aligned}$$

(17)

From Eq. (17), three different front properties are possible. As $x \rightarrow d_0 h(t)$, we observe: (i) a sharp decreasing function with non-negative gradient, for $0< r < 1$, as in Fig. 2e–h; (ii) a sharp front with finite negative slope, for $r = 1$, as in Fig. 2i–l, with $\partial C /\partial x \rightarrow -2K/(d_0^2h(t)^3)$; and (iii) a sharp front with unbounded negative slope, for $r > 1$, as in Fig. 2m–p, with $\partial C /\partial x \rightarrow -\infty$.

To explore the concavity of the density profile, C(x, t), at the contact point, it is sufficient to show how the sign of $\partial ^2 C/\partial x^2$ at the contact point depends on r. The second derivative with respect to x, for $r > 0$, is

$$\begin{aligned} \frac{\partial ^2 C}{\partial x^2}= & {} -\frac{2 K}{d_0^2 h(t)^3r}\left[ 1 - \left( \frac{x}{d_0 h(t)}\right) ^2\right] ^{1/{r}}\left\{ \left[ 1 - \left( \frac{x}{d_0 h(t)}\right) ^2\right] ^{-1} \right. \\&\quad \left. -\,\frac{2x^2(1-r)}{d_0^2 h(t)^2 r} \left[ 1 - \left( \frac{x}{d_0 h(t)}\right) ^2\right] ^{-2}\right\} . \end{aligned}$$

We have, for $0< r < 1$, that $\partial ^2 C/\partial x^2 > 0$ as $x \rightarrow d_0 h(t)$. For $r \ge 1$, $\partial ^2C/\partial x^2 < 0$ as $x \rightarrow d_0 h(t)$. Hence, at the contact point, $x = d_0 h(t)$, the solution is concave down for $r \ge 1$ and concave up otherwise.

Appendix B: Numerical Scheme

Here we describe our numerical scheme for the computational solution to the following reaction–diffusion equation:

$$\begin{aligned} \frac{\partial {C}}{\partial {t}} = \frac{\partial }{\partial x} \left[ D(C) \frac{\partial {C}}{\partial {x}}\right] + S(C), \quad 0< t< T,\quad 0< x < L, \end{aligned}$$

(18)

with the initial condition,

$$\begin{aligned} C(x,t) = C_0(x), \quad t = 0, \end{aligned}$$

and boundary conditions,

$$\begin{aligned} \frac{\partial {C}}{\partial {x}} = 0, \quad x = 0\text { and } x = L. \end{aligned}$$

Consider N points in space, $\{x_i\}_{i=1}^N$, with $x_1 = 0$, $x_N = L$ and $\varDelta x = x_{i+1} - x_{i}$ for all $i = [1,2,\ldots , N]$. Similarly, define M temporal points, $\{t_j\}_{j=1}^M$, with $t_1 = 0$, $t_M = T$ and $\varDelta t = t_{j+1} - t_{j}$ for all $j = [1,2, \ldots , T]$. Next, define the notation, $C_{i+k} = C(x_i + k\varDelta x,t)$, and $C_{i+k}^{j+s}~=~C(x_i + k \varDelta x, t_j + s \varDelta t)$.

Let $J(C) = - D(C)\partial C/ \partial x$ and substitute into Eq. (18) to yield

$$\begin{aligned} \frac{\partial {C}}{\partial {t}} = -\frac{\partial {J}}{\partial {x}} + S(C). \end{aligned}$$

At the ith point, apply a first-order central difference to $\partial J / \partial x$ with step $\varDelta x/2$. The result is the system of ODEs

$$\begin{aligned} \frac{\text {d} C_i}{\text {d} t} = -\frac{1}{\varDelta x}\left[ J(C_{i+1/2}) - J(C_{i-1/2})\right] + S(C_i), \quad i=1,2,\ldots N. \end{aligned}$$

(19)

