Cancer arises from the accumulation of somatic genomic and epigenomic alterations throughout an organism's life. These alterations, such as mutations or epigenetic marks, occur during DNA replication at a low rate. Multi-step models of cancer have been employed to understand age-incidence relationships, where progression through multiple steps ultimately leads to malignancy. The number of steps isn't necessarily equivalent to the number of mutations; multiple rapid changes can appear as one step, and a single impactful mutation might manifest as multiple steps in the age-incidence curve. Long-lived species like elephants have evolved mechanisms to suppress cancer, such as multiple copies of the p53 tumor suppressor gene. Conversely, companion animals, with their improved living conditions, have significantly increased lifespans, leading to a higher prevalence of cancer. This raises the question of how short-term responses to increased longevity (due to environmental factors) differ from long-term evolutionary adaptations that reduce cancer risk. This study uses a mathematical model to address this.

The paper reviews existing multi-step models of cancer, highlighting their use in analyzing age- and sex-dependent cancer incidence. It cites examples of long-lived species, such as elephants, which have evolved mechanisms like multiple copies of the p53 gene to suppress cancer. The increased longevity and cancer rates in companion animals, particularly dogs, are presented as a contrasting case, where environmental improvements lead to extended lifespan and increased cancer prevalence. The existing literature on allometric relationships between longevity, body size, cell divisions, and cancer risk is also reviewed, laying the groundwork for the authors' mathematical modeling approach.

The authors develop a multi-step cancer model where an individual progresses through 'n' steps before developing cancer. The rate of transition between steps (cᵢ) is assumed to be proportional to the genomic error rate (x), representing the accumulation of somatic mutations or epigenetic changes. The model incorporates a constant non-cancerous mortality rate (μ). The differential equations describing the probability of being in each step (Pᵢ(a) at age 'a') are formulated and solved, particularly for the case where all transition rates are equal (cᵢ = kx, where k is a constant). The model calculates the total mortality due to cancer (M_C), mean longevity (T), and age-dependent cancer mortality (g_c(a)). The authors also incorporate an evolutionary component, considering the genomic error rate (x) as a parameter subject to natural selection. The fitness (F(x)) is defined as the expected number of offspring, considering both survival (dependent on x) and a cost function for reducing the genomic error rate (f₀/x^q). The evolutionarily stable strategy (ESS) is determined by maximizing the fitness function. The model analyzes both the direct effect of environmental improvement (reduction in μ) and the indirect effect through the evolution of x.

The model reveals that improving the environment (reducing μ) directly enhances the total cancer mortality (M_C) and increases the relative fraction of cancer mortality at each age, even without changes in the genomic error rate (x). This explains the observed increase in cancer in companion animals. The mean longevity (T) increases with reduced μ. However, over many generations, the genomic error rate (x) evolves to a lower value (x*), minimizing cancer risk. This evolutionary adaptation reduces the total cancer mortality (M_C) and age-specific cancer mortality (g_c(a)). The authors find that the direct effect of environmental improvement on increasing cancer mortality is stronger than the long-term evolutionary effect to reduce it. The magnitude of the evolutionary response depends on the cost function's shape, influencing how x* adjusts to changes in μ. The effect of environmental improvement is stronger for solid cancers (larger 'n') than leukemia (smaller 'n'). The model shows that accelerating transition rates in later stages of cancer progression have a similar effect to reducing the step number 'n'. When fertility is age-dependent, as in dogs, the evolutionary response to environmental improvement is smaller, resulting in less reduction of cancer mortality.

The study's findings directly address the observed paradox of increased cancer in companion animals with longer lifespans. The model demonstrates that short-term environmental changes have a dominant effect on cancer incidence, while long-term evolutionary adaptations play a mitigating but less impactful role. This highlights the importance of considering both direct environmental effects and long-term evolutionary pressures when evaluating cancer risk. The study's multi-step model provides a useful framework for understanding the complex interplay between environment, longevity, and cancer risk, applicable not only to companion animals but potentially also to human populations.

This study successfully demonstrates the complex interaction of environmental factors and evolution in shaping cancer risk. While environmental improvements leading to increased longevity directly increase cancer prevalence in the short term, long-term evolutionary adaptations can partially mitigate this increase. Future research should focus on more refined models incorporating specific factors like diet, lifestyle, and breed-specific genetic predispositions to provide a more nuanced understanding of cancer risk in companion animals and its implications for human health.

The model simplifies several aspects of cancer biology and evolution. The assumption of constant non-cancerous mortality might not hold in reality, and the cost function for reducing genomic error rates could be more complex. Additionally, the model does not explicitly account for factors like breed-specific genetic differences or the influence of specific environmental carcinogens, which could further refine the predictions.