Fig 1.
Cellular automata based rules of proliferation, death and conversion.
Healthy cells are colored in green, stressed cells in magenta, empty spaces are in black, unmodeled CV is colored in orange. (A) Initial conditions for a 1D sinusoidal cord or a 2D hexagonal lobule. (B) Output of the CA model at intermediate time, at any point in time total number of sites N is made of healthy (H), stressed (S) and empty spaces (D) left by dead cells. (C) Output of CA model of tissue survival with H = N, S = 0, D = 0. (D-F) CA rules of updating the model with α as the proliferation timescale, β as the timescale of conversion of a healthy to a stressed cell and γ as the timescale of stressed cell death. The cell to which the rule is being applied is marked by an X. The Conversion Rule converts a cell from normal to stressed and is dependent on the number of stressed neighbors around the healthy cell. Two random numbers are chosen - r1 has to be lesser than the conversion parameter and r2 has to be lesser than the total number of stressed neighbors (Ns) normalized over the total number of the cells’ neighbors (T). The Death Rule kills a stressed cell if a random number is less than the death parameter Pd. The Proliferation Rule fills in an empty location (created by a dead cell) and will only occur if the space is next to a healthy cell. For each healthy cell, a random number rp is chosen and proliferation occurs at that time step if it is less than the proliferation parameter Pp and one of its six immediate neighbors contains an empty site (Nd>1).
Fig 2.
Dose response, pattern and timing of hepatic necrosis and recovery after APAP.
Dose Response: (A-E) H&E staining of mouse liver 24 hours after APAP at 0, 10, 100, 250 and 500 mg/kg. Bar = 300 μm. (F) Serum AST and ALT 24 hours after APAP at 0, 10, 100, 250 and 500 mg/kg (n = 3 animals in each group). No necrosis is evident at 10, or 100 mg/kg APAP although modest increases in AST/ALT were observed. Pericentral necrosis, visible as regions of hepatocytes with lighter pink cytoplasm and condensation and loss of hepatic nuclei, has developed at 250 and 500 mg/kg and is accompanied by major increases in AST/ALT (mean ± s.e.). Time Course: (G-J, D, K) H&E staining of mouse liver at 0, 2, 6, 12, 24 and 48 hours after APAP at 250 mg/kg. (G-K) Bar = 300 μm. (L) Serum AST and ALT at 0, 2, 6, 12 and 48 hours after APAP at 250 mg/kg (n = 3 animals in each group). Data re-visualized as in Dunn et al [16]. At 250 mg/kg the cytoplasm of cells around the CVs begins to appear lighter by H&E staining as early as 30 minutes after administration of APAP. The lightened region becomes easily discernible around 2 hours after APAP treatment, but hepatic nuclei remain intact and AST/ALT is unchanged from control. Pericentral necrosis has begun by 4 hours after APAP treatment and is accompanied by increased AST/ALT. Pericentral necrosis is prominent at 6, 12 and 24 hours and is accompanied by further increased AST/ALT. (K) By 48 hours pericentral region shows signs of recovery.
Fig 3.
Levels and localization of resident and infiltrating Kupffer cells and cell proliferation after APAP induced centrilobular necrosis.
(A-C) F4/80 labeling (yellow) identifies resident and infiltrating Kupffer cells. (F-H) H3 labeling (yellow) identifies proliferating cells. (A-C, F-H) Sections were counterstained to visualize cells using Lens Culinaris Agglutinin (LCA, green), nuclei using Sytox Green (red) and additionally in (A-C) plasma membranes of endothelial cells and hepatocytes using Tomato Lectin (TL, blue). The TL label also allows visualization of the prominent intracellular membrane bound regions in hydropic hepatocytes. (D,E) and (I,J) F4/80 and H3 positive cell counts are expressed as mean ± s.d. per 100,000 μm2. (D) Macrophage counts greatly increase by 6 hours after APAP. Cell counts increase further by 48 hours after APAP. Kupffer cell counts return to normal by 72 hours after APAP. (E,J) Labeled cells (yellow) are counted within 1,2,3, and 4 cell diameters from central veins. (E) Kupffers are concentrated at the periphery of the necrotic region (4 cell diameters from the CV) at 6 and 12 hours after APAP. By 48 hours after APAP, the centrilobular damaged region has been almost filled with hepatocytes and macrophages are concentrated at the CV (within 1 cell diameter from the CV). (J) Dividing cells are concentrated at the periphery of the necrotic region at 6 hours after APAP. By 48 hours after APAP dividing cells are concentrated near the CV.
Fig 4.
Parameter space of tissue survival.
