Chemo Secrets From a Breast Cancer Survivor

Breast Cancer Survivors

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2.1 Natural History of Breast Cancer

The basis of our investigation of optimal combinations of screening modalities is a simulation model that can generate individual health histories. It is useful to distinguish the natural history model, which refers to the health histories of women without early detection screening, from the intervention model, which refers to the effects of screening. For a patient with preclinical disease, the natural history model provides a way of simulating age of onset and preclinical sojourn time, or, equivalently, growth rate. Conditional on these, it then simulates the age of the woman and the tumor size at the time of diagnosis. These variables can then be used in turn as covariates in predicting a woman's survival and quality adjusted survival. This multi-stage prediction can be repeated for various screening strategies, by superimposing a history of examinations to the natural history, appropriately simulating results of screening tests based on assumed sensitivity, and appropriately adjusting age and size at detection when early detection takes place. Thus, given women's risk factors, a decision model using Monte Carlo simulations can be employed to jointly model the disease histories and screening interventions, and predict the outcomes of interest.

In the natural history model, breast cancer events are simulated according to the age-specific incidence of preclinical disease and mortality from other causes. For a woman with breast cancer, the natural history model also provides a way of generating her history of disease over time. The natural history of the disease over time requires a description of the transition between different states of the disease. Using the same notation as in Parmigiani [PARM02], we assume that there are four relevant states: H, women who are either disease-free or asymptomatic; P, women who have detectable preclinical disease; C, women with clinical manifestation of the disease; and D, women who have died. For women in the cohort who have breast cancer, we generate their ages at the onset of pre-clinical breast cancer, P, ages at the onset of clinical breast cancer, C, via tumor growth, and ages at death D according to corresponding models.

Because the age-specific incidence of pre-clinical disease cannot be directly observed, we have to estimate such a quantity from the sojourn time distribution and age-specific incidence of the clinical disease. Specifically, we can derive the incidence of pre-clinical breast cancer backward from the following deconvolution formula:

■Jo where Ic(y) is the age-specific incidence of clinical breast cancer, whp is the instantaneous probability of making a transition from H to P, and wpc is age-specific sojourn time density.

Note that the age-specific incidence of clinical breast cancer can be observed and is often well documented in cancer registries or from the control arms of early detection trials. We use the age-cohort-specific breast cancer incidence estimates developed by Moolgavkar et al. [MSL79]. With a given distribution for the sojourn time of the pre-clinical disease state, the age-specific incidence of pre-clinical breast cancer (whp) can be estimated using the method of Parmigiani and Skates [PS01].

Fig. 1. Summary of states, possible transitions, and transition densities for the natural history model. This scheme describes the progress of breast cancer in the absence of screening. All instantaneous probabilities of transition are indicated next to the corresponding transition. The two subscripts correspond to the origin and destination states, respectively.

Fig. 1. Summary of states, possible transitions, and transition densities for the natural history model. This scheme describes the progress of breast cancer in the absence of screening. All instantaneous probabilities of transition are indicated next to the corresponding transition. The two subscripts correspond to the origin and destination states, respectively.

However, the estimation of the sojourn time distribution is not straightforward in general [AGL78, DW84, BDM86, ES97, SPV97, SZ99]. In this study, we focus on three commonly used parametric distributions for the sojourn time of the preclinical disease state, which are further modified to incorporate the effect of age at the onset of the preclinical disease.

We first consider a smoothed age-specific exponential sojourn time distribution:

wpc(xI\(t)) = A-1 (t) exp(—A-1 (t)x), where the mean sojourn time A(t) depends on the woman's age. To incorporate the uncertainty of the parameter into the model, we introduce an inverse gamma prior to the parameter A, where the two parameters of the inverse gamma distribution are age-specific and are chosen to match the mean and standard deviation of sojourn times estimated from the Canadian National Breast Screening Studies (CNBSS) trials in Shen and Zelen [SZ01].

