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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Evaluation Summary:
This work develops a multistage/component mathematical model to analyze advanced colorectal adenomas and the impact that aspirin therapy has on adenoma formation rates. This study will be interesting to the cancer evolution community and in particular those interested in colorectal cancer incidence. While the model is mainly focused on aspirin chemoprevention, the model could be adapted to test other putative preventative agents, and thus could have a broad impact.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #2 and Reviewer 3 agreed to share their names with the authors.)
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Abstract
Aspirin intake has been shown to lead to significant protection against colorectal cancer, for example with an up to twofold reduction in colorectal adenoma incidence rates at higher doses. The mechanisms contributing to protection are not yet fully understood. While aspirin is an antiinflammatory drug and can thus influence the tumor microenvironment, in vitro and in vivo experiments have recently shown that aspirin can also have a direct effect on cellular kinetics and fitness. It reduces the rate of tumor cell division and increases the rate of cell death. The question arises whether such changes in cellular fitness are sufficient to significantly contribute to the epidemiologically observed protection. To investigate this, we constructed a class of mathematical models of in vivo evolution of advanced adenomas, parameterized it with available estimates, and calculated population level incidence. Fitting the predictions to age incidence data revealed that only a model that included colonic crypt competition can account for the observed ageincidence curve. This model was then used to predict modified incidence patterns if cellular kinetics were altered as a result of aspirin treatment. We found that changes in cellular fitness that were within the experimentally observed ranges could reduce advanced adenoma incidence by a sufficient amount to account for age incidence data in aspirintreated patient cohorts. While the mechanisms that contribute to the protective effect of aspirin are likely complex and multifactorial, our study demonstrates that direct aspirininduced changes of tumor cell fitness can significantly contribute to epidemiologically observed reduced incidence patterns.
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Author Response
Reviewer #1 (Public Review):
Wang et al. use a multistage mathematical model to analyze incidence of advanced colorectal adenomas and to investigate if the protective effect of aspirin on adenoma incidence could be due to its effect on cellular fitness. The advanced adenoma incidence model is similar to the recently published Paterson et al. PNAS 2020 model, in that it includes the first three of the five steps to colorectal cancer from the Paterson et al. model. Interestingly, Wang et al. find that adding crypt competition to the previous model is needed to account for the observed advanced adenoma incidence curve. This is a nice contribution to the study of colorectal tumor incidence. The authors are also able to confirm previous findings that the order of mutations of the way to an advanced colorectal adenoma is …
Author Response
Reviewer #1 (Public Review):
Wang et al. use a multistage mathematical model to analyze incidence of advanced colorectal adenomas and to investigate if the protective effect of aspirin on adenoma incidence could be due to its effect on cellular fitness. The advanced adenoma incidence model is similar to the recently published Paterson et al. PNAS 2020 model, in that it includes the first three of the five steps to colorectal cancer from the Paterson et al. model. Interestingly, Wang et al. find that adding crypt competition to the previous model is needed to account for the observed advanced adenoma incidence curve. This is a nice contribution to the study of colorectal tumor incidence. The authors are also able to confirm previous findings that the order of mutations of the way to an advanced colorectal adenoma is determined by the crypt fission rates, and not the mutation rates.
In the second part of the paper, Wang et al. use the advanced adenoma incidence model to study the effects of aspirin on reduction of adenoma incidence. This part of the paper would benefit from more precise explanation of the assumptions used, especially how exactly is the effect of aspirin implemented at the level of individual crypts and/or crypt cells. Adding these explanations to the main text would significantly increase the clarity of the second half of the manuscript.
We agree with this comment and have now revised the main text significantly to describe in more detail how aspirin intervention was implemented in the model at the level of individual crypts (intracrypt dynamics) and crypt fission/turnover (intercrypt dynamics). See heavily revised text on pages 1315 of the main text (and also the new Appendix 1 Section 6 for furrher detals), as well as a new Fig 4 of the main text for the model structure. The main text now also includes Table 1 showing aspirin dosage in mice, the human equivalent, and the resulting fold differences in the division and death rates of cells that were previously measured by us and implemented in this study. Further, model assumptions and parameter definitions were explained in a more consistent way in Appendix 1 – Section 2.
