Parameter Identifiability and Redundancy, with Applications to a General Class of Stochastic Carcinogenesis Models

Abstract

Models for complex biological systems may involve a large number of parameters. It may well be that some of these parameters (in particular any value of those parameters) cannot be derived from observed data via regression techniques. Such parameters are said to be unidentifiable, the remaining parameters being identifiable. Closely related to this idea is that of redundancy, that a set of parameters can be expressed in terms of some smaller set. Before data is analysed it is critical to determine which model parameters are identifiable or redundant to avoid ill-defined and poorly-convergent regression. This problem has been considered from a number of points of view in the literature. One distinct recent application has been to biologically-based cancer models. Heidenreich et al. (Risk Anal 1997 17 391–399) considered parameter identifiability in the context of the two-mutation cancer model and demonstrated that combinations of all but two of the model parameters are identifiable. Here we outline general considerations on parameter identifiability, and introduce the notion of weak local identifiability and gradient weak local identifiability. These are based on local properties of the likelihood, in particular the rank of the Hessian matrix. We relate these to the notions of parameter identifiability and redundancy previously introduced by Rothenberg (Econometrica 1971 39 577–591) and Catchpole and Morgan (Biometrika 1997 84 187–196). Within the widely-used exponential family, parameter irredundancy, local identifiability, gradient weak local identifiability and weak local identifiability are shown to be largely equivalent. We consider applications to a recently developed class of cancer models of Little and Wright (Math Biosciences 2003 183 111–134) and Little et al. (J Theoret Biol 2008 254 229–238) that generalize a large number of other recently used quasi-biological cancer models, in particular the two-mutation model of Heidenreich et al. (Risk Anal 1997 17 391–399). We show that in the simpler model proposed by Little and Wright (Math Biosciences 2003 183 111–134) the number of identifiable combinations of parameters is at most two less than the number of biological parameters, thereby generalizing previous results of Heidenreich et al. (Risk Anal 1997 17 391–399) for the two-mutation model. For the more general model of Little et al. (J Theoret Biol 2008 254 229–238) the number of identifiable combinations of parameters is at most $r + 1$ less than the number of biological parameters, where $r$ is the number of destabilization types (types of genomic instability), thereby also generalizing all these results. Numerical evaluations suggest that these bounds are sharp. We also identify particular combinations of identifiable parameters.

Appendices

Appendix A

Proof of Theorem 1

In this Section we outline a proof of Theorem 1 in the main text. To prove this result we need the following lemma of Rudin [18] (p.229).

Lemma A1

Suppose $m,n,r$ are non-negative integers such that (s.t.) $m \ge r$, $n \ge r$ and $F$ is a $C^{1}$ function $E \subset R^{n} \to R^{m}$ where $E$ is an open set. Suppose that $rk(F^{\prime } (x)) = r$ $\forall x \in E$. Fix $a \in E$ and put $A = F^{\prime } (a)$, and let $Y_{1} = A(R^{n} )$ and let $P:R^{m} \to R^{m}$ be a linear projection operator ($P^{2} = P$) s.t. $Y_{1} = P(R^{m} )$ and $Y_{2} = null(P)$. Then $\exists U,V \subset R^{n}$, open sets and a bijective $C^{1}$ function $H:V \to U$ whose inverse is also $C^{1}$ and s.t. $F(H(x)) = Ax + \varphi (Ax),$ $\forall x \in V$ where $\varphi :AV \subset Y_{1} \to Y_{2}$ is a $C^{1}$ function.

We now restate Theorem 1 here.

Theorem A2

Suppose that the log-likelihood $L(x|\theta )$ is $C^{2}$ as a function of the parameter vector $\theta \in \Upomega \subset R^{p}$, and for all $x = (x_{1} , \ldots ,x_{n} ) \in \Upsigma^{n}$.

• 1. Suppose that for some $x$ and $\theta \in \text{int} (\Upomega )$ it is the case that $rk\left[ {\left( {\frac{{\partial^{2} L(x|\theta )}}{{\partial \theta_{i} \partial \theta_{j} }}} \right)_{i,j = 1}^{p} } \right] = p$. Then turning points of the likelihood in the neighborhood of $\theta$ are isolated, i.e., there is an open neighborhood $N \in \aleph_{\theta } \subset \Upomega$ for which there is at most one $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\theta } \in N$ that satisfies $\left. {\left( {\frac{\partial L(x|\theta )}{{\partial \theta_{i} }}} \right)_{i = 1}^{p} } \right|_{{\theta = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\theta } }} = 0$.

• 2. Suppose that for some $x$ and $\theta \in \text{int} (\Upomega )$ it is the case that $rk\left[ {\left( {\frac{{\partial^{2} L(x|\theta )}}{{\partial \theta_{i} \partial \theta_{j} }}} \right)_{i,j = 1}^{p} } \right] = p$ then local maxima of the likelihood in the neighborhood of $\theta$ are isolated, i.e., there is an open neighborhood $N \in \aleph_{\theta } \subset \Upomega$ for which there is at most one $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\theta } \in N$ that is a local maximum of $L(x|\theta )$.

• 3. Suppose that for some $x$ and all $\theta \in \text{int} (\Upomega )$ it is the case that $rk\left[ {\left( {\frac{{\partial^{2} L(x|\theta )}}{{\partial \theta_{i} \partial \theta_{j} }}} \right)_{i,j = 1}^{p} } \right] = r < p$ then all local maxima of the likelihood in $\text{int} (\Upomega )$ are not isolated, as indeed are all $\theta \in \text{int} (\Upomega )$ for which $\left( {\frac{\partial L(x|\theta )}{{\partial \theta_{i} }}} \right)_{i = 1}^{p} = 0$.

Proof

• 1. Let $F:\Upomega \subset R^{p} \to R^{p}$ be defined by $F(\theta_{1} ,\theta_{2} , \ldots ,\theta_{p} ) = \left( {\frac{\partial L(x|\theta )}{{\partial \theta_{1} }},\frac{\partial L(x|\theta )}{{\partial \theta_{2} }}, \ldots ,\frac{\partial L(x|\theta )}{{\partial \theta_{p} }}} \right)$. Since $L$ is $C^{2}$, $F$ is $C^{1}$ on $\text{int} (\Upomega ) \subset R^{p}$. By assumption $\frac{{\partial F(\theta )_{i} }}{{\partial \theta_{j} }} = \left( {\frac{{\partial^{2} L(x|\theta )}}{{\partial \theta_{j} \partial \theta_{i} }}} \right)_{i,j = 1}^{p}$ is of full rank at $\theta$. By the inverse function theorem [18] (p.221–223) there are open $N,M \subset R^{p}$ such that $\theta \in N$ and a $C^{1}$ bijective function $G:M \to N$ such that $G(F(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\theta } )) = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\theta }$ for all $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\theta } \in N$. In particular there can be at most a single $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\theta } \in N$ for which $F(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\theta } ) = 0$. QED.

• 2. By (1) there is an open neighborhood $N \in \aleph_{\theta } \subset \Upomega$ for which if $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\theta } \in N$ is such that $\left. {\left( {\frac{\partial L(x|\theta )}{{\partial \theta_{i} }}} \right)_{i = 1}^{p} } \right|_{{\theta = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\theta } }} = 0$ then for $\theta ' \ne \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\theta } \in N$ $\left. {\left( {\frac{\partial L(x|\theta )}{{\partial \theta_{i} }}} \right)_{i = 1}^{p} } \right|_{\theta = \theta '} \ne 0$. Suppose now that $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\theta } \in N$ is a local maximum of $L(x|\theta )$. Any member of this neighborhood other than $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\theta }$ cannot be a turning point, and so by the Mean Value Theorem [18] (p.107) cannot be a local maximum. QED.

• 3. Let $F:\Upomega \subset R^{p} \to R^{p}$ be defined by $F(\theta_{1} ,\theta_{2} , \ldots ,\theta_{p} ) = \left( {\frac{\partial L(x|\theta )}{{\partial \theta_{1} }},\frac{\partial L(x|\theta )}{{\partial \theta_{2} }}, \ldots ,\frac{\partial L(x|\theta )}{{\partial \theta_{p} }}} \right)$. Since $L$ is $C^{2}$, $F$ is $C^{1}$ on $\text{int} (\Upomega ) \subset R^{p}$. By assumption $rk(F) = rk\left[ {\left( {\frac{{\partial^{2} L(x|\theta )}}{{\partial \theta_{i} \partial \theta_{j} }}} \right)_{i,j = 1}^{p} } \right] = r$ for all $\theta \in \text{int} (\Upomega ) \subset R^{p}$. Suppose that $\theta_{0} \in \text{int} (\Upomega )$ is a local maximum of $L$. Let $A = \left. {\frac{\partial F}{\partial \theta }} \right|_{{\theta = \theta_{0} }} :R^{p} \to R^{p}$ ($A \in L(R^{p} ,R^{p} )$), and choose some arbitrary projection $P \in L(R^{p} ,R^{p})$ s.t. $P(R^{p} ) = Y_{1} = A(R^{p} )$, and let $Y_{2} = null(P)$. By Lemma A1 there are open sets $U,V \subset R^{p}$ with $\theta_{0} \in U \subset \text{int} (\Upomega )$ and a bijective $C^{1}$ mapping with $C^{1}$ inverse $H:V \to U$ s.t. $F(y) = AH^{ - 1} y + \varphi (AH^{ - 1} y),$ $\forall y \in V$ where $\varphi :AV \subset Y_{1} \to Y_{2}$ is a $C^{1}$ function.