Similarly, a first-order central difference is applied to $J(C_{i+1/2})$ and $J(C_{i-1/2})$ using the step $\varDelta x/2$ yields

$$\begin{aligned} J(C_{i+1/2})&= -\frac{1}{\varDelta x}D(C_{i+1/2})\left( C_{i+1} - C_i\right) , \end{aligned}$$

(20)

$$\begin{aligned} J(C_{i-1/2})&= -\frac{1}{\varDelta x}D(C_{i-1/2})\left( C_{i} - C_{i-1}\right) . \end{aligned}$$

(21)

It is important to note that we will only obtain a solution for $C_{i+k}$ at integer values of k; therefore, the evaluation of the diffusion terms in Eqs. (20) and (21) cannot be directly computed since $k = \pm 1/2$. We thus approximate with an averaging scheme,

$$\begin{aligned} D(C_{i+1/2})&= \frac{1}{2}\left( D(C_{i+1}) + D(C_{i})\right) , \end{aligned}$$

(22)

$$\begin{aligned} D(C_{i-1/2})&= \frac{1}{2}\left( D(C_{i}) + D(C_{i-1})\right) . \end{aligned}$$

(23)

After substitution of Eqs.  (20), (21), (22) and (23) into Eq. (19), we have the coupled system of nonlinear ODEs defined in terms of our spatial discretisation,

$$\begin{aligned} \frac{\text {d} C_i}{\text {d} t}&= \frac{1}{2 \varDelta x^2}\left[ \left( D(C_{i+1}) + D(C_i)\right) \left( C_{i+1} - C_i\right) \right. \\&\quad - \left. \left( D(C_{i}) + D(C_{i-1})\right) \left( C_{i} - C_{i-1}\right) \right] + S(C_i), \quad i = 1, 2, \ldots , N. \end{aligned}$$

The no-flux boundaries are enforced using first-order forward differences

$$\begin{aligned} \frac{C_1 - C_0}{\varDelta x} = 0 \quad \text {and}\quad \frac{C_{N+1} - C_N}{\varDelta x} = 0, \end{aligned}$$

where $C_0$ and $C_{N+1}$ represent the solution at “ghost nodes” that are not a part of the domain.

The ODEs are discretised in time using a first-order backward difference method leading to the backward-time, centred-space (BTCS) scheme,

$$\begin{aligned} \frac{C_1^{j+1} - C_0^{j+1}}{\varDelta x}&= 0, \nonumber \\ \frac{C_i^{j+1} - C_i^j}{\varDelta t}&= \frac{1}{2 \varDelta x^2}\left[ \left( D(C_{i+1}^{j+1}) + D(C_i^{j+1})\right) \left( C_{i+1}^{j+1} - C_i^{j+1}\right) \right. \nonumber \\&\quad - \left. \left( D(C_{i}^{j+1}) + D(C_{i-1}^{j+1})\right) \left( C_{i}^{j+1} - C_{i-1}^{j+1}\right) \right] + S(C_i^{j+1}),\quad i = 1,\ldots , N, \nonumber \\ \frac{C_{N+1}^{j+1} - C_{N}^{j+1}}{\varDelta x}&= 0. \end{aligned}$$

(24)

While this scheme is first order in time and space, it has the advantage of unconditional stability.

Since the scheme is implicit, a nonlinear root finding solver is required to compute solution at $t_{j+1}$ given a previously computed solution at time $t_j$. To achieve this, we apply fixed-point iteration. We re-arrange the system to be of the form $\mathbf {C}^{j+1} = \mathbf {G}(\mathbf {C}^{j+1})$ where $\mathbf {C}^{j+1} = \left[ C_0^{j+1}, C_1^{j+1},\ldots ,C_{N+1}^{j+1}\right] ^\text {T}$. That is,

