Revisiting Dosing Regimen Using Pharmacokinetic/Pharmacodynamic Mathematical Modeling: Densification and Intensification of Combination Cancer Therapy

Abstract

Controlling effects of drugs administered in combination is particularly challenging with a densified regimen because of life-threatening hematological toxicities. We have developed a mathematical model to optimize drug dosing regimens and to redesign the dose intensification–dose escalation process, using densified cycles of combined anticancer drugs. A generic mathematical model was developed to describe the main components of the real process, including pharmacokinetics, safety and efficacy pharmacodynamics, and non-hematological toxicity risk. This model allowed for computing the distribution of the total drug amount of each drug in combination, for each escalation dose level, in order to minimize the average tumor mass for each cycle. This was achieved while complying with absolute neutrophil count clinical constraints and without exceeding a fixed risk of non-hematological dose-limiting toxicity. The innovative part of this work was the development of densifying and intensifying designs in a unified procedure. This model enabled us to determine the appropriate regimen in a pilot phase I/II study in metastatic breast patients for a 2-week-cycle treatment of docetaxel plus epirubicin doublet, and to propose a new dose-ranging process. In addition to the present application, this method can be further used to achieve optimization of any combination therapy, thus improving the efficacy versus toxicity balance of such a regimen.

Appendix

1.1 Pharmacokinetic Modeling

The PKs of the two drugs were described by an L—compartment model:

$$ \begin{array}{*{20}l} {\dot{c}_{1} \left( t \right) = - k_{e} \cdot c_{1} \left( t \right) - \sum\limits_{s = 2}^{L} {k_{1s} \cdot \left[ {c_{1} \left( t \right) - c_{s} \left( t \right)} \right]} + {{u\left( t \right)} \mathord{\left/ {\vphantom {{u\left( t \right)} {V_{1} }}} \right. \kern-0pt} {V_{1} }}} \hfill & {\quad c_{1} \left( 0 \right) = 0} \hfill & {} \hfill \\ {\dot{c}_{s} \left( t \right) = k_{s1} \cdot \left[ {c_{1} \left( t \right) - c_{s} \left( t \right)} \right]} \hfill & {\quad c_{s} \left( 0 \right) = 0} \hfill & {\quad s = 2:L} \hfill \\ \end{array} $$

(1)

where $c_{1} \left( t \right)$ and $c_{s} \left( t \right)$ are concentrations in the first and $s - th$ peripheral compartment, respectively. The function $u\left( t \right)$ describes the administration protocol corresponding to intermittent intravenous infusions:

$$ u\left( t \right) = \sum\limits_{j = 1}^{{n_{\text{D}} }} {\sum\limits_{i = 1}^{{n_{\text{A}} }} {\frac{{d_{ij} }}{{T_{0} }} \cdot \left[ {H\left( {t - t_{ij} } \right) - H\left( {t - t_{ij} - T_{0} } \right)} \right]} } $$

(2)

with

$$ t_{ij} = t_{0} + \tau_{1} \cdot \left( {i - 1} \right) + \tau_{2} \cdot \left( {j - 1} \right), $$

where index $i$ is the administration number ($n_{\text{A}}$ administrations per day), index $j$ is the day number ($n_{\text{D}}$ days of treatment per cycle), $H\left( t \right)$ is the Heaviside step function (equal to 1 for $t \ge 0$ and 0 otherwise), $T_{0}$ is the infusion duration, $t_{0}$ is a time lag, and $\tau_{1}$ and $\tau_{2}$ are periods of repeating the administration within a day and repeating the days of treatment per cycle, respectively. The $n_{\text{A}} \times n_{\text{D}}$ drug doses $d_{ij}$ were compiled in vector form $\underline{d}$. The schedule characteristics were also compiled in the vector form $\underline{t}^{ * } = \left[ {\begin{array}{*{20}c} {T_{0} } & {t_{0} } & {\tau_{1} } & {\tau_{2} } \\ \end{array} } \right]^{T}$. The input function $u\left( t \right)$ could then be stated as $u\left( {t,\underline{d} ,\underline{t}^{ * } } \right)$ since depending on $t$, $\underline{d}$ and $\underline{t}^{ * }$. To specify PKs for docetaxel and epirubicin implied in the combination, doses and concentrations were superscripted $\underline{d}^{\left( D \right)}$, $c^{\left( D \right)} \left( t \right)$ and $\underline{d}^{\left( E \right)}$, $c^{\left( E \right)} \left( t \right)$, respectively.

