Fig 1.
Left: Compartmental representation of the whole patient by QSP model. Right: Spatial representation of the tumor by ABM. Some deterministic species and reactions from the QSP tumor compartment are replaced with equivalent agents and stochastic reactions in ABM. The antigen module, however, is not explicitly represented in ABM. It consists of antigen-presenting cells that mature after taking antigens from the tumor compartment to transport them through lymphatic vessels to the tumor-draining lymph node compartment. Once there, they prime naïve cytotoxic T lymphocytes (CTL) and regulatory T cells (Treg) that clonally expand in the tumor draining lymph nodes, intravasate, circulate through the central compartment (blood system), and extravasate into the tumor microenvironment as described in [1]. The figure has been created with Biorender.
Fig 2.
Workflow and description of the spatial QSP model explaining how the QSP model and ABM are coupled.
Left: Workflow of the calculation steps in the spQSP algorithm. Right: Breakdown of each step and list of the main C++ files used in the step. The figure has been created with Biorender.
Fig 3.
Scaled spatio-temporal evolution of the tumor.
The relation between CSC and PC migration and proliferation rates, R, the asymmetric division probability, k, and the maximum number of progenitor cell divisions, dmax, define the evolution in time of the tumor shape and size. In panels A-C, the turquoise scale bar represents the normalized cancer cell density of slices at the center of the tumors with 0.2 mm thickness every 4 months; the dark blue dots are CSCs. In panel D, dark brown, light brown, and grey dots represent CSCs, PCs, and senescent cells, respectively. Thus, three scenarios are presented: R < 1, high probability of asymmetric division, k = 0.95, and a large maximum number of progenitor cell divisions, dmax = 18 (panel A); R ~ 1, low probability of asymmetric division, k = 0.75, and dmax = 18 (panel B); R ~ 1, k = 0.95, and a small maximum number of progenitor cell divisions, dmax = 9 (panel C). The spatial QSP algorithm calculated the evolution of three-dimensional tumors for 16 months starting from one CSC located at the center of the grid. Panel D shows the three-dimensional spatial representation of the tumors from panels A-C after 16 months of growth. All simulations were performed in a 3x3x3 mm grid. Cases with spherical shapes are included in section B.1 of the S1 Supplementary Material. The scaling factor is γ=1 in all cases.
Table 1.
Spatio-temporal characterization of tumors in terms of the non-dimensional cancer cell parameters R, k, and dmax.
Cases A-C correspond to panels A-C, respectively, in Fig 3. CSCs refers to cancer stem-like cells.
Fig 4.
Spatial distributions of cell densities under different tumor growth conditions.
Cancer cell, CD8+ T cell, and FoxP3+ T cell densities are normalized and represented in turquoise, purple, and blue scale bars, respectively. CSC density distributions are represented in the first column of contour plots as yellow-to-blue lines. ξp,y = y/yref and ξp,x = x/xref are non-dimensional spatial coordinates with yref = xref = 2up/rp. Four different scenarios are presented: similar migration and proliferation effects, R ~ 1, 10% of the grid occupied by voxels with cell recruitment sources, and dmax = 9 (panel A); fast migration of CSCs and proliferation of PCs, R < 1, 10% of the grid occupied by voxels with cell recruitment sources, and dmax = 9 (panel B); R ~ 1, 10% of the grid occupied by voxels with cell recruitment sources, and dmax = 4 (panel C); R < 1, 30% of the grid occupied by voxels with cell recruitment sources, and dmax = 9 (panel D). The spatial QSP algorithm calculated the evolution of three-dimensional tumors starting from a normal distribution of cancer cells located at the center of the grid. QSP model and ABM are coupled before reaching the point where T cells are recruited and also before the initial tumor diameter condition from the QSP model is met. Thus, no initial T cell spatial distribution is enforced. The figures show the cancer cell density in a two-dimensional slice at the center of the tumor 10 months after the initial tumor diameter condition is met. The scaling factor is γ=50000 in all cases.
Table 2.
Spatial characterization of the immune response under different tumor growth conditions.
R.V. stands for recruitment volume and refers to the percentage of spatial domain occupied by voxels with cell recruitment points (where only one recruitment point is assumed per voxel); CSCs denotes cancer stem-like cells.
Fig 5.
Spatial distributions of cell densities under different tumor growth conditions and immunotherapy.
Cancer cell, CD8+ T cell, and FoxP3+ T cell densities are normalized and represented in turquoise, purple, and blue scale bars, respectively. CSC density distributions are represented in the first column of contour plots as yellow-to-blue lines. ξp,y = y/yref and ξp,x = x/xref are non-dimensional spatial coordinates with yref = xref = 2up/rp. Four different scenarios are presented: similar migration and proliferation effects, R ~ 1, 10% of the grid occupied by voxels with cell recruitment sources, and dmax = 9 (panel A); fast migration of CSCs and proliferation of PCs, R < 1, 10% of the grid occupied by voxels with cell recruitment sources, and dmax = 9 (panel B); R ~ 1, 10% of the grid occupied by voxels with cell recruitment sources, and dmax = 4 (panel C); R < 1, 30% of the grid occupied by voxels with cell recruitment sources, and dmax = 9 (panel D). The spatial QSP algorithm calculated the evolution of three-dimensional tumors starting from a normal distribution of cancer cells located at the center of the grid. QSP model and ABM are coupled before reaching the point where T cells are recruited and also before the initial tumor diameter condition from the QSP model is met. Thus, no initial T cell spatial distribution is enforced. The figures show the cancer cell density of a two-dimensional slice at the center of the tumor 10 months after the initial tumor diameter condition is met and 3 mg/kg nivolumab is administered every two weeks. The scaling factor is γ=50000 in all cases.