An alternative assumption for the sojourn time distribution is the lognormal assumption. We consider a modified version taking into account the womanSs age at the onset of the preclinical disease, while generalizing the model by Spratt et al [SGH86]:

w where the logarithm of the mean, ¡j,(t) is specified to be a linear function of the womanSs age, t. An inverse gamma prior is used to incorporate the uncertainty for the variance, which does not show an age effect [PARM02].

The parameters of the inverse gamma are chosen to match the moments of the reported age-specific variances in [SGH86].

A modified tumor growth distribution of Peer et al [PVH93, PVS96] is also used in our simulation for sensitivity analyses. Specifically, the sojourn time of the preclinical duration is modeled by the tumor growth rate or, equivalently, by the tumor doubling time. In particular, the relationship between sojourn time and tumor doubling time can be expressed as

X = \n(Vp/Vc)DT/ ln(2), where X is the sojourn time, DT is the tumor doubling time, and Vp and Vc are the volumes of a tumor at onset of detectable preclinical disease and at onset of clinical disease, respectively. We assume that the smallest tumor detectable by screening exam is 5mm, and that the average diameter at which breast cancer manifests is 20 mm [PARM02]. Tumor doubling times are assumed to follow an age-dependent log-normal distribution. Note that there is a direct relationship between the tumor growth rate (or doubling time) and the sojourn time in the preclinical duration. The parameters in the sojourn time distribution are estimated to match with the median and 95% quantile of the tumor doubling time based on findings from the Nijmegen trial [PVH93].

2.2 Survival Distributions and Mortality

One primary interest of the study is to evaluate the length of survival after diagnosis of breast cancer with various screening strategies. The survival distribution for women with breast cancer is determined by their age and tumor characteristics at diagnosis, and by the treatment they receive following diagnosis. Women in the cohort who receive periodic screenings are more likely to have breast tumors detected early and thus are more likely to have better prognoses than women who do not receive such screening. However, due to imperfect screening sensitivities and heterogeneity in pre-clinical durations, some breast cancer may still be clinically diagnosed between exams (interval cases). The survival distribution depends on screening only through the tumor characteristics and age at diagnosis.

Based on the natural history model, the tumor size and age at diagnosis are generated for a woman diagnosed to have breast cancer in the cohort. It is well known that lymph node involvement (nodal status) and the estogen receptor (ER) status of the tumor (positive or negative) are also important risk factors, and are related to treatment options and survival. To estimate the number of positive nodes at diagnosis, a predictive model was developed using the data of a womanSs age and tumor size at diagnosis from the SEER registries [PARM02, SEER98]. A constraint via the truncated Poisson distribution is given to ensure that the number of positive nodes for a screening- detected breast tumor is less than or equal to that for the same woman if her tumor is clinically detected. Without enough evidence to connect ER status with other risk factors, the ER status of a womanSs breast tumor is simulated independently of the other risk factors, but according to the distribution for the general population. It is estimated that roughly 70% of breast tumors are ER positive [NIH02].

As expected, the tumor characteristics at diagnosis will determine the treatment received thereafter. We assume that women in the cohort are treated according to the guidelines established by the NIH Consensus Conference on Early Breast Cancer (1991), given their risk factors including age, tumor ER status, tumor size, and nodal status at diagnosis. Whether a woman receives tamoxifen depends on her age and tumor ER status. The survival distribution for length with quality of life adjustment after diagnosis of breast cancer is estimated using a Cox regression model with covariates of treatment, age, tumor ER status, primary tumor size, and number of nodes involved. The predictive survival model was established based on a combined analysis of four CALGB trials [PARM02, WWT85, PNK96, WBK94], as described in [PBW99].

For a woman in the cohort, her age-specific mortality due to causes other than breast cancer is obtained from actuarial tables, using a 1960 birth cohort from the census database. If the breast-cancer-specific survival time for a woman is shorter than her simulated natural lifetime, then we assume that she died from breast cancer and contributed to the breast cancer mortality. Otherwise, we assume that she died from a competing cause.