Since $\theta_{0} \in \text{int} (\Upomega )$ is a local maximum of $L(x;\theta )$, by the Mean Value Theorem [18] (p.107) $F(\theta_{0} ) = 0$. Now choose some non-trivial vector $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k} \in null(A)$ and define a function, as we can, on some interval $\delta :( - \varepsilon ,\varepsilon ) \to R^{p}$ by $\delta (t) = H(H^{ - 1} (\theta_{0} ) + t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k} )$. Because $H:V \to U$ is bijective and $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k}$ is non-trivial $\delta (t) = \delta (t^{\prime } ) \Leftrightarrow t = t^{\prime }$. Also, it is the case that:

$$ \begin{array}{*{20}l} {F(\delta (t)) = AH^{{ - 1}} (H[H^{{ - 1}} (\theta _{0} ) + t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k} ]) + \varphi (AH^{{ - 1}} (H[H^{{ - 1}} (\theta _{0} ) + t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k} ])) = } \hfill \\ {A[H^{{ - 1}} (\theta _{0} ) + t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k} ] + \varphi (A[H^{{ - 1}} (\theta _{0} ) + t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k} ]) = A[H^{{ - 1}} (\theta _{0} )] + \varphi (A[H^{{ - 1}} (\theta _{0} )]) = F(\theta _{0} ) = 0} \hfill \\ \end{array}$$

(11.A1)

Define $G:( - \varepsilon ,\varepsilon ) \to R$ by $G(t) = L(\delta (t)) = L((\delta_{1} (t),\delta_{2} (t), \ldots ,\delta_{n} (t)))$. By the Chain Rule [18] (p.215) $\frac{dG}{dt} = \sum\limits_{i = 1}^{p} {\frac{\partial L(x|\theta )}{{\partial \theta_{i} }}\frac{{d\delta_{i} }}{dt}} = 0$ $\forall t \in ( - \varepsilon ,\varepsilon )$. Finally, by the Mean Value Theorem [18] (p.107) $G$ must be constant; in particular $L(x|\delta (t)) = L(x|\delta (0)) = L(x|\theta_{0} )$ $\forall t \in ( - \varepsilon ,\varepsilon )$ and so all points $\delta (t)$ must also be local maxima of $L(x|\theta )$. Therefore $\theta_{0}$ is not an isolated local maximum. Since all we used about $\theta_{0} \in \text{int} (\Upomega )$ was that $F(\theta_{0} ) = 0$, $F((\theta_{0i} )_{i = 1}^{p} ) = \left( {\frac{{\partial L(x|\theta_{0} )}}{{\partial \theta_{01} }},\frac{{\partial L(x|\theta_{0} )}}{{\partial \theta_{02} }}, \ldots ,\frac{{\partial L(x|\theta_{0} )}}{{\partial \theta_{0p} }}} \right) = 0$, the above argument also shows that turning points cannot be isolated: $F(\delta (t)) = 0$. QED.

Appendix B

2.1 Specification of Embedded Exponential Family Model

In this Section we outline the specification of an embedding of a stochastic cancer model in a general class of statistical models, the so-called exponential family [17]. This is often done in fitting cancer models to epidemiological and biological data (e.g., see references [8, 9, 16, 21]). Recall that a model is a member of the exponential family if the observed data $x = (x_{l} )_{l = 1}^{n} \in \sum^{n}$ is such that the log-likelihood is given by $L(x|\theta ) = \sum\limits_{l = 1}^{n} {\left[ {\frac{{x_{l} \varsigma_{l} - b(\varsigma_{l} )}}{a(\phi )} + c(x_{l} ,\phi )} \right]}$ for some functions $a(\phi ),b(\varsigma ),c(x,\phi )$. We assume that the natural parameters $\varsigma_{l} = \varsigma_{l} [(\theta_{i} )_{i = 1}^{p} ,z_{l} ]$ are functions of the model parameters $(\theta_{i} )_{i = 1}^{p}$ and some auxiliary data $(z_{l} )_{l = 1}^{n}$, and that $\mu_{l} = b^{\prime } (\varsigma_{l} [(\theta_{i} )_{i = 1}^{p} ,z_{l} ]) = z_{l} \cdot h[(\theta_{i} )_{i = 1}^{p} ,y_{l} ]$. Here $h[(\theta_{i} )_{i = 1}^{p} ,y_{l} ]$ is the cancer hazard function (for example, that of Little et al. [9], as also specified in Appendix C), $(y_{l} )_{l = 1}^{n}$ are some further auxiliary data, and we assume that the $(z_{l} )_{l = 1}^{n}$ are all non-zero. [Note: this is not necessarily a GLM.] In this case it is seen that

$$ \frac{{\partial ^{2} L(x|\theta )}}{{\partial \theta _{i} \partial \theta _{j} }} = \sum\limits_{{l = 1}}^{n} {\left[ {\begin{array}{*{20}l} {\frac{{[x_{l} - b^{\prime}(\varsigma _{l} )]z_{l} }}{{a(\phi )b^{\prime\prime}(\varsigma _{l} )}}\frac{{\partial ^{2} h(\theta ,y_{l} )}}{{\partial \theta _{i} \partial \theta _{j} }}} \hfill \\ { - \frac{{z_{l} ^{2} }}{{a(\phi )}}\frac{{\partial h(\theta ,y_{l} )}}{{\partial \theta _{i} }}\frac{{\partial h(\theta ,y_{l} )}}{{\partial \theta _{j} }}\left\{ {\frac{{[b^{\prime\prime}(\varsigma _{l} )]^{2} + b^{\prime\prime\prime}(\varsigma _{l} )[x_{l} - b^{\prime}(\varsigma _{l} )]}}{{[b^{\prime\prime}(\varsigma _{l} )]^{3} }}} \right\}} \hfill \\ \end{array} } \right]} $$

(11.B1)

so that the Fisher information matrix is given by

$$ I(\theta )_{ij} = - E_{\theta } \left[ {\frac{{\partial^{2} L(x|\theta )}}{{\partial \theta_{i} \partial \theta_{j} }}} \right] = \frac{1}{a(\phi )}\sum\limits_{l = 1}^{n} {\frac{{z_{l}^{2} }}{{b^{\prime \prime } (\varsigma_{l} )}}\frac{{\partial h(\theta ,y_{l} )}}{{\partial \theta_{i} }}\frac{{\partial h(\theta ,y_{l} )}}{{\partial \theta_{j} }}} $$

(11.B2)

Appendix C

3.1 Derivation of Hazard Function in Terms of Specific Parameter Combinations for the Cancer Model of Little et al. [9]

In this Appendix we derive the hazard function for the cancer model of Little et al. [9] and show that it can be written in terms of certain combinations of parameters, given in Table 11.2. The hazard function is defined as:

$$ h(t) = - \frac{d}{dt}\ln \psi \left( {1,1, \ldots ,1,0;t,0} \right) $$

(11.C1)

where

$$ \begin{array}{*{20}l} {\psi (y_{1,0,0} ,y_{2,0,0} , \ldots ,y_{k - 1,0,0} ,y_{0,1,1} , \ldots ,y_{k - 1,1,1} ,y_{0,2,1} ,y_{1,2,1} , \ldots ,y_{{k - 1,m_{r} ,r}} ,y_{k} ;t,0)} \hfill \\ { \equiv \psi (t)} \hfill \\ { = \sum\limits_{n} {y_{1,0,0}^{{n_{1,0,0} }} \cdot \ldots \cdot y_{k - 1,0,0}^{{n_{k - 1,0,0} }} \cdot y_{0,1,1}^{{n_{0,1,1} }} \cdot \ldots \cdot y_{k - 1,1,1}^{{n_{k - 1,1,1} }} \cdot \ldots \cdot y_{{k - 1,m_{r} ,r}}^{{n_{{k - 1,m_{r} ,r}} }} \cdot y_{k}^{{n_{k} }} \,\times\,} } \hfill \\ {\quad \quad P\left( {Y_{1,0,0} (t) = n_{1,0,0} , \ldots ,Y_{k} (t) = n_{k} \left| {N(0) = X(0),Y_{1,0,0} (0) = \ldots = Y_{k} (0) = 0} \right.} \right)} \hfill \\ \end{array} $$

(11.C2)

is the full probability generating function (PGF) starting with X(0) cell(s) in the normal compartment at time 0. The number of biological parameters in this specific model is summarized in Table 11.1.