To prevent DLT events specific to long infusions [24], the two drugs were administered according to the following time schedule, as in the work of Viens et al. [18]: epirubicin was administered first ($t_{0}^{\left( E \right)} = 0{\text{ h}}$) as a 15-min ($T_{0}^{\left( E \right)} = 0.25{\text{ h}}$) constant rate infusion, and 1 h later ($t_{0}^{\left( D \right)} = 1{\text{ h}}$) docetaxel was administered as a 1-h ($T_{0}^{\left( D \right)} = 1{\text{ h}}$) constant rate infusion. This scheme, separated by 2-h discontinuation intervals ($\tau_{1} = 2{\text{ h}}$), was allowed to be repeated three times a day ($n_{\text{A}} = 3$) for the first 3 days of the cycle ($\begin{array}{*{20}c} {\tau_{2} = 1{\text{ d}}} & {n_{\text{D}} = 3} \\ \end{array}$). These data completed the schedule component $\underline{t}^{ * }$ in the input function $u\left( {t,\underline{d} ,\underline{t}^{ * } } \right)$ (Eq. 2). Therefore, the number of doses $\underline{d} = d_{ij}$ to be optimized was $n_{A} \times n_{D} \times \left( {2{\text{ drugs}}} \right) = 18$.

In population analyses involving these drugs either as single agents [25–27] or in combination [22], three-compartment models ($L = 3$) were used to describe the docetaxel and epirubicin kinetic profiles. Assuming no PK interaction [28], these models were also used to describe PKs in the present combination. Typical values of the two drugs are presented in Table 1.

1.2 PK/PD Interface Modeling

The interface model first introduced by Meille et al. [29] has been used to dynamically connect PKs and PDs. The interface model receives as input the PK levels of the administered drug in the central compartment, $c_{1} \left( t \right)$, and supplies as output the exposure variable, $y\left( t \right)$, linked to the PD response. This model was described by the following non-linear differential equation:

$$ \begin{array}{*{20}l} {\dot{y}\left( t \right) = - \alpha \cdot \exp \left[ { - y\left( t \right)} \right] \cdot y\left( t \right) + \beta \cdot \left[ {c_{1} \left( t \right) - \zeta } \right] \cdot H\left[ {c_{1} \left( t \right) - \zeta } \right]} \hfill & {} \hfill & {y\left( 0 \right) = 0} \hfill \\ \end{array} $$

(3)

where parameters $\alpha$, $\beta$, and $\zeta$ are positive adjustable parameters. Exposures and parameters were superscripted by $\left( D \right)$ and $\left( E \right)$ for docetaxel and epirubicin, respectively.

1.3 Dynamic Safety Modeling

The dynamic model is semi-mechanistic and describes the toxicity induced by anticancer drugs on the hematopoietic chain. Because of the drugs used and the background knowledge from other studies [1, 5], only neutropenia was considered in the dynamic safety model. Initially proposed by Meille et al. [29], this model was slightly modified to incorporate the cytotoxic effect of drugs in combination and the concomitant administration of G-CSF. The dynamic safety model is described by the equations:

$$ \begin{array}{*{20}l} {\dot{w}_{1} \left( t \right) = \gamma \cdot {\lambda \mathord{\left/ {\vphantom {\lambda \omega }} \right. \kern-0pt} \omega } \cdot w_{0} \cdot \varPhi \left[ {w\left( t \right),\varphi } \right] - \gamma \cdot w_{1} \left( t \right) - N\left[ {y\left( t \right),\nu } \right] \cdot w_{1} \left( t \right)} \hfill & {\quad w_{1} \left( 0 \right) = {\lambda \mathord{\left/ {\vphantom {\lambda \omega }} \right. \kern-0pt} \omega } \cdot w_{0} } \hfill \\ {\dot{w}_{2} \left( t \right) = \gamma \cdot \left[ {w_{1} \left( t \right) - w_{2} \left( t \right)} \right] - N\left[ {y\left( t \right),\nu } \right] \cdot w_{2} \left( t \right)} \hfill & {\quad w_{2} \left( 0 \right) = {\lambda \mathord{\left/ {\vphantom {\lambda \omega }} \right. \kern-0pt} \omega } \cdot w_{0} } \hfill \\ {\dot{w}_{3} \left( t \right) = \omega \cdot w_{2} \left( {t - \tau } \right) - \lambda \cdot w_{3} \left( t \right)} \hfill & {\quad w_{3} \left( { - \tau \le t \le 0} \right) = w_{0} } \hfill \\ {\dot{w}\left( t \right) = \lambda \cdot \left\{ { \, M\left[ {z\left( t \right),\mu } \right] \cdot w_{3} \left( t \right) - w\left( t \right) \, } \right\}} \hfill & {\quad w\left( 0 \right) = w_{0}} \hfill \\ \end{array} $$

