Pharmacodynamic modeling of combined chemotherapeutic effects predicts synergistic activity of gemcitabine and trabectedin in pancreatic cancer cells

Abstract

Purpose

This study investigates the combined effects of gemcitabine and trabectedin (ecteinascidin 743) in two pancreatic cancer cell lines and proposes a pharmacodynamic (PD) model to quantify their pharmacological interactions.

Methods

Effects of gemcitabine and trabectedin upon the pancreatic cancer cell lines MiaPaCa-2 and BxPC-3 were investigated using cell proliferation assays. Cells were exposed to a range of concentrations of the two drugs, alone and in combination. Viable cell numbers were obtained daily over 5 days. A model incorporating nonlinear cytotoxicity, transit compartments, and an interaction parameter ψ was used to quantify the effects of the individual drugs and combinations.

Results

Simultaneous fitting of temporal cell growth profiles for all drug concentrations provided reasonable cytotoxicity parameter estimates (the cell killing rate constant K _max and the sensitivity constant KC ₅₀) for each drug. The interaction parameter ψ was estimated as 0.806 for MiaPaCa-2 and 0.843 for BxPC-3 cells, suggesting that the two drugs exert modestly synergistic effects.

Conclusions

The proposed PD model enables quantification of the temporal profiles of drug combinations over a range of concentrations with drug-specific parameters. Based upon these in vitro studies, trabectedin may have augmented benefit in combination with gemcitabine. The PD model may have general relevance for the study of other cytotoxic drug combinations.

Appendix

The following provides a proof of Eqs. (5)–(10), assuming that drug concentrations are constant during experiments. The stationary equations of the proposed PD model Eqs. (3) and (4) are as below:

$$K_{\text{d}} = \frac{{K_{\text{max,d}} C_{\text{d}}^{{\gamma_{\text{d}} }} }}{{KC_{{50,{\text{d}}}}^{{\gamma_{\text{d}} }} + C_{\text{d}}^{{\gamma_{\text{d}} }} }}$$

(11a)

$$0 = \frac{1}{{\tau_{\text{d}} }}\left( {K_{\text{d}} - K_{{1,{\text{d}}_{\text{ss}} }} } \right)$$

(11b)

$$0 = \frac{1}{{\tau_{\text{d}} }}\left( {K_{{1,{\text{d}}_{\text{ss}} }} - K_{{2,{\text{d}}_{\text{ss}} }} } \right)$$

(11c)

$$0 = \frac{1}{{\tau_{\text{d}} }}\left( {K_{{2,{\text{d}}_{\text{ss}} }} - K_{{3,{\text{d}}_{\text{ss}} }} } \right)$$

(11d)

$$0 = \frac{1}{{\tau_{\text{d}} }}\left( {K_{{3,{\text{d}}_{\text{ss}} }} - K_{{4,{\text{d}}_{\text{ss}} }} } \right)$$

(11e)

$$0 = k_{\text{g}} R_{\text{ss}}^{{\prime }} \left( {1 - \frac{{R_{\text{ss}}^{{\prime }} }}{{R_{\text{ss}} }}} \right) - \mu_{\text{ss}} R_{\text{ss}}^{{\prime }}$$

(11f)

with

$$\mu_{\text{ss}} = \left\{ \begin{array}{ll} {K_{{4,{\text{d}}_{\text{ss}} }} } &\quad {{\text{for}}\,{\text{single}}\,{\text{agent}}} \\ {K_{{4,{\text{Gem}}_{\text{ss}} }} + K_{{4,{\text{Et}}743_{\text{ss}} }} } &\quad {\text{for}}\,{\text{combination}}\end{array} \right.$$

From Eqs. (11a)–(11e) we have

$$K_{\text{d}} = K_{{1,{\text{d}}_{\text{ss}} }} = \cdots = K_{{4,{\text{d}}_{\text{ss}} }}$$