By straightforward generalizations of material in Little and Wright [8] (given in Appendix D) it is seen that $\psi$ satisfies a Kolmogorov forward equation:

$$ \begin{array}{*{20}l} {\frac{{d\psi }}{{dt}} = \psi \cdot [y_{{1,0,0}} - 1] \cdot X(t) \cdot M(0,0,0)(t) + \psi \cdot \sum\limits_{{d = 1}}^{r} {[y_{{0,1,d}} - 1] \cdot X(t) \cdot A(0,0,d)(t)} \,+ } \hfill \\ {\sum\limits_{{1 \le \alpha \le k - 1}} {\left[ {\begin{array}{*{20}l} { - y_{{\alpha ,0,0}} \cdot [D(\alpha ,0,0)(t) + G(\alpha ,0,0)(t) + M(\alpha ,0,0)(t)} \hfill \\ { + \sum\limits_{{d' = 1}}^{r} {A(\alpha ,0,d')(t)} ] + y_{{\alpha ,0,0}} ^{2} \cdot G(\alpha ,0,0)(t) + D(\alpha ,0,0)(t)} \hfill \\ { + y_{{\alpha ,0,0}} \cdot y_{{\alpha + 1,0,0}} \cdot M(\alpha ,0,0)(t) + \sum\limits_{{d' = 1}}^{r} {y_{{\alpha ,0,0}} \cdot y_{{\alpha ,1,d'}} \cdot A(\alpha ,0,d')(t)} } \hfill \\ \end{array} } \right] \cdot \frac{{\partial \psi }}{{\partial y_{{\alpha ,0,0}} }}} } + \hfill \\ {\sum\limits_{{\begin{array}{*{20}l} {0 \le \alpha \le k - 1} \hfill \\ {1 \le \beta \le m_{d} } \hfill \\ {1 \le d \le r} \hfill \\ \end{array} }} {\left[ {\begin{array}{*{20}l} { - y_{{\alpha ,\beta ,d}} \cdot [D(\alpha ,\beta ,d)(t) + G(\alpha ,\beta ,d)(t) + M(\alpha ,\beta ,d)(t)} \hfill \\ { + A(\alpha ,\beta ,d)(t)] + y_{{\alpha ,\beta ,d}} ^{2} \cdot G(\alpha ,\beta ,d)(t) + D(\alpha ,\beta ,d)(t)} \hfill \\ { + y_{{\alpha ,\beta ,d}} \cdot y_{{\alpha + 1,\beta ,d}} \cdot M(\alpha ,\beta ,d)(t) + y_{{\alpha ,\beta ,d}} \cdot y_{{\alpha ,\beta + 1,d}} \cdot A(\alpha ,\beta ,d)(t)} \hfill \\ \end{array} } \right] \cdot \frac{{\partial \psi }}{{\partial }}} } \hfill \\ \end{array} $$

(11.C3)

with the conventions that $y_{\alpha ,\beta ,d} \equiv 0$ for $\beta > m_{d}$, $A(\alpha ,\beta ,d) \equiv 0$ for $\beta \ge m_{d}$ and $y_{k,\beta ,d} \equiv y_{k}$ for all defined $\beta$ and $d$. We solve the equation by means of Cauchy’s method of characteristics. Suppose $y_{\alpha ,\beta ,d} \equiv y_{\alpha ,\beta ,d} (u)$ and $t \equiv t(u)$, then $\psi \equiv \psi (y_{\alpha ,\beta ,d} (u),t(u))$. This implies that:

$$ \begin{array}{*{20}l} {\frac{{\partial \psi }}{{\partial u}} = \frac{{\partial \psi }}{{\partial t}} \cdot \frac{{dt}}{{du}} + \frac{{\partial \psi }}{{\partial y_{k} }} \cdot \frac{{dy_{k} }}{{du}} + \sum\limits_{{1 \le \alpha \le k - 1}} {\frac{{\partial \psi }}{{\partial y_{{\alpha ,0,0}} }} \cdot \frac{{dy_{{\alpha ,0,0}} }}{{du}}} + \sum\limits_{{\begin{array}{*{20}l} {0 \le \alpha \le k - 1} \hfill \\ {1 \le \beta \le m_{d} } \hfill \\ {1 \le d \le r} \hfill \\ \end{array} }} {\frac{{\partial \psi }}{{\partial y_{{\alpha ,\beta ,d}} }} \cdot \frac{{dy_{{\alpha ,\beta ,d}} }}{{du}}} } \hfill \\ { = \frac{{dt}}{{du}} \cdot \left[ {\begin{array}{*{20}l} {\psi \cdot [y_{{1,0,0}} - 1] \cdot X(t) \cdot M(0,0,0)(t) + \psi \cdot \sum\limits_{{d = 1}}^{r} {[y_{{0,1,d}} - 1] \cdot X(t) \cdot A(0,0,d)(t)} + } \hfill \\ {\sum\limits_{{1 \le \alpha \le k - 1}} {\left[ {\begin{array}{*{20}l} { - y_{{\alpha ,0,0}} \cdot [D(\alpha ,0,0)(t) + G(\alpha ,0,0)(t) + M(\alpha ,0,0)(t)} \hfill \\ { + \sum\limits_{{d' = 1}}^{r} {A(\alpha ,0,d')(t)} ] + y_{{\alpha ,0,0}} ^{2} \cdot G(\alpha ,0,0)(t) + D(\alpha ,0,0)(t)} \hfill \\ { +\, y_{{\alpha ,0,0}} \cdot y_{{\alpha + 1,0,0}} \cdot M(\alpha ,0,0)(t) + \sum\limits_{{d' = 1}}^{r} {y_{{\alpha ,0,0}} \cdot y_{{\alpha ,1,d'}} \cdot A(\alpha ,0,d')(t)} } \hfill \\ \end{array} } \right] \cdot \frac{{\partial \psi }}{{\partial y_{{\alpha ,0,0}} }}} } + \hfill \\ {\sum\limits_{{\begin{array}{*{20}l} {0 \le \alpha \le k - 1} \hfill \\ {1 \le \beta \le m_{d} } \hfill \\ {1 \le d \le r} \hfill \\ \end{array} }} {\left[ {\begin{array}{*{20}l} { - y_{{\alpha ,\beta ,d}} \cdot [D(\alpha ,\beta ,d)(t) + G(\alpha ,\beta ,d)(t) + M(\alpha ,\beta ,d)(t)} \hfill \\ { + A(\alpha ,\beta ,d)(t)] + y_{{\alpha ,\beta ,d}} ^{2} \cdot G(\alpha ,\beta ,d)(t) + D(\alpha ,\beta ,d)(t)} \hfill \\ { + y_{{\alpha ,\beta ,d}} \cdot y_{{\alpha + 1,\beta ,d}} \cdot M(\alpha ,\beta ,d)(t) + y_{{\alpha ,\beta ,d}} \cdot y_{{\alpha ,\beta + 1,d}} \cdot A(\alpha ,\beta ,d)(t)} \hfill \\ \end{array} } \right] \cdot \frac{{\partial \psi }}{{\partial y_{{\alpha ,\beta ,d}} }}} } \hfill \\ \end{array} } \right]} \hfill \\ { + \frac{{\partial \psi }}{{\partial y_{k} }} \cdot \frac{{dy_{k} }}{{du}} + \sum\limits_{{1 \le \alpha \le k - 1}} {\frac{{\partial \psi }}{{\partial y_{{\alpha ,0,0}} }} \cdot \frac{{dy_{{\alpha ,0,0}} }}{{du}}} + \sum\limits_{{\begin{array}{*{20}l} {0 \le \alpha \le k - 1} \hfill \\ {1 \le \beta \le m_{d} } \hfill \\ {1 \le d \le r} \hfill \\ \end{array} }} {\frac{{\partial \psi }}{{\partial y_{{\alpha ,\beta ,d}} }} \cdot \frac{{dy_{{\alpha ,\beta ,d}} }}{{du}}} } \hfill \\ \end{array} $$

(11.C4)

A solution is therefore given by:

$$ \begin{aligned} \frac{\partial \psi }{\partial u} &= \psi \cdot [y_{1,0,0} (u) - 1] \cdot X(u) \cdot M(0,0,0)(u)\\ &+ \psi \cdot \sum\limits_{d = 1}^{r} {[y_{0,1,d} (u) - 1] \cdot X(u) \cdot A(0,0,d)(u)}\end{aligned} $$

(11.C5)

$$ \frac{{\partial t}}{{\partial u}} = 1 $$

(11.C6)

$$ \frac{{\partial y_{k} }}{{\partial u}} = 0 $$

(11.C7)

and for $d = 0:$

$$ \begin{array}{*{20}l} {\frac{{dy_{{\alpha ,0,0}} }}{{du}} = y_{{\alpha ,0,0}} \cdot [D(\alpha ,0,0)(t) + G(\alpha ,0,0)(t) + M(\alpha ,0,0)(t)} \hfill \\ \quad \quad \quad { + \sum\limits_{{d' = 1}}^{r} {A(\alpha ,0,d')(t)} ] - y_{{\alpha ,0,0}} ^{2} \cdot G(\alpha ,0,0)(t) - D(\alpha ,0,0)(t)} \hfill \\ \quad \quad \quad { - y_{{\alpha ,0,0}} \cdot y_{{\alpha + 1,0,0}} \cdot M(\alpha ,0,0)(t) - \sum\limits_{{d' = 1}}^{r} {y_{{\alpha ,0,0}} \cdot y_{{\alpha ,1,d'}} \cdot A(\alpha ,0,d')(t)} } \hfill \\ \end{array} $$

