Enhanced risk of cancer in companion animals as a response to the longevity

The study addresses how improved environmental conditions that extend lifespan alter cancer risk in the short term versus after evolutionary adaptation. Cancer progression is often modeled as an n-step process in which individuals transition through states until malignancy. Long-lived taxa such as elephants appear to have evolved enhanced tumor suppression (e.g., expanded TP53 and LIF6 activation), consistent with selection on error checking and surveillance in lineages with long life histories. In companion animals, especially dogs, longevity has increased due to better food, shelter, hygiene, and healthcare, and cancer has become a leading mortality cause in older ages. This suggests that immediate improvements reducing noncancer mortality can elevate the fraction of deaths due to cancer without any change in intrinsic cancer processes, whereas long-term evolution may lower replication error rates and cancer risk. The paper aims to formalize and clarify these direct (ecological) versus indirect (evolutionary) effects using a parsimonious multi-step model.

The work builds on classic multistage carcinogenesis (Armitage-Doll) linking age-incidence to step number n and recognizes that observed steps may not map one-to-one to mutations, due to variable timescales and processes (e.g., single mutations appearing as multiple steps in incidence curves). Comparative biology suggests evolved cancer suppression in long-lived or large-bodied species: elephants with multiple TP53 copies and LIF6-mediated apoptosis; capybaras potentially suppressing fast-growing cells. Prior allometric analyses have modeled leukemia risk across mammals. These lines motivate treating replication error rates and surveillance as evolvable traits shaped by life-history trade-offs.

Model structure: An n-step (n>1) multistate model. Individuals start at step 0 at birth; each event increments the step by 1; reaching step n yields malignant cancer. Let P_i(a) be the probability of being in state i at age a. Transition rates c_i from i to i+1 are proportional to the heritable genomic replication error rate x: c_i = k_i x for i = 0,...,n−1. Differential equations: dP_0/da = −c_0 P_0; for i=1,...,n−1, dP_i/da = c_{i−1} P_{i−1} − c_i P_i; and dP_n/da = c_{n−1} P_{n−1}, with initial condition P_0(0)=1, others 0. For the special case k_i = k (all equal), solutions include P_i(a) = ((k x a)^i / i!) e^{−k x a} for i=0,...,n−1, and P_n(a) = (1/Γ(n)) γ(n, k x a) (incomplete gamma). Demography: Noncancer mortality is modeled as a constant hazard u (Type II survivorship). Survivorship to age a is l(a) = (1−P_n(a)) e^{−u a}. Outcomes: Total noncancer mortality fraction M_N = ∫_0^∞ u (1−P_n(a)) e^{−u a} da, and total cancer mortality M_C = 1 − M_N = ∏_{j=0}^{n−1} (k_j x)/(u + k_j x). Mean longevity T = ∫_0^∞ l(a) da = (1/u) (1 − M_C). Age-specific instantaneous cancer mortality g_c(a) = (∂P_n/∂a) / (1 − P_n(a)); when k_i = k, g_c(a) = [k x (k x a)^{n−1} e^{−k x a}] / [(n−1)! − γ(n, k x a)]. Age-specific noncancer mortality is g_N(a)=u; the relative fraction due to cancer is g_c(a)/(g_c(a)+u). Evolution: Fitness is lifetime reproductive output F(x) = ∫_0^∞ l(a) m(a,x) da. Error reduction incurs a fecundity cost parameterized as m(a,x) = m_0(a) − f_0 / x^q, with typical analyses assuming age-independent baseline fertility m_0(a)=m_0. An evolutionarily stable error rate x* maximizes F(x), balancing reduced cancer risk (lower x slows progression) against reproduction costs (lower x reduces fecundity). Comparative statics explore direct effects of u on outcomes holding x fixed versus indirect evolutionary effects via x*(u). Extensions analyzed in appendices include: varying step number n (e.g., smaller for leukemias), accelerating transition rates across steps, and age-dependent fertility patterns mimicking dogs (fertile roughly ages 2–6, zero <1 and >10), and different cost exponents q in f_0/x^q.

