Analysis of lung cancer incidence in the nurses’ health and the health professionals’ follow-up studies using a multistage carcinogenesis model

Abstract

We analyzed lung cancer incidence among non-smokers, continuing smokers, and ex-smokers in the Nurses Health Study (NHS) and the Health Professionals Follow-Up Study (HPFS) using the two-stage clonal expansion (TSCE) model. Age-specific lung cancer incidence rates among non-smokers are identical in the two cohorts. Within the framework of the model, the main effect of cigarette smoke is on the promotion of partially altered cells on the pathway to cancer. Smoking-related promotion is somewhat higher among women, whereas smoking-related malignant conversion is somewhat lower. In both cohorts the relative risk for a given daily level of smoking is strongly modified by duration. Among smokers, the incidence in NHS relative to that in HPFS depends both on smoking intensity and duration. The age-adjusted risk is somewhat larger in NHS, but not significantly so. After smokers quit, the risk decreases over a period of many years and the temporal pattern of the decline is similar to that reported in other recent studies. Among ex-smokers, the incidence in NHS relative to that in HPFS depends both on previous levels of smoking and on time since quitting. The age-adjusted risk among ex-smokers is somewhat higher in NHS, possibly due to differences in the age-distribution between the two cohorts.

Appendix

In this section we show how to calculate explicitly the individual likelihoods (Eq. 2) as function of the model parameters and the individual’s smoking history. For simplicity, we assume a constant lag time between the appearance of the first malignant cell and diagnosis, so the survival at any age t, S(t), is equal to the TSCE model survival at age t − t _lag , S ₂(t − t _lag ) (see Eq. 4). We use the expressions derived in Heidenreich et al. [11] for the TSCE model survival in case of piecewise-constant parameters. From Heidenreich et al. [11], we have that the survival function and its derivative at any age t are

$$ S(t)=S_2(t-t_{lag})=\exp\left\{\sum\limits_{j=1}^k\frac{\mu_{0,j}X} {\alpha_{j}}\ln\left(\frac{q_j-p_j} {f_j(t_{j-1},t_k)}\right)\right\}, $$

(5)

$$ \begin{aligned} S^\prime(t)= &S^\prime_2(t-t_{lag})\\ = &S_2(t-t_{lag})\times \sum\limits_{j=1}^k-\frac{\mu_{0,j}X} {\alpha_{j}}\frac{\partial}{\partial t_k}\ln{(f_j(t_{j-1},t_k)}),\\ \end{aligned} $$

(6)

where k is the number of age-periods with different smoking-dose before age t _k ≡ t − t _lag , [t _j−1,t _j ] denote the end-points of the j − th age-period, d _j is the smoking-dose during the j − th age-period, μ_0,j , α _j , g _j , μ_{1, j} denote the parameter values during the j − th age-period and

$$ g_{j}=g(1+g_{c}d_j^{g_{e}}),\quad \alpha_{j}=\alpha(1+g_{c}d_j^{g_{e}}), $$

(7)

$$ \mu_{1,j}=\mu_{1}(1+\mu_{1c}d_j^{\mu_{1e}}),\quad \mu_{0,j}=\mu_{0}(1+\mu_{0c}d_j^{\mu_{0e}}), $$

(8)

$$ p_j,q_j=\frac{1}{2}\left(-g_{j} \mp \sqrt{g_{j}^2+4\alpha_{j}\mu_{1,j}}\right) $$

(9)

$$ \tilde{y}_k=0,\quad \tilde{y}_{j-1}=\frac{\alpha_{j-1}} {\alpha_{j}}\frac{(\tilde{y}_j-p_j)q_je^{q_j(t_{j-1}-t_j)}+(q_j-\tilde{y}_j)p_j e^{p_j(t_{j-1}-t_j)}} {f_j(t_{j-1},t_k)}, $$

(10)

$$ f_j(t_{j-1},t_k)=(\tilde{y_j}-p_j) \exp\left\{q_j(t_{j-1}-t_j)\right\}+ (q_j-\tilde{y_j})\exp\left\{p_j(t_{j-1}-t_j)\right\}, $$

(11)

$$ \frac{\partial}{\partial t_k}f_k(t_{k-1},t_k)=[\exp\left\{q_k(t_{k-1}-t_k)\right\} -\exp\left\{p_k(t_{k-1}-t_k)\right\}] p_kq_k, $$

(12)

$$ \frac{\partial}{\partial t_k}f_j(t_{j-1},t_k)=[\exp\left\{q_j(t_{j-1}-t_j)\right\} -\exp\left\{p_j(t_{j-1}-t_j)\right\}] \frac{\partial}{\partial t_k}\tilde{y}_{j}, $$

(13)

$$ \frac{\partial}{\partial t_k}\tilde{y}_{k-1}=\frac{\alpha_{k-1}} {\alpha_k}\frac{(q_k-p_k)^2e^{(p_k+q_k)(t_{k-1}-t_k)}} {(f_k(t_{k-1},t_k))^2}p_kq_k, $$

(14)

$$ \frac{\partial}{\partial t_k}\tilde{y}_{j-1}=\frac{\alpha_{j-1}} {\alpha_j}\frac{(q_j-p_j)^2e^{(p_j+q_j)(t_{j-1}-t_j)}} {(f_j(t_{j-1},t_k))^2}\frac{\partial}{\partial t_k}\tilde{y}_{j}. $$

(15)

To calculate the individual likelihood (Eq. 2), we only need to evaluate expressions (5) and (6) at the age at entry (t = a _e ) and at the age at last (t = a _l ) accordingly.

