Fig 1.
Experimental design and pharmacokinetic model: A. Schematic outline of the experiment. B. Plasma (C1) and tumor site (C2) drug concentrations for the two 5-FU dosages administered in the experiment.
The figures show the concentrations from the time when the drug was administered until 24 hours later and are calculated according to Eq (5).
Table 1.
Normalized mean squared error (NMSE) values for the fitted models with and without treatment.
The values are percentages (%).
Table 2.
Numerical estimates of the tumor doubling time parameter, τg, for the 8 mice in the DMSO (untreated) group.
In each mouse three tumors were monitored. The doubling parameter is measured in days [d]. The values are more similar within mice (across columns) than between mice (across lines).
Fig 2.
The variability of tumor growth in vivo can be captured with the Gompertz model: Indicative growth curves for a mouse with fast growing tumors (CM.37 –Left panel) and another with slow growing tumors (CM.53 –Right panel).
The mice belong to the DMSO group and received no drug treatment. The black squares are the measured tumor volumes and the red line is the fitted model output using Eq (2) without treatment. The model provides an overall satisfactory fit to the data. The NMSE values are 17.4%, 5.3%, 10.7% for the three tumors of CM.37 and 16.1%, 27.1%, 19.8% for the three tumors of CM.53. Additionally tumor growth rate appears to be mouse specific.
Fig 3.
Tumor growth in vivo is mouse-specific: A. Top panel–Estimates of the tumor doubling time parameter τg. For each of the eight mice in the DMSO group we obtain one estimate for each of the three monitored tumors. The figure suggests that tumor growth is mouse specific. Four mice (CM.37, 38, 60 & 78) exhibit three fast growing tumors while three mice (CM.53, 76 & 77) exhibit three slow growing tumors. The only exception is CM.79, for which the first tumor is slow growing, unlike the other two tumors. B. Bottom panel–The existence of two discrete growth rate groups is further illustrated by the distribution of all three tumor doubling times for the eight mice, shown here in a three-dimensional scatter plot.
Mice CM.37, 38, 60 & 78 comprise one group, while the other four comprise a second group.
Fig 4.
The response to 5-FU treatment can be captured with the combined Gompertz/two-compartment pharmacokinetic model: Indicative figures from two treated mice, one from each treatment group.
Left panel–CM.41 from 5-FU 1; Right panel–CM.43 from 5-FU 2. The black squares indicate the measured tumor volumes while the green line is the fitted model based on Eqs (2–5). The red dashed line is the prediction of what would have happened if the tumor was left untreated based on the pre-treatment data estimates of the tumor growth rate. Our model appears to accurately capture, both the growth and treatment dynamics of the tumor. The NMSE values for the three tumors of CM.41 are 44.7%, 26.8% and 35.4% and for the three tumors of CM.43 these are 18.5%, 6.7% and 13.0% respectively.
Table 3.
Numerical estimates of the tumor doubling time τg and the drug kill rate keff for the mice tumors in the two treatment groups.
For some tumors the estimation of τg and subsequently keff was not possible due to the lack of sufficient pre-treatment data. Further analysis showed that keff is less subject-dependent and more tumor-specific compared to τg.
Fig 5.
Drug efficiency is variable and depends on tumor growth rate.
A. Left Panel–Estimates of the drug kill rate parameter, keff, for the two treatment groups in our experiment (A.1.: 5-FU 1; A.2.: 5-FU 2). Unlike the tumor growth rate, which is appears mouse specific, response to treatment that is reflected through keff is less dependent on mouse. B. Top-right Panel: Scatterplot of τg against keff and non-parametric regression fit from kernel regression (red line). A negative correlation between the two parameters or equivalently, a positive correlation between growth and drug kill rates is observed, further supported by the value of Spearman’s correlation coefficient (r = -0.726). C. Bottom-right Panel: Scatter plot of the estimated values of the drug kill rate parameter (keff) from all three tumors per mouse, for both treatment groups, Two distinct groups are formed, whereby the first group (black–more effective per unit drug volume) comprises four out of the five 5FU-1 mice and the second group (red–less effective per unit drug volume) comprises the remaining 5FU-1 mouse (CM.62) and all five 5-FU2 mice.