(11.C8)

while for $d \ne 0$:

$$\begin{aligned}\frac{{dy_{\alpha ,\beta ,d} }} {{du}} & = y_{\alpha ,\beta ,d} \cdot [D(\alpha ,\beta ,d)(t) + G(\alpha ,\beta ,d)(t) + M(\alpha ,\beta ,d)(t) \\ & \quad + A(\alpha ,\beta ,d)(t)] - y_{\alpha ,\beta ,d}^2 \cdot G(\alpha ,\beta ,d)(t) - D(\alpha ,\beta ,d)(t) \\ & \quad- y_{\alpha ,\beta ,d} \cdot y_{\alpha + 1,\beta ,d} \cdot M(\alpha ,\beta ,d)(t) - y_{\alpha ,\beta ,d} \cdot y_{\alpha ,\beta + 1,d} \cdot A(\alpha ,\beta ,d)(t) \\ \end{aligned} $$

(11.C9)

For the hazard, a solution is required for $\psi (1,1,1, \ldots ,1,0;t,0)$, i.e., $\psi (y_{1,0,0} = 1,y_{2,0,0} = 1,y_{3,0,0} = 1, \ldots ,y_{{k - 1,m_{r} ,r}} = 1,y_{k} = 0;t,s = 0)$, so that a particular characteristic must have the boundary value $y_{\alpha ,\beta ,d} (t) = 1$ and $y_{k} (t) = 0$ [implying by (11.C7) $y_{k} (u) \equiv 0$], so that $y_{\alpha ,\beta ,d} (u)$ is a function of both $u$ and $t$, i.e., $y_{\alpha ,\beta ,d} (u) \equiv y_{\alpha ,\beta ,d} (u,t)$. Integrating (11.C5) over $u \in [0,t]$ yields

$$ \psi (t) = \exp \left\{ {\int\limits_{0}^{t} {\left[ {\begin{array}{*{20}l} {\left\{ {y_{{1,0,0}} (u,t)-1} \right\} \cdot X(u) \cdot M(0,0,0)(u)} \hfill \\ { + \sum\limits_{{d = 1}}^{r} {\left\{ {y_{{0,1,d}} (u,t) - 1} \right\}} \cdot X(u) \cdot A(0,0,d)(u)} \hfill \\ \end{array} } \right]\;du} } \right\} $$

(11.C10)

Assume now that the model parameters $G(\alpha ,\beta ,d)(t)$, $D(\alpha ,\beta ,d)(t)$, $M(\alpha ,\beta ,d)(t)$, $A(\alpha ,\beta ,d)(t)$ and $X(t)$ are constant over time. By substituting $z_{\alpha ,\beta ,d} (u,t) = [y_{\alpha ,\beta ,d} (u,t) - 1] \cdot G(\alpha ,\beta ,d)$ into (11.C8) and (11.C9), the following can be obtained

$$ \left\{ {\begin{array}{*{20}l} {\frac{{dz_{{\alpha ,\beta ,d}} }}{{ds}} = - z_{{\alpha ,\beta ,d}} ^{2} + z_{{\alpha ,\beta ,d}} \cdot N\left[ {\alpha ,\beta ,d,z_{{\alpha ,\beta + 1,d}} ,z_{{\alpha + 1,\beta ,d}} } \right]\quad \quad when\;d \ne 0} \hfill \\ {\quad \quad +\,P\left[ {\alpha ,\beta ,d,z_{{\alpha ,\beta + 1,d}} ,z_{{\alpha + 1,\beta ,d}} } \right]} \hfill \\ {\frac{{dz_{{\alpha ,0,0}} }}{{ds}} = - z_{{\alpha ,0,0}} ^{2} + z_{{\alpha ,0,0}} \cdot N'\left[ {\alpha ,z_{{\alpha ,1,1}} , \cdots ,z_{{\alpha ,1,r}} ,z_{{\alpha + 1,0,0}} } \right]\quad when\;d = 0} \hfill \\ {\quad \quad +\,P'\left[ {\alpha ,z_{{\alpha ,1,1}} , \cdots ,z_{{\alpha ,1,r}} ,z_{{\alpha + 1,0,0}} } \right]} \hfill \\ \end{array} } \right. $$

(11.C11)

where

$$N\left[{\alpha ,\beta ,d,v,w} \right] = \left\{\begin{array}{ll} D(\alpha ,\beta ,d) - G(\alpha ,\beta ,d) - w \cdot \frac{{M(\alpha,\beta ,d)}}{{G(\alpha + 1,\beta ,d)}} \hfill \\ - v \cdot \frac{{A(\alpha ,\beta ,d)}}{{G(\alpha ,\beta + 1,d)}} & {(\alpha < k - 1,\beta < m_{d} )}\hfill \\ {D(\alpha ,\beta ,d) - G(\alpha,\beta ,d) + M(\alpha ,\beta ,d)} \hfill \\ { - v \cdot \frac{{A(\alpha ,\beta ,d)}}{{G(\alpha ,\beta + 1,d)}}} & {(\alpha = k - 1,\beta < m_{d} )}\\ {D(\alpha ,\beta ,d) - G(\alpha ,\beta ,d) - w \cdot \frac{{M(\alpha ,\beta ,d)}}{{G(\alpha + 1,\beta ,d)}}} & {(\alpha < k - 1,\beta = m_{d} )} \hfill \\ D(\alpha,\beta ,d) - G(\alpha ,\beta ,d) + M(\alpha ,\beta ,d) & {(\alpha = k - 1,\beta = m_{d} )}\\ \end{array}\right. $$

(11.C12)

$$ P\left[ {\alpha ,\beta ,d,v,w} \right] = \left\{ {\begin{array}{*{20}c} { - G(\alpha ,\beta ,d) \cdot \left[ {w \cdot \frac{{M(\alpha ,\beta ,d)}}{{G(\alpha + 1,\beta ,d)}} + v \cdot \frac{{A(\alpha ,\beta ,d)}}{{G(\alpha ,\beta + 1,d)}}} \right]} \hfill & {(\alpha < k - 1,\beta < m_{d} )} \hfill \\ {G(\alpha ,\beta ,d) \cdot \left[ {M(\alpha ,\beta ,d) - v \cdot \frac{{A(\alpha ,\beta ,d)}}{{G(\alpha ,\beta + 1,d)}}} \right]} \hfill & {(\alpha = k - 1,\beta < m_{d} )} \hfill \\ { - w \cdot \frac{{G(\alpha ,\beta ,d) \cdot M(\alpha ,\beta ,d)}}{{G(\alpha + 1,\beta ,d)}}} \hfill & {(\alpha < k - 1,\beta = m_{d} )} \hfill \\ {G(\alpha ,\beta ,d) \cdot M(\alpha ,\beta ,d)} \hfill & {(\alpha = k - 1,\beta = m_{d} )} \hfill \\ \end{array} } \right. $$

(11.C13)

$$ N^{\prime}\left[ {\alpha ,v_{1} , \cdots ,v_{r} ,w} \right] = \left\{ {\begin{array}{*{20}l} {D(\alpha ,0,0) - G(\alpha ,0,0) - w \cdot \frac{{M(\alpha ,0,0)}}{{G(\alpha + 1,0,0)}}} \hfill \\ { - \sum\limits_{{d = 1}}^{r} {v_{d} \cdot \frac{{A(\alpha ,0,d)}}{{G(\alpha ,1,d)}}\quad \quad \quad \quad \quad \quad \left( {0 < \alpha < k - 1} \right)} } \hfill \\ {D(\alpha ,0,0) - G(\alpha ,0,0) + M(\alpha ,0,0)} \hfill \\ { - \sum\limits_{{d = 1}}^{r} {v_{d} \cdot \frac{{A(\alpha ,0,d)}}{{G(\alpha ,1,d)}}\quad \quad \quad \quad \quad \quad \left( {\alpha = k - 1} \right)} } \hfill \\ \end{array} } \right. $$

(11.C14)

and

$$ P^{\prime}\left[ {\alpha ,v_{1} , \cdots ,v_{r} ,w} \right] = \left\{ {\begin{array}{*{20}l} { - G(\alpha ,0,0) \cdot \left[ {w \cdot \frac{{M(\alpha ,0,0)}}{{G(\alpha + 1,0,0)}} + \sum\limits_{{d = 1}}^{r} {v_{d} \cdot \frac{{A(\alpha ,0,d)}}{{G(\alpha ,1,d)}}} } \right]\;\;\;\; (0 < \alpha < k - 1)} \hfill \\ {G(\alpha ,0,0) \cdot \left[ {M(\alpha ,0,0) - \sum\limits_{{d = 1}}^{r} {v_{d} \cdot \frac{{A(\alpha ,0,d)}}{{G(\alpha ,1,d)}}} } \right]\quad \;\;\;\;\; (\alpha = k - 1)} \hfill \\ \end{array} } \right. $$

(11.C15)

Likewise, with the same trick, (11.C10) can be rewritten as

$$ \psi (t) = \exp \left\{ {\int\limits_{0}^{t} {z_{1,0,0} (s,t) \cdot \frac{X \cdot M(0,0,0)}{G(1,0,0)} + \sum\limits_{d = 1}^{r} {z_{0,1,d} (s,t) \cdot \frac{X \cdot A(0,0,d)}{G(0,1,d)}\;} ds} } \right\} $$