As an example, we calculate the likelihood contribution of the individual whose smoking history is illustrated in the top panel of Fig. 1 at the maximum likelihood estimates of the joint-model (Table 2). The individual entered the NHS study at age 43 (approx) and died at age 63 from other causes. Hence, we must calculate the TSCE model survival at ages 43 − t _lag  = 38 and 63 − t _lag  = 58 to obtain her individual likelihood (Eq. 2). This individual started smoking at age 20, at a constant rate of 30 cigarettes per day, and continued with this pattern until age 42. Hence, to calculate the TSCE model survival at age 38, we have to consider two different age-periods, namely [0, 20]≡[t ₀, t ₁] and [20, 38]≡[t ₁, t ₂]. From expression (5), we have that the survival at age 43 is

$$ S(43)=S_2(38)=\exp\left\{\sum\limits_{j=1}^2\frac{\mu_{0,j}X} {\alpha_{j}}\ln\frac{q_j-p_j}{f_j(t_{j-1},38)}\right\} $$

(16)

$$ =\exp\left\{\frac{\mu_{0,1}X} {\alpha_{1}}\ln\frac{q_1-p_1}{f_1(0,38)}+\frac{\mu_{0,2}X} {\alpha_{2}}\ln\frac{q_2-p_2}{f_2(20,38)}\right\} $$

(17)

In this particular example, the parameter values during the first age-period are equal to the background parameters (i.e., d ₁ = 0). In contrast, during the second period, we have to multiply the promotion rate g and the malignant conversion rate μ₁ by their corresponding dose–responses (with d ₂ = 30). We also multiply the initiated cells’ division rate, α, by the dose–response of g in order to keep them consistent. In this model, there is no dose–response on the initiation rate, μ₀, so it is constant between all age-periods. Using (Eqs. 5, 7–15) and the parameter estimates from Table 2, we obtain that

$$ \begin{aligned} \ln[S_2(38)]=&\frac{(8.14\times 10^{-8})(10^7)}{3} \ln\frac{2.55 \times10^{-6}-(-0.0960)}{0.0959}\\ &+\frac{(8.14\times 10^{-8})(10^7)}{5.54} \ln\frac{5.18\times 10^{-6}-(-0.1765)}{0.1766}\\ =&-0.00116. \\ \end{aligned} $$

(18)

So the survival at age 43 is S(43) = S ₂(38) = 0.9988. To calculate the TSCE model survival at age 58, we need to consider the following age-periods: $[0, 20]\equiv[t_0,t_1], [20, 42]\equiv[t_1,t_2], [42,44]\equiv[t_2,t_3], [44,46]\equiv[t_3,t_4], [46,48]\equiv[t_4,t_5], [48,55]\equiv[t_5,t_6]$ and [55, 57]≡[t ₆, t ₇], [57, 58]≡[t ₇, t ₈]. At each age-period, we must adjust the model parameters by the corresponding dose–response. Similarly as before, we calculate the survival at age 63 using (Eqs. 5, 7–15)

$$ \begin{aligned} S(63)=S_2(58)=&\exp\left\{\sum\limits_{j=1}^8\frac{\mu_{0,j}X} {\alpha_{j}}\ln\frac{q_j-p_j}{f_j(t_{j-1},58)}\right\}\\ = &0.9768.\\ \end{aligned} $$

(19)

So the likelihood contribution of this individual is equal to

$$ L_i(63,43;\bar{\theta}(d_i))=\frac{S(63)}{S(43)}=\frac{0.9768} {0.9988}=0.9779. $$

(20)

As a final example, let’s assume that an individual with the same smoking history, is diagnosed with lung cancer at age 63. In this case, we must obtain S′(63) = S′₂(58) to calculate its individual likelihood (Eq. 2). Using (Eqs. 5–15) we have that

$$ \begin{aligned} S{^\prime}(63)= S{^\prime}_2(58)=&S_2(58)\times \sum\limits_{j=1}^8-\frac{\mu_{0, j}X}{\alpha_{j}}\frac{\partial} {\partial t_k}\ln{(f_j(t_{j-1},58))}\\ =&-3.1171\times 10^{-3}.\\ \end{aligned} $$

(21)

So the likelihood contribution of this individual is equal to

$$ L_i(63,43;\bar{\theta}(d_i))=-\frac{S^\prime(63)}{S(43)}= \frac{3.1171\times 10^{-3}}{0.9988}=3.1208\times 10^{-3}. $$

(22)

Fortran and R code to compute the TSCE hazard and survival functions in case of piecewise-constant parameters are available from the authors by request.