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. The first, second, and third rows show the results for Cases II, III, and IV, respectively. The first and second columns show the results for clumped and random initial 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 the results under CT-MTD and AT, respectively. (TIF)</p

Temporal change of tumor configurations for different initial R-cell configurations.

The video shows the evolution of tumor for clumped (first column), random (second column), and uniform (third column) initial cell configurations with time under CT-MTD (first row) and AT (second row). The red, blue, and white dots correspond to R-cells, S-cells, and empty sites respectively. (MP4)</p

Temporal change of cell configuration of different metastatic lesions for case I (Fig 11) with clumped invisible metastasis.

The red, blue, and white dots correspond to R-cells, S-cells, and empty sites respectively. The gray dots show the sites that are accompanied by CAFs. The first and second row show results for CT-MTD and AT respectively. (MP4)</p

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.</p

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%.</p

Increased carrying capacity reduces the benefit of adaptive therapy by reducing spatial competition.

(A) The blue and red boxplots show the TG from the 30 couple realizations (for both AT and CT-MTD) with respect to carrying capacities of K = 1 and 2, respectively. The triple asterisk (***) signifies that increasing the carrying capacity significantly reduced the TG (p − value B) 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 for both CT-MTD (solid lines) and AT (dashed lines); K = 1 (blue) and 2 (red). K = 2 offers a greater number of empty sites in the VNHDs of R-cells than K = 1. For the clumped initial cell distribution, we investigated the effect of the spatial carrying capacity on the TG. The spatial carrying capacity was characterized as K = 1 (each lattice point could hold one cell) or K = 2 (each lattice point could hold, at most, two cells, regardless of their sensitivity or resistance). When K = 1 was used, a total of four cells could occupy the VNHD of each cell (i.e., ). For each cell in K = 2, a total of eight cells could occupy a VNHD, and one additional cell could be located in the respective cell’s site (i.e., ). S2A Fig shows that increasing the carrying capacity significantly decreased the TG (p − value K = 2, whereas it was below 2 for K = 1. Due to this ample space in their neighborhoods, R-cells hardly experienced any spatial competition and grew at a higher pace when K = 2 under both AT and CT-MTD. As the total cell population grew, decreased abruptly and tended to settle below 1. For K = 1, a similar trend was observed; however, the number of empty sites was lower than that for K = 2 (). Comparing the number of empty sites in each R-cell’s VNHD () for AT in the case of K = 1 with that in the case of K = 2 (S2B Fig, dotted lines)), we observed that, for K = 1, went through ups and downs several times, which suggested spatial competition with neighboring cells. On the other hand, for K = 2, this value monotonically decreased, and there was a very slight difference due to AT and CT-MTD. Therefore, we concluded that the short TG with K = 2 was due to the lack of spatial competition. We observed that the probabilities of having a negative TG were 0.03 and 0.4 for K = 1 and 2, respectively, i.e., an increase in carrying capacity reduces the benefit of AT over CT-MTD. (TIF)</p

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.</p

Fibroblast mediated growth for <i>α</i> = 4 with clumped initial R-cell distribution.

(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 fibroblast configurations. (B) Boxplot of the TTP (time to progression) under CT-MTD in the 30 realizations. (C) Boxplot of the time gain under AT in the 30 realizations. (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. And the solid and dashed lines are the respective regression lines. 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 fibroblast structures. (TIF)</p

The parameter values are listed in the following table.

The parameter values are listed in the following table.</p

Temporal change of cell configuration for different fibroblast distributions.

The video shows the evolution of tumor for different fibroblast structures (column wise) for clumped initial cell configuration under CT-MTD (first row) and AT (second row). The red, blue, and white dots correspond to R-cells, S-cells, and empty sites respectively. (MP4)</p