A Mathematical Model to Estimate Chemotherapy Concentration at the Tumor-Site and Predict Therapy Response in Colorectal Cancer Patients with Liver Metastases

Abstract

Chemotherapy remains a primary treatment for metastatic cancer, with tumor response being the benchmark outcome marker. However, therapeutic response in cancer is unpredictable due to heterogeneity in drug delivery from systemic circulation to solid tumors. In this proof-of-concept study, we evaluated chemotherapy concentration at the tumor-site and its association with therapy response by applying a mathematical model. By using pre-treatment imaging, clinical and biologic variables, and chemotherapy regimen to inform the model, we estimated tumor-site chemotherapy concentration in patients with colorectal cancer liver metastases, who received treatment prior to surgical hepatic resection with curative-intent. The differential response to therapy in resected specimens, measured with the gold-standard Tumor Regression Grade (TRG; from 1, complete response to 5, no response) was examined, relative to the model predicted systemic and tumor-site chemotherapy concentrations. We found that the average calculated plasma concentration of the cytotoxic drug was essentially equivalent across patients exhibiting different TRGs, while the estimated tumor-site chemotherapeutic concentration (eTSCC) showed a quadratic decline from TRG = 1 to TRG = 5 (p < 0.001). The eTSCC was significantly lower than the observed plasma concentration and dropped by a factor of ~5 between patients with complete response (TRG = 1) and those with no response (TRG = 5), while the plasma concentration remained stable across TRG groups. TRG variations were driven and predicted by differences in tumor perfusion and eTSCC. If confirmed in carefully planned prospective studies, these findings will form the basis of a paradigm shift in the care of patients with potentially curable colorectal cancer and liver metastases.

Keywords: FOLFOX; chemotherapy; colorectal cancer; liver metastases; mathematical model.

Figure 1

Schematic description of the mathematical model and its variables. The model presented here takes as input the treatment, patient, and tumor-related variables to predict tumor response to chemotherapy. (a) Calculated systemic drug concentration, as obtained through pharmacokinetic analysis of patient-specific dosage regimen, is scaled by the (b) frequency of drug administration and (c) total tumor burden (surrogate: serum carcinoembryonic antigen (CEA)) to provide an estimate for (d) tumor vascular drug concentration, which upon further scaling with the tumor blood volume fraction (BVF), estimates drug concentration in the (e) tumor interstitium, which correlates directly with response. Note: Normalized area under the curve (AUC) of contrast enhancement kinetics in abdominal CT (computed tomography) scan provides a measure of tumor BVF. Δt is the duration of a single therapy cycle. Color gradients denote level of drug concentration and drug diffusion barriers in the tumor. Illustration is not to scale.

Figure 2

Correlation analysis of tumor, treatment and outcomes variables. Statistically significant correlations are observed between tumor and treatment-related variables and pathological TRG (n = 33 patients). (A) Correlation of serum CEA with tumor size (largest dimension). (B) Correlation of therapy cycle rate per unit tumor burden (i.e., number of treatment cycles per month per unit CEA) with TRG. (C) Correlation between tumor perfusion and TRG. This measurement is a surrogate for the blood volume fraction (BVF) parameter in the model. (D) Tumor perfusion correlates negatively with tumor size, indicating reduced perfusion in larger lesions.

Figure 3

Estimated tumor-site chemotherapy concentration correlation with diffusion-and clinical-based variables. Statistically significant correlations are observed between tumor-site chemotherapy concentration and (A) tumor perfusion, (B) tumor size, (C) serum CEA level, and (D) chemotherapy dose overtime per unit tumor burden.

Figure 4

Calculated chemotherapy concentration and response to therapy. Calculated time-averaged plasma concentration ${\stackrel{=}{\sigma }}_{\mathrm{B},i}$ (red squares) and estimated time-averaged tumor-site concentration of chemotherapy $\langle {\sigma }_{\mathrm{T}}\rangle$ (blue squares with best fit, p value < 0.01, $R^2=0.98$) throughout the course of multiple cycles of treatment is predicted by PK analysis and the mathematical model. Note that $\langle {\sigma }_{\mathrm{T}}\rangle$ < ${\stackrel{=}{\sigma }}_{\mathrm{B},i}$, and in particular for TRG = 5, the average tumor concentration is ~5.7 times smaller than the corresponding average plasma concentration, nearly matching the lower bound of IC₅₀ for 5-FU. Data represents mean ± standard deviation (SD). (Patient cohort size, n = 33).

Figure 5

Logistic regression-based binary classification and cross-validation. Receiver operating characteristic (ROC) curve to evaluate the classification ability of eTSCC into responders and non-responders (A). Complementary cumulative distribution function (CCD) of patients shows the accuracy of binary classification at a discrimination threshold of 0.51 µg mL⁻¹ (B). ROC curves generated for multiple training data sets obtained through the leave-one-out cross validation technique (C). Results of cross validation in correctly classifying the test data point (D).