(11.C16)

Appendix D

4.1 Derivation of the Kolmogorov Forward Differential Equation for the Cancer Model of Little et al. [9]

In this Appendix we derive the Kolmogorov forward differential equation defining the generating function of the cancer model of Little et al. [9]. The generating function $\psi$ is given by

$$ \begin{array}{*{20}l} {\psi (y_{1,0,0} ,y_{2,0,0} , \ldots ,y_{k - 1,0,0} ,y_{0,1,1} , \ldots ,y_{k - 1,1,1} ,y_{0,2,1} ,y_{1,2,1} , \ldots ,y_{{k - 1,m_{r} ,r}} ,y_{k} ;t,s)} \hfill \\ { \equiv \psi (t,s)} \hfill \\ { = \sum\limits_{n} {y_{1,0,0}^{{n_{1,0,0} }} \cdot \ldots \cdot y_{k - 1,0,0}^{{n_{k - 1,0,0} }} \cdot y_{0,1,1}^{{n_{0,1,1} }} \cdot \ldots \cdot y_{k - 1,1,1}^{{n_{k - 1,1,1} }} \cdot \ldots \cdot y_{{k - 1,m_{r} ,r}}^{{n_{{k - 1,m_{r} ,r}} }} \cdot y_{k}^{{n_{k} }} \times } } \hfill \\ {\quad \quad P[Y_{1,0,0} (t) = n_{1,0,0} , \ldots ,Y_{k} (t) = n_{k} \left| {N(s) = X(0),Y_{1,0,0} (s) = \ldots = Y_{k} (s) = 0} \right.]} \hfill \\ \end{array} $$

(11.D1)

By differentiating term by term (justified by the absolute convergence of the derivative power series) $\psi$ satisfies:

$$ \begin{array}{*{20}l} {\frac{{\partial \psi }}{{\partial t}}[t,s] = \sum\limits_{{n_{{1,0,0}} ,n_{{2,0,0}} , \ldots ,n_{k} }} {y_{{1,0,0}} ^{{n_{{1,0,0}} }} \cdot \ldots \cdot y_{{k - 1,m_{r} ,r}} ^{{n_{{k - 1,m_{r} ,r}} }} \cdot y_{k} ^{{n_{k} }} \cdot \frac{{dP}}{{dt}}[Y_{{1,0,0}} (t) = n_{{1,0,0}} , \ldots ,Y_{{k - 1,m_{r} ,r}} (t) = n_{{k - 1,m_{r} ,r}} ,Y_{k} (t) = n_{k} ]} } \hfill \\ { = \sum\limits_{{n_{{1,0,0}} ,n_{{2,0,0}} , \ldots ,n_{k} }} {y_{{1,0,0}} ^{{n_{{1,0,0}} }} \cdot \ldots \cdot y_{{k - 1,m_{r} ,r}} ^{{n_{{k - 1,m_{r} ,r}} }} \cdot y_{k} ^{{n_{k} }} \cdot X(t) \cdot M(0,0,0)(t) \cdot P[Y_{{1,0,0}} (t) = n_{{1,0,0}} - 1, \ldots ,Y_{{k - 1,m}} (t) = n_{{k - 1,m_{r} ,r}} ,Y_{k} (t) = n_{k} ]} + } \hfill \\ {\sum\limits_{{d = 1}}^{r} {\sum\limits_{{n_{{1,0,0}} ,n_{{2,0,0}} , \ldots ,n_{k} }} {y_{{1,0,0}} ^{{n_{{1,0,0}} }} \cdot \ldots \cdot y_{{k - 1,m_{r} ,r}} ^{{n_{{k - 1,m_{r} ,r}} }} \cdot y_{k} ^{{n_{k} }} \cdot X(t) \cdot A(0,0,d)(t) \cdot P[Y_{{1,0,0}} (t) = n_{{1,0,0}} , \ldots ,Y_{{0,1,d}} (t) = n_{{0,1,d}} - 1, \ldots ,Y_{{k - 1,m_{r} ,r}} (t) = n_{{k - 1,m_{r} ,r}} ,Y_{k} (t) = n_{k} ]} - } } \hfill \\ {\sum\limits_{{n_{{1,0,0}} ,n_{{2,0,0}} , \ldots ,n_{k} }} {y_{{1,0,0}} ^{{n_{{1,0,0}} }} \cdot \ldots \cdot y_{{k - 1,m_{r} ,r}} ^{{n_{{k - 1,m_{r} ,r}} }} \cdot y_{k} ^{{n_{k} }} \cdot X(t) \cdot M(0,0,0)(t) \cdot P[Y_{{1,0,0}} (t) = n_{{1,0,0}} , \ldots ,Y_{{k - 1,m_{r} ,r}} (t) = n_{{k - 1,m_{r} ,r}} ,Y_{k} (t) = n_{k} ]} - } \hfill \\ {\sum\limits_{{d = 1}}^{r} {\sum\limits_{{n_{{1,0,0}} ,n_{{2,0,0}} , \ldots ,n_{k} }} {y_{{1,0,0}} ^{{n_{{1,0,0}} }} \cdot \ldots \cdot y_{{k - 1,m_{r} ,r}} ^{{n_{{k - 1,m_{r} ,r}} }} \cdot y_{k} ^{{n_{k} }} \cdot X(t) \cdot A(0,0,d)(t) \cdot P[Y_{{1,0,0}} (t) = n_{{1,0,0}} , \ldots ,Y_{{k - 1,m_{r} ,r}} (t) = n_{{k - 1,m_{r} ,r}} ,Y_{k} (t) = n_{k} ]} + } } \hfill \\ {\sum\limits_{{1 \le \alpha \le k - 1}} {\sum\limits_{{n_{{1,0,0}} ,n_{{2,0,0}} , \ldots ,n_{k} }} {y_{{1,0,0}} ^{{n_{{1,0,0}} }} \cdot \ldots \cdot y_{{k - 1,m_{r} ,r}} ^{{n_{{k - 1,m_{r} ,r}} }} \cdot y_{k} ^{{n_{k} }} \cdot \left[ {\begin{array}{*{20}l} {G(\alpha ,0,0)(t) \cdot [n_{{\alpha ,0,0}} - 1] \cdot P[Y_{{1,0,0}} (t) = n_{{1,0,0}} , \ldots ,Y_{{\alpha ,0,0}} (t) = n_{{\alpha ,0,0}} - 1, \ldots ,Y_{k} (t) = n_{k} ] + } \hfill \\ {D(\alpha ,0,0)(t) \cdot [n_{{\alpha ,0,0}} + 1] \cdot P[Y_{{1,0,0}} (t) = n_{{1,0,0}} , \ldots ,Y_{{\alpha ,0,0}} (t) = n_{{\alpha ,0,0}} + 1, \ldots ,Y_{k} (t) = n_{k} ] + } \hfill \\ {M(\alpha ,0,0)(t) \cdot n_{{\alpha ,0,0}} \cdot P[Y_{{1,0,0}} (t) = n_{{1,0,0}} , \ldots ,Y_{{\alpha ,0,0}} (t) = n_{{\alpha ,0,0}} ,Y_{{\alpha + 1,0,0}} (t) = n_{{\alpha + 1,0,0}} - 1, \ldots ,Y_{k} (t) = n_{k} ] + } \hfill \\ {\sum\limits_{{d' = 1}}^{r} {A(\alpha ,0,d')(t) \cdot n_{{\alpha ,0,0}} \cdot P[Y_{{1,0,0}} (t) = n_{{1,0,0}} , \ldots ,Y_{{\alpha ,0,0}} (t) = n_{{\alpha ,0,0}} , \ldots ,Y_{{\alpha ,1,d'}} (t) = n_{{\alpha ,1,d'}} - 1, \ldots ,Y_{k} (t) = n_{k} ]} - } \hfill \\ {n_{{\alpha ,0,0}} \cdot P[Y_{{1,0,0}} (t) = n_{{1,0,0}} , \ldots ,Y_{k} (t) = n_{k} ] \cdot \left[ {G(\alpha ,0,0)(t) + D(\alpha ,0,0)(t) + M(\alpha ,0,0)(t) + \sum\limits_{{d' = 1}}^{r} {A(\alpha ,0,d')(t)} } \right]} \hfill \\ \end{array} } \right] + } } } \hfill \\ {\sum\limits_{{\begin{array}{*{20}l} {0 \le \alpha \le k - 1} \hfill \\ {1 \le \beta \le m_{d} } \hfill \\ {1 \le d \le r} \hfill \\ \end{array} }} {\sum\limits_{{n_{{1,0,0}} ,n_{{2,0,0}} , \ldots ,n_{k} }} {y_{{1,0,0}} ^{{n_{{1,0,0}} }} \cdot \ldots \cdot y_{{k - 1,m_{r} ,r}} ^{{n_{{k - 1,m_{r} ,r}} }} \cdot y_{k} ^{{n_{k} }} \cdot \left[ {\begin{array}{*{20}l} {G(\alpha ,\beta ,d)(t) \cdot [n_{{\alpha ,\beta ,d}} - 1] \cdot P[Y_{{1,0,0}} (t) = n_{{1,0,0}} , \ldots ,Y_{{\alpha ,\beta ,d}} (t) = n_{{\alpha ,\beta ,d}} - 1, \ldots ,Y_{k} (t) = n_{k} ] + } \hfill \\ {D(\alpha ,\beta ,d)(t) \cdot [n_{{\alpha ,\beta ,d}} + 1] \cdot P[Y_{{1,0,0}} (t) = n_{{1,0,0}} , \ldots ,Y_{{\alpha ,\beta ,d}} (t) = n_{{\alpha ,\beta ,d}} + 1, \ldots ,Y_{k} (t) = n_{k} ] + } \hfill \\ {M(\alpha ,\beta ,d)(t) \cdot n_{{\alpha ,\beta ,d}} \cdot P[Y_{{1,0,0}} (t) = n_{{1,0,0}} , \ldots ,Y_{{\alpha ,\beta ,d}} (t) = n_{{\alpha ,\beta ,d}} ,Y_{{\alpha + 1,\beta ,d}} (t) = n_{{\alpha + 1,\beta ,d}} - 1, \ldots ,Y_{k} (t) = n_{k} ] + } \hfill \\ {A(\alpha ,\beta ,d)(t) \cdot n_{{\alpha ,0,0}} \cdot P[Y_{{1,0,0}} (t) = n_{{1,0,0}} , \ldots ,Y_{{\alpha ,\beta ,d}} (t) = n_{{\alpha ,\beta ,d}} , \ldots ,Y_{{\alpha ,\beta + 1,d}} (t) = n_{{\alpha ,\beta + 1,d}} - 1, \ldots ,Y_{k} (t) = n_{k} ] - } \hfill \\ {n_{{\alpha ,\beta ,d}} \cdot P[Y_{{1,0,0}} (t) = n_{{1,0,0}} , \ldots ,Y_{k} (t) = n_{k} ] \cdot \left[ {G(\alpha ,\beta ,d)(t) + D(\alpha ,\beta ,d)(t) + M(\alpha ,\beta ,d)(t) + A(\alpha ,\beta ,d)(t)} \right]} \hfill \\ \end{array} } \right]} } } \hfill \\ { = \psi \cdot [y_{{1,0,0}} - 1] \cdot X(t) \cdot M(0,0,0)(t) + \psi \cdot \sum\limits_{{d = 1}}^{r} {[y_{{0,1,d}} - 1] \cdot X(t) \cdot A(0,0,d)(t)} + } \hfill \\ {\sum\limits_{{1 \le \alpha \le k - 1}} {\left[ {\begin{array}{*{20}l} { - y_{{\alpha ,0,0}} \cdot [D(\alpha ,0,0)(t) + G(\alpha ,0,0)(t) + M(\alpha ,0,0)(t)} \hfill \\ { + \sum\limits_{{d' = 1}}^{r} {A(\alpha ,0,d')(t)} ] + y_{{\alpha ,0,0}} ^{2} \cdot G(\alpha ,0,0)(t) + D(\alpha ,0,0)(t)} \hfill \\ { + y_{{\alpha ,0,0}} \cdot y_{{\alpha + 1,0,0}} \cdot M(\alpha ,0,0)(t) + \sum\limits_{{d' = 1}}^{r} {y_{{\alpha ,0,0}} \cdot y_{{\alpha ,1,d'}} \cdot A(\alpha ,0,d')(t)} } \hfill \\ \end{array} } \right] \cdot \frac{{\partial \psi }}{{\partial y_{{\alpha ,0,0}} }}} } \;+ \hfill \\ {\sum\limits_{{\begin{array}{*{20}l} {0 \le \alpha \le k - 1} \hfill \\ {1 \le \beta \le m_{d} } \hfill \\ {1 \le d \le r} \hfill \\ \end{array} }} {\left[ {\begin{array}{*{20}l} { - y_{{\alpha ,\beta ,d}} \cdot [D(\alpha ,\beta ,d)(t) + G(\alpha ,\beta ,d)(t) + M(\alpha ,\beta ,d)(t)} \hfill \\ { + A(\alpha ,\beta ,d)(t)] + y_{{\alpha ,\beta ,d}} ^{2} \cdot G(\alpha ,\beta ,d)(t) + D(\alpha ,\beta ,d)(t)} \hfill \\ { + y_{{\alpha ,\beta ,d}} \cdot y_{{\alpha + 1,\beta ,d}} \cdot M(\alpha ,\beta ,d)(t) + y_{{\alpha ,\beta ,d}} \cdot y_{{\alpha ,\beta + 1,d}} \cdot A(\alpha ,\beta ,d)(t)} \hfill \\ \end{array} } \right] \cdot \frac{{\partial \psi }} {{\partial y_{{\alpha ,\beta ,d}} }}} } \hfill \\ \end{array} $$