so that

$$\mu_{\text{ss}} = \left\{ {\begin{array}{ll} {\frac{{K_{{\hbox{max} ,{\text{d}}}} C_{\text{d}}^{{\gamma {\text{d}}}} }}{{KC_{{50,{\text{d}}}}^{{\gamma {\text{d}}}} + C_{\text{d}}^{{\gamma {\text{d}}}} }}} &\quad {{\text{for}}\,{\text{single}}\,{\text{agent}}} \\ {\frac{{K_{{\hbox{max} ,{\text{Gem}}}} C_{\text{Gem}}^{{\gamma {\text{Gem}}}} }}{{\left (\varphi KC_{{50,{\text{Gem}}}} \right ) ^{{\gamma {\text{Gem}}}} + C_{\text{Gem}}^{{\gamma {\text{Gem}}}} }} + \frac{{K_{{\hbox{max} ,{\text{Et}}743}} C_{{{\text{Et}}743}}^{{\gamma {\text{Et}}743}} }}{{KC_{{50,{\text{Et}}743}}^{{\gamma {\text{Et}}743}} + C_{{{\text{Et}}743}}^{{\gamma {\text{Et}}743}} }}} &\quad {{\text{for}}\,{\text{combination}}} \\ \end{array} } \right.$$

(12)

To calculate the new steady state for single-agents, we solve Eqs. (11f) and (12) with respect to $R_{\text{ss}}^{{\prime }}$ and obtain Eq. (6). In order to achieve that the number of cells decreases for t > 0, one needs $R_{\text{ss}}^{'} \le R_{0}$, i.e., the threshold concentration is defined via

$$R_{\text{ss}}^{{\prime }} = \left( {k_{\text{g}} - \frac{{K_{{{ \hbox{max} },{\text{d}}}} C_{\text{d}}^{{\gamma_{\text{d}} }} }}{{KC_{{50,{\text{d}}}}^{{\gamma_{\text{d}} }} + C_{\text{d}}^{{\gamma_{\text{d}} }} }}} \right)\frac{{R_{\text{ss}} }}{{k_{\text{g}} }} = R_{0}$$

which is equivalent to

$$k_{\text{g}} \left( {R_{\text{ss}} - R_{0} } \right)KC_{{50,{\text{d}}}}^{{\gamma_{\text{d}} }} + k_{\text{g}} \left( {R_{\text{ss}} - R_{0} } \right)C_{\text{d}}^{{\gamma_{\text{d}} }} = K_{\text{max,d}} C_{\text{d}}^{{\gamma_{\text{d}} }} R_{\text{ss}} .$$

This leads to

$$C_{\text{T,d}}^{{\gamma_{\text{d}} }} = \frac{{k_{\text{g}} \left( {R_{\text{ss}} - R_{0} } \right)KC_{{50,{\text{d}}}}^{{\gamma_{\text{d}} }} }}{{K_{{{ \hbox{max} },{\text{d}}}} R_{\text{ss}} - k_{\text{g}} \left( {R_{\text{ss}} - R_{0} } \right)}}$$

as Eq. (5) is shown. To obtain total cell eradication, the concentration

$$\frac{{K_{\text{max,d}} C_{\text{d}}^{{\gamma_{\text{d}} }} }}{{KC_{{50,{\text{d}}}}^{{\gamma_{\text{d}} }} + C_{\text{d}}^{{\gamma_{\text{d}} }} }} \ge k_{\text{g}}$$

is needed, i.e., C _E,d is defined via

$$\frac{{K_{\text{max,d}} C_{\text{E,d}}^{{\gamma_{\text{d}} }} }}{{KC_{{50,{\text{d}}}}^{{\gamma_{\text{d}} }} + C_{\text{E,d}}^{{\gamma_{\text{d}} }} }} = k_{\text{g}}$$

resulting in Eq. (7).

For combination effects, Eqs. (11f) and (12) provide

$$\frac{{K_{{{ \hbox{max} },{\text{Gem}}}} C_{\text{Gem}}^{{\gamma_{\text{Gem}} }} }}{{\left( {\psi KC_{{50,{\text{Gem}}}}^{ } } \right)^{{\gamma_{\text{Gem}} }} + C_{\text{Gem}}^{{\gamma_{\text{Gem}} }} }} + \frac{{K_{{{ \hbox{max} },{\text{Et}}743}} C_{{{\text{Et}}743}}^{{\gamma_{{{\text{Et}}743}} }} }}{{KC_{{50,{\text{Et}}743}}^{{\gamma_{{{\text{Et}}743}} }} + C_{{{\text{Et}}743}}^{{\gamma_{{{\text{Et}}743}} }} }} = k_{\text{g}} \left( {1 - \frac{{R_{\text{ss}}^{{\prime }} }}{{R_{\text{ss}} }}} \right) .$$

(13)

and Eq. (9) is obtained.