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Modeling breast cancer metastasis
- Molecular biology of breast metastasis: The use of mathematical models to determine relapse and to predict response to chemotherapy in breast cancer

Using the various models of the natural history of breast cancer, researchers have then modeled the initiation time of metastatic growth; the effects of screening on metastases; the effects of surgery on recurrence; and methods of effective adjuvant chemotherapy. These models are used to help address the questions posed in the introduction to this paper.

Estimated initiation time of metastatic growth

The ability to determine the initiation time of metastatic growth would enable us to determine the likelihood of a patient having metastatic recurrence. As should be expected, the predicted initiation time relies significantly on the model of the cancer's natural history. For example, if the model predicts there will be metastatic growth with an early initiation time relative to the diagnosis of the primary (ie there is a high probability that the metastasis has already occurred by the time the primary is diagnosed), then the treatment strategy would be designed differently than if the metastasis were unlikely to have been initiated.

One particular model that considers this question was developed by Koscielny et al [19**]. They developed a model to determine the age of metastases at primary tumor diagnosis; the volume of the primary tumor when the metastasis is initiated; the duration of metastatic growth; and the delay between primary diagnosis and appearance of the metastasis. Two main hypotheses play a role in how they developed their model. The first hypothesis is that the metastatic growth rate is proportional to the size of the tumor from which it was derived. The second is that the probability of metastasis is related to the primary tumor doubling time. Using their model, they observed that their data showed that the metastatic initiation occurred at a tumor volume of the primary lesion that was only slightly smaller than that at the time of tumor diagnosis. This is in contrast to the findings of others who used an exponential growth model, with the doubling time of the primary tumor equal to that of the metastatic growth, and predicted that metastatic initiation occurs very early in the development of the primary tumor.

Koscielny et al [19**] concluded that the metastatic doubling time is faster than that of the primary (specifically, metastatic growth is about 2.2 times faster than that of the primary); the number of cells needed for metastatic initiation is greater than a single cell; tumor growth is not exponential over the life of the tumor; and growth duration of metastases is approximately 3.8 years, which is much shorter than the previously determined duration of about 17 years (determined using exponential growth and equivalent doubling times for the primary and metastasis). Following these conclusions, they estimated that approximately 30% of patients have metastases that are less than 1 year old, and therefore annual screening would be expected to result in a 30% decrease in the incidence of metastasis. This prediction is in accord with the results of breast cancer screening trials including the Health Insurance Plan trial, the Swedish Centers Combined and Edinburgh Trial ([31] and references therein), which showed a decrease in mortality of approximately 30% for women aged 50-69 years who underwent screening mammography.

Effects of surgery on metastasis

Retsky et al [12**] observed a statistically significant bimodal distribution for local and distant, but not contralateral breast cancer relapse using the Milan National Cancer Institute database of 1173 breast cancer patients. This distribution includes a sharp peak of relapse at 18 months and another broad peak at 60 months after surgery. They attributed this bimodal distribution to the effects of surgery on promoting metastatic growth.

To help understand this previously unobserved bimodal recurrence pattern, they developed a stochastic model to attempt to simulate (by Monte Carlo simulations) the clinical results. The model consists of a component to describe the primary tumor growth, based on the model of Speer et al [13*] (see the previous section). This is used to describe the release, via a stochastic mechanism, of metastatic cells once the primary tumor is vascularized. The other main component of the model describes metastatic growth and detection, and has three main growth stages. The first stage is a dormant single metastatic cell phase. The second is an avascular stage modeled by Gompertzian growth, with a limiting size of approximately 105 cells (or about 0.1-0.5 mm in diameter). The size is limited by the fact that the cells must be nourished by diffusion of nutrients from the existing vasculature. Cells in this stage may remain viable but nongrowing indefinitely. Proangiogenic factors elaborated by the tumor or a downregulation of antiangiogenic factors produced in the stroma, or a combination of both, may result in the induction of a neovasculature that will nourish the metastatic deposit and enable regrowth. This change accounts for entry into the third stage - a vascular stage that is also modeled by Gompertzian growth with a limiting size of approximately 1012 cells. The transition between these three phases is considered stochastic.

