Developing a better understanding of the natural history of breast cancer via mathematical models may suggest more effective methods of screening and treatment, and may enable us to answer some of the above questions. A variety of models have been proposed for the natural history of breast cancer. They include models by Speer et al [13*], Norton and Simon [14*,15**,16], Spratt et al [17,18], and Koscielny et al [19**], to list just a few.
The Gompertz model has been the mainstay for models of solid tumors, including breast cancers, for a considerable period of time. The Gompertz model is a modification of exponential growth, with the addition of a decreasing growth rate over time. This decelerated growth causes the cancer to asymptotically approach a limiting size, referred to as its carrying capacity. This limited growth is attributed to several factors, including hypoxia and the lack of nutrients. The origin of this model is a variety of in vivo studies in which the Gompertz equation most accurately describes the growth dynamics of the tumor [20]. Using data from Bloom et al [21] on the natural history of breast cancer in untreated women admitted to the Middlesex Hospital, London, UK, from 1805 to 1933, Norton and Simon [14*,15**] and Spratt et al [17] used this model to describe the data.
Speer et al [13*] observed that the subclinical duration of growth given by the original Gompertz growth equation, using a range of parameter values similar to those used by Sullivan and Salmon [22], is too short (approximately 4 months). Also, Heuser et al [23] reported that clinical data derived from serial mammograms indicated that nine out of 109 untreated breast cancers measured over a 1-year period showed no growth, and the original Gompertz equation could not account for this observed dormant phase. Thus, they developed a modified Gompertzian model with a stochastic growth rate. This allows for a stepwise growth pattern, with the possibility of dormant phases. In a continued effort to verify this modified model of Gompertz growth with dormant stages and growth spurts, Retsky et al [24] reviewed the literature and described a variety of clinical cases in which the traditional exponential or Gompertz model was not consistent with the data.
If the current hypotheses regarding angiogenesis and the development of a tumor microvasculature are correct (see Holmgren et al [25] and Folkman [26,27,28]), then models will need to include some type of dormant phase if they are to accurately account for the complete natural history of the cancer. In fact, Spratt et al [17] indicated that, although the original Gompertz model can give a good approximation to clinical tumor growth over the short term, the growth rate of the cancer is more likely to be stochastic over the full history of the cancer. This allows for various growth patterns, including dormancy.
Although initially it may seem that it will make little difference whether either the original Gompertz model or the modified Gompertz model is used to describe the natural history of breast cancer, a quick comparison of the modeling results in the different papers indicates that there are significant differences. For example, using the same data sets Norton [29] (using the original Gompertz growth model) predicted a 2.25-year median preclinical growth phase, whereas Speer et al [13*] (using the modified Gompertz growth model) determined it to be approximately 8 years. (Also see the letter by Retsky et al [30] in reference to the above-mentioned paper by Norton.)
One of the major impediments to the modeling of breast cancer is our lack of data on its natural history. This makes it difficult to determine whether the growth of a primary breast tumor is a continuous function, or whether it is interrupted by periods of dormancy. Most of the data available comes from serial mammograms. The problem with this is that there are usually only two data points (the estimated size of the cancer when it was diagnosed and the estimated size from a previous mammogram that is reviewed retrospectively) [17,18]. Such a sparse amount of data leads to an indistinguishability problem between the various proposed models (ie biologically different models can describe the same data set equally well). Consider, for example, the exponential equation. Spratt et al [17] determined from the mammogram data that the median doubling time is 260 days for breast tumors at detectable levels. Extrapolating back from the approximate detectable size of 109 cells, this would indicate that the tumor was initiated about 21 years prior, which seems rather long. In fact, if we consider one of the slower growing breast cancers that they observed, with a doubling time of 7051 days, then the tumor would have been initiated 578 years prior, which is of course absurd. On the basis of this, we should rule out exponential growth as a viable model of the full natural history of breast cancer.
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