(11.D2)

Appendix E

5.1 Derivation of System of Differential Equations Defining the Hessian of the Hazard Function for the Cancer Model of Little and Wright [8] and Little et al. [9]

In this Section we derive the set of differential equations defining the Hessian (with respect to the model parameters) for the cancer model of Little and Wright [8] and Little et al. [9] in the case when all model parameters are constant. For simplicity we present only the derivation for the simpler model of Little and Wright [8]; the derivation for the more complex model of Little et al. [9] is straightforward but lengthy. This allows us to drop the final identifying label in each of $G(\alpha ,\beta ,d),D(\alpha ,\beta ,d),M(\alpha ,\beta ,d),A(\alpha ,\beta ,d)$, which we will henceforth write as $G(\alpha ,\beta ),D(\alpha ,\beta ),M(\alpha ,\beta ),A(\alpha ,\beta )$, respectively. The hazard function of the cancer model with $k$ cancer-stage mutations and $m$ destabilizing mutations developed by Little and Wright [8] may be written as:

$$ h(t) = - \int\limits_{0}^{t} {\left\{ {\frac{{\partial \phi_{1,0} [t,s]}}{\partial t} \cdot M(0,0) + \frac{{\partial \phi_{0,1} [t,s]}}{\partial t} \cdot A(0,0)} \right\} \cdot Xds} $$

(11.E1)

where the PGFs $\phi _{i,j}$ also satisfy the following Kolmogorov backward equations (for $0 \le i \le k - 1$, $0 \le j \le m$, $(i,j) \ne (0,0)$):

$$ \begin{array}{*{20}l} {\frac{{\partial \phi _{i,j} }}{\partial s}[t,s] = [D(i,j) + G(i,j) + M(i,j) + A(i,j)] \cdot \phi _{i,j} [t,s]} \hfill \\\qquad\quad\;\;{ - G(i,j) \cdot \phi _{i,j} [t,s]^{2} - M(i,j) \cdot \phi _{i,j} [t,s] \cdot \phi _{i + 1,j} [t,s]} \hfill \\ \qquad\quad\;\;{ - A(i,j) \cdot \phi _{i,j} [t,s] \cdot \phi _{i,j + 1} [t,s] - D(i,j)} \hfill \\ \end{array} $$

(11.E2)

Differentiating (11.E1) gives:

$$ \begin{array}{*{20}l} {\frac{\partial h(t)}{\partial X}= - \int\limits_{0}^{t} {\left\{ {\frac{{\partial \phi_{1,0} [t,s]}}{\partial t} \cdot M(0,0) + \frac{{\partial \phi_{0,1} [t,s]}}{\partial t} \cdot A(0,0)} \right\}ds} } \hfill \\ \quad\;\;{ =[1 - \phi_{1,0} [t,0]] \cdot M(0,0) + [1 - \phi_{0,1} [t,0]] \cdot A(0,0)} \hfill \\ \end{array} $$

(11.E3)

$$ \frac{\partial h(t)}{\partial M(0,0)} = - \int\limits_{0}^{t} {\frac{{\partial \phi_{1,0} [t,s]}}{\partial t} \cdot Xds} = [1 - \phi_{1,0} [t,0]] \cdot X $$

(11.E4)

$$ \frac{\partial h(t)}{\partial A(0,0)} = - \int\limits_{0}^{t} {\frac{{\partial \phi_{0,1} [t,s]}}{\partial t} \cdot Xds} = [1 - \phi_{0,1} [t,0]] \cdot X $$

(11.E5)

and for all other model parameters, $\beta_{k}$:

$$ \begin{array}{*{20}l} {\frac{\partial h(t)}{{\partial \beta_{k} }} = - \int\limits_{0}^{t} {\left\{ {\frac{{\partial^{2} \phi_{1,0} [t,s]}}{{\partial \beta_{k} \partial t}} \cdot M(0,0) + \frac{{\partial^{2} \phi_{0,1} [t,s]}}{{\partial \beta_{k} \partial t}} \cdot A(0,0)} \right\}Xds} } \hfill \\ \quad\;\; { = - \frac{{\partial \phi_{1,0} [t,0]}}{{\partial \beta_{k} }} \cdot M(0,0) - \frac{{\partial \phi_{0,1} [t,0]}}{{\partial \beta_{k} }} \cdot A(0,0)} \hfill \\ \end{array} $$

(11.E6)

Likewise, we can evaluate the second derivatives by differentiating (11.E3–11.E5) further:

$$ \frac{{\partial^{2} h(t)}}{\partial X\partial M(0,0)} = 1 - \phi_{1,0} [t,0] $$