Reviewer #2 (Public Review):
This mathematical model of the development of advanced colorectal adenoma (polyps) stands out in terms of its biological realism and inclusion of quantitative information related to stem cell turnover and crypt biology. Early detection and prevention of colorectal cancer, be it via endoscopic screening, stool DNA tests, or use of COX2 inhibitors (Aspirin, Sulindac etc), continues to be an important public health goal, especially in view of stressed health care services under COVID and a rising incidence of CRC among younger individuals in the US. The findings are interesting and motivate further experiments to generate data that inform the model and predictions for effective chemoprevention of colorectal cancer using Aspirin.
The authors recognize that only about ~40% of nonhypermutated CRC are KRAS+. Although KRAS oncoactivation may well occur after APC/ crypts have begun to proliferate (after a 36 transition in the model) forming an advanced adenoma, all nonhypermutated CRCs should be carrying detectable KRAS mutations  according to this model. However, Brenner et al did not ascertain KRAS status of their adenomas and cancers. It appears that one way of finessing this problem is to assume that the prevalences of advanced adenomas used to fit this multipathway model can be represented effectively by fractions of those observed clinically. Did the authors consider/test such an adjustment? If not, why not?
Thank you for this comment, these important issues indeed need clarification. Instead of KRAS+, other mutations can be present in advanced adenomas, for example BRAF. In terms of the model, the identity of the mutations is not the defining issue, but the types of mutations are. One of the mutations is assumed to be the inactivation of a tumor suppressor gene (i.e. lossoffunction, most likely APC/), while the other mutation is assumed to be gain of function (which can be KRAS+ or alternative gain of function mutations). As long as these assumptions are fulfilled, the predicted dynamics would remain the same. It is unlikely that adenomas are characterized by the presence of two inactivated tumor suppressor genes, which would significantly reduce the rate of their emergence in the model. Further tumor suppressor gene inactivations tend to occur at more advanced stages of the disease. We have now explained this in the revised version of the manuscript by adding the following:
“While the model assumptions about the pathways to adenoma formation are clearly defined in our model, it is important to point out that there are uncertainties in those assumptions, and that there is heterogeneity in the types of mutations that can lead to colorectal carcinogenesis. For example, it has been reported that among nonhypermutated colorectal tumors, KRAS was mutated in only about 43% of patient samples [39], indicating the importance of a variety of evolutionary pathways. Our model, however, does not depend on the identity of particular mutations, but assumes the occurrence of mutation types, which are the inactivation of a tumor suppressor gene (which is a lossoffunction mutation, e.g. APC/), and a gainoffunction mutation, which can be in KRAS or an alternative gene, such as BRAF [4043]. Our model predictions hold as long as the evolutionary pathway to advanced adenomas involves these two types of mutational events, regardless of their identity.”
To model multiple pathways efficiently the authors resort to a deterministic crypt proliferation model, forgoing a stochastic treatment. Thus, in the deterministic model, there will always be a nonzero fraction of type 6 crypts at any given time > 0. The paper does not reveal how large type 3 (APC/, no KRAS) crypt clones are on average when an advanced adenoma (type 6 crypt) occurs in the colon. A discussion of how exactly the advanced adenoma is defined (1 type 6 crypt or more?) and what observation thresholds were applied to model the endoscopic observations by Brenner et al, would be greatly helpful. Typically, in clinical practice, an advanced adenoma is > 1cm in caliper size, or shows signs of dysplasia.
We thank the referee for these comments, as they allowed further refinements of the model that we found very valuable. The model that is presented in Appendix 1 (system (1923), see also the equations in the main text) is a model that describes the expected behavior of the system. We used the meanfield approximation for the probability P(t) that at least one crypt of type 6 is present by time t. Since this is an approximation, in the new version of the paper (see new section 5.1 of Appendix 1) we showed that it works very well by checking it against fully stochastic Gillespie simulations. The ODE model (which is computationally very inexpensive) was then used to fit the advanced adenoma incidence curve.
In the new version of the manuscript we have also taken fuller advantage of Gillespie simulations: once a set of the best fitting parameters was determined, we studied the effect of aspirin in reducing the adenoma risk by using stochastic simulations. To this end, we have also provided more details of the model behavior. The new figures 12 and 13 of Appendix 1 show the probability distributions of the populations of the crypts of different sizes at the time when the first crypt of type 6 is created.
