Foundations of Adaptive High-Level Tight Control of Prostate Cancer: A Path from From Terminal Disease to Chronic Condition

TL;DR

This work develops an adaptive HLTC framework for resistant metastatic prostate cancer by combining a Stackelberg game-theoretic model with Bayesian optimization to tailor Abiraterone therapy to individual PSA dynamics. It advances from a simple two-population carrying-capacity model to a patient-aware formulation, estimates common and patient-specific parameters via NLME with empirical Bayes, and demonstrates that HLTC—high on-drug levels with tight on/off control—often maximizes time to progression and can convert the disease into a chronic state under certain PSA thresholds. The study provides analytical and numerical evidence that HLTC can yield substantial clinical benefits and outlines pathways for extending the approach to more realistic, stochastic, and spatial cancer models. These insights offer a principled route to refine adaptive chemotherapy in hormone-sensitive cancers and guide future experimental validation.

Abstract

Metastatic prostate cancer is one of the leading causes of cancer-related morbidity and mortality worldwide. It is characterized by a high mortality rate and a poor prognosis. In this work, we explore how a clinical oncologist can apply a Stackelberg game-theoretic framework to prolong metastatic prostate cancer survival, or even make it chronic in duration. We utilize a Bayesian optimization approach to identify the optimal adaptive chemotherapeutic treatment policy for a single drug (Abiraterone) to maximize the time before the patient begins to show symptoms. We show that, with precise adaptive optimization of drug delivery, it is possible to significantly prolong the cancer suppression period, potentially converting metastatic prostate cancer from a terminal disease to a chronic disease for most patients, as supported by clinical and analytical evidence. We suggest that clinicians might explore the possibility of implementing a high-level tight control (HLTC) treatment, in which the trigger signals (i.e. biomarker levels) for drug administration and cessation are both high and close together, typically yield the best outcomes, as demonstrated through both computation and theoretical analysis. This simple insight could serve as a valuable guide for improving current adaptive chemotherapy treatments in other hormone-sensitive cancers.

Figures (10)

• Figure 1: Time series of PSA levels for all $N=32$ patients, for alternative normalizations. (A) The PSA level in ng/L. (B) The PSA level as normalized in Zhang et al. zhang2022evolution, in which the maximum value of any given PSA time series after the normalization is always at $1$. (C) The normalized PSA level is adjusted so that the value at the start of the first treatment in any given PSA time series is always set to $1$ after normalization.
• Figure 2: Time series of PSA levels for the first 16 patients.. Each plot title shows the patient's name, SSE for Zhang et al. zhang2022evolution fit, and the LBEB fit. The cyan line marks drug use periods, orange dots represent normalized PSA Eq. \ref{['normalized_PSA']}, the light-blue dashed line shows Zhang et al. zhang2022evolution best fit, and the blue line shows the LBEB best fit.
• Figure 3: Time series of PSA levels for the next 16 patients. Each plot title shows the patient's name, SSE for Zhang et al. zhang2022evolution fit, and the LBEB fit. The cyan line marks drug use periods, orange dots represent normalized PSA Eq. \ref{['normalized_PSA']}, the light-blue dashed line shows Zhang et al. zhang2022evolution best fit, and the blue line shows the LBEB best fit.
• Figure 4: The relations between patient-specific parameters and the PSA level at the first drug treatment. Here we show the identified resistant cell growth rate $r_R$, drug effectiveness $\gamma$, and initial population values $\{ x_S(0), x_R(0)\}$, and the PSA level when the total cancel population reaches its carrying capacity $\text{PSA}_K = \tilde{\lambda}$ for all patients. (A) The resistant cell growth rate $r_R$ and the drug ineffectiveness$1-\gamma$. The green continuous line shows a slight positive trend between these parameters in log-log plot ($R^2=0.10$montgomery2021introduction), indicating a power-law relation $1-\gamma \propto r_R^{0.5\pm 0.5}$. (B) The initial values $\{ x_S(0), x_R(0)\}$. The yellow dash line divides this two-dimensional parameter space into two: sensitive cell dominated (above) and resistant cell dominated (below), based on their influences on the growth of resistant cells at $t=0$. (C) The PSA levels at first drug treatment $\text{PSA}(t_{m^{\text{treat}}})$ and at carrying capacity $\text{PSA}_K$. The green continuous line shows a linear regression (that passes through the origin) between these parameters ($R^2=0.59$montgomery2021introduction), indicating a linear relation $\text{PSA}(t_{m^{\text{treat}}}) = \left( 0.25 \pm 0.05 \right)\text{PSA}_K$. The yellow dash line divides this two-dimensional space into two: $\text{PSA}(t_{m^{\text{treat}}})/\text{PSA}_K > 1/\alpha_{RS}$ (above) and $\text{PSA}(t_{m^{\text{treat}}})/\text{PSA}_K < 1/\alpha_{RS}$ (below).
• Figure 5: Bayesian optimization using a GPR model to search for the maximum total drug response time $\tau_{\text{tot}}$ for patient P1007 under adaptive chemotherapy. We consider different PSA threshold values, i.e. $\text{PSA}_\text{thr}/\text{PSA}_K=0.14$ in (A) and $\text{PSA}_\text{thr}/\text{PSA}_K=0.19$. Here we show the Gaussian extrapolated surfaces of the total achievable time, as given in Eq. \ref{['GPR_tau']}. The circles represent the points where the total achievable time is evaluated by numerical simulation. The star indicates the point of convergence for the search, which corresponds to the global maxmimum.