(11.E7)

$$ \frac{{\partial^{2} h(t)}}{\partial X\partial A(0,0)} = 1 - \phi_{0,1} [t,0] $$

(11.E8)

$$ \frac{{\partial^{2} h(t)}}{{\partial^{2} M(0,0)}} = \frac{{\partial^{2} h(t)}}{{\partial^{2} A(0,0)}} = \frac{{\partial^{2} h(t)}}{\partial A(0,0)\partial M(0,0)} = 0 $$

(11.E9)

and for all model parameters, $\beta_{k} ,\beta_{l} \notin \left\{ {X,M(0,0),A(0,0)} \right\}$:

$$ \frac{{\partial^{2} h(t)}}{{\partial \beta_{k} \partial X}} = - \frac{{\partial \phi_{1,0} [t,0]}}{{\partial \beta_{k} }} \cdot M(0,0) - \frac{{\partial \phi_{0,1} [t,0]}}{{\partial \beta_{k} }} \cdot A(0,0) $$

(11.E10)

$$ \frac{{\partial^{2} h(t)}}{{\partial \beta_{k} \partial M(0,0)}} = - \frac{{\partial \phi_{1,0} [t,0]}}{{\partial \beta_{k} }} \cdot X $$

(11.E11)

$$ \frac{{\partial^{2} h(t)}}{{\partial \beta_{k} \partial A(0,0)}} = - \frac{{\partial \phi_{0,1} [t,0]}}{{\partial \beta_{k} }} \cdot X $$

(11.E12)

$$ \frac{{\partial^{2} h(t)}}{{\partial \beta_{k} \partial \beta_{l} }} = - \frac{{\partial^{2} \phi_{1,0} [t,0]}}{{\partial \beta_{k} \partial \beta_{l} }} \cdot M(0,0) - \frac{{\partial^{2} \phi_{0,1} [t,0]}}{{\partial \beta_{k} \partial \beta_{l} }} \cdot A(0,0) $$

(11.E13)

We can evaluate $\frac{{\partial \phi_{i,j} [t,s]}}{{\partial \beta_{k} }}$ by differentiating (11.E2), for $\beta_{k} \notin \left\{ {D(i,j),G(i,j),M(i,j),A(i,j)} \right\}$:

$$ \begin{array}{*{20}l} {\frac{{\partial^{2} \phi_{i,j} }}{{\partial \beta_{k} \partial s}}[t,s] = [D(i,j) + G(i,j) + M(i,j) + A(i,j)] \cdot \frac{{\partial \phi_{i,j} }}{{\partial \beta_{k} }}[t,s]} \hfill \\ \quad \quad \quad\;{ - 2 \cdot G(i,j) \cdot \phi_{i,j} [t,s] \cdot \frac{{\partial \phi_{i,j} }}{{\partial \beta_{k} }}[t,s] - M(i,j) \cdot \left[ {\frac{{\partial \phi_{i,j} }}{{\partial \beta_{k} }}[t,s] \cdot \phi_{i + 1,j} [t,s] + \frac{{\partial \phi_{i + 1,j} }}{{\partial \beta_{k} }}[t,s] \cdot \phi_{i,j} [t,s]} \right]} \hfill \\ \quad\quad\quad\;{ - A(i,j) \cdot \left[ {\frac{{\partial \phi_{i,j} }}{{\partial \beta_{k} }}[t,s] \cdot \phi_{i,j + 1} [t,s] + \frac{{\partial \phi_{i,j + 1} }}{{\partial \beta_{k} }}[t,s] \cdot \phi_{i,j} [t,s]} \right]} \hfill \\ \quad\qquad\; { = \Upomega_{i,j} \left[ {\beta_{k} ,t,s} \right]} \hfill \\ \end{array} $$

(11.E14)

with appropriate initial conditions (discussed later). For $\beta_{k} \in \left\{ {D(i,j),G(i,j),M(i,j),A(i,j)} \right\}$ we have:

$$ \frac{{\partial ^{2} \phi _{{i,j}} }}{{\partial \beta _{k} \partial s}}[t,s] = \left\{ {\begin{array}{*{20}l} {\Omega _{{i,j}} \left[ {\beta _{k} ,t,s} \right] + \phi _{{i,j}} [t,s] - 1} \hfill & {\beta _{k} = D(i,j)} \hfill \\ {\Omega _{{i,j}} \left[ {\beta _{k} ,t,s} \right] + \phi _{{i,j}} [t,s] - \phi _{{i,j}} [t,s]^{2} } \hfill & {\beta _{k} = G(i,j)} \hfill \\ {\Omega _{{i,j}} \left[ {\beta _{k} ,t,s} \right] + \phi _{{i,j}} [t,s] - \phi _{{i,j}} [t,s]\phi _{{i + 1,j}} [t,s]} \hfill & {\beta _{k} = M(i,j)} \hfill \\ {\Omega _{{i,j}} \left[ {\beta _{k} ,t,s} \right] + \phi _{{i,j}} [t,s] - \phi _{{i,j}} [t,s]\phi _{{i,j + 1}} [t,s]} \hfill & {\beta _{k} = A(i,j){\text{ }}} \hfill \\ \end{array} } \right. $$

(11.E15)

Likewise, we can evaluate $\frac{{\partial^{2} h(t)}}{{\partial \beta_{k} \partial \beta_{l} }}$ by differentiating (11.E14), for $\beta_{k} ,\beta_{l} \notin \left\{ {D(i,j),G(i,j),M(i,j),A(i,j)} \right\}$:

$$ \begin{array}{*{20}l} {\frac{{\partial ^{3} \phi _{{i,j}} }}{{\partial \beta _{k} \partial \beta _{l} \partial s}}[t,s] = [D(i,j) + G(i,j) + M(i,j) + A(i,j)] \cdot \frac{{\partial ^{2} \phi _{{i,j}} }}{{\partial \beta _{k} \partial \beta _{l} }}[t,s]} \hfill \\ \quad\quad\quad \quad\;{ - 2 \cdot G(i,j) \cdot \left[ {\frac{{\partial \phi _{{i,j}} }}{{\partial \beta _{k} }}[t,s] \cdot \frac{{\partial \phi _{{i,j}} }}{{\partial \beta _{l} }}[t,s] + \phi _{{i,j}} [t,s] \cdot \frac{{\partial ^{2} \phi _{{i,j}} }}{{\partial \beta _{k} \partial \beta _{l} }}[t,s]} \right]} \hfill \\ \quad\quad\quad\quad\; { - M(i,j) \cdot \left[ {\begin{array}{*{20}l} {\frac{{\partial ^{2} \phi _{{i,j}} }}{{\partial \beta _{k} \partial \beta _{l} }}[t,s] \cdot \phi _{{i + 1,j}} [t,s] + \frac{{\partial \phi _{{i,j}} }}{{\partial \beta _{k} }}[t,s] \cdot \frac{{\partial \phi _{{i + 1,j}} }}{{\partial \beta _{l} }}[t,s] + \frac{{\partial \phi _{{i,j}} }}{{\partial \beta _{l} }}[t,s] \cdot \frac{{\partial \phi _{{i + 1,j}} }}{{\partial \beta _{k} }}[t,s] + } \hfill \\ {\frac{{\partial ^{2} \phi _{{i + 1,j}} }}{{\partial \beta _{k} \partial \beta _{l} }}[t,s] \cdot \phi _{{i,j}} [t,s]} \hfill \\ \end{array} } \right]} \hfill \\ \quad\quad\quad\quad\; { - A(i,j) \cdot \left[ {\begin{array}{*{20}l} {\frac{{\partial ^{2} \phi _{{i,j}} }}{{\partial \beta _{k} \partial \beta _{l} }}[t,s] \cdot \phi _{{i,j + 1}} [t,s] + \frac{{\partial \phi _{{i,j}} }}{{\partial \beta _{k} }}[t,s] \cdot \frac{{\partial \phi _{{i,j + 1}} }}{{\partial \beta _{l} }}[t,s] + \frac{{\partial \phi _{{i,j}} }}{{\partial \beta _{l} }}[t,s] \cdot \frac{{\partial \phi _{{i,j + 1}} }}{{\partial \beta _{k} }}[t,s] + } \hfill \\ {\frac{{\partial ^{2} \phi _{{i,j + 1}} }}{{\partial \beta _{k} \partial \beta _{l} }}[t,s] \cdot \phi _{{i,j}} [t,s]} \hfill \\ \end{array} } \right]} \hfill \\ \quad\quad\quad\;\; { = \Uppsi_{{i,j}} \left[ {\beta _{k} ,\beta _{l} ,t,s} \right]} \hfill \\ \end{array} $$

(11.E16)

