Drug response experiments <i>in vitro</i>.

<p>(<i>Left</i>) Measurement of <i>in vitro</i> cell kill in cell culture for <i>Eμ-myc/Arf-/-</i> and <i>Eμ-myc/p53-/-</i> cells after 48 hours at 50 nM Dox concentration (N.S.: not statistically significant). (Right) Results from a flow cytometry study were used to measure apoptotic cells. <i>Eμ-myc/p53-/-</i> cells are displayed along the top row with <i>Eμ-myc/Arf-/-</i> cells along the bottom; controls (no drug) are in the left column, and drug-treated cells (Dox) are in the right column. For each block, lower left quadrant represents live (proliferating) cells; lower right quadrant shows apoptotic cells; upper right quadrant shows dead cells.</p

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

Mathematical model predicts lymphoma tumor death due to chemotherapy drug treatment.

<p>Comparison of histopathology measurements with mathematical model predictions (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0129433#pone.0129433.e003" target="_blank">Eq 2</a>, solid lines) based on estimates of two parameters <i>r</i>_<i>b</i> / <i>L</i> and <math><mrow><msubsup><mi>f</mi><mrow>kill</mrow>M</msubsup></mrow></math>. Data points for drug-resistant cells (blue) were scaled by 3.5 (see <b><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0129433#pone.0129433.g002" target="_blank">Fig 2A</a></b>) to be comparable with data for drug-sensitive cells (green). Obtained <i>R</i>² = 0.86; estimated <math><mrow><msubsup><mi>f</mi><mrow>kill</mrow>M</msubsup></mrow></math> = 0.25, and <i>r</i>_<i>b</i> / <i>L</i> = 0.068. Diffusion distance of drug from the vessels (40 ± 20 μm) was assumed in the best case not to exceed half that of O₂. Each point represents measurements from one tumor Set; 5 data points for the drug-sensitive cell line (green) and 6 data points for the drug-resistant cell line (blue).</p

Necrotic cell fraction in murine lymphoma tumors after treatment with Dox.

<p>Data are shown for tumor slices S1 through S5. Most of the necrosis is a result of the drug treatment since necrosis measured in untreated tumors was negligible (<b><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0129433#pone.0129433.t002" target="_blank">Table 2</a></b>). Note that the drug-sensitive tumors shrank in size after treatment and thus had one less histological slice than the drug-resistant tumors (to account for this, two slices of the drug-resistant tumor are included in the central region S3, i.e., five total slices for <i>Eμ-myc/Arf-/-</i> and six for <i>Eμ-myc/p53-/-</i>). All error bars represent standard deviation from at least n = 3 measurements in each section. Asterisks show level of statistical significance determined by student’s <i>t</i>-test with α = 0.05 (asterisk, <i>P</i> < 0.05).</p

Whole-tumor measurement of lymphoma characteristics.

<p>Measurements from the IHC data after treatment with Dox shows cell fractions for: (<b>A</b>) apoptosis, (<b>B</b>) endothelium, (<b>C</b>) hypoxia, (<b>D</b>) proliferation. Note that the drug-sensitive tumors shrank in size after treatment and thus had one less histological slice than the drug-resistant tumors in the middle Set (S3). Error bars represent standard deviation (n = 3 regions of interest per slice).</p

Average of tumor measurements from IHC used for model calibration.

<p>Average of tumor measurements from IHC used for model calibration.</p

Sensitivity analysis results.

<p>Plots of absolute values of sensitivity coefficients for the three parameters for <b>(A)</b> the drug-sensitive cell line, <i>Eμ-myc/Arf-/-</i> and <b>(B)</b> the drug-resistant cell line, <i>Eμ-myc/p53-/-</i>. The range of variation for each parameter is listed in <b><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0129433#pone.0129433.s001" target="_blank">S1 Table</a></b>. <i>S</i> represents sensitivity coefficient.</p

Strategy for model calibration and validation.

<p>Values for mathematical model input parameters are initially calibrated from experimental data obtained from untreated subjects and cell culture, yielding blood volume fraction, diffusion penetration distance, radius of blood sources, and fraction of cells killed in culture. Based on these parameter values, the model then calculates the fraction of tumor volume that would be killed <i>in vivo</i>, which can be compared to experimental data obtained from treated subjects.</p

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