(4)

States $w_{1} \left( t \right)$ and $w_{2} \left( t \right)$ are two forms of cells corresponding to two consecutive steps of the maturation chain. Both disappear with loss rate $\gamma$. States $w_{3} \left( t \right)$ and $w\left( t \right)$ represent circulating ANC, which disappear from the blood with rate constant $\lambda$. Because of the maturation times in the hematopoietic chain, a change in the progenitor levels resulted in a change in ANC $w_{3} \left( t \right)$ only after a time $\tau$. The rate constant $\omega$ is not structurally identifiable and therefore was set at $\omega = 1{\text{ day}}^{ - 1}$.

The production rate of cells from the progenitor cells was assumed to be regulated by a homeostatic mechanism $\varPhi \left[ {w\left( t \right),\varphi } \right]$, controlled by the mature neutrophil levels, $w\left( t \right)$ with parameter $\varphi$:

$$ \varPhi \left[ {w\left( t \right),\varphi } \right] = \left[ {{{w\left( t \right)} \mathord{\left/ {\vphantom {{w\left( t \right)} {w_{0} }}} \right. \kern-0pt} {w_{0} }}} \right]^{ - \varphi } $$

(5)

Circulating in the central compartment, anticancer drug levels $c_{1} \left( t \right)$ were assumed to result in exposure $y\left( t \right)$ that disturbs the neutrophil lineage by a direct hematological toxicity mechanism with parameters $\nu$:

$$ N\left[ {y\left( t \right),\nu } \right] = \nu^{\left( D \right)} \cdot y^{\left( D \right)} \left( t \right) + \nu^{\left( E \right)} \cdot y^{\left( E \right)} \left( t \right) + \nu_{0} \cdot y^{\left( D \right)} \left( t \right) \cdot y^{\left( E \right)} \left( t \right) $$

(6)

Finally, the stimulation mechanism of G-CSF (4) is represented by the function $M\left[ {z\left( t \right),\mu } \right]$ with parameters $\mu$.

Typical parameter values were obtained by modeling the ANC profiles from the phase I study of Viens et al. [18]. This study was conducted on 65 patients receiving the drug combination every 3 weeks without G-CSF support. Hematological toxicity profiles from the above dataset were used to estimate parameters involved in the interface model and the dynamic model. For example, Fig. 2 illustrates the fitted model of a given profile. The averages of estimated parameters for all available profiles were considered as typical values (shown in Table 1). The simulated ANC profiles using these typical values were in agreement with profiles reported in the literature by several investigations [18, 24, 26, 30–41].

Fig. 2

ANC simulated profile fitted to experimental data for one patient receiving combination chemotherapy. ANC profile (solid line) and the associated 95 % confidence bounds (dashed lines) for three consecutive cycles for one patient receiving 75 mg/m² of docetaxel and 90 mg/m² of epirubicin from the phase I study of Viens et al. [18]. Full circles represent the experimental data. ANC absolute neutrophil count

1.4 PK/PD of Granulocyte Colony-Stimulating Factor

The dynamic safety model (4) includes the effect of concomitant administration of G-CSF growth factors [42, 43] administered by subcutaneous route. The PKs of G-CSF were described by the following compartment model:

$$ \begin{array}{*{20}c} {\dot{z}_{1} \left( t \right) = - h_{F} \cdot z_{3} \left( t \right) \cdot z_{1} \left( t \right) - h_{12} \cdot \left[ {z_{1} \left( t \right) - z_{2} \left( t \right)} \right] + {{\upsilon \left( t \right)} \mathord{\left/ {\vphantom {{\upsilon \left( t \right)} {V_{Z1} }}} \right. \kern-0pt} {V_{Z1} }}}&{z_{1} \left( 0 \right) = 0} \\ {\dot{z}_{2} \left( t \right) = h_{21} \cdot \left[ {z_{1} \left( t \right) - z_{2} \left( t \right)} \right]}&{z_{2} \left( 0 \right) = 0} \\ {\dot{z}_{3} \left( t \right) = w\left( t \right) - h_{0} \cdot z_{3} \left( t \right) - h_{F} \cdot z_{3} \left( t \right) \cdot z_{1} \left( t \right) + h_{B} \cdot z_{4} \left( t \right)} & {z_{3} \left( 0 \right) = {{w_{0} } \mathord{\left/ {\vphantom {{w_{0} } {h_{0} }}} \right. \kern-0pt} {h_{0} }}} \\ {\dot{z}_{4} \left( t \right) = h_{F} \cdot z_{3} \left( t \right) \cdot z_{1} \left( t \right) - h_{B} \cdot z_{4} \left( t \right)} & {z_{4} \left( 0 \right) = 0} \\ \end{array} $$

