Figures and data in Aspirin’s effect on kinetic parameters of cells contributes to its role in reducing incidence of advanced colorectal adenomas, shown by a multiscale computational study

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Yifan Wang

C Richard Boland

Ajay Goel

Dominik Wodarz

Natalia L Komarova

(2022)
Aspirin’s effect on kinetic parameters of cells contributes to its role in reducing incidence of advanced colorectal adenomas, shown by a multiscale computational study

eLife 11:e71953.

https://doi.org/10.7554/eLife.71953
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Figures
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Additional files

22 figures, 4 tables and 4 additional files

Figures

Figure 1

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A schematic illustrating the mathematical model.

The six cell types are denoted by circles, the mutation rates that give rise to different types are marked by the straight arrows. Type 6 (late adenoma) is marked in red. Crypt conversion rates are …

Figure 2

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Fitting the nonlinear model to late adenoma incidence data.

(a) The fitting error as a function of r₁, for the best fitting pairs (γ₃ and γ₄) with K_A = K_R = 1000; see Figure 2—source data 1. (b) The fitting error as a function of r₁, for the best fitting …

Figure 2—source data 1 Data for Figure 2a.: https://cdn.elifesciences.org/articles/71953/elife-71953-fig2-data1-v1.txt
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Figure 2—source data 2 Data for Figure 2b.: https://cdn.elifesciences.org/articles/71953/elife-71953-fig2-data2-v1.txt
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Figure 3

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Pathways to adenoma.

(a) A schematic representation of the two pathways. (b) For the two groups of fits in Figure 2(b), the pairs (K_A,K_R) are shown. The two groups are characterized by K_A >K_R and K_A <K_R, respectively. …

Figure 3—source data 1 Date for Figure 3b.: https://cdn.elifesciences.org/articles/71953/elife-71953-fig3-data1-v1.txt
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Figure 4

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The effect of aspirin on the advanced adenoma incidence curve.

(a) Modeling assumptions and references to figures that present the results. (b) A schematic showing the timing of the model with aspirin treatment. (c) Predicted advanced adenoma incidence in the …

Figure 4—source data 1 Data for Figure 4c.: https://cdn.elifesciences.org/articles/71953/elife-71953-fig4-data1-v1.txt
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Figure 4—source data 2 Data for Figure 4d.: https://cdn.elifesciences.org/articles/71953/elife-71953-fig4-data2-v1.txt
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Appendix 1—figure 1

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Appendix 1—figure 2

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Model fitting to the incidence data.

(a) The best fitting curves corresponding to increasing SC division rates, r₁, are plotted together with the epidemiological data. (b) The best fitting parameters $γ_{3}$ and $γ_{4}$ are shown for each value …

Appendix 1—figure 3

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Fitting the linear model.

Each panel shows fitting results for a particular parameter combination, ( $R_{A P C^{+, / -}}, R_{K R A S}$ ); the values of these two fitness parameters are indicated, and $F_{A P C^{+, / +}} = 2 F_{A P C^{+, / -}}$ . The green lines show the *log*₁₀(fitting error) as a …

Appendix 1—figure 4

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Fitting the linear system: for each parameter combination, ( $R_{A P C^{+, / -}}, R_{K R A S}$ ), of Appendix 1—figure 3, the error landscape is shown that corresponds to the best fitting r₁ (see the minima of the green lines in Appendix 1—figure 3).

The contour plot represents *log*₁₀(fitting error) as a function of $γ_{3}$ and $γ_{4}$ (darker colors correspond to lower values). The rest of the parameters are as in Table Appendix 1—table 2.

Appendix 1—figure 5

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Nonlinear model fitting to the incidence data (fixing the crypt carrying capacity, varying crypt fission rates).

(a) The best fitting curves corresponding to increasing SC division rates, r₁, are plotted together with the epidemiological data; inset: the best fitting curves corresponding to the values of r₁ …

Appendix 1—figure 6

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Nonlinear model fitting to the incidence data (fixing crypt fission rates, varying the crypt carrying capacity).

Compared are the cases in the presence of a non-zero crypt death rate, $δ = 0.05$ yrs^-1 (**a–c**), and in the absence of crypt death rate, $δ = 0.0$ yrs^-1 (**d-f**). (**a,d**) The heatplot of the fitting error, where all …

Appendix 1—figure 7

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Comparing deterministic solutions (system (LABEL:n1non-23, 12), solid lines) with the stochastic Gillespie simulations (symbols of the same color).