$$\begin{aligned} \mathbf {G}(\mathbf {C}^{j+1}) = \left[ g_0(\mathbf {C}^{j+1}),g_1(\mathbf {C}^{j+1}),\ldots ,g_{N+1}(\mathbf {C}^{j+1})\right] , \end{aligned}$$

where

$$\begin{aligned} g_0(\mathbf {C}^{j+1})&= C_1^{j+1}, \nonumber \\ g_i(\mathbf {C}^{j+1})&= C_i^j + \frac{\varDelta t}{2 \varDelta x^2}\left[ \left( D(C_{i+1}^{j+1}) + D(C_i^{j+1})\right) \left( C_{i+1}^{j+1} - C_i^{j+1}\right) \right. \nonumber \\&\quad - \left. \left( D(C_{i}^{j+1}) + D(C_{i-1}^{j+1})\right) \left( C_{i}^{j+1} - C_{i-1}^{j+1}\right) \right] + S(C_i^{j+1}),\quad i = 1,\ldots , N, \nonumber \\ g_{N+1}(\mathbf {C}^{j+1})&= C_{N}^{j+1}. \end{aligned}$$

(25)

We then define the sequence $\{\mathbf {X}^k\}_{k\ge 0}$, generated through the nonlinear recurrence relation $\mathbf {X}^{k+1} = \mathbf {G}(\mathbf {X}^{k})$ with $\mathbf {X}^{0} = \mathbf {C}^j$. This sequence is iterated until $\Vert \mathbf {X}^{k+1} - \mathbf {X}^k\Vert _2 < \tau$, where $\tau$ is the error tolerance and $\Vert \cdot \Vert _2$ is the Euclidean vector norm. Once the sequence has converged, we set $\mathbf {C}^{j+1} = \mathbf {X}^{k+1}$ and continue to solve for the next time step.

For a given set of model parameters, the spatial and temporal step sizes, $\varDelta x$ and $\varDelta t$, need to be selected. In particular, the following condition must hold to ensure accuracy, $\max _{C \in [0,K]} D(C) < \varDelta x ^2 / \varDelta t$. We then refine $\varDelta x$ and $\varDelta t$ together to ensure solutions are independent of the discretisation. Note that as r increases, higher values of D(C) become valid; therefore, particular attention is required to generate Fig. 5 in the main text. The values of $\varDelta x$, $\varDelta t$ and $\tau$ used for the simulations in this work are shown in Table 2. Note that in all cases the discretisation is more refined than required to solve the given problem accurately.

Table 2 Discretisation and tolerance for numerical simulations

Appendix C: Computational Inference

The Bayesian inference problems described in the main text all require the computation of the posterior PDF. Up to a normalisation constant, the posterior PDF is given by

$$\begin{aligned} p(\varvec{\theta } \mid \mathscr {D}) \propto \mathscr {L}(\varvec{\theta } ; \mathscr {D})p(\varvec{\theta }). \end{aligned}$$

(26)

If the posterior distribution can be sampled, the posterior PDF may be determined by using Monte Carlo integration. Thus, the main requirement is a method of generating N independent, identically distributed (i.i.d.) samples from the posterior distribution.

For many applications of practical interest, Eq. (26) cannot be used directly to generate the samples required since the likelihood is often intractable. Approximate Bayesian computation (ABC) techniques resolve this complexity through the approximation (Sunnåker et al. 2013)

$$\begin{aligned} p(\varvec{\theta } \mid d( \mathscr {D}, \mathscr {D}_s)< \epsilon ) \propto \mathbb {P}(d( \mathscr {D}, \mathscr {D}_s) < \epsilon \mid \varvec{\theta }) p(\varvec{\theta }), \end{aligned}$$