One of the interesting features of the model is that it allows for an increased progression of metastatic cells to the avascular and the vascular stages immediately after surgery. This was hypothesized to be due to a reduction in levels of tumor angiogenesis factor (produced by the primary tumor) after surgery, allowing angiogenesis to occur at the metastasis and thus allowing rapid growth of the metastatic lesion. The model without the possibility of stimulation due to the removal of the primary tumor could not produce the bimodal distribution of relapses observed clinically, and only the second peak was observed. However, when the model did account for stimulation of the metastatic cells to stage 2 (avascular) or 3 (vascular) due to the surgical removal of the primary tumor, then the bimodal distribution of relapses similar to that seen in the clinical data was observed. Retsky et al attributed the first peak of the bimodal recurrence distribution to metastases in the first two stages before surgery that are then promoted (due to angiogenesis) to the second or third stages.

Adjuvant therapy

Kinetic resistance

One of the early models that attempted to describe effective adjuvant chemotherapy is that of Norton and Simon [14*,15**]. They assumed that all tumor growth, tumor regression, and tumor regrowth is Gompertzian. The initial response of a tumor to chemotherapy is cell death or depopulation. According to Gompertzian kinetics, as the tumor becomes smaller its growth fraction increases, and it regrows at a faster rate. At some point the rate of cell kill may equal the rate of cell repopulation, and the cell population will approach an asymptotic limit. If the asymptotic limit after chemotherapy is always greater than one cell, a cure will never be effected. However, if it is less than one cell, a cure can reasonably be expected. The implication for treatment is that the only way to effect a cure is the absolute eradication of every viable micrometastatic cell. Therefore, Norton and Simon suggested that, when treating micrometastases, high-dose, short-duration treatment (single drug or combination) followed sequentially by a non-cross-resistant treatment may be preferable to prolonged duration, low-dose therapy. This treatment strategy challenges the log-kill hypothesis of Skipper et al [32*], which suggests that cell kill is proportional to the tumor size.

Clonal resistance

Rather than kinetic resistance, the resistance of cancer cells to chemotherapeutics may be a consequence of clonal selection. The resistance of bacteria to bacteriophages is a random event, as demonstrated by Luria and Delbruck [33**]. A study reported by Law [34], which used an adaptation of the Luria and Delbruck method, concluded that the resistance of the murine leukemia cell line L1210 to A-methopterin, a folic acid antagonist, was also secondary to mutations occurring at random. The Goldie-Coldman model [35] was the first major attempt to place the theory of the evolution of drug resistance by clonal selection into a sound mathematical basis. One of the assumptions of their model is that at each cell division of a nonmutant tumor cell, there exists a fixed nonzero probability that any new daughter cell will be a resistant mutant. By the time a tumor is detected, that is, by the time it reaches 109 cells, the Goldie-Coldman model estimates that drug-resistant mutants are present. Also, with an increasing population, the probability that a double mutant is created at random is also increased. Given these assumptions, the optimal way to administer chemotherapy would be to alternate two or more equally effective drugs as rapidly as possible to prevent clonal resistance [36**].

In a similar manner, Panetta [37] defined a model that describes a heterogeneous tumor population and the effects of chemotherapy. The model is used to determine conditions on when treatment should be switched from one drug (or combination of drugs) to a second noncross-resistant drug (or combination of drugs). This condition is related to the ratio of resistant to sensitive cells. The model indicates that the more effective the treatment is, the sooner it will be necessary to switch to the second noncross-resistant treatment.

Tumor dormancy

More recently Retsky and coworkers [12**,38], Demicheli et al [39], and Swartzendruber et al [40] hypothesized that adjuvant therapy may only benefit those patients who are not cured by local therapy and who would relapse under the first peak of the frequency of the relapse curve (at around 18 months). Smaller tumors with good prognostic factors tend to relapse under the second peak of the frequency of the relapse curve (at around 5 years). Therefore, early detection of tumors could make adjuvant chemotherapy less effective because the metastatic growth has a good chance of being in an avascular dormant stage that is kinetically resistant to adjuvant chemotherapy administered immediately after an operation. Demicheli et al [41*] demonstrated the above hypothesis by showing that cyclophosphamide, methotrexate, fluorouracil (CMF) therapy can reduce the risk of metastatic recurrence when compared with surgery alone, but only for patients with recurrences that occur within about the first 3 years. This supports the dormancy hypothesis of Retsky et al [12**]. They suggested that either an angiogenesis-inhibiting drug such as angiostatin or the reintroduction of chemotherapy at a later time need to be discussed as alternative treatment strategies.

Another interesting observation made by Demicheli et al [41*] is that CMF does not shift the bimodal hazard function to the right (thus just postponing relapse), but rather lowers the peaks. This implies that CMF adjuvant therapy can 'cure' some patients rather than just postpone the recurrence. This idea challenges the view that adjuvant chemotherapy only prolongs survival transiently, and no or very few patients are cured.

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