$$ \newcommand{\esp}[1]{\mathbb{E}\left(#1\right)} \newcommand{\var}[1]{\mbox{Var}\left(#1\right)} \newcommand{\deriv}[1]{\dot{#1}(t)} \newcommand{\prob}[1]{ \mathbb{P}\!(#1)} \newcommand{\eqdef}{\mathop{=}\limits^{\mathrm{def}}} \newcommand{\by}{\boldsymbol{y}} \newcommand{\bc}{\boldsymbol{c}} \newcommand{\bpsi}{\boldsymbol{\psi}} \def\pmacro{\texttt{p}} \def\like{{\cal L}} \def\llike{{\cal LL}} \def\logit{{\rm logit}} \def\probit{{\rm probit}} \def\one{{\rm 1\!I}} \def\iid{\mathop{\sim}_{\rm i.i.d.}} \def\simh0{\mathop{\sim}_{H_0}} \def\df{\texttt{df}} \def\res{e} \def\xomega{x} \newcommand{\argmin}[1]{{\rm arg}\min_{#1}} \newcommand{\argmax}[1]{{\rm arg}\max_{#1}} \newcommand{\Rset}{\mbox{$\mathbb{R}$}} \def\param{\theta} \def\setparam{\Theta} \def\xnew{x_{\rm new}} \def\fnew{f_{\rm new}} \def\ynew{y_{\rm new}} \def\nnew{n_{\rm new}} \def\enew{e_{\rm new}} \def\Xnew{X_{\rm new}} \def\hfnew{\widehat{\fnew}} \def\degree{m} \def\nbeta{d} \newcommand{\limite}[1]{\mathop{\longrightarrow}\limits_{#1}} \def\ka{k{\scriptstyle a}} \def\ska{k{\scriptscriptstyle a}} \def\kel{k{\scriptstyle e}} \def\skel{k{\scriptscriptstyle e}} \def\cl{C{\small l}} \def\Tlag{T\hspace{-0.1em}{\scriptstyle lag}} \def\sTlag{T\hspace{-0.07em}{\scriptscriptstyle lag}} \def\Tk{T\hspace{-0.1em}{\scriptstyle k0}} \def\sTk{T\hspace{-0.07em}{\scriptscriptstyle k0}} \def\thalf{t{\scriptstyle 1/2}} \newcommand{\Dphi}[1]{\partial_\pphi #1} \def\asigma{a} \def\pphi{\psi} \newcommand{\stheta}{{\theta^\star}} \newcommand{\htheta}{{\widehat{\theta}}}$$

Delbene

Mathematical model: \[ \begin{aligned} \deriv{N_1} & = k_2\, C \, N_p(t) - k_1 \, N_1(t) \\ \deriv{N_2} & = k_1 \, N_1(t) - k_1 \, N_2(t) \\ \deriv{N_3} & = k_1\, N_2(t) - k_1 \, N_3(t) \\ \deriv{N_p} & = \lambda_0 \, N_p(t) - k_2\, C \, N_p(t) \\ N_t(t) & = N_1(t) + N_2(t) + N_3(t) + N_p(t) \end{aligned} \]

Initial conditions: \[ \begin{aligned} N_1(0) &= N_2(0) = N_3(0) = 0 \\ N_p(0) &= N_0 \end{aligned} \]



Del Bene F, Germani M, De Nicolao G, Magni P, Re CE, Ballinari D, Rocchetti M A model-based approach to the in vitro evaluation of anticancer activity, Cancer chemotherapy and pharmacology, 4/2009, Volume 63, Issue 5, pages: 827-836

Ribba

Mathematical model: \[ \begin{aligned} P^\star(t) &= P_T(t) + Q(t) + Q_P(t) \\ \deriv{C} &= - k_{de} C(t) \\ \deriv{P_T} &= \lambda P_T(t)(1- P^{\star}(t)/K) + k_{QPP}Q_P(t) -k_{PQ} P_T(t) -\gamma \, k_{de} P_T(t)C(t) \\ \deriv{Q} &= k_{PK} P_T(t) -\gamma \, k_{de} Q(t)C(t) \\ \deriv{Q_P} &= \gamma \, k_{de} Q(t)C(t) - k_{QPP} Q_P(t) -\delta_{QP} Q_P(t) \end{aligned} \]

Initial conditions: \[ \begin{aligned} C(0) &= Q_P(0) = 0 \\ P_T(0) &= p_{T0} \\ Q(0) &= q_0 \end{aligned} \]

Treatment:


Ribba, B., Kaloshi, G., Peyre, M., Ricard, D., Calvez, V., Tod, M., … & Ducray, F. (2012). A tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy. Clinical Cancer Research, 18(18), 5071-5080.