(A) Probability of tissue recovery as a function of γ, α with fixed β = 5 (hours). Metric plotted is the average fraction of healthy cells at steady state over 100 replicas, ⟨H⟩/N, using the colorbar as shown. Bistable states of survival are seen where the tissue sometimes survives (H = N, S = 0, D = 0) and sometimes dies (H = 0, S = 0, D = N). A critical ratio of γ/β* = 2.25± 0.37 marks a transition (indicated by the red line) above which recovery probability falls down below 0.95. (B-C) Showing the variation in recovery probability by keeping α/β fixed at 0.25, 3.25, 5.25 (lines in blue, black, and magenta respectively) across values of γ/β in (B) and by setting γ/β = 0.25, 3.25, 5.25 (lines in blue, black, magenta respectively) across values of α/β in (C). (D) Probability of tissue survival as a function of β, γ with the value of fixed α = 5 (hours). Red line indicates a ratio of γ/β* = 1.69±0.24. Survival probabilities for fixed values of γ/α = 0.15, 3.25, 5.25 (blue, black and magenta, respectively) across values of β/α (E) and fixed values of β/α = 0.05, 1.25, 3.25 (blue, black and magenta, respectively) across values of γ/α (F).
Fig 5.
Number of new hepatocytes created.
Average number of new hepatocytes (shown by the colorbars) created for fixed β = 5 (hours) (A, C) and fixed α = 5 (B, D). Lines indicate γ/β* = 2.25±0.37 in (A), and γ/β* = 1.69±0.24 in (B). (C) shows this as a function of γ/β for fixed α/β = 0.25 (blue), 3.25 (black), 5.25 (magenta) and (D) as a function of β/α with fixed γ/α = 0.15 (blue), 3.25 (black), 5.25 (magenta).
Fig 6.
Critical thresholds of healthy cell populations.
Loss of healthy tissue resulting in a second wave of damage, as a function of the parameters γ and β for fixed α = 5 (hours). Statistics are measured only for simulations where the cell population recovers and the red line shows γ/β = 1.69 (A). Average size of the minimum healthy population ⟨min(H)⟩ plotted against final length (B). ⟨min(H)⟩ is now rescaled against the tissue length (N) and plotted against ⟨H⟩/N (C). ⟨min(H)⟩/N plotted with different lengths of initial damage (D).
Fig 7.
Variation of the initial damage pattern and tissue size.
(A-C) Fixed tissue length (N = 10) with Hi = 2 (A), Hi = 4 (B) and Hi = 8 (C) with α = 5 (hours). Lines are drawn through points where survival probability drops to 0.95±0.01. Values of γ/β* = 0.38±0.11, 1.07±0.18, 2.53±0.35 (A-C). (D-F) Fixed damage fraction (≅40%) with different tissue lengths of N = 8, 16, 24. Approximate slopes are . (G) Combined curves showing the slope boundaries for different fractions of damage with different tissue lengths. (H) Rescaling the slopes by the tissue lengths collapses the curves.
Fig 8.
2D Parameter space of tissue repair.
(A) Simulations on a hexagonal lobule with fixed β = 5 (hours) show presence of coexisting states shown in red (marked with C). Other regions show parameters for which the tissue always survives (shown in yellow, marked with S) and regions where the tissue always dies (shown in green, marked with D). The other colors are combinations of these states where different outcomes are possible for the same parameter set. (B) Average size of the number of healthy states. Sample trajectories along the dashed line shown for γ/β = 0.05 (C), 1.45 (D), 2.45 (E), 4.45 (F), 9.05 (G).Coexisting states also show a gradient in their steady state population as shown for (E,F). (H) Average healthy population is plotted along fixed ratios of α/β = 1.65 (blue), 5.95 (black), 10.25 (magenta) across γ/β. (F) Average healthy population is plotted along fixed ratios of γ/β = 1.65 (blue), 5.95 (black) and 10.5 (magenta) across α/β.
Fig 9.
Second order neighbor effects and fixing number of divisions.
(A) Simulations on a hexagonal lobule with fixed β = 5 (hours), including second order neighbor effects. Coexisting states shown in red (marked with C). Other regions show parameters for which the tissue always survives (shown in yellow, marked with S) and regions where the tissue always dies (shown in green, marked with D). The other colors are combinations of these states where different outcomes are possible for the same parameter set. (B) Average size of the number of healthy states. (C) Healthy population fraction across γ/β for fixed values of α/β = 1.65 (blue), 4.95 (black), 10.25 (magenta). (D) Simulations on a hexagonal lobule with fixed β = 5 (hours) and fixing the number of maximum divisions per hepatocyte removes the coexistence states. Criteria of tissue survival is the presence of healthy cells at the end. Red dashed line showing the transition from the survival states at γ/β* = 5.25±0.29 (E) Average size of the number of healthy states. (F) Healthy population fraction across γ/β for fixed values of α/β = 1.65 (blue), 5.95 (black), 10.25 (magenta) with the red dashed line showing γ/β* = 5.25±0.29. Visualization output showing a representative state at the final simulation time point for α/β = 1 and γ/β = 1.45 (G), 2.45 (H), 4.45 (I) and 9.05 (J) at the final time.