Age effects and temporal trends in adenocarcinoma of the esophagus and gastric cardia (United States)

Abstract

A number of hypotheses have been advanced to explain the rapid increase of the incidence of esophageal adenocarcinoma in the US. A major problem in identifying and understanding the nature of this increase is the difficulty in untangling age effects from temporal trends due to cohort and period effects. To address this problem, we have developed multi-stage carcinogenesis models that describe the age-specific incidence of adenocarcinoma of the esophagus and of the gastric cardia with separate adjustments for temporal trends. These models explicitly incorporate important features of the cancers, such as the metaplastic conversion of normal esophagus to Barrett’s esophagus (BE). We fit these models separately to the incidence of adenocarcinoma of the esophagus and of the gastric cardia reported in the Surveillance Epidemiology and End Results (SEER) registry over the period 1973–2000. We conclude that the incidence of both cancers is consistent with a sequence that posits a tissue conversion step in the target organ followed by a multi-stage process with three rate-limiting events, the first two leading to an initiated cell that can expand clonally into a premalignant lesion, and the third converting an initiated cell into a malignant cell. Temporal trends in the incidence of both cancers are dominated by dramatically increasing period effects.

Appendix

BE/multi-stage clonal expansion model

We briefly summarize the mathematical development for the model depicted in Fig. 2. The model consists of two parts: the random occurrence of BE in the population, and the multi-stage carcinogenesis process arising in BE. Thus we derive the hazard function for the composition model of two random processes: tissue conversion from normal esophagus to BE, and multi-stage carcinogenesis process after onset of BE. The probability density function (pdf) for our model is defined by

$$\begin{aligned} f(t) &=\int_{0}^{\infty}f_{\rm BE}(u)f_{\rm MS}(t-u)\hbox{d}u \\ &=\int_{0}^{t}f_{\rm BE}(u)f_{\rm MS}(t-u)\hbox{d}u, \quad \hbox {if }u > t,\hbox{ then }f_{\rm MS}=0,\end{aligned} $$

(1)

where f _BE and f _MS are the pdfs of tissue conversion process and MSCE model, respectively, in Fig. 2. Let T be a random variable of appearance time of a malignant tumor, and let

$$ P(t)=\hbox{Pr}\{ T \leq t \}. $$

The hazard function h(t) is defined by

$$\begin{aligned}h(t) &=\lim_ {\Delta t \rightarrow 0} \frac{1}{\Delta t}\hbox{Pr}\{t\leq T < t+\Delta t | T \geq t\} \\ & = \frac{P'(t)}{1-P(t)}.\end{aligned} $$

(2)

The survival function S(t) is obtained by

$$\begin{aligned} S(t)=1-P(t)&=1-\int_0^t\int_0^s f_{\rm BE}(u)f_{\rm MS}(s-u)\hbox{d}u\hbox{d}s\\ &=1-\int_0^t f_{\rm BE}(u)(1-S_{\rm MS}(t-u))\hbox{d}u. \end{aligned} $$

Thus the hazard function h(t) is the following:

$$\begin{aligned} h(t) &= -\frac{S^{\prime}(t)}{S(t)}=\frac{f(t)}{S(t)}\\ &=\frac{\int_0^tf_{\rm BE}(u)f_{\rm MS}(t-u)\hbox{d}u}{1-\int_0^t f_{\rm BE}(u)(1-S_{\rm MS}(t-u))\hbox{d}u},\end{aligned} $$

(3)

where S _MS is the survival function for MSCE model in Fig. 2. Note that f(t)=−S′(t) and thus f(t)=h(t)S(t). Since $h(t)=-{\partial}/{\partial t} \hbox{ln} S(t)$, we have the following equation:

$$ S(t)=\hbox{e}^{-\int_{0}^t h(s){\rm d}s}. $$

(4)

Since we assume that the age to onset of BE follows an exponential distribution with rate ν, f _BE(u)=ν e^{−ν u}. For the details about h _MS and S _MS, refer to [29]. Let X be the number of stem cells in BE, and α and β be the cell division rate and the cell differentiation/death rate of an initiated stem cell, respectively. For k=3 in Fig. 2, which was considered as the best model in our study, the hazard function is given by (5) below.

$$ h(t)=\frac{\int_0^t \nu \hbox{e}^{-\nu u}\mu_0 X\left(1-\left[\frac{q-p}{q{\rm e}^{-p(t-u)}-p{\rm e}^{-q(t-u)}}\right]^{\mu_1/\alpha}\right)\exp\left[{-\mu_0 X \int_0^{t-u}1-\left(\frac{q-p}{q{\rm e}^{-ps}-p{\rm e}^{-qs}}\right)^{\mu_1/\alpha}\hbox{d}s}\right]\hbox{d}u}{1-\int_0^t \nu {\rm e}^{-\nu u}\left(1-\exp\left[{-\mu_0 X \int_0^{t-u} 1-\left(\frac{q-p}{q{\rm e}^{-ps}-p{\rm e}^{-qs}}\right)^{\mu_1/\alpha}\hbox{d}s}\right]\right)\hbox{d}u}, $$

(5)

and p and q are given by

$$ \begin{aligned} p &= \frac{1}{2}\left((-\alpha +\beta +\mu_2)-\sqrt{(\alpha+\beta+\mu_2)^2-4\alpha\beta}\right)\\ q&=\frac{1}{2}\left((-\alpha +\beta+\mu_2)+\sqrt{(\alpha+\beta+\mu_2)^2-4\alpha\beta}\right).\end{aligned} $$

(6)