(7)

where $z_{1} \left( t \right)$ and $z_{2} \left( t \right)$ are concentrations in the central and peripheral compartment, respectively. The G-CSF elimination rate $h_{F} \cdot z_{3} \left( t \right)$ was controlled by the $z_{3} \left( t \right)$ state of the third differential equation. This equation uses the mature ANC levels $w\left( t \right)$ as input, a constant rate $h_{0}$ elimination, and an enzymatic-like mechanism describing the association of $z_{3} \left( t \right)$ with $z_{1} \left( t \right)$. This association yields a complex $z_{4} \left( t \right)$ with rate $h_{F}$, which is dissociated to $z_{3} \left( t \right)$ and $z_{1} \left( t \right)$ with rate $h_{B}$. The distribution volume of the central compartment was denoted $V_{Z1}$. The input function $\upsilon \left( t \right)$ was a fractional Weibull-type absorption with parameters $g_{1}$ and $g_{2}$, and was given by the integral form:

$$ \int\limits_{0}^{t} {\upsilon \left( {t'} \right)dt'} = \sum\limits_{k = 1}^{{n_{G} }} {\delta_{k} \cdot \left[ {1 - \exp \left[ { - \left[ {g_{1} \cdot \left( {t - \bar{t}_{k} } \right)} \right]^{{g_{2} }} } \right]} \right]} $$

(8)

implying $n_{G}$ G-CSF doses $\delta_{k}$ administered at times $\bar{t}_{k}$. Finally, the stimulation mechanism (4) was:

$$ M\left[ {z\left( t \right),\mu } \right] = 1 + \mu_{0} \cdot \left[ { \, 1 - \exp \left[ { - \left( {{{z_{1} \left( t \right)} \mathord{\left/ {\vphantom {{z_{1} \left( t \right)} {\mu_{1} }}} \right. \kern-0pt} {\mu_{1} }}} \right)^{{\mu_{2} }} } \right] \, } \right] $$

(9)

Therefore, the chain $w\left( t \right) \to \left[ {h_{F} \cdot z_{3} \left( t \right)} \right] \to z_{1} \left( t \right) \to M\left[ {z\left( t \right),\mu } \right]$ resulted in an homeostatic mechanism on $w\left( t \right)$ levels, externally controlled by the G-CSF administration scheme $\upsilon \left( t \right)$; when $w\left( t \right)$ decreases or increases, G-CSF levels are high or low because its clearance is increased or decreased, respectively.

Typical PK/PD parameter values for G-CSF were obtained from unpublished data on file (Chugai Pharmaceutical Co.). The data came from two phase I studies, where the granocyte was administered as single or repeated doses in 40 healthy volunteers. For both studies, several blood samples were drawn and G-CSF concentrations and ANC were evaluated. By pooling these raw data, a single average profile was obtained for G-CSF and ANC. The typical values of the G-CSF model were then estimated (Table 2). Figure 3 illustrates the fit of the G-CSF model on the naively pooled profiles.

Fig. 3

G-CSF and ANC simulated profiles fitted to experimental data obtained in healthy volunteers. G-CSF (upper panel) and ANC (lower panel) experimental median profiles (full circles) from the data on file (Chugai Pharmaceutical Co.), fitted by the mathematical model (solid lines) and the associated 95 % confidence bounds (dashed lines). From time zero to day 10, the single-dose administration, and from day 10 to day 18, the repeated dose administration of G-CSF. G-CSF granulocyte colony-stimulating factors, ANC absolute neutrophil count

1.5 Hematological Constraints

The following clinical constraints on acceptable ANC levels were applied to the dynamic model:

• 1. “ANC must be maintained higher than the fixed level $W_{\text{D}}$”, translated by:

  $$ w\left( t \right) \ge W_{\text{D}} $$

  (10)

• 2. “Neutropenia should not exceed the period of time $T_{\text{U}}$”. To prevent the ANC from staying too long below a fixed level $W_{\text{U}}$, the time $t_{\text{U}}$ over which the ANC remains lower than $W_{\text{U}}$ should comply with:

  $$ t_{\text{U}} \left[ {w\left( t \right) \le W_{\text{U}} } \right] \le T_{\text{U}} $$