Finally, in the new version of the model we included a growth phase of type 6 crypts, which is outlined in the model description in the Materials and Methods section of the main text, and also discussed throughout the main text. This is further described in detail in the new Section 5.2 of Appendix 1, as well as the new figure 7 (Appendix 1) that introduces this aspect of modeling, and the new figures 818 of Appendix 1, which all include the expansion phase of type 6 crypts. The simulations presented in the main text now also include the expansion phase. Because of uncertainties existing in the literature in terms of the exact numbers of crypts at detection, we explored a large range of threshold sizes to which the population of type 6 crypts expands. It is very encouraging that the effects of aspirin are very consistent throughout this range.
We would like to note that initially we did not realize the extent of the difference the expansion phase could make for the numerical values of the fitted parameters. After implementing this part of the model however we became convinced that this part of the dynamics which often remains left out of the model (e.g. in [32]) has to be included explicitly for better accuracy of predictions. All of the results in the main text are now updated to include the expansion phase of the modified type 6 crypts.
Related to the above, the assumption of zero crypt death/fusion appears unrealistic given the findings by Baker AM, et al. Gut 2019;0:18. Furthermore, given the small number of SCs in human colonic crypts, the sporadic loss of crypts cannot be zero and normal crypt fissions are clearly compensated for by crypt loss (or fusion?) in normal colon. Of further note, a recent modeling paper by Birtwell et al in (Evolutionary Applications 13, 17711783 (2020) also argues for crypt turnover as an important aspect of the metapopulation and stem cell dynamics in intestinal crypts and other tissue structures in multicellular bodies.
Spontaneous crypt loss is part of the model that is developed here. It is incorporated in the crypt death rate, δ, in the system of equations presented in the main text (as well as system (1923) in Appendix 1). All the simulations in the main text (e.g. Figs 24) correspond to a nonzero death rate. We have also researched the effect of crypt death rate, by performing simulations with both zero and nonzero values of d (see Appendix 1 Figure 6). Including a nonzero death rate was not found to result in qualitatively different conclusions when fitting the model to the advanced adenoma incidence data. Since the new version of the model also takes into account the expansion phase of type 6 crypts, a nonzero crypt death rate associated with type 6 crypts is also included. It plays a role in the stochastic model as sometimes, a newly generated crypt may spontaneously disappear. It is also assumed to be affected by aspirin, when we consider the intercrypt dynamics.
In the new version of the manuscript we added a discussion of these points including the published research that the referee pointed out (thank you!).
Finally, it is not clear whether the model recapitulated the relatively short Aspirin exposures used in trials. In other words, were the equations solved for the situation encountered in trials where the interventions (Aspirin) only last for a few years (typically < 10) unless longterm users are included?
We thank the referee for bringing up this aspect of the problem. In the new version of the manuscript we have included a more detailed study of the timing of aspirin administration (see e.g. the schematic in Fig4(b)). The relatively strong effect of aspirin on the advanced adenoma risk (see e.g. fig.4 of the main text) was obtained when the patients were assumed to receive aspirin for a 10year period prior to the risk assessment. Three different doses were tested, which corresponded to the three doses used in our previous experiments, for which the kinetic parameters were measured. These mouse doses were converted to the human equivalent, please see the new Table 1 of the main text and the surrounding explanations (as well as new section 6 of Appendix 1), where the conversion is explained and the factors modifying cell division and death rates are calculated. The highest dose that was used for the modeling was the strongest dose in our previous experiments [19,20], which roughly translates to the second strongest dose (614 pills a week) in the study by [14]. The effect of aspirin decreases with a decrease in the dose and the duration of aspirin treatment. In particular, the role of the duration is examined in Appendix 1 – Figure 18. A number of different comparisons are now presented in the paper (see new Figures 1518 on Appendix 1) that investigate different assumptions of aspirin modeling, as well as the age at which the drug was taken, the followup time, etc.
Reviewer #3 (Public Review):
The work by Wang et al. demonstrates a computational approach to modeling populationlevel tumor incidence using a cryptlevel algorithm to predict modified incidence based on differential effects on cellular dynamics. While the manuscript makes ultimate conclusions about the impact of aspirin chemoprevention and how this can be parametrized for the model, the work demonstrates a potential in silico approach towards testing putative preventive agents as well as potential factors that may accelerate tumorigenesis by accounting for epithelial cell growth dynamics under different mutation conditions. The base model operates on many relatively basic and broadsweeping assumptions about colorectal tumorigenesis that will need to be considered for individual downstream applications.