Simeoni

Mathematical model: \[ \begin{aligned} C &= Q_1 / V_1 \\ \deriv{Q_{1}} & = k_{21}Q_{2}(t) - (k_{10}+k_{12})Q_{1}(t) \\ \deriv{Q_{2}} & = k_{12}Q_{1}(t) -k_{21}Q_{2}(t) \\ \deriv{Z_0} &= \frac{\lambda_0 Z_0(t)}{\left(1 + \left(W\, \lambda_0 / \lambda_1 \right)^\psi \right)^{\frac{1}{\psi}} } - k_{2} C \, Z_0 \\ \deriv{Z_1} &= k_{2} C \, Z_0(t) - k_1 Z_1(t) \\ \deriv{Z_2} &= k_{1}Z_1(t) - k_1 Z_2(t) \\ \deriv{Z_3} &= k_{1}Z_2(t) - k_1 Z_3(t) \\ W(t) & = Z_0(t) + Z_1(t)+ Z_2(t) +Z_3(t) \end{aligned} \]

Initial conditions: \[ \begin{aligned} Q_{1}(0) &= Q_{2}(0) = Z_1(0) = Z_2(0) = Z_3(0) = 0 \\ Z_0(0) &= w_0 \end{aligned} \]

Treatment:



Simeoni M, Magni P, Cammia C, De Nicolao G, Croci V, Pesenti E, Germani M, Poggesi I, Rocchetti, Predictive pharmacokinetic-pharmacodynamic modeling of tumor growth kinetics in xenograft models after administration of anticancer agents. M Cancer research, 2/2004, Volume 64, Issue 3, pages: 1094-1101


Simeoni revisited

Simeoni et al. (2004) presented a model for tumor growth and anti-cancer effects where a transit compartment model describes the apoptosis of tumor cells attacked by a drug. Following Koch et al. (2014), this model can be rewritten with a lifespan model.

Mathematical model: \[ \begin{aligned} \dot{A_d}(t) &= -k_a \, A_d(t) \\ \dot{A_c}(t) &= k_a \, A_d(t) - k_e \, A_c(t) \\ C_c(t) & = A_c(t)/V \\ \dot{p}(t) &= \frac{2 l_0 l_1 p(t)^2}{(l_1+2 l_0 p(t))(p(t)+d(t))} - k_{\rm pot} C_c(t) p(t) \\\ \dot{d}(t) &= k_{\rm pot} C_c(t) p(t) - k_{\rm pot} p(t-\tau) \frac{A_c(t-\tau)}{V} \\ w(t) &= p(t) + d(t) \end{aligned} \]

Initial conditions: \[ \begin{aligned} A_c(t) &= 0 \\\ p(t) &= w_0 \\\ d(t) &= 0 \end{aligned} \]

The PK parameters \((k_a, V, k_e)\) are fixed: \[ k_a = 20 \quad ; \quad V = 2.8 \quad ; \quad k_e = 2.4 \]

Treatment:



Koch G, Krzyzanski W, Pérez-Ruixo JJ, Schropp J. (2014) Modeling of delays in PKPD: classical approaches and a tutorial for delay differential equations, J Pharmacokinet Pharmacodyn. 41(4):291-318.


Rocchetti

Mathematical model: \[ \begin{aligned} c_{1A} &= FV_{1A} Q_{1A} \\ c_{1B} &= FV_{1B} Q_{1B} \\ k_{2h} &= k_2 \left(1-\frac{c_{1A}}{IC_{50c}+c_{1A}}\right) \\ \deriv{Q_{0A}} & = -k_{aA}Q_{0A}(t) \\ \deriv{Q_{1A}} & = k_{aA}Q_{0A}(t) -k_{eA}Q_{1A}(t) \\ \deriv{Q_{0B}} & = -k_{aB}Q_{0B}(t) \\ \deriv{Q_{1B}} & = k_{aB}Q_{0B}(t) - (k_{12}+k_{eB})Q_{1B}(t)-k_{21}Q_{2B}(t) \\ \deriv{Q_{2B}} & = k_{12}Q_{1B}(t) -k_{21}Q_{2B}(t) \\ \deriv{Z_0} &= \frac{\lambda_0 Z_0(t)}{\left(1 + \left(W\, \lambda_0 / \lambda_1 \right)^\psi \right)^{\frac{1}{\psi}} } \left(1 - E_{\rm max}\frac{c_{1A}}{IC_{50}+c_{1A}} \right) - k_{2h}c_{1B} \\ \deriv{Z_1} &= k_{2h}c_{1B}Z_0(t) - k_1 Z_1(t) \\ \deriv{Z_2} &= k_{1}Z_1(t) - k_1 Z_2(t) \\ \deriv{Z_3} &= k_{1}Z_2(t) - k_1 Z_3(t) \\ W(t) & = Z_0(t) + Z_1(t)+ Z_2(t) +Z_3(t) \end{aligned} \]

Initial conditions: \[ \begin{aligned} Q_{0A}(0) &= Q_{1A}(0) = Q_{0B}(0) = Q_{1B}(0) = Q_{2B}(0) = 0 \\ Z_0(0) &= w_0 \\ Z_1(0) &= Z_2(0) = Z_3(0) = 0 \end{aligned} \]

Treatment:



Rocchetti M, Germani M, Del Bene F, Poggesi I, Magni P, Pesenti E, De Nicolao G, Predictive pharmacokinetic-pharmacodynamic modeling of tumor growth after administration of an anti-angiogenic agent, bevacizumab, as single-agent and combination therapy in tumor xenografts, Cancer chemotherapy and pharmacology, 5/2013, Volume 71, Issue 5, pages: 1147-1157