(27)

where $d( \mathscr {D}, \mathscr {D}_s)$ is a discrepancy metric between the true data, $\mathscr {D}$, and simulated data, $\mathscr {D}_s\sim \mathscr {L}(\varvec{\theta } ; \mathscr {D}_s)$ and $\epsilon$ is the discrepancy threshold. ABC methods have the property that $p(\varvec{\theta } \mid d( \mathscr {D}, \mathscr {D}_s) < \epsilon ) \rightarrow p(\varvec{\theta } \mid \mathscr {D})$ as $\epsilon \rightarrow 0$. This leads directly to the ABC rejection sampling algorithm (Algorithm C.1). For deterministic models, under the assumption of Gaussian observation errors, $\epsilon /\sigma \ll 1$, and $d( \mathscr {D}, \mathscr {D}_s)$ taken as the sum of the squared errors, it can be shown that ABC methods are equivalent to exact posterior sampling (Wilkinson 2013).

In some cases, the acceptance probability in Algorithm C.1 is computationally prohibitive for small $\epsilon$. In such situations, an ABC extension to Markov Chain Monte Carlo sampling may be applied (Marjoram et al. 2003). The resulting ABC MCMC sampling method (Algorithm C.2), under reasonable conditions on the proposal kernel $K(\varvec{\theta }^{i} \mid \varvec{\theta }^{i-1})$, simulates a Markov chain with $p(\varvec{\theta } \mid d( \mathscr {D}, \mathscr {D}_s) < \epsilon )$ (Eq. (27)) as its stationary distribution. It is essential to simulate the Markov chain for a sufficiently long time such that the $N_T$ dependent samples are effectively equivalent to the required N i.i.d. samples.

Using either ABC rejection sampling (Algorithm C.1) or ABC MCMC sampling (Algorithm C.2), we can apply Monte Carlo integration to compute the posterior PDF as given in Eq. (27). For simplicity, we focus on the approximation of the jth marginal posterior PDF (Silverman 1986),

$$\begin{aligned} p(\theta _j \mid \mathscr {D}) \approx \frac{1}{Nb} \sum _{i=1}^N K\left( \frac{\theta _j - \theta _j^{(i)}}{b}\right) , \end{aligned}$$

(28)

where $\theta _j$ is the jth element of $\varvec{\theta }$, $\theta _j^{(i)}$ are the jth elements of $\varvec{\theta }^{(i)} \overset{\text {i.i.d}}{\sim } p(\varvec{\theta } \mid \mathscr {D})$, b is the smoothing parameter and K(x) is the smoothing kernel with property $\displaystyle {\int _{-\infty }^{\infty } K(x) \, \text {d}x = 1}$.

Appendix D: Additional Results

In this appendix, we present extended results that are excluded from the main text for brevity. We provide more detailed information on the Bayesian analysis presented in Sects. 4.2 and 4.3. Furthermore, we extend the Bayesian inference problem, as provided in Section 4.2, to account for the treatment of uncertainty in the initial condition.

1.1 D.1 Joint Posterior Features

Here we report various descriptive statistics for the joint posterior PDFs computed in Sect. 4. For each posterior distribution, we report the posterior mode, the posterior mean, the variance/covariance matrix and the correlation coefficient matrix.

Given cell density data, $\mathscr {D}$, a set of continuum model parameters, $\varvec{\theta }$, in parameter space $\varvec{\varTheta }\subseteq \mathbb {R}^k$ with $k > 0$, and a model implied through a likelihood function, $\mathscr {L}(\varvec{\theta } ; \mathscr {D})$, then summary statistics can be computed from the joint posterior, $p(\varvec{\theta } \mid \mathscr {D})$, to obtain estimates and uncertainties on the true parameters. The maximum a posteriori (MAP) parameter estimate is the parameter set with the greatest posterior probability density as given by the posterior mode,

$$\begin{aligned} \hat{\varvec{\theta }}_{\text {mode}} = \mathop {{{\,\mathrm{argmax}\,}}}\limits _{\varvec{\theta }\in \varvec{\varTheta }} p(\varvec{\theta } \mid \mathscr {D}). \end{aligned}$$