Table 3.
Spatial characterization of the immune response under different tumor growth conditions and immunotherapy.
R.V. stands for recruitment volume as defined in Table 2; CSCs denotes cancer stem-like cells.
Fig 6.
Temporal evolution of cells in the tumor microenvironment under different tumor growth conditions without and with immunotherapy.
Panel A: variation in time of number of cancer cells, T cells in central and tumor compartments, and CD8+/FoxP3+ ratio. Panel B: variation in time of cancer cell and CD8+ T cell subtypes. Thick and thin red lines represent cases (a) and (a*) without and with immunotherapy, respectively (R ~ 1, 10% of the grid occupied by voxels with cell recruitment sources, and dmax = 9); thick and thin blue lines represent cases (b) and (b*) without and with immunotherapy, respectively (R < 1, 10% of the grid occupied by voxels with cell recruitment sources, and dmax = 9); thick and thin green lines represent cases (c) and (c*) without and with immunotherapy, respectively (R ~ 1, 10% grid occupied by voxels with cell recruitment sources, and dmax = 4); thick and thin purple lines represent cases (d) and (d*) without and with immunotherapy, respectively (R < 1, 30% of the grid occupied by voxels with cell recruitment sources, and dmax = 9).
Fig 7.
Cell distribution and cell density at the IF.
Panel A: Spatial representation of cancer cell subtypes (left), CD8+ T cells subtypes, FoxP3+ T cells, and MDSCs (center), and all cells (right) in a section of a tumor slice. The IF region is depicted in pale turquoise. Slow tumor growth case (kC1,growth = 0.005 day-1) and R << 1 (R = 1/50). Panel B: 3D representation of CD8+ T cell and FoxP3+ T cell densities in the central tumor (CT; yellow), the invasive front (IF; red), and the normal tissue (N; green). Panel C: CD8+ T cell and FoxP3+ T cell density profiles along the direction perpendicular to the IF averaged over the circumference of the IF. 95% confidence intervals are calculated (grey areas). Two definitions of IF are introduced here and are indicated as vertical lines, blue: width wpathol=0.5 mm; red: width wpathol=1 mm. Panel D: CD8+ T cell and FoxP3+ T cell density profiles along the direction perpendicular to the IF averaged over the circumference of the IF. In panels C and D every row is a case with a different combination of ratios kC1,growth/kC,T1, kT1/kTreg, and the parameter Treg,max, where kC1,growth, kC,T1, kT1, kTreg, and Treg,max are the cancer cell growth rate, the rate of cancer cell death by T cells, the exhaustion rate of cytotoxic T cells by all cells that express PD-L1, the inhibition rate of cytotoxic T cells by regulatory T cells, and the maximal regulatory T cell density in the tumor, respectively. The spatial QSP algorithm calculated the evolution of a tumor slice starting from a fraction of a normal distribution of cancer cells. QSP model and ABM are coupled before reaching the point where T cells are recruited and also before the initial tumor diameter condition from the QSP model is met. Thus, no initial T cell spatial distribution is enforced. The figures here show cell distributions and densities 6 months after the initial tumor diameter condition is met.
Fig 8.
Cell distribution and cell density at the IF under treatment.
Panel A: Spatial representation of cancer cell subtypes (left), CD8+ T cells subtypes, FoxP3+ T cells, and MDSCs (center), and all cells (right) in a section of a tumor slice. The IF region is depicted in pale turquoise. Slow tumor growth case (kC1,growth = 0.005 day-1) and R << 1 (R = 1/50). Panel B: 3D representation of CD8+ T cell and FoxP3+ T cell densities in the central tumor (CT; yellow), the invasive front (IF; red), and the normal tissue (N; green). Panel C: Average of CD8+ T cell and FoxP3+ T cell density profiles that are perpendicular to the IF. 95% confidence intervals are calculated upon the profile (grey areas). Two definitions of IF are introduced here and are indicated as vertical lines, blue: width wpathol=0.5 mm; red: width wpathol=1 mm. Panel D: Average of CD8+ T cell and FoxP3+ T cell density profiles along the IF. In panels C and D every row is a case with a different combination of ratios kC1,growth/kC,T1, kT1/kTreg, and the parameter Treg,max, where kC1,growth, kC,T1, kT1, kTreg, and Treg,max are the cancer cell growth rate, the rate of cancer cell death by T cells, the exhaustion rate of cytotoxic T cells by all cells that express PD-L1, the inhibition rate of cytotoxic T cells by regulatory T cells, and the maximal regulatory T cell density in the tumor, respectively. The spatial QSP algorithm calculated the evolution of a tumor slice starting from a fraction of a normal distribution of cancer cells. QSP model and ABM are coupled before reaching the point where T cells are recruited and also before the initial tumor diameter condition from the QSP model is met. Thus, no initial T cell spatial distribution is enforced. The figures here presented show cell distributions and densities 6 months after the initial tumor diameter condition is met with 3 mg/kg nivolumab administered every two weeks.
Table 4.
Combinations of ratios kC1,growth/kC,T1, kT1/kTreg, and the parameter Treg,max, where kC1,growth, kC,T1, kT1, kTreg, and Treg,max are the cancer cell growth rate, the rate of cancer cell death by T cells, the exhaustion rate of cytotoxic T cells by all cells that express PD-L1, the inhibition rate of cytotoxic T cells by regulatory T cells, and the maximal regulatory T cell density in the tumor, respectively.
G.R. stands for growth rate.