Table 1 Basic structure of the CRC-SPIN Model

Adenoma incidence: non-homogeneous Poisson process

   Log-risk for the i th individual at time t, ℓn(Ψ _i (t)) =

   ${\alpha }_{0i}+{\alpha }_{1}se{x}_i+\sum _{k=1}^4\delta \left(A_k<ag{e}_i\left(t\right)\le A_{k+1}\right)\left\{ag{e}_i\left(t\right){\alpha }_{2k}+\sum _{j=2}^k{A}_j\left({\alpha }_{2,j-1}-{\alpha }_{2j}\right)\right\}$

• Baseline log-risk, α_0i, is Normally distributed, mean Λ, standard deviation σ

• δ(.) is an indicator function with δ(x) = 1 when the condition x is true and δ(x) = 0 otherwise

• age _i (t) is the i th individual's age at time t

• A₁ = 20, A₂ = 50, A₃ = 60, A₄ = 70, A₅=∞ (effectively 100 years old)

   Adenoma location probabilities: cecum: 0.08; ascending colon: 0.23; transverse colon: 24; descending colon: 0.12; sigmoid colon: 0.24; rectum: 0.09.

Adenoma growth: Janoshek growth curve

_dij (t) = d _∞ - (d _∞ - d ₀) exp (-λ _ij t)

• d _ij(t) is the maximum diameter of the j th adenoma in the i th individual at time t after initiation.

• d ₀ = 1 mm, minimal detectable adenoma size

• d _∞ = 50 mm, maximum adenoma size

• time to reach 10 mm: -In((d _∞ - 10)/(d _∞ - d ₀))/λ.

Time to reach 10 mm: type 2 extreme value distribution

   Adenomas in the colon: distribution parameterized by β_1c and β_2c

   Adenomas in the rectum: distribution parameterized by β_1r and β_2r

Transition to preclinical cancer: normal cumulative distribution

   Probability of transition, male colon

Ф({In (γ _{1 cm} size) + γ _{2 cm} (a-50)}/γ ₃

   Probability of transition, male rectum

Ф({In (γ _{1 rm} size) + γ _{2 rm} (a-50)}/γ ₃

   Probability of transition, female colon

Ф({In (γ _{1 cf} size) + γ _{2 cf} (a-50)}/γ ₃

   Probability of transition, female rectum

Ф({In (γ _{1 rf} size) + γ _{2 rf} (a-50)}/γ ₃

   Where Ф(.) is the standard Normal cumulative distribution function, size is adenoma size in mm, and a is age at adenoma initiation.

Sojourn time: lognormal distribution

   Preclinical colon cancer, lognormal with mean μ_c, standard deviation τ_cμ_c

   Preclinical rectal cancer, lognormal with mean μ_r, standard deviation τ_rμ_r