The posterior mean is the central tendency of the parameters,

$$\begin{aligned} \bar{\varvec{\theta }} = \mathbb {E}\left[ \varvec{\theta }\right] = \int _{\varvec{\varTheta }} \varvec{\theta }p(\varvec{\theta } \mid \mathscr {D}) \, \text {d} \varvec{\theta }. \end{aligned}$$

The variance/covariance, matrix $\varSigma \in \mathbb {R}^{k \times k}$, provides information on the multivariate uncertainties, that is the spread of parameters. The (i, j)th element of $\varSigma$, denoted by $\sigma _{i,j}$, is given by

$$\begin{aligned} \sigma _{i,j}= & {} \mathbb {C}\left[ \theta _i,\theta _j\right] =\mathbb {E}\left[ (\theta _i - \mathbb {E}\left[ \theta _j\right] )(\theta _j - \mathbb {E}\left[ \theta _i\right] )\right] \\= & {} \int _{\varvec{\varTheta }} (\theta _i - \mathbb {E}\left[ \theta _j\right] )(\theta _j - \mathbb {E}\left[ \theta _i\right] ) p(\varvec{\theta } \mid \mathscr {D}) \, \text {d} \varvec{\theta }, \end{aligned}$$

where $\theta _i$ and $\theta _j$ are the ith and jth elements of $\varvec{\theta }$. Note that $\mathbb {C}\left[ \theta _i,\theta _i\right] = \mathbb {V}\left[ \theta _i\right]$ and $\sigma _{i,j} = \sigma _{j,i}$. Lastly, the correlation coefficient matrix $R \in \mathbb {R}^{k \times k}$ measures the linear dependence between parameter pairs. The (i, j)th element of R, denoted by $\rho _{i,j}$, is given by

$$\begin{aligned} \rho _{i,j} = \frac{\sigma _{i,j}}{\left( \sigma _{i,i},\sigma _{j,j}\right) ^{1/2}}. \end{aligned}$$

Note $\rho _{i,i} = 1$ for all $i \in [1,k]$, and $\rho _{i,j} = \rho _{j,i}$. The results of all these statistics, for the inference problems considered in the main text, are presented in Tables 3, 4, 5, and 6.

Table 3 MAP parameter estimates (posterior modes) from posterior PDFs using initial conditions of 12,000 cells, 16,000 cells and 20,000 cells

Table 4 Joint posterior means for posterior PDFs using initial conditions of 12,000 cells, 16,000 cells and 20,000 cells

Table 5 Variance/covariance matrices for posterior PDFs using initial conditions of 12,000 cells, 16,000 cells and 20,000 cells

Table 6 Correlation coefficient matrices for posterior PDFs using initial conditions of 12, 000 cells, 16, 000 cells and 20, 000 cells

1.2 D.2 Bivariate Marginal Posterior PDFs

In the main text, we computed only univariate marginal posterior PDFs, and we extend this analysis by providing bivariate marginal PDFs here. For the Fisher–KPP and Porous Fisher models, we have three bivariate marginal posterior PDFs,

$$\begin{aligned} p(D_0,\lambda \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})&= \int _{\mathbb {R}}p(D_0,\lambda ,K \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})\, \text {d} K,\\ p(\lambda ,K \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})&= \int _{\mathbb {R}}p(D_0,\lambda ,K \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})\, \text {d}D_0, \\ p(D_0,K \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})&= \int _{\mathbb {R}}p(D_0,\lambda ,K \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})\, \text {d}\lambda . \end{aligned}$$

