Predicting Outcomes of Prostate Cancer Immunotherapy by Personalized Mathematical Models

Therapeutic vaccination against disseminated prostate cancer (PCa) is partially effective in some PCa patients. We hypothesized that the efficacy of treatment will be enhanced by individualized vaccination regimens tailored by simple mathematical models.We developed a general mathematical model encompassing the basic interactions of a vaccine, immune system and PCa cells, and validated it by the results of a clinical trial testing an allogeneic PCa whole-cell vaccine. For model validation in the absence of any other pertinent marker, we used the clinically measured changes in prostate-specific antigen (PSA) levels as a correlate of tumor burden. Up to 26 PSA levels measured per patient were divided into each patient's training set and his validation set. The training set, used for model personalization, contained the patient's initial sequence of PSA levels; the validation set contained his subsequent PSA data points. Personalized models were simulated to predict changes in tumor burden and PSA levels and predictions were compared to the validation set. The model accurately predicted PSA levels over the entire measured period in 12 of the 15 vaccination-responsive patients (the coefficient of determination between the predicted and observed PSA values was R(2) = 0.972). The model could not account for the inconsistent changes in PSA levels in 3 of the 15 responsive patients at the end of treatment. Each validated personalized model was simulated under many hypothetical immunotherapy protocols to suggest alternative vaccination regimens. Personalized regimens predicted to enhance the effects of therapy differed among the patients.Using a few initial measurements, we constructed robust patient-specific models of PCa immunotherapy, which were retrospectively validated by clinical trial results. Our results emphasize the potential value and feasibility of individualized model-suggested immunotherapy protocols

The parameters of the model and their evaluated values for WT APC.

<p>The parameters of the model and their evaluated values for WT APC.</p

Model–predicted effects of sFRP1 and Dkk1 on cells with APC truncation mutations.

<p>Model predictions of TCF activity under inhibition by sFRP1 or Dkk1 in differently mutated cells are shown; the grey area marks the level of inhibition by each inhibitor. In APC^min/min mutant cells, neither of the inhibitors restores normal TCF activity level (Fig 3A and 3B). Restoration of normal TCF activity level (i.e., equal to that of WT) is predicted to be achievable in APC^1638N/1572T mutant cells (grey area), by addition of either sFRP1 (~160nM) or Dkk1 (~5nM) (Fig 3A and 3B, respectively). For comparison, simulations of the effects of inhibitors on WT APC under the same conditions are shown in all four panels (dashed lines). TCF activity levels are relative to WT.</p

Model validation by comparison of the predicted effects of attenuated full-length APC on TCF activity levels, to experimental data.

<p>Simulation results (black line) for cells with reduced APC expression are presented in comparison with the experimental results for cells with the mutations 1638N/1638N, 1638N/1638T and 1638T/1638T, which express APC at 2%, 51% and 100% of the WT level, respectively. The observed average TCF activity levels are taken from [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0179888#pone.0179888.ref034" target="_blank">34</a>] (magenta circles), [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0179888#pone.0179888.ref029" target="_blank">29</a>] (blue circles) and [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0179888#pone.0179888.ref033" target="_blank">33</a>] (green circles); error bars are reproduced from the original publications. The discrepancies between the experimental results from different sources may be due to differences in the time of exposure to Wnt. The simulated and experimental values of TCF activity level are detailed in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0179888#pone.0179888.s004" target="_blank">S1 Table</a> for each specific mutation.</p

Extracellular inhibitors can attenuate tumorigenic Wnt pathway activity in adenomatous polyposis coli mutants: Predictions of a validated mathematical model - Fig 4

<p><b>Effective doses of sFRP1 (Fig 4A) and Dkk1 (Fig 4B), required to restore normal TCF activity level, predicted for hypothetical mutations of different severity.</b> Shown are simulation results for cells bearing different mutations: heterozygous mutations, with one APC allele truncated and the other practically inactive (blue dots), homozygous truncation mutations (red dots), and mutations with reduced amount of normally-functioning APC (black stars). The severity of mutation is expressed by the increase in TCF activity level. The range of inhibitor doses which are effective for each TCF activity level is due to the fact that the same level of increase in TCF activity can result from different truncation mutations (having different parameter values within the biologically plausible range).</p

Different APC mutations and their effect on values of model parameters.

<p>Different APC mutations and their effect on values of model parameters.</p

Effective doses of sFRP1 and Dkk1 for different mutants as predicted by the model.

<p>Effective doses of sFRP1 and Dkk1 for different mutants as predicted by the model.</p

A schematic description of the mathematical model for the Wnt signaling pathway.

<p>(A) The central part of the scheme shows the regulation of the β-catenin level in the cell. This process is described by the following cascade of reactions. The Wnt ligand binds to the Frizzled receptor (reaction labeled k_±2). The resulting receptor–ligand complex may recruit an unoccupied LRP receptor and create a ternary complex consisting of Wnt, Frizzled and LRP (k_±3). The latter complex transduces the signal inside the cell and interferes with the destruction cycle of β-catenin, by binding a specific destruction complex comprising Axin, APC and GSK3β (k_±5). This complex regulates the intracellular level of β-catenin; when unbound to the Wnt/Frizzled/LRP ternary complex, the destruction complex binds β-catenin and causes its phosphorylation (k_±6). Phosphorylated β-catenin dissociates from the destruction complex and is rapidly degraded. The reverse rate constants of these reactions are denoted by a minus in the subscript. Production and degradation of β-catenin, independent of the destruction complex, are also modelled and their rates are labelled k₇ and k₈, respectively, and the circled part shows β-catenin function inside the nucleus as TCF activator. The greyed part describes the formation of the destruction complex (GSK/Axin/APC); K_D1 and K_D2 are dissociation constants of APC from the Axin/APC dimer, and of the dimer from the destruction complex, respectively. The boxes show reactions between SFRP and Wnt and Dkk1 and LRP receptor, with the rates k_±1 and k_±4, respectively. The circled part shows β-catenin function inside the nucleus as TCF activator. (B) A schematic illustration of the APC gene, showing the location of the 1572T mutation, truncated close to the MCR, and of the <i>min</i> mutation. These mutations form a truncated APC protein, lacking more functional binding sites as the gene is truncated closer to the translation starting site. Note that all other mutations referred to in this work are not known to form a truncated APC protein, but rather attenuate the expression of the full length APC protein.</p