Modeling the synergistic properties of drugs in hormonal treatment for prostate cancer

Abstract

Prostate cancer is one of the most prevalent cancers in men, with increasing incidence worldwide. This public health concern has inspired considerable effort to study various aspects of prostate cancer treatment using dynamical models, especially in clinical settings. The standard of care for metastatic prostate cancer is hormonal therapy, which reduces the production of androgen that fuels the growth of prostate tumor cells prior to treatment resistance. Existing population models often use patients' prostate-specific antigen levels as a biomarker for model validation and for finding optimal treatment schedules; however, the synergistic effects of drugs used in hormonal therapy have not been well-examined. This paper describes the first mathematical model that explicitly incorporates the synergistic effects of two drugs used to inhibit androgen production in hormonal therapy. The drugs are cyproterone acetate, representing the drug family of anti-androgens that affect luteinizing hormones, and leuprolide acetate, representing the drug family of gonadotropin-releasing hormone analogs. By fitting the model to clinical data, we show that the proposed model can capture the dynamics of serum androgen levels during intermittent hormonal therapy better than previously published models. Our results highlight the importance of considering the synergistic effects of drugs in cancer treatment, thus suggesting that the dynamics of the drugs should be taken into account in optimal treatment studies, particularly for adaptive therapy. Otherwise, an unrealistic treatment schedule may be prescribed and render the treatment less effective. Furthermore, the drug dynamics allow our model to explain the delay in the relapse of androgen the moment a patient is taken off treatment, which supports that this delay is due to the residual effects of the drugs.

Keywords: Adaptive therapy; Androgen dynamics; Drug effects; Hormonal therapy; Intermittent androgen deprivation therapy; Optimal treatment schedule; Pharmacokinetics; Prostate cancer modeling.

Figure 1:

(a) Time series of serum androgen levels in a representative patient (Patient 15). (b) Corresponding on- and off-treatment timeline.

Figure 2:

Simulation results for Patient 15. Blue curve: serum androgen level simulated by the New Androgen Model using a parameter vector obtained by minimizing Eq. (21) with clinical data (black circles) for the first 837 days of treatment. The simulation is continued to Day 1369 using the same parameters and the Day 837 state vector as the initial condition.

Figure 3:

New Androgen Model results for Patient 15. (a) Black curve: simulated CPA effect f_C, Eq. (12). Red curve: simulated LEU effect f_L, Eq. (15). Green curve: the combined drug effect F, Eq. (11). Values 0 < F < 1 correspond to reductions in the androgen production rate in the testes; smaller values denote more potent effects. (b) Zoomed-in version of (a) over an initial 20-day treatment period. (c) Total serum mass of each drug, Eqs. (9) and (10). The spike in LEU mass at Day 606 reflects an additional injection. CPA therapy begins before LEU injection in each successive treatment cycle. (d) Intake and clearance of each drug for a portion of a single drug cycle. CPA is taken daily with rapid degradation; LEU, monthly with slow degradation. These daily fluctuations are present but not visible in subsequent plots.

Figure 4:

Androgen fitting and forecasting results for the Improved BK model (red curves) and New PSA Model (blue). Black circles denote clinical measurements for each patient. The fitting and forecasting procedure is similar for each patient, as described in the text.

Figure 5:

New PSA Model simulated (blue curves) and clinically measured (black circles) PSA levels for four representative patients. The parameter fitting and model forecast intervals are as indicated for each patient.

Figure 6:

Simulated androgen-dependent and -independent cell populations using the same parameters and fitting intervals in the New PSA Model as in Fig. 5.

Figure 7:

Simulation results of a hypothetical treatment regimen in which CPA therapy halts after Day 1140 for Patient 15. (a) Simulated serum masses of CPA and LEU; administration of the latter is assumed to continue as documented in the clinical record. (b) Simulated serum androgen levels, using the New Androgen Model, at the original (green curve) and reduced (black) CPA dosages. (c) Simulated effects of CPA (green) and LEU (black) at the new dosages. (d) Simulated PSA levels, using the New PSA Model, at the original (green curve) and reduced (black) CPA dosages. (e) Estimated androgen-dependent and -independent cell populations, by the New PSA model, under the reduced dosage regimen. (f) Absolute differences between the androgen and PSA levels for the New Androgen Model (red curve) and Improved BK Model (black) for the two treatment regimens.

Figure 8:

Total sensitivity T_p, Eq. (25), for the New Androgen Model, Eqs. (8)–(15), for output model variables representing serum androgen levels (A, blue bars), total CPA drug mass (C. red), and total LEU drug mass (L, green), as a function of each of the first 10 components p of the parameter vector ${\widehat{\mathbf{q}}}_A$ obtained for Patient 15 from the fitting procedure described in Section 2.

Figure 9:

Simulated androgen levels, according to the New Androgen Model, for Patient 15 for three different CPA and LEU dosage levels over 1.5 cycles of therapy. Red curve: large doses (600 mg CPA daily and 24 mg LEU monthly). Blue curve: 200 mg CPA daily and 7.5 mg LEU monthly (closest to those actually given). Green curve: small doses (50 mg CPA daily and 3.25 mg LEU monthly).