Similarly, for the Generalised Porous Fisher Model, we have six bivariate marginal posterior PDFs,

$$\begin{aligned} p(D_0,\lambda \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})&= \iint _{\mathbb {R}^2}p(D_0,\lambda ,K,r \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})\,\text {d}r\, \text {d} K,\\ p(D_0, K \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})&= \iint _{\mathbb {R}^2}p(D_0,\lambda ,K,r \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})\,\text {d}r\, \text {d}\lambda , \\ p(D_0, r \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})&= \iint _{\mathbb {R}^2}p(D_0,\lambda ,K,r \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})\,\text {d}K \, \text {d}\lambda , \\ p(\lambda , K \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})&= \iint _{\mathbb {R}^2}p(D_0,\lambda ,K,r \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})\,\text {d}r\,\text {d}D_0, \\ p(\lambda ,r \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})&= \iint _{\mathbb {R}^2}p(D_0,\lambda ,K,r \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})\,\text {d}K\, \text {d}D_0, \\ p(K, r \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})&= \iint _{\mathbb {R}^2}p(D_0,\lambda ,K,r \mid \mathbf {C}_{\text {obs}}^{1:N,1:M})\,\text {d}\lambda \, \text {d}D_0. \end{aligned}$$

The resulting PDFs using the three initial density conditions are shown for: the Fisher–KPP model (Fig. 6); the Porous Fisher model (Fig. 7); and the Generalised Porous model (Fig. 8).

Fig. 6

Plot matrix of univariate and bivariate marginal posterior probability densities obtained through Bayesian inference on the PC-3 scratch assay data under the Fisher–KPP model for the three different initial conditions; 12,000 cells (solid green), 16,000 cells (solid orange) and 20,000 cells (solid purple). Univariate marginal densities, on the main plot matrix diagonal, demonstrate the degree of uncertainty in the diffusivity, $D_0$, the proliferation rate, $\lambda$, and the carrying capacity, K. Off-diagonals are contour plots of the pairwise bivariate posterior PDFs; these demonstrate the relationships between parameters (Color figure online)

Fig. 7

Plot matrix of univariate and bivariate marginal posterior probability densities obtained through Bayesian inference on the PC-3 scratch assay data under the Porous Fisher model for the three different initial conditions: 12,000 cells (solid green), 16,000 cells (solid orange) and 20,000 cells (solid purple). Univariate marginal densities, on the main plot matrix diagonal, demonstrate the degree of uncertainty in the diffusivity, $D_0$, the proliferation rate, $\lambda$, and the carrying capacity, K. Off-diagonals are contour plots of the pairwise bivariate posterior PDFs; these demonstrate the relationships between parameters (Color figure online)

Fig. 8

Plot matrix of univariate and bivariate marginal posterior probability densities obtained through Bayesian inference on the PC-3 scratch assay data under the Porous Fisher model for the three different initial conditions: 12,000 cells (solid green), 16,000 cells (solid orange) and 20,000 cells (solid purple). Univariate marginal densities, on the main plot matrix diagonal, demonstrate the degree of uncertainty in the diffusivity, $D_0$, the proliferation rate, $\lambda$, the carrying capacity, K, and the power, r. Off-diagonals are contour plots of the pairwise bivariate posterior PDFs; these demonstrate the relationships between parameters (Color figure online)

1.3 D.3 Uncertainty in Initial Condition

In the main text, the assumption was made that $C_{\text {obs}}(x,0) = C(x,0 ; \varvec{\theta })$. That is, we use initial observations as the initial density profile to simulate the model given parameters $\varvec{\theta }$. Since the model is deterministic, the final form of the likelihood is a multivariate Gaussian distribution, which simplifies calculations considerably. Both Jin et al. (2016b) and Warne et al. (2017) indicate that such an assumption could result in underestimation of the uncertainties in parameter estimates.