The three lines correspond to the different stopping conditions: (0) 1 crypt of type 6; (1) $N_{1} = 10^{2}$ crypts of type 6, (2) $N_{2} = 10^{5}$ crypts of type 6. The rest of the parameters are as in Appendix 1—figure 6a-c. …

Appendix 1—figure 8

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Fitting the adenoma incidence curve to the nonlinear model with an explicit expansion phase.

Under a fixed $K_{A} = K_{R} = 1000$ , for each value of r₁ the best fitting values of $γ_{3}$ and $γ_{4}$ were found. (a) The fitting error is shown as a function of r₁ for the original model (blue line, same as in Appendix …

Appendix 1—figure 9

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Fitting the adenoma incidence curve to the nonlinear model with an explicit expansion phase.

Under fixed crypt fission rates, for each value of r₁ the best fitting value of $K_{A} = K_{R}$ was found. (a) The fitting error is shown as a function of r₁ for the original model (blue line, same as in Appendix…

Appendix 1—figure 10

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Pathways to adenoma: Fitting the incidence curve to the nonlinear model with an explicit expansion phase.

Under fixed crypt fission rates, for each value of r₁ the best fitting pair ( $K_{A}, K_{R})$ was found. (a) The fitting error is shown as a function of r₁; 3 groups of parameter combinations are marked by blue, …

Appendix 1—figure 11

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Same as Appendix 1—figure 10, except expansion to 10⁵ crypts is assumed.

Appendix 1—figure 12

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Compartment dynamics for the best fitting models with $K_{A} = K_{R}$ .

(a) The expected crypt numbers (using equations (LABEL:n1non-23)) for the 5 compartments are shown for the 2nd parameter set in table Appendix 1—figure 9(c) (expansion to 10² crypts). (b) For the …

Appendix 1—figure 13

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Same as Appendix 1—figure 12(b-c), but without the restriction $K_{A} = K_{R}$ .

(a) Parameter set #2 from Table Appendix 1—table 3 was used. (b) Parameter set #4 from Table Appendix 1—table 3 was used.

Appendix 1—figure 14

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extends the results of main text Figure 4(a) and shows that relative adenoma risk predicted for each decade of treatment is very similar for all parameter sets (#1-4, Table Appendix 1—table 3).

Appendix 1—figure 15

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Relative incidence of advanced adenoma, where (a) parameter set #1 (Table Appendix 1—table 3) and (b) parameter sets #2 is used.

In each panel, a comparison is presented between the case where aspirin affects crypt fission/death rates (yellow) and where it does not affect crypt fission/death rates (blue). Aspirin affects …

Appendix 1—figure 16

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Predicted advanced adenoma incidence in the absence of aspirin treatment (thin black lines are the fitted curves and black dots are incidence data); under aspirin treatment where the drug affects types 2–6 (yellow lines), under aspirin treatment where the drug affects type 6 only (blue lines).

Each panel corresponds to aspirin treatment administered during one decade (20-30 years, 30–40 years, etc). The treatment period is shaded light green. It was assumed that type 6 crypts grow to 10⁵, …

Appendix 1—figure 17

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A comparison of predicted relative advanced adenoma risk under the assumption that cell types 2–6 (yellow bars) and only type 6 cells (blue bars) are affected by aspirin.

Aspirin is administered during different decades of patients’ life (as marked under each pair of bars), and different panels correspond to different durations of the follow-up period. (a) Parameter …

Appendix 1—figure 18

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A comparison of predicted relative advanced adenoma risk under the assumption that aspirin treatment lasts 10 years (yellow) or 5 years (blue).

(a) Predicted advanced adenoma incidence in the absence of aspirin treatment (thin black lines are the fitted curves and black dots are incidence data); the incidence curve fo patients treated for …

Tables

Table 1

Aspirin doses in xenograft experiments, the equivalent human dose and the resulting changes in kinetic rates.

Dose in xeno- grafts (mg/kg)	Equivalent dose in humans (325 cm pills per week)	Fold difference in division rate, F_r	Fold difference Un death rate, F_d	Fitness factor
15	1.8	0.9	1.5	0.86
50	6.1	0.75	1.75	0.70
100	12.2	0.5	2.0	0.45

Appendix 1—table 1

Enumeration of the different genotypes.

Mutations in APC	Mutations in KRAS	Type number
0	0	1
1	0	2
2	0	3
0	1	4
1	1	5
2	1	6

Appendix 1—table 2

Parameters, notations, and their values.