- Direct effect of improved environment (lower u): As u decreases, total cancer mortality M_C increases (approaches 1 as u→0), while age-specific cancer hazard g_c(a) is unchanged (independent of u). The fraction of age-specific mortality due to cancer g_c(a)/(g_c(a)+u) increases with age and with improved environment (smaller u). Mean longevity T increases as u decreases despite cancer. - Example quantitative results (baseline parameters k=1.8, x=0.1, n=5, m_0=10, f_0=0.044, q=1.1): In the original environment u=0.2, M_C=0.0238 (mean longevity about 5 years ignoring units). If u is suddenly reduced to 0.0667 without evolutionary change in x, M_C rises to 0.2068 (≈8.7× increase) and g_c(a) remains the same. - Evolutionary (indirect) effect over many generations: Lower u selects for reduced replication error rate x*. For u=0.0667, x* evolves from 0.1 to 0.0507, lowering cancer risk. After adaptation, M_C drops from 0.2068 to 0.0644, and age-specific g_c(a) is reduced at all ages compared to pre-adaptation. - Net effect: Even after evolutionary adjustment at lower u, total cancer mortality remains higher than in the original high-u environment (0.0644 versus 0.0238), indicating the direct effect dominates the indirect effect in this parameterization. - Sensitivity to cost exponent q: Smaller q (cheaper error reduction) yields stronger evolutionary reduction in x* and larger decreases in M_C after adaptation (e.g., for u=0.0667, x*=0.0424 and M_C=0.0433 when q=0.5; x*=0.0599 for q=2 gives weaker reduction). Nonetheless, post-adaptation M_C remains above original M_C at high u. - Dependence on step number n and step acceleration: Smaller n (e.g., leukemia-like) yields weaker direct and indirect responses to improved environment than larger n (solid tumors), implying a larger proportional increase in solid tumors with longevity extension. Accelerating step rates act similarly to reducing effective n. - Age-dependent fertility: When reproduction occurs mainly at younger ages (e.g., dogs), selection to reduce late-life cancer is weak; evolutionary reductions in x* and M_C are small, so elevated cancer burden persists even after adaptation.

The model demonstrates that immediate improvements in noncancer mortality (u) increase the proportion of deaths due to cancer and the relative age-specific importance of cancer, even though the intrinsic age-specific cancer hazard g_c(a) is unchanged. Over evolutionary timescales, organisms can mitigate cancer risk by evolving lower replication error rates x*, reducing g_c(a) and total cancer mortality. This reconciles observations that long-lived species evolve cancer suppression (e.g., elephants) with the contemporaneous rise of cancer in companion animals experiencing rapid environmental improvements without evolutionary change. The framework also explains why cancers with larger effective step numbers (solid tumors) are expected to increase more with longevity extension than those with small step numbers (leukemias). In species with reproduction concentrated at younger ages, selection against late-life cancer is weak, limiting evolutionary mitigation of cancer risk under improved environments.

This study integrates a simple multistep carcinogenesis model with demography and life-history trade-offs to separate direct ecological effects of improved environments from indirect evolutionary responses. Key contributions include closed-form expressions for total cancer mortality and mean longevity, demonstration that decreasing noncancer mortality increases cancer’s relative and total burden without changing age-specific cancer hazard, and prediction that over long timescales the genomic error rate will evolve downward to partially offset increased cancer risk. The model anticipates more pronounced increases in solid tumors than leukemias with longevity extension and highlights weak selection on late-life cancer when fertility is confined to early life. Future research could calibrate model parameters with species-specific incidence and survivorship data, incorporate more realistic age-dependent mortality and fertility, heterogeneous tissues and step-specific rates, explicit cellular population dynamics and genomic instability, and test predictions in companion animals and humans as environments change.

- Demographic simplification: Noncancer mortality u is assumed constant (Type II survivorship), neglecting age-dependent hazards and early-life mortality spikes. - Cancer progression simplifications: Step transitions represented by constant or equal proportional rates (c_i = k_i x; often k_i=k), omitting explicit modeling of cell population dynamics, angiogenesis, metastasis, immune surveillance heterogeneity, and genomic instability except in extensions. - Mapping of steps to mutations: The step number n may not correspond to a fixed number of mutations; observed incidence patterns can reflect combined processes. - Cost function form: Fertility cost of error reduction is modeled as f_0/x^q with age-independent baseline m_0 in the main analysis; real trade-offs and age dependence may differ. - Evolutionary scope: Adaptation assessed via lifetime reproductive success assumes selection acts on x with sufficient heritability; in species with reproduction restricted to early ages, selection against late-life cancer is weak, limiting evolutionary change. - Empirical validation: No datasets were analyzed; parameter values and scenarios are illustrative. Environmental factors like diet, activity, and obesity are not modeled explicitly. - Generalizability: Results rely on model structure and parameter ranges; other mortality processes or step accelerations can alter quantitative outcomes.