An obvious limitation of chemoprevention studies in humans is the long timescale necessary to perform such experiments/trials and the relatively large subject numbers required to measure incremental effect sizes on relatively low incidence outcomes. Similarly, the availability of appropriate and broadly translatable in vitro (e.g. cell lines that recapitulate precancers or even 'normal' tissue within which to model protective vs. 'anticancer' effects) and preclinical models is limited. The strength of this approach towards being able to computationally model crypt dynamics using our best understanding of intestinal cellular proliferation is an important step towards a tool for identification and testing of putative chemoprevention agents which may assist basic and translational researchers as they consider interventions towards intercepting colorectal cancer. The authors specifically understand the constraints and limitations of their modeling approach and transparently discuss the assumptions that feed into the model. Of course, while the manuscript penultimately focuses on the effects of aspirin, the real strength is the identification of a computational model that may similarly behave to biologically relevant crypt dynamics that is able to not only consider different inputs for accelerating effects, but can also model different effects caused by preventive measures and begin to disentangle the underlying biology that is mathematically likely to be occurring in vivo. These assumptions however limit the generalizability of the individual findings as they pertain to our broader understanding of tumorigenesis and aspirin chemoprevention. As an oversimplification of this critique, the authors focus on APC/KRAS mutations, which to be fair are the most common CRC/adenoma mutations, but may not be critical to understanding aspirin's chemopreventive mechanisms, where APC/KRAS mutations have not been shown to predict the variable protective response to aspirin observed in humans. Similarly, the effect estimates on cell dynamics are derived from in vitro and preclinical work using aspirin that use high, albeit appropriately acknowledged as 'physiologically attainable', doses that do not quite represent circulating doses likely achieved through regular use of lowdose aspirin. Nonetheless, using aspirin as a 'model preventative exposure' is suitable in this setting as a way to demonstrate the adenoma models' responsiveness to this type of parameter. Appropriately, the authors do not overinterpret their findings in light of these assumptions, but readers should be similarly cautious to avoid overinterpreting the conclusions.
Thank you very much for the positive assessment and the constructive input, which we have now incorporated into the revised version of the manuscript. As we point out in the revised text, the predicted dynamics and the effect of aspirin do not depend on the particular identity of the mutations involved. We do frame the model of advanced adenoma evolution in terms of APC/ and KRAS+ mutations, as these are often documented in the literature. Alternative mutations, however, can characterize this process, as shown by tumors that lack KRAS+ mutations and carry alternative mutations, e.g. in BRAF. The revised manuscript now points out that instead of specific identities of mutants, the model behavior depends on “types” of mutations. Hence, the model assumes that the evolution to advanced adenoma involves the inactivation of a tumor suppressor gene (e.g. APC/) and the activation of an oncogene, which can be KRAS or an alternative. Aspirin is assumed to change the kinetics of these evolutionary processes, and our predictions about incidence hold as long as the assumed evolutionary process involves the inactivation of a tumor suppressor gene and the activation of an oncogene.
We now also discuss in more detail the relationship between the in vitro and in vivo data that were collected in our experiments and the processes that occur in the colorectal tissue in humans that develop disease. We focus this discussion around the hierarchical organization in the colorectal tissue, the cell of origins of colorectal carcinogenesis, the dynamics of crypt fission and crypt death, and also discuss in more detail the aspirin dosing used in our experiments in relation to physiological doses in humans. We draw attention to limitations of our work, including alternative processes that can determine the response to aspirin in the colorectal tissue that have not been included in the model (due to lack of quantitative information), such as the effect of the microbiome composition. Further details are given below.

Evaluation Summary:
This work develops a multistage/component mathematical model to analyze advanced colorectal adenomas and the impact that aspirin therapy has on adenoma formation rates. This study will be interesting to the cancer evolution community and in particular those interested in colorectal cancer incidence. While the model is mainly focused on aspirin chemoprevention, the model could be adapted to test other putative preventative agents, and thus could have a broad impact.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #2 and Reviewer 3 agreed to share their names with the authors.)

