Fig 1.
Three types of initial cell configurations. Orange dots: R-cells, blue dots: S-cells, white dots: empty sites. Clumped R cells (first), random R-cells (center), and uniform R-cells (right).
Fig 2.
The pink shaded regions show the five types of representative CAF distribution (FC, FCp, FSq, FSp and FR). The orange square at the centre is the clump of R-cells. The dimensions have been provided in the text.
Fig 3.
Flow chart of the cells’ life cycle. In each time step, all of the cells follow the steps in the flow chart.
Table 1.
The parameter values are listed in the following table.
Fig 4.
Effects of initial R-cell distribution on the TTP under CT-MTD.
(A) The cell configurations on days 1, 120, and 2000 are shown. The blue, orange, and white dots are S-cells, R-cells, and empty sites, respectively. (B) The average temporal evolution of the average number of S-cell and R-cell populations over 30 realizations with clumped (upper panel), random (middle panel), and uniform (bottom panel) initial cell configurations (blue: S-cell, orange: R-cell). Black solid line: average total cell population (). Vertical cyan line: TTP in each case; horizontal cyan line: the 120% level of the initial tumor volume (tumor progression threshold). The average numbers of empty sites (
) and R-cells (
) in the VNHD of an R-cell in the 30 realizations are shown as boxplots in (C) and (D), respectively, for i = c, r, u. The blue, orange, and yellow boxes are for the clumped (c), random (r), and uniform (u) cases, respectively.
Fig 5.
Effect of the initial R-cell distribution on the TTP under AT.
(A) Cell configurations on days 1, 120, and 2000. The square signifies the domain representative of the tumor. The blue, orange, and white dots are S-cells, R-cells, and empty sites, respectively. (B) Average temporal evolution of the S-cell and R-cell populations for the clumped (upper panel), random (middle panel), and uniform (bottom panel) cases of the initial cell configurations (blue dots: S-cells, orange dots: R-cells). Black solid line: the total population (N(t) = S(t) + R(t)). Vertical solid cyan line: TTP; horizontal solid cyan line: 120% level of the initial tumor volume. Vertical dotted cyan line: time for the R-cells to reach 50% of the initial tumor volume (TR50); horizontal dotted cyan line: 50% level of the initial tumor volume. The average numbers of empty sites (), S-cells (
), and R-cells (
) in the VNHD of an R-cell in the 30 realizations are shown as boxplots in (C), (D), and (E), respectively, for i = c, r, u. The blue, red, and yellow boxes are for the clumped (c), random (r), and uniform (u) cases, respectively.
Fig 6.
Role of the initial R-cell distribution on the benefit of AT over CT-MTD.
(A) The blue and red bars show the TTP for N(0) = 5000 and 7000 respectively under both treatment strategies and different initial R-cell configurations of c (clumped), r(random) and u(uniform) for f0 = 10%. (B) Similar results to those in A are shown for f0 = 1%. (C) The boxplots show the TG in the 30 realizations for the clumped, random, and uniform initial cell configurations for f0 = 10% (***: p − value < 0.001; n.s.: not significant). The blue shaded and orange shaded regions show results for N(0) = 5000 and 7000 respectively. (D) Similar results to those in C are shown for f0 = 1%.
Fig 7.
Effects of cell migration on treatment responses.
(A) The time evolution of the mean of the total cell population () in the 30 simulations is shown for m = 0 (blue), 2rS (red), and 4rS (yellow) under AT (dashed line) and CT-MTD (solid line) with N(0) = 5000, f0 = 10%. The vertical solid lines show the time to tumor progression (TTP) under CT-MTD. The vertical dashed line shows the TTP under AT. The horizontal cyan line shows the 120% level of the initial cell population (progression threshold). (B) The same result as in A is shown for N(0) = 7000, f0 = 1%. (C) The time evolution of the mean of the average number of empty sites in the VNHD of each R-cell in the 30 realizations (
) is shown with the same line styles as in A. (D) The same result as in C is shown for N(0) = 7000, f0 = 1%. (E) The boxplots of the TG for m = 0, rS, 2rS, 3rS, 4rS are shown. (F) The same result as in D is shown for N(0) = 7000, f0 = 1%.