Following from Warne et al. (2017), we take $C_{\text {obs}}(x,0) = C(x,0 ; \varvec{\theta }) + \eta _0$, where $\eta _0$ is a Gaussian random variable with mean $C(x,0 ; \varvec{\theta })$ and variance $\sigma _0^2$. Note that we do not require $\sigma _0 = \sigma$, in fact, there are reasons to consider $\sigma _0 > \sigma$; for example, experimental protocols for seeding cell culture plates can be an additional source of variation in initial cell densities (Jin et al. 2016b; Warne et al. 2017). Since $C_{\text {obs}}(x,0) \sim \mathscr {N}(C(x,0 ; \varvec{\theta }),\sigma _0^2)$, it is also true that $C(x,0 ; \varvec{\theta }) \sim \mathscr {N}(C_{\text {obs}}(x,0),\sigma _0^2)$. Therefore, our models are to be treated as random PDEs with deterministic dynamics, but random initial conditions.

Since the initial conditions are random, the initial condition is a latent variable that must be integrated out. Thus, the likelihood becomes

$$\begin{aligned}&\mathscr {L}(\mathbf {C}_{\text {obs}}^{1:N,1:M} ; \varvec{\theta }) = \\&\int _{\mathbb {R}^N}\frac{1}{(\sigma \sqrt{2\pi })^{NM}}\prod _{i=1}^{N} \prod _{j=1}^M \exp {\left( -\dfrac{(C_{\text {obs}}(x_i,t_j) - C(x_i,t_j ; \varvec{\theta }))^2}{2\sigma ^2}\right) } p(C(x_i,0 ; \varvec{\theta }) \mid \sigma _0) \, \text {d}C(x_i,0 ; \varvec{\theta }), \end{aligned}$$

where $\sigma _0$ is assumed to be known and $p(C(x_i,0 ; \varvec{\theta }) \mid \sigma _0)$ is a Gaussian PDF with mean $C_{\text {obs}}(x_i,0)$ and variance $\sigma _0$. This likelihood integral must be computed using Monte Carlo methods. Computationally, we apply directly the ABC MCMC method as given in Algorithm C.2. The only algorithmic difference being that simulated data, $\mathscr {D}_s$, is generated though solving the model PDE after a realisation of the initial density profile has been generated. Overall, this leads to slower convergence in the Markov chain and hence longer computation times.

The inference problem using random initial density profiles was solved using ABC MCMC under the Fisher–KPP model and the Porous Fisher model for initial densities based on 16,000 initial cells only. We take $\sigma _0 = 2\sigma$. Univariate and bivariate marginal posterior PDFs are shown in Figs. 9 and 10. In the Fisher–KPP model, the additional uncertainty seems to have a significant effect on the uncertainty in the carrying capacity, K, in agreement with Warne et al. (2017). However, the diffusion coefficient, $D_0$, and proliferation rate, $\lambda$, are not affected as significantly.

Fig. 9

Plot matrix of univariate and bivariate marginal posterior probability densities obtained through Bayesian inference on the PC-3 scratch assay data under the Fisher–KPP model for the fixed initial conditions (solid green) and random initial conditions (solid orange). Univariate marginal densities, on the main plot matrix diagonal, demonstrate the degree of uncertainty in the diffusivity, $D_0$, the proliferation rate, $\lambda$, and the carrying capacity, K. Off-diagonals are contour plots of the pairwise bivariate posterior PDFs; these demonstrate the relationships between parameters (Color figure online)

Fig. 10

Plot matrix of univariate and bivariate marginal posterior probability densities obtained through Bayesian inference on the PC-3 scratch assay data under the Porous Fisher model for the fixed initial conditions (solid green) and random initial conditions (solid orange). Univariate marginal densities, on the main plot matrix diagonal, demonstrate the degree of uncertainty in the diffusivity, $D_0$, the proliferation rate, $\lambda$, and the carrying capacity, K. Off-diagonals are contour plots of the pairwise bivariate posterior PDFs; these demonstrate the relationships between parameters (Color figure online)

For the Porous Fisher model, both $D_0$ and K are greatly affected. This is not surprising, since motility is density dependent for the Porous Fisher model. By contrast, the Fisher–KPP model is almost unaffected in the marginal posterior PDF of $D_0$, since it is independent of initial cell density.