Theory and Experimental Validation of a Spatio-temporal Model of Chemotherapy Transport to Enhance Tumor Cell Kill

<div><p>It has been hypothesized that continuously releasing drug molecules into the tumor over an extended period of time may significantly improve the chemotherapeutic efficacy by overcoming physical transport limitations of conventional bolus drug treatment. In this paper, we present a generalized space- and time-dependent mathematical model of drug transport and drug-cell interactions to quantitatively formulate this hypothesis. Model parameters describe: perfusion and tissue architecture (blood volume fraction and blood vessel radius); diffusion penetration distance of drug (i.e., a function of tissue compactness and drug uptake rates by tumor cells); and cell death rates (as function of history of drug uptake). We performed preliminary testing and validation of the mathematical model using <i>in vivo</i> experiments with different drug delivery methods on a breast cancer mouse model. Experimental data demonstrated a 3-fold increase in response using nano-vectored drug <i>vs</i>. free drug delivery, in excellent quantitative agreement with the model predictions. Our model results implicate that therapeutically targeting blood volume fraction, e.g., through vascular normalization, would achieve a better outcome due to enhanced drug delivery.</p><p>Author Summary</p><p>Cancer treatment efficacy can be significantly enhanced through the elution of drug from nano-carriers that can temporarily stay in the tumor vasculature. Here we present a relatively simple yet powerful mathematical model that accounts for both spatial and temporal heterogeneities of drug dosing to help explain, examine, and prove this concept. We find that the delivery of systemic chemotherapy through a certain form of nano-carriers would have enhanced tumor kill by a factor of 2 to 4 over the standard therapy that the patients actually received. We also find that targeting blood volume fraction (a parameter of the model) through vascular normalization can achieve more effective drug delivery and tumor kill. More importantly, this model only requires a limited number of parameters which can all be readily assessed from standard clinical diagnostic measurements (e.g., histopathology and CT). This addresses an important challenge in current translational research and justifies further development of the model towards clinical translation.</p></div

Parameter calibration from patient data demonstrates model predictivity.

<p>(<i>A</i>) Nonlinear regression analysis of Eq. S2 (coefficient of determination <i>R</i>² = 0.86) to the measurements of kill fraction and blood volume fraction BVF from histopathology images of 21 patients with CRC metastatic to liver (standard deviations reflect variability of measured values across 20 slides per patient). Inset: parameter values obtained from fit. (<i>B</i>) Linear regression analysis of Hounsfield Unit measurements from pre-treatment arterial-phase contrast-enhanced CT data from 18 patients and blood volume fraction (BVF) measurements from histopathology leads to calibration of BVF parameter (inset). (<i>C</i>) Side-by-side boxplots of <i>f</i>_kill values measured from histopathology and predicted by mathematical model Eq. S2 based on calibration in <i>A</i> and <i>B</i> (18 data points in each set, symbols). In each boxplot, the thick horizontal line is the median; the box is defined by the 25th and 75th percentiles (lower and upper quartile); the diamond is the mean. A paired t-test at the 0.05 significance level resulted in <i>P</i> = 0.44, indicating that the observed difference between the two data sets is not significant. (<i>D</i>) Predictions of Eq. S2 (open circles, average relative error ≈ 24%) compared, for each patient, to the direct measurements from histopathology post-treatment and resection (filled circles, with standard deviation of multiple measurements per patient).</p

Numerical simulations of the general integro-differential model (Eqs 6 and 7) in a cylindrically symmetric domain.

<p>As cell kill ensues over several cell cycles, (<i>A</i>) successive cell layers next to the blood vessel (<i>r</i> = <i>r</i>_b) die out, i.e., tumor volume fraction <i>φ</i> decreases; (<i>B</i>) local drug concentration <i>σ</i> increases due to an enhancement of drug penetration; and (<i>C</i>) cell kill accelerates further from the vessel and deep into the tumor. Input parameters: <i>r</i>_b / <i>L</i> = 0.102 and BVF = 0.01. The duration of the entire simulation was 10 (<i>λ</i>_<i>k</i><i>λ</i>_<i>u</i><i>φ</i>₀<i>σ</i>₀)^−1/2, where time unit is a characteristic cell apoptosis time. Drug concentration and tumor volume fraction were normalized by their initial values, and radial distance by the diffusion penetration distance <i>L</i>. The fraction of tumor kill <i>f</i>_kill is calculated from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004969#pcbi.1004969.e012" target="_blank">Eq 12</a> (<b>Methods</b>).</p

Testing the efficacy of drug-loaded nano-carriers in mice.

<p>Comparison of fraction of tumor killed measured across three different treatment BALB/c mice groups (n = 10 per group) over a period of 17 days (from day 14 to day 31 after 4T1 tumor cell inoculation, see <b><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004969#sec003" target="_blank">Methods</a></b>) showing a roughly 3-fold increase in kill from nano-vectored drug vs. free drug. At each time point, tumor volume measurements from the three drug treatment groups were first normalized to the measurement from the control (PBS) group (no drug treatment), and then to the initial tumor volume for each group; <i>f</i>_kill was then calculated as (1 –normalized tumor volume).</p

Illustration of transport-based hypothesis.

<p>By diffusion, a blood vessel supplies substrates to the cylindrical tissue volume surrounding the vessel. We hypothesize that at each position inside the tissue, the substrate supply is supported by the closest blood vessel. Thus, the influenced tissue surrounding a vessel can be estimated to be between a cylinder of radius <i>r</i>_b / (<i>L</i> BVF^1/2) in dimensionless form, and the vessel itself with dimensionless radius <i>r</i>_b / <i>L</i>. Theoretically, chemotherapeutic drugs delivered by a blood vessel kill the tissues immediately adjacent to the vessel, leaving some viable tissues on the far end. Here, we propose that through drug-loaded nano-carriers that can accumulate within tumors and continuously release drugs for a longer time (e.g., lasting several cell cycles), the drugs can penetrate further into the surrounding tissue volume and thus achieve a higher tumor killing ratio.</p

Drug-loaded nano-carriers lead to cell-kill enhancement over bolus delivery.

<p>(<i>A</i>) Time-evolution curves of chemotherapeutic efficacy <i>f</i>_kill (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004969#pcbi.1004969.e012" target="_blank">Eq 12</a>) of nano-carriers releasing drug compared to the estimated efficacy (symbols) of conventional chemotherapy (Eq. S2), for parameter values: <i>r</i>_b / <i>L</i> = 0.05 (dashed curves, upper triangles), 0.1 (solid curves, diamonds), and 0.5 (dotted curves, lower triangles), paired with BVF = 0.005 (red curves and symbols), 0.01 (blue curves and symbols), and 0.05 (green curves and symbols). (<i>B</i>) Same as (<i>A</i>), but normalized to the corresponding bolus values of tumor kill, <i>f</i>_kill,bolus.</p