Fig 8.
Cell configurations for CAF-mediated tumor growth.
Cell configurations at different times under CT-MTD (continuous MTD therapy) for all type of fibroblast configurations. Each row shows results for one type of fibroblast configuration. The blue and orange dots show the S-cells and R-cells respectively and the purple shaded areas show the fibroblast regions. Specification of the fibroblast region is shown in the section Initial and boundary conditions.
Fig 9.
Consequences of CAF-mediated resistance with clumped initial R-cell distribution.
(A) The time evolution of the average of the total cell population () in the 30 simulations is shown for CT-MTD for all types of fibroblast configurations. NoF: no fibroblast, FC: fibroblast overlapping with the R-cell clump at the center of domain, FCp: fibroblast partially overlapping with the R-cell clump at the center, FSq: fibroblast completely surrounding the R-cell clump. FSp: fibroblast over a “L” shaped region, partially surrounding the R-cell clump, FR: fibroblast clumps randomly placed in the domain. (B) Boxplot of the time to progression (TTP) in the 30 realizations under CT-MTD. (C) Boxplot of the time to progression (TTP) in the 30 realizations under AT is shown. (D) Correlation of TTP with the area of impact is shown. The circles and the asterisks (the colors are similar to the legend in A) show the TTP under CT-MTD and AT respectively. The solid and dashed lines are the respective regression lines (sold line: CT-MTD, dashed line: AT). The p − values for AT and CT-MTD indicate significance of the correlation. (E) Time gain (TG) is shown as boxplots for all six types of fibroblast distributions.
Fig 10.
CAF-mediated growth with clumped initial R-cell distribution and cell migration (α = 2, m = 1).
(A) The time evolution of the average of the total cell population () under CT-MTD in the 30 simulations is shown for all types of CAF configurations. (B) Boxplot of the time to progression (TTP) under CT-MTD in the 30 realizations. (C) Boxplot of the time to progression under AT in the 30 realizations. (D) Correlation of TTP with the perimeter of impact. The circles and the asterisks (the colors are similar to the legend in A) show the TTP under CT-MTD and AT respectively. solid line: CT-MTD and dashed line: AT regression line. The p − values for AT and CT-MTD indicate significance of the correlation. (E) Time gain (TG) is shown as boxplots for all type of CAF distributions.
Fig 11.
Combinations of the four metastasis scenarios.
The green and red colors correspond to detected and undetected metastases, respectively. F and NF correspond to the existence and absence of fibroblasts, respectively.
Fig 12.
Initial cell configurations for the four cases with an invisible clumped metastasis.
The red, blue, and white dots correspond to R-cells, S-cells, and empty sites. The gray dots shows the sites accompanied by CAFs.
Fig 13.
Initial cell configurations for the four cases with a random invisible metastasis.
The red, blue, and white dots correspond to R-cells, S-cells, and empty sites. The gray dots show the sites that are accompanied by CAFs.
Fig 14.
Complex dynamics of multiple metastases under AT and CT-MTD.
The time evolution of the total cell population in the four metastases is shown in the sub-figures for Case I. The first and second columns show the results for the initial clumped and random cell configurations in the invisible metastasis, respectively. In each sub-figure, the blue, red, yellow, and black colors show the total cell populations in metastasis 1, metastasis 2, metastasis 3, and metastasis 4, respectively; the vertical cyan lines show the emergence time (ET) of the invisible metastasis, and the red line shows the TTP. The solid and dashed lines show results under CT-MTD and AT, respectively.
Fig 15.
Cell configurations at the ET and TTP under CT-MTD and AT for Case I with the clumped invisible metastasis.
The red, blue, and white dots correspond to R-cells, S-cells, and empty sites. The gray dots show the sites that are accompanied by fibroblasts.
Fig 16.
Cell configurations at the ET and TTP under CT-MTD and AT for Case I with the random invisible metastasis.
The red, blue, and white dots correspond to R-cells, S-cells, and empty sites. The gray dots show the sites that are accompanied by fibroblasts.
Fig 17.
Bar chart of the ET and TTP under CT-MTD and AT for Cases I to IV.
(A) clumped invisible metastases, (B) random invisible metastases.