Reviewer #1 (Public Review):
Wang et al. use a multistage mathematical model to analyze incidence of advanced colorectal adenomas and to investigate if the protective effect of aspirin on adenoma incidence could be due to its effect on cellular fitness. The advanced adenoma incidence model is similar to the recently published Paterson et al. PNAS 2020 model, in that it includes the first three of the five steps to colorectal cancer from the Paterson et al. model. Interestingly, Wang et al. find that adding crypt competition to the previous model is needed to account for the observed advanced adenoma incidence curve. This is a nice contribution to the study of colorectal tumor incidence. The authors are also able to confirm previous findings that the order of mutations of the way to an advanced colorectal adenoma is determined by the …
Reviewer #1 (Public Review):
Wang et al. use a multistage mathematical model to analyze incidence of advanced colorectal adenomas and to investigate if the protective effect of aspirin on adenoma incidence could be due to its effect on cellular fitness. The advanced adenoma incidence model is similar to the recently published Paterson et al. PNAS 2020 model, in that it includes the first three of the five steps to colorectal cancer from the Paterson et al. model. Interestingly, Wang et al. find that adding crypt competition to the previous model is needed to account for the observed advanced adenoma incidence curve. This is a nice contribution to the study of colorectal tumor incidence. The authors are also able to confirm previous findings that the order of mutations of the way to an advanced colorectal adenoma is determined by the crypt fission rates, and not the mutation rates.
In the second part of the paper, Wang et al. use the advanced adenoma incidence model to study the effects of aspirin on reduction of adenoma incidence. This part of the paper would benefit from more precise explanation of the assumptions used, especially how exactly is the effect of aspirin implemented at the level of individual crypts and/or crypt cells. Adding these explanations to the main text would significantly increase the clarity of the second half of the manuscript.

Reviewer #2 (Public Review):
This mathematical model of the development of advanced colorectal adenoma (polyps) stands out in terms of its biological realism and inclusion of quantitative information related to stem cell turnover and crypt biology. Early detection and prevention of colorectal cancer, be it via endoscopic screening, stool DNA tests, or use of COX2 inhibitors (Aspirin, Sulindac etc), continues to be an important public health goal, especially in view of stressed health care services under COVID and a rising incidence of CRC among younger individuals in the US. The findings are interesting and motivate further experiments to generate data that inform the model and predictions for effective chemoprevention of colorectal cancer using Aspirin.
The authors recognize that only about ~40% of nonhypermutated CRC are KRAS+. …
Reviewer #2 (Public Review):
This mathematical model of the development of advanced colorectal adenoma (polyps) stands out in terms of its biological realism and inclusion of quantitative information related to stem cell turnover and crypt biology. Early detection and prevention of colorectal cancer, be it via endoscopic screening, stool DNA tests, or use of COX2 inhibitors (Aspirin, Sulindac etc), continues to be an important public health goal, especially in view of stressed health care services under COVID and a rising incidence of CRC among younger individuals in the US. The findings are interesting and motivate further experiments to generate data that inform the model and predictions for effective chemoprevention of colorectal cancer using Aspirin.
The authors recognize that only about ~40% of nonhypermutated CRC are KRAS+. Although KRAS oncoactivation may well occur after APC/ crypts have begun to proliferate (after a 36 transition in the model) forming an advanced adenoma, all nonhypermutated CRCs should be carrying detectable KRAS mutations  according to this model. However, Brenner et al did not ascertain KRAS status of their adenomas and cancers. It appears that one way of finessing this problem is to assume that the prevalences of advanced adenomas used to fit this multipathway model can be represented effectively by fractions of those observed clinically. Did the authors consider/test such an adjustment? If not, why not?
To model multiple pathways efficiently the authors resort to a deterministic crypt proliferation model, forgoing a stochastic treatment. Thus, in the deterministic model, there will always be a nonzero fraction of type 6 crypts at any given time > 0. The paper does not reveal how large type 3 (APC/, no KRAS) crypt clones are on average when an advanced adenoma (type 6 crypt) occurs in the colon. A discussion of how exactly the advanced adenoma is defined (1 type 6 crypt or more?) and what observation thresholds were applied to model the endoscopic observations by Brenner et al, would be greatly helpful. Typically, in clinical practice, an advanced adenoma is > 1cm in caliper size, or shows signs of dysplasia.
