A dynamic K-PD joint model for the kinetics of CTC (Circulating Tumor Cell) count and PSA concentration during treatment in metastatic castration-resistant prostate cancer
Mélanie Wilbaux (1), Michel Tod (1), Johann De Bono (2), David Lorente (2), Joaquin Mateo (2), Gilles Freyer (1), Benoit You (1), Emilie Hénin (1)
(1) EMR 3738 CTO, UCBL - HCL Faculté de Médecine Lyon-Sud, Université Lyon 1, Oullins, France, (2) Royal Marsden Hospital, London, United Kingdom
Prostate cancer is one of the most common cancers and the second leading cause of death. The PSA (Prostate-Specific Antigen) is a serum tumor marker currently used to evaluate treatment effect in prostate cancer; however its validity remains controversial. Therefore new markers are emerging, such as the count of Circulating Tumor Cells (CTCs), measured as the number of cells per sample of 7.5mL blood (aliquot). CTCs are tumor cells that have been released into blood and potentially prone to the development of new metastases. De Bono et al. reported that patients with a baseline CTC count greater or equal to 5 had a shorter survival than patients with a baseline CTC count lower than 5 . They also showed that CTC count (< 5 or ≥5 per aliquot) was a better predictor of survival than PSA decrease (of 30% or 50%). As a consequence, the Food and Drug Administration (FDA) approved the use of CTC counts in the evaluation of metastatic castration-resistant prostate cancer (CRPC) p atients. The kinetics of CTC and their relationships with other markers such as PSA and tumor size need to be addressed.
The main objective of the present study was to quantify the dynamic relationships between the kinetics of PSA and CTC count during treatment in metastatic CRPC patients, linked by a latent variable.
The data from 224 patients, enrolled in IMMC38 trial meant to assess the relationships between categorized CTC count (-1[0–17800], were observed at different time-points along treatment. A median of 5 CTC and 5 PSA values were available per subject until 6 months after treatment initiation. Different types of kinetic profiles were observed, some with parallel PSA and CTC kinetics, others with faster or slower CTC kinetics compared to PSA kinetics.
A semi-mechanistic model was built to describe CTC count and PSA kinetics during treatment in CRPC patients. The model required to consider 4 levels of complexity:
1) The kinetics of the effects of the 3 treatment types: chemotherapy, hormonotherapy, or both simultaneously;
2) The dynamic relationships between PSA and CTC kinetics: these 2 variables had no clear direct relationships, but were triggered by a common unobserved variable;
3) The joint modeling of 2 dependent variables of different types: count (CTCs) and continuous data (PSA);
4) The sampling statistics of blood collection: the observed CTC count from a 7.5mL aliquot of blood (CTCAliquot) is a sample of a true CTC counts in the total blood volume (CTCTotal). For instance, CTC equal to 0 does not imply that no CTCs are produced.
In order to take into account the inter-individual variability in the kinetic profiles, the population analysis was performed with a non-linear mixed effects model using NONMEM 7.3. Selection and evaluation of the best model were achieved using criteria based on the likelihood, goodness-of-fit plots and simulation-based diagnostics.
The 4 levels of complexity were considered in the final model:
1) Since no drug concentration data were available, a K-PD approach has been used for the kinetics of treatment effects. Chemotherapy and hormonotherapy administrations were assigned to 2 K-PD compartments, allowing the estimation of different kinetic and efficacy parameters for chemotherapy and hormonotherapy.
2) Their respective effects on both PSA and CTCs were mediated through a latent variable, defined as an underlying, non-observed variable. Each treatment was acting as an inhibitor of the latent variable production.
3) The PSA kinetics was described by a non-steady-state 1-compartment model with 0-order production and 1st order elimination rates.
The CTC kinetics was characterized by a cell life span model, commonly used for the modeling of blood cell maturation . The main assumption of the cell life span model is that each CTC has the same life span (LS), so that the rate of CTC loss at time t+LS is equal to the production rate at time t.
4) The CTCs being counted from a 7.5mL blood sample, a Poisson process for the CTCAliquot was considered for the discrete sampling techniques, applying a scaling factor (α=(aliquot volume)/(total blood volume)). Different Poisson related models were tested. The best one was the negative binomial distribution, which allowed taking into account the overdispersion (variance>mean) of observed CTCAliquot counts.
According to goodness-of-fit plots, PSA kinetics in treated CRPC patients were properly fit over the 6-month period, and Visual Predictive Check (VPC) showed good agreement between the distribution of observed and simulated values.
The predictive performance for the CTCs was assessed with simulation-based diagnostics: categorical VPCs, CTC count distributions and overdispersion plots (variance vs mean).
Relative Standard Errors of typical mean parameters and inter-individual variability, representative of estimation precision, were all less than 40%.
Finally, the model enabled simulations of different types of kinetics profiles, similarly to those observed.
Parameter estimates showed an inhibitory effect for latent variable of the chemotherapy superior to the hormonotherapy, as expected. The CTC life span was estimated at 97 days, and its production rate at 160 CTC.day-1.AU-1. The PSA half-life was assessed at 65 days, and its production rate at 2 ng.mL-1.day-1.AU-1. The ratio of production rates was equal to 0.01 ng.mL-1.CTC.
The proposed semi-mechanistic K-PD model is the first to quantify the dynamic relationships between the kinetics of PSA and CTC count in metastatic CRPC patients. This is an atypical model combining several advanced features in pharmacometrics: K-PD modeling, joint modeling of count and continuous data, both driven by a latent variable and the discrete processes for CTC modeled by a cell life span model combined to random sampling statistics. The latent variable could be interpreted as the non-measured tumor burden producing CTCs and PSA.
This model allowed simulations of different kinetic profiles, and enabled to obtain information about PSA and CTC production and CTC life span and its inter-individual variability.
It will be challenged for evaluating the prognostic value of CTC count and PSA: i) to identify some covariates explaining the variability; ii) to establish a link between a CTC kinetic parameter and survival; iii) to test the sensitivity and specificity of the early modeled decrease in marker for predicting survival.
The proposed model would potentially help oncologists in the evaluation of CRPC patients' response to chemotherapy and hormonotherapy.
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