2.3 Sensitivities of Mammography and Clinical Breast Examinations

The sensitivity of a screening program for the early detection of breast cancer plays a critical role in its potential for the reduction of disease-specific mortality. When a screening program involves more than one modality, it is important to obtain the sensitivity of each individual screening modality and the dependence structure among the multiple diagnostic tests [SZ99, SWZ01]. This knowledge provides a basis to guide health policy makers in designing optimal and cost-effective screening programs.

Some recent studies reveal that the sensitivity of a screening exam is likely to depend on tumor size and age at the time of diagnosis [PVS96, SZ01]. Based on literature in the area of breast cancer screening and the estimates of screening sensitivities for both MM and CBE, we consider a model to relate the sensitivity of each modality with age and tumor size at diagnosis, respectively [PARM02, SZ01]. In particular, a logit function is employed to model the effects of age and tumor size at diagnosis on the sensitivities of mammography and clinical breast exam, respectively. We assume the sensitivity of each modality satisfying the following equation:

g (t d)= exP{"k0 + ak1(t - 45) + ak2(d - 2)} ' ( , )= 1 + exp{ak0 + ak1 (t - 45) + ak2(d - 2)}, where t is the age at diagnosis, d is the diameter in centimeters of the primary tumor at diagnosis, k = 1 corresponds to mammography, and 2 is for CBE.

The coefficients in the logit models are determined based on the corresponding sensitivity estimates from the CNBSS trials [SZ01] as follows. A sensitivity of mammography of 0.61 corresponds to a woman at age 45 with a tumor diameter of 2cm; a sensitivity of 0.1 corresponds to a woman at the same age but with a tumor size of 0.1 cm; and a sensitivity of 0.66 corresponds to a woman of age 55 with a tumor size of 2cm: ¡31(45, 2) = 0.61, fh(45,0.05) = 0.1 and f31(55, 2) = 0.66. Thus, the coefficients in the logit model are solved to be, a.10 = 0.447, an = 0.216 and ai2 = 1.36 for mam-mography. In the same vein, we can solve the coefficients for the sensitivity of CBE: a20 = 0.364, a21 = —0.077 and a22 = 1.31. Moreover, because the sensitivity can vary from subject to subject even when given the same age and tumor size [KGB98], we use a beta distribution to reflect such a random variation for each sensitivity, while matching the corresponding mean and variance for the estimated sensitivity from the CNBSS trials, as reported in Shen and Zelen [SZ01].

The Health Insurance Plan of Greater New York (HIP) trial and the CNBSS both offered independent annual clinical breast exams and mammo-grams to women in their study arms, which gave us an opportunity to assess the dependence between the two screening modalities. The analyses based on data from these trials indicate that mammography and clinical breast examinations contribute independently to the detection of breast cancer [SWZ01]. Therefore, given the sensitivity of each individual screening modality, the overall sensitivity of a screening program using both MM and CBE is as follows:

f3(t,d)= /3i(t,d) + fo(t,d) — f3i(t,d)p2(t,d), when the two modalities are independent to each other.

2.4 Costs of Screening Programs

As expected in screening practices, the primary costs of a screening program is proportional to the total number of mammograms and clinical breast examinations given. Although there are additional costs related to follow-up confirmative tests such as a biopsy, and costs for the treatment of breast cancer at various stages after diagnosis, we will focus only on the cost of screening examinations in the current study. On its website, the National Cancer Institute lists the estimated cost of mammography in 2002 at $100-200, and acknowledges that the cost can vary widely among different centers and hospitals. Since it is frequently part of a routine physical examination, the cost of a CBE is often less than that of mammography. In a public website promoting cancer prevention, the estimated cost for an annual CBE is $45-55, whereas the cost of MM is $75-150 [PRE02]. In the decision analysis, it is clear that the cost ratio of MM and CBE determines the results in the comparison of different screening strategies. Therefore, we investigate the effects of two cost ratios (1.5 and 2) between MM and CBE, and allow the cost for a CBE to be $100. For simplification, we will not adjust for the type of currency, or for inflation over the years.

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