Parameter	Notation	Value/Range
Number of crypts	$N_{c r y p t}$	10⁷
Number of SCs per crypt	K	7
Rate of inactivation of APC (per cell division)	u	10^-7
Rate of inactivation of APC (per cell division)	μ	10^-9
Division rate of WT SCs (per year)	r₁	(18,365)
Relative fitness of APC +/- cells	$F_{2} = F_{A P C^{+, / -}}$	1.6
Relative fitness of APC-/- cells	$F_{3} = F_{A P C^{-, / -}}$	3.76
Relative fitness of KRAS+ cells	$F_{4} = F_{K R A S}$	3.54
Division rate of APC-/- crypts (per year)	$γ_{3}$	0.2
Division rate of KRAS+ crypts (per year)	$γ_{4}$	0.07

Appendix 1—table 3

Parameters obtained from the fitting procedures in Appendix 1—figure 10 and Appendix 1—figure 11.

Parameter set #	$N_{i}$ (crypts)	$Δ T_{i}$ (yrs)	r₁(yrs-1)	$K_{A}$	$K_{R}$	Comment
1	10²	4.79	204	1000	17	best fit
2	10²	4.79	198	1039	171	2nd best fit
3	10⁵	11.97	300	607	10	best fit
4	10⁵	11.97	288	631	68	2nd best fit

Additional files

Transparent reporting form: https://cdn.elifesciences.org/articles/71953/elife-71953-transrepform1-v1.pdf
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Source code 1 Mathematica code for Figures 2 and 3.: https://cdn.elifesciences.org/articles/71953/elife-71953-code1-v1.zip
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Source code 2 Mathematica code for Figure 4.: https://cdn.elifesciences.org/articles/71953/elife-71953-code2-v1.zip
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Source code 3 A Fortran code for stochastic simulations (e.g. Figure 4).: https://cdn.elifesciences.org/articles/71953/elife-71953-code3-v1.zip
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A schematic illustrating the mathematical model.

Fitting the nonlinear model to late adenoma incidence data.

Figure 2—source data 1

Figure 2—source data 2

Pathways to adenoma.

Figure 3—source data 1

The effect of aspirin on the advanced adenoma incidence curve.

Figure 4—source data 1

Figure 4—source data 2

Three pathways to adenoma.

Model fitting to the incidence data.

Fitting the linear model.

Nonlinear model fitting to the incidence data (fixing the crypt carrying capacity, varying crypt fission rates).

Nonlinear model fitting to the incidence data (fixing crypt fission rates, varying the crypt carrying capacity).

Comparing deterministic solutions (system (LABEL:n1non-23, 12), solid lines) with the stochastic Gillespie simulations (symbols of the same color).

Fitting the adenoma incidence curve to the nonlinear model with an explicit expansion phase.

Fitting the adenoma incidence curve to the nonlinear model with an explicit expansion phase.

Pathways to adenoma: Fitting the incidence curve to the nonlinear model with an explicit expansion phase.

Same as Appendix 1—figure 10, except expansion to 105 crypts is assumed.

Compartment dynamics for the best fitting models with KA=KR<math id="inf184"><mrow><msub><mi>K</mi><mi>A</mi></msub><mo>=</mo><msub><mi>K</mi><mi>R</mi></msub></mrow></math>.

Same as Appendix 1—figure 12(b-c), but without the restriction KA=KR<math id="inf186"><mrow><msub><mi>K</mi><mi>A</mi></msub><mo>=</mo><msub><mi>K</mi><mi>R</mi></msub></mrow></math>.

extends the results of main text Figure 4(a) and shows that relative adenoma risk predicted for each decade of treatment is very similar for all parameter sets (#1-4, Table Appendix 1—table 3).

Relative incidence of advanced adenoma, where (a) parameter set #1 (Table Appendix 1—table 3) and (b) parameter sets #2 is used.

Predicted advanced adenoma incidence in the absence of aspirin treatment (thin black lines are the fitted curves and black dots are incidence data); under aspirin treatment where the drug affects types 2–6 (yellow lines), under aspirin treatment where the drug affects type 6 only (blue lines).

A comparison of predicted relative advanced adenoma risk under the assumption that cell types 2–6 (yellow bars) and only type 6 cells (blue bars) are affected by aspirin.

A comparison of predicted relative advanced adenoma risk under the assumption that aspirin treatment lasts 10 years (yellow) or 5 years (blue).

Aspirin doses in xenograft experiments, the equivalent human dose and the resulting changes in kinetic rates.

Enumeration of the different genotypes.

Parameters, notations, and their values.

Parameters obtained from the fitting procedures in Appendix 1—figure 10 and Appendix 1—figure 11.

Transparent reporting form

Source code 1

Source code 2

Source code 3

Same as Appendix 1—figure 10, except expansion to 10⁵ crypts is assumed.

Compartment dynamics for the best fitting models with $K_{A} = K_{R}$ .

Same as Appendix 1—figure 12(b-c), but without the restriction $K_{A} = K_{R}$ .