Related to the above, the assumption of zero crypt death/fusion appears unrealistic given the findings by Baker AM, et al. Gut 2019;0:18. Furthermore, given the small number of SCs in human colonic crypts, the sporadic loss of crypts cannot be zero and normal crypt fissions are clearly compensated for by crypt loss (or fusion?) in normal colon. Of further note, a recent modeling paper by Birtwell et al in (Evolutionary Applications 13, 17711783 (2020) also argues for crypt turnover as an important aspect of the metapopulation and stem cell dynamics in intestinal crypts and other tissue structures in multicellular bodies.
Finally, it is not clear whether the model recapitulated the relatively short Aspirin exposures used in trials. In other words, were the equations solved for the situation encountered in trials where the interventions (Aspirin) only last for a few years (typically < 10) unless longterm users are included?

Reviewer #3 (Public Review):
The work by Wang et al. demonstrates a computational approach to modeling populationlevel tumor incidence using a cryptlevel algorithm to predict modified incidence based on differential effects on cellular dynamics. While the manuscript makes ultimate conclusions about the impact of aspirin chemoprevention and how this can be parametrized for the model, the work demonstrates a potential in silico approach towards testing putative preventive agents as well as potential factors that may accelerate tumorigenesis by accounting for epithelial cell growth dynamics under different mutation conditions. The base model operates on many relatively basic and broadsweeping assumptions about colorectal tumorigenesis that will need to be considered for individual downstream applications.
An obvious limitation of …
Reviewer #3 (Public Review):
The work by Wang et al. demonstrates a computational approach to modeling populationlevel tumor incidence using a cryptlevel algorithm to predict modified incidence based on differential effects on cellular dynamics. While the manuscript makes ultimate conclusions about the impact of aspirin chemoprevention and how this can be parametrized for the model, the work demonstrates a potential in silico approach towards testing putative preventive agents as well as potential factors that may accelerate tumorigenesis by accounting for epithelial cell growth dynamics under different mutation conditions. The base model operates on many relatively basic and broadsweeping assumptions about colorectal tumorigenesis that will need to be considered for individual downstream applications.
An obvious limitation of chemoprevention studies in humans is the long timescale necessary to perform such experiments/trials and the relatively large subject numbers required to measure incremental effect sizes on relatively low incidence outcomes. Similarly, the availability of appropriate and broadly translatable in vitro (e.g. cell lines that recapitulate precancers or even 'normal' tissue within which to model protective vs. 'anticancer' effects) and preclinical models is limited. The strength of this approach towards being able to computationally model crypt dynamics using our best understanding of intestinal cellular proliferation is an important step towards a tool for identification and testing of putative chemoprevention agents which may assist basic and translational researchers as they consider interventions towards intercepting colorectal cancer. The authors specifically understand the constraints and limitations of their modeling approach and transparently discuss the assumptions that feed into the model. Of course, while the manuscript penultimately focuses on the effects of aspirin, the real strength is the identification of a computational model that may similarly behave to biologically relevant crypt dynamics that is able to not only consider different inputs for accelerating effects, but can also model different effects caused by preventive measures and begin to disentangle the underlying biology that is mathematically likely to be occurring in vivo. These assumptions however limit the generalizability of the individual findings as they pertain to our broader understanding of tumorigenesis and aspirin chemoprevention. As an oversimplification of this critique, the authors focus on APC/KRAS mutations, which to be fair are the most common CRC/adenoma mutations, but may not be critical to understanding aspirin's chemopreventive mechanisms, where APC/KRAS mutations have not been shown to predict the variable protective response to aspirin observed in humans. Similarly, the effect estimates on cell dynamics are derived from in vitro and preclinical work using aspirin that use high, albeit appropriately acknowledged as 'physiologically attainable', doses that do not quite represent circulating doses likely achieved through regular use of lowdose aspirin. Nonetheless, using aspirin as a 'model preventative exposure' is suitable in this setting as a way to demonstrate the adenoma models' responsiveness to this type of parameter. Appropriately, the authors do not overinterpret their findings in light of these assumptions, but readers should be similarly cautious to avoid overinterpreting the conclusions.