  (11)

  with $W_{\text{U}} > W_{\text{D}}$ usually. Chronometer $t_{\text{U}} \left( t \right)$ was evaluated from the following differential equation:

  $$ \begin{array}{*{20}l} {\frac{{{\text{d}}t_{\text{U}} \left( t \right)}}{{{\text{d}}t}} = H\left[ {W_{\text{U}} - w\left( t \right)} \right]} \hfill & {t_{\text{U}} \left( 0 \right) = 0} \hfill \\ \end{array} $$

  (12)

• 3. “ANC levels at the end of the cycle, just before the next administration, should be higher than the minimum level $W_{0}$”:

  $$ w\left( T \right) \ge W_{0} $$

  (13)

1.6 Modeling Antitumor Efficacy

A disturbed Gompertzian tumor growth model described tumor fate in the presence of anticancer agents [44]:

$$ \begin{array}{*{20}c} {\frac{{{\text{d}}n\left( t \right)}}{{{\text{d}}t}} = \rho \cdot n\left( t \right) \cdot \ln \left[ {{\theta \mathord{\left/ {\vphantom {\theta {n\left( t \right)}}} \right. \kern-0pt} {n\left( t \right)}}} \right] - \kappa \cdot f\left( {c_{L}^{\left( D \right)} ,\;c_{L}^{\left( E \right)} } \right) \cdot n\left( t \right)} & {\quad n\left( 0 \right) = n_{0} } \\ \end{array} $$

(14)

where $n\left( t \right)$ is the number of tumor cells at time $t$, and $n_{0}$ the initial tumor size. Parameters $\rho$ and $\theta$ are the tumor proliferation rate and the largest tumor mass, respectively. In the cell-loss term, constant $\kappa$ and term $f\left( {c_{L}^{\left( D \right)} ,c_{L}^{\left( E \right)} } \right)$ represent the global potency and effective concentration of combined drugs in the tumor site, respectively. The effective concentration term has a Weibull-like structure:

$$ f\left( {c_{L}^{\left( D \right)} ,c_{L}^{\left( E \right)} } \right) = 1 - \exp \left[ { - \left( {\underline{q}^{T} \cdot B \cdot \underline{q} } \right)^{{{\xi \mathord{\left/ {\vphantom {\xi 2}} \right. \kern-0pt} 2}}} } \right] $$

(15)

with ${q}^{\left({D}\right)} = {\left( {c_{L}^{\left( D \right)} - C_{MIN}^{\left( D \right)} } \right) \cdot H\left( {c_{L}^{\left( D \right)} - C_{MIN}^{\left( D \right)} } \right)}, \, {q}^{\left({E}\right)} = {\left( {c_{L}^{\left( E \right)} - C_{MIN}^{\left(E \right)} } \right) \cdot H\left( {c_{L}^{\left( E \right)} - C_{MIN}^{\left( E \right)} } \right)}$ and $\underline{q} = {\left[ {q^{\left( D \right)}\, q^{\left( E \right)} } \right]}^T$. In this formulation, $\xi$ is a shape parameter, the array $B$ weights the relative potency of combined drugs, and concentrations $C_{MIN}$ are thresholds below which no tumor cells were killed. Concentrations $c_{L} \left( t \right)$ in the deepest PK compartment were considered as the effective drug concentrations. As $n\left( t \right)$ depends on concentration of drugs, it also depends on the administration protocol and could therefore be stated by $n\left( {t,\underline{d} ,\underline{t}^{ * } } \right)$.

Typical values of these parameters are reported in Table 1. The values of $\rho$ and $\theta$ correspond to a doubling time of 2 months and a maximum tumor weight of 1 kg in human, respectively. The initial tumor size $n_{0}$ was set according to the evaluation of tumor mass before cancer therapy, usually near 30 g.

The parameters involved in $f\left( {c_{L}^{\left( D \right)} ,c_{L}^{\left( E \right)} } \right)$ were obtained from in vitro cytotoxicity experiments. These experiments performed with the MCF7 human breast cancer line generated data on the impact of docetaxel and epirubicin, used either alone or as a combination, on cancer cell proliferation. Tumor cells were incubated with 60 crossed concentration pairs of docetaxel (from $10^{ - 3}$ to 30 nM) and epirubicin (from $10^{ - 3}$ to 100 µM); the relative viability of cells was assessed at 72 h. The fit of the Weibull-like structure (15) to these data led to calculating $B$, $\xi$, and $C_{MIN}$. These values are reported in Table 1. The off-diagonal element in $B$ corresponds to a correlation coefficient equal to 0.6, indicating synergistic effect. Global potency $\kappa$ was calibrated by fitting the simulated treatment responses by the above models to the responses reported by previously published clinical work [26, 30, 32, 33, 35, 36, 38, 40, 45, 46].