Modeling growth kinetics and statistical distribution of oligometastases

The kinetics of development of micrometastases, and especially of small numbers of metastases (oligometastases), was explored by using simple assumptions to develop concepts that may be useful for framing future research. The conclusions depend on the assumptions and hence must be considered speculative. It is assumed that beyond a threshold size for initiation of metastatic spread, which varies widely from tumor to tumor, the rate at which a primary tumor sheds new metastases increases exponentially, in parallel with its exponential growth. This increasing rate of release of new metastatic clonogens from the primary tumor is accompanied by a similar exponential growth of each of the micrometastases newly established at a secondary site. This creates a log-log linear relationship between the volume distribution of metastases and number of metastases, there being one largest metastasis followed by an exponentially expanding number of logarithmically smaller micrometastases. For example, if the micrometastases and the primary tumor grew at the same rate for 6 doublings after initiation of the first metastasis, then the primary tumor would have increased its volume by a factor of 64 (2(6)) and would be shedding metastatic clonogens at 64 times the initial rate. The first metastasis would undergo 6 doublings and contain 64 cells; the succeeding 2 metastases, released as the primary doubled in volume, would undergo 5 doublings and each would contain 32 cells; and so forth down to the 64 most recently developed single-cell metastases. However, the growth rate of metastases is expected to be faster than that of the primary tumor so that the rate of increase in volume of the micrometastases would be faster than the rate of increase in their numbers (through release of new metastases from the primary). Thus, although the log-log linear relationship is maintained, the slope of the volume frequency curve is changed; if the micrometastases grew 5 times faster than the primary, the slope would change by a factor of 5. Removal of the primary tumor as a source of new metastases truncates the expansion in numbers of metastases without affecting the growth rate of existing micrometastases, with the result that the volume-frequency relationship is maintained but the whole curve is shifted to larger volumes as micrometastases grow toward clinical detectability. The development of an oligometastatic distribution requires that the exponential expansion in the number of new metastases be stopped by eliminating the primary tumor soon after the first metastasis is shed. A cell destined to become part of an oligometastatic distribution had just been newly deposited at its metastatic site at the time the primary tumor was removed and must undergo about 30 doublings to become clinically detectable as an overt metastasis (2(30) or 10(9) cells). Thus, the time interval between removal of the primary and subsequent appearance of oligometastases will be toward the upper end of a distribution of "metastasis-free" intervals for its particular class of tumor. The actual time to appearance of a solitary metastasis, or of oligometastases, in any particular patient will depend on the growth rate of the metastases in that individual but will always require about 30 volume doublings. An apparently solitary metastasis appearing synchronously with the primary tumor is unlikely to be solitary because, to do so, it would have to have undergone about 30 doublings without further release of metastatic clonogens from the primary that is, in our model, within 1 doubling in volume of the primary tumor. For the same reason, a synchronous or early appearing oligometastatic distribution is unlikely, but if it were to exist, there would be a steep gradient between the volumes of largest and smallest metastases because the growth rate of the micrometastases to produce synchronous metastases, without having further metastases shed from the primary, would have to be fast (up to 30x) relative to the growth rate of the primary. Conversely, a steep gradient in volumes of successive echelons of metastases reflects fast growth of metastases relative to the primary and favors the possibility of an oligometastatic distribution. This ratio of growth rates of metastases to primary is defined by the slope of the log-log curve for the volume-frequency distribution of metastases, which, in clinical practice, is difficult to determine over a wide range and is, by definition, essentially impossible for oligometastases. However, the volume-frequency relationship, measured over a wide range, is the same as the ratio of the volume of the largest to second-largest metastases in an oligometastatic situation. For example, if the metastasis doubled 5 times faster than the primary, the largest metastasis would be larger by 5 doublings than its closest follower(s), that is, by a factor of 2(5) or 32, equivalent to a 3.2-fold difference in diameter if the metastases were spherical. Alternatively, if an initially solitary and measurable metastasis is subsequently joined by measurable followers, the number of volume doublings separating successive echelons in the series can be determined directly, and the larger the difference (measured in doublings), the greater the probability that there will be a limited, oligometastatic condition (ie, in clinical terms, subsequent metastases will stop appearing after the large leader metastasis and a short succession of followers have been removed at 1 or more operations). In summary, the probability of there being an oligometastatic distribution is increased as the interval between removal of the primary tumor and appearance of metastases lengthens. It is also more likely the faster the metastases are growing relative to the growth rate of the primary tumor before its removal. Effective systemic cytotoxic treatment (eg, chemotherapy, hormonal manipulation, biological agents) given in the perioperative period, or concomitantly with radiation therapy for the primary tumor, would truncate the volume-frequency distribution toward an oligometastatic one by eliminating the smallest, most recently formed "tail-ender" metastases. That process, which only occurs at the threshold volume of the primary at which metastases are first initiated, would not be influenced by whether surgery or radiation therapy was chosen to treat the primary tumor, regardless of the overall duration of radiation therapy. Chemotherapy adjuvant to surgery is not usually indicated in the curative treatment of solitary or oligometastases because they represent a truncated distribution with few or no stragglers. If subclinical stragglers exist, they would usually be relatively large and, even though subclinical, too large to be cured by chemotherapy. Exceptions would be early rapidly growing oligometastases, especially from a slowly growing primary, or solitary metastases from an unknown primary where second echelon metastases, if they exist, may still be small. Otherwise chemotherapy could be postponed and used for palliative growth restraint of unusually large and/or numerous stragglers.