For $\beta_{k} \notin \left\{ {D(i,j),G(i,j),M(i,j),A(i,j)} \right\}$ we have:

$$ \frac{{\partial^{3} \phi_{i,j} }}{{\partial \beta_{k} \partial \beta_{l} \partial s}}[t,s] = \left\{ {\begin{array}{*{20}l} {\Uppsi_{i,j} \left[ {\beta_{k} ,\beta_{l} ,t,s} \right] + \frac{{\partial \phi_{i,j} [t,s]}}{{\partial \beta_{k} }}\quad \quad \quad \quad \quad \quad \beta_{l} = D(i,j)} \hfill \\ {\Uppsi_{i,j} \left[ {\beta_{k} ,\beta_{l} ,t,s} \right] + \frac{{\partial \phi_{i,j} [t,s]}}{{\partial \beta_{k} }}\quad \quad \quad \quad \quad \quad \beta_{l} = G(i,j)} \hfill \\ {\quad \quad - 2 \cdot \frac{{\partial \phi_{i,j} [t,s]}}{{\partial \beta_{k} }} \cdot \phi_{i,j} [t,s]} \hfill \\ {\Uppsi_{i,j} \left[ {\beta_{k} ,\beta_{l} ,t,s} \right] + \frac{{\partial \phi_{i,j} [t,s]}}{{\partial \beta_{k} }} - \frac{{\partial \phi_{i,j} [t,s]}}{{\partial \beta_{k} }} \cdot \phi_{i + 1,j} [t,s]\; \beta_{l} = M(i,j)} \hfill \\ {\quad \quad - \frac{{\partial \phi_{i + 1,j} [t,s]}}{{\partial \beta_{k} }} \cdot \phi_{i,j} [t,s]} \hfill \\ {\Uppsi_{i,j} \left[ {\beta_{k} ,\beta_{l} ,t,s} \right] + \frac{{\partial \phi_{i,j} [t,s]}}{{\partial \beta_{k} }} - \frac{{\partial \phi_{i,j} [t,s]}}{{\partial \beta_{k} }} \cdot \phi_{i,j + 1} [t,s]\; \beta_{l} = A(i,j)} \hfill \\ {\quad \quad - \frac{{\partial \phi_{i,j + 1} [t,s]}}{{\partial \beta_{k} }} \cdot \phi_{i,j} [t,s]\;} \hfill \\ \end{array} } \right. $$

(11.E17)

Finally, we have that:

$$ \frac{{\partial^{3} \phi_{i,j} }}{{\partial D(i,j)^{2} \partial s}}[t,s] = \Uppsi_{i,j} \left[ {D(i,j),D(i,j),t,s} \right] + 2 \cdot \frac{{\partial \phi_{i,j} [t,s]}}{\partial D(i,j)} $$

(11.E18)

$$ \frac{{\partial^{3} \phi_{i,j} }}{\partial D(i,j)\partial G(i,j)\partial s}[t,s] = \Uppsi_{i,j} \left[ {D(i,j),G(i,j),t,s} \right] + \frac{{\partial \phi_{i,j} [t,s]}}{\partial D(i,j)} + \frac{{\partial \phi_{i,j} [t,s]}}{\partial G(i,j)} - 2 \cdot \frac{{\partial \phi_{i,j} [t,s]}}{\partial D(i,j)} \cdot \phi_{i,j} [t,s] $$

(11.E19)

$$ \frac{{\partial^{3} \phi_{i,j} }}{\partial D(i,j)\partial M(i,j)\partial s}[t,s] = \Uppsi_{i,j} \left[ {D(i,j),M(i,j),t,s} \right] + \frac{{\partial \phi_{i,j} [t,s]}}{\partial D(i,j)} + \frac{{\partial \phi_{i,j} [t,s]}}{\partial M(i,j)} - \frac{{\partial \phi_{i,j} [t,s]}}{\partial D(i,j)} \cdot \phi_{i + 1,j} [t,s] $$

(11.E20)

$$ \frac{{\partial^{3} \phi_{i,j} }}{\partial D(i,j)\partial A(i,j)\partial s}[t,s] = \Uppsi_{i,j} \left[ {D(i,j),A(i,j),t,s} \right] + \frac{{\partial \phi_{i,j} [t,s]}}{\partial D(i,j)} + \frac{{\partial \phi_{i,j} [t,s]}}{\partial A(i,j)} - \frac{{\partial \phi_{i,j} [t,s]}}{\partial D(i,j)} \cdot \phi_{i,j + 1} [t,s] $$

(11.E21)

$$ \frac{{\partial^{3} \phi_{i,j} }}{{\partial G(i,j)^{2} \partial s}}[t,s] = \Uppsi_{i,j} \left[ {G(i,j),G(i,j),t,s} \right] + 2 \cdot \frac{{\partial \phi_{i,j} [t,s]}}{\partial G(i,j)} - 4 \cdot \frac{{\partial \phi_{i,j} [t,s]}}{\partial G(i,j)} \cdot \phi_{i,j} [t,s] $$

(11.E22)

$$ \begin{array}{*{20}l} {\frac{{\partial ^{3} \phi _{{i,j}} }}{{\partial G(i,j)\partial M(i,j)\partial s}}[t,s] = \Uppsi_{{i,j}} \left[ {G(i,j),M(i,j),t,s} \right] + \frac{{\partial \phi _{{i,j}} [t,s]}}{{\partial G(i,j)}} + \frac{{\partial \phi _{{i,j}} [t,s]}}{{\partial M(i,j)}} - 2 \cdot \frac{{\partial \phi _{{i,j}} [t,s]}}{{\partial M(i,j)}} \cdot \phi _{{i,j}} [t,s]} \hfill \\ { - \frac{{\partial \phi _{{i,j}} [t,s]}}{{\partial G(i,j)}} \cdot \phi _{{i + 1,j}} [t,s]} \hfill \\ \end{array} $$

(11.E23)

$$ \begin{array}{*{20}l} {\frac{{\partial ^{3} \phi _{{i,j}} }}{{\partial G(i,j)\partial A(i,j)\partial s}}[t,s] = \Uppsi _{{i,j}} \left[ {G(i,j),A(i,j),t,s} \right] + \frac{{\partial \phi _{{i,j}} [t,s]}}{{\partial G(i,j)}} + \frac{{\partial \phi _{{i,j}} [t,s]}}{{\partial A(i,j)}} - 2 \cdot \frac{{\partial \phi _{{i,j}} [t,s]}}{{\partial A(i,j)}} \cdot \phi _{{i,j}} [t,s]} \hfill \\ { - \frac{{\partial \phi _{{i,j}} [t,s]}}{{\partial G(i,j)}} \cdot \phi _{{i,j + 1}} [t,s]} \hfill \\ \end{array} $$

(11.E24)

$$ \frac{{\partial^{3} \phi_{i,j} }}{{\partial M(i,j)^{2} \partial s}}[t,s] = \Uppsi_{i,j} \left[ {M(i,j),M(i,j),t,s} \right] + 2 \cdot \frac{{\partial \phi_{i,j} [t,s]}}{\partial M(i,j)} - 2 \cdot \frac{{\partial \phi_{i,j} [t,s]}}{\partial M(i,j)} \cdot \phi_{i + 1,j} [t,s] $$

(11.E25)

$$ \begin{array}{*{20}l} {\frac{{\partial ^{3} \phi _{{i,j}} }}{{\partial M(i,j)\partial A(i,j)\partial s}}[t,s] = \Uppsi _{{i,j}} \left[ {M(i,j),A(i,j),t,s} \right] + \frac{{\partial \phi _{{i,j}} [t,s]}}{{\partial M(i,j)}} + \frac{{\partial \phi _{{i,j}} [t,s]}}{{\partial A(i,j)}} - \frac{{\partial \phi _{{i,j}} [t,s]}}{{\partial M(i,j)}} \cdot \phi _{{i,j + 1}} [t,s]} \hfill \\ { - \frac{{\partial \phi _{{i,j}} [t,s]}}{{\partial A(i,j)}} \cdot \phi _{{i + 1,j}} [t,s]} \hfill \\ \end{array} $$

(11.E26)

$$ \frac{{\partial^{3} \phi_{i,j} }}{{\partial A(i,j)^{2} \partial s}}[t,s] = \Uppsi_{i,j} \left[ {A(i,j),A(i,j),t,s} \right] + 2 \cdot \frac{{\partial \phi_{i,j} [t,s]}}{\partial A(i,j)} - 2 \cdot \frac{{\partial \phi_{i,j} [t,s]}}{\partial A(i,j)} \cdot \phi_{i,j + 1} [t,s] $$

(11.E27)

As in Little and Wright [8], the following boundary conditions must be satisfied, for all $i,j,\beta_{k} ,\beta_{l}$:

$$ \phi_{i,j} [t,t] = 1\;\quad 0 \le i \le k - 1 $$

(11.E28)

$$ \frac{{\partial \phi_{i,j} [t,t]}}{{\partial \beta_{k} }} = \frac{{\partial^{2} \phi_{i,j} [t,t]}}{{\partial \beta_{k} \partial \beta_{l} }} = 0 $$

(11.E29)

This system of ordinary differential equations (in the variable $s$) for $\phi_{i,j} [t,s]$, $\frac{{\partial \phi_{i,j} [t,s]}}{{\partial \beta_{k} }}$, $\frac{{\partial^{2} \phi_{i,j} [t,s]}}{{\partial \beta_{k} \partial \beta_{l} }}$ were integrated using the Boerlisch-Stoer algorithm with adaptive stepsize control [24]. Very similar results were obtained using a Runge-Kutta integrator with adaptive stepsize control [24].