.
Gompertzian model of breast cancer growth.
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The correlations between cytokinetics and clinical behavior support the concept that cell proliferation is intimately associated with the generation of tumor heterogeneity. Cell proliferation is, in addition, the primary mechanism for tumor growth. Anticancer therapy is, of course, intended to reverse growth by killing or removing cancer cells. A type of mathematic function called a growth curve describes increases and decreases in the number of cells over time. These curves not only summarize clinical course, but they also relate to the rate of emergence of mutations toward clinically relevant cellular diversity. Through both these attributes, growth curves are proving to be useful in explaining human cancer and in indicating new directions for therapeutic research.
This model of tumor growth, formulated and popularized by investigators at the Southern Research Institute, is commonly called the log-kill model. It was the original, and is still the preeminent, model of tumor growth and therapeutic regression.250–252 The model is based on the observation that leukemia L1210 in BDF1 or DBA mice grows exponentially until it reaches a lethal tumor volume of 109 cells (1 cubic centimeter).253 Ninety percent of the leukemia cells divide every 12 to 13 hours. This percentage is the same for both a tiny tumor and a tumor close to the lethal volume. As a result, the doubling time is always constant: if it takes 11 hours for 100 cells to grow into 200 cells, it will take 11 hours for 107 cells to grow into 2 × 107 cells. This pattern generalizes for any constant fractional increase: if it takes 40 hours for 103 cells to grow into 104 cells (an increase by a factor of 10), it will take 40 hours for 107 cells to grow into 108 cells.
Exponential growth and its associated concept of the doubling time are clinically relevant.254 Different histologic types of cancer display a great variety of doubling times within the observable range of tumor sizes.255 The most therapeutically responsive human cancers, such as testicular cancer and choriocarcinoma, tend to have doubling times that are < 1 month long. Less responsive cancers, such as squamous cell cancer of the head and neck, seem to double in about 2 months. The relatively unresponsive cancers, such as colon adenocarcinoma, tend to double every 3 months. Clearly, this clinical observation may relate to the higher chemosensitivity of proliferating cells (see below), that is, if a tumor has a high fraction of dividing cells, it will tend to grow faster and will also tend to be more responsive to drugs that kill dividing cells. Alternatively, tumors with a higher rate of cell loss tend to have a relatively slower growth rate and also a higher rate of mutations toward drug resistance. A combination of factors may be relevant, in that slower growth due to fewer mitoses may impede therapeutic response because of kinetics, whereas slower growth due to a high rate of apoptosis may impede response due to drug resistance.
An unspoken assumption in these theoretic considerations is that the doubling time remains fixed and thereby accurately summarizes the proliferative behavior of a given tumor. This assumption may not be realistic, as will be examined in some detail below. Nevertheless, we may use exponential growth to illustrate some important properties, which are also relevant to more complex growth patterns.
Let us consider a hypothetical tumor that is growing exponentially and is also homogeneous in drug sensitivity. When such a tumor is treated with a specific chemotherapy regimen, the fraction of cells killed is always the same, regardless of the initial size of the malignant population. This has been demonstrated in experimental animal cancers that do indeed grow exponentially, L1210 being the major example.256 If a given dose of a given drug reduces 106 cells to 105, the same therapy applied against 104 cells will result in 103 survivors. These two cytoreductions are both examples of a one-log kill, which means a 90% decrease in cell number. It was shown quite early in the development of this field that for many drugs, the log kill increases with increasing dose. Hence, it requires higher drug dosages to eradicate larger inoculum sizes of transplanted tumors.257 In addition, if two or more drugs are used, the log kills are multiplicative, that is, imagine that a given dose of drug A kills 90% of a population of cells (a one log kill) when administered as a single agent. Imagine as well that were we to treat the same population of cells with a given dose of drug B as a single agent, we would also kill 90%. Then drug B added to therapy with drug A should kill 90% of the 10% of cells left after drug A alone, resulting in a kill of 99% of the cells (a two-log kill). In other words, two one-log kills equal one two-log kill. As a numeric example, if treatment A given alone leaves 105 cells out of 106, and if treatment B given alone would accomplish the same, the combination A + B (at full doses of each) should be able to reduce 106 cells to 104. If treatment C is also a one-log kill therapy, A + B + C against 106 cells should leave only 103 cells. If A + B + C is used to treat 103 cells, only 100, or 1, cell should remain. Thus, if enough drugs at adequate doses were applied against a tumor of sufficiently small size, the number of cells left after treatment should be smaller than 1, which means that the tumor is destroyed. This concept, the fundamental concept underlying combination chemotherapy, was first demonstrated to be of major value in the design of early curative approaches to childhood leukemia.258 Other applications will be discussed below.
When the concept of fractional kill was first applied to the postoperative adjuvant treatment of micrometastases, say from breast cancer, it engendered enormous optimism.259,260 After all, micrometastases are very small collections of cancer cells. Indeed, very small solid tumors in the laboratory contain a higher percentage of actively dividing cells than do their larger counterparts.110,111 It is thought, as mentioned above, that most chemotherapeutic agents preferentially damage mitotic cells. Hence, the fraction of cells killed in a small tumor should actually be even greater than the fraction of cells killed in a histologically identical tumor of larger size. As a consequence, according to the Skipper-Schabel-Wilcox model, if the log-kill estimate is wrong, the error should be in the direction of underestimating the impact of therapy against micrometastases. Putting this all together, tumors of small volume should be easily cured by aggressive combination chemotherapy, even more readily than would be predicted by the model.
Clinical trials, unfortunately, have not entirely confirmed these optimistic predictions. An illustration is the postoperative adjuvant chemotherapy of early-stage breast cancer. After quality surgery very few cells should be left, largely disseminated in multiple micrometastatic sites. By Skipper's model, these should be easily reduced to below the volume of a single cell by appropriate drug therapy.261,262 The adjuvant chemotherapy of breast cancer with active agents at conventional doses does indeed reduce the probability of patients developing stage IV disease and does result in improved survival. However, in composite, this effect is relatively modest.263,264 Is this because the duration of the therapy is not long enough? Assume that a given drug combination causes a one-log kill with each application. Six cycles of that combination should cure tumors of fewer than 106 cells. For tumors of exactly 106 cells, the six cycles would leave just 1 cell to regrow. If this were the case, then merely extending the duration of treatment beyond six cycles should kill the remaining cell and thereby increase the cure rate. From a modeling perspective, this same argument generalizes for higher degrees of cell kill and higher tumor cell burdens. Yet, durations of exposure to the same chemotherapeutic regimen longer than 4 to 6 months have not improved results in adjuvant chemotherapy.264 Hence, the predictions of the model—that cure in this setting should be easy and that duration of therapy should increase that likelihood—do not match actual observations.
What is wrong? If we accept the basic tenets of the Skipper-Schabel-Wilcox model, the failure of adjuvant chemotherapy to cure all cases of early breast cancer can only be due to cellular biochemical drug resistance. Skipper and colleagues were aware of the divergence between their theory and actual experience and, for this reason, hypothesized that some cells in the tumor must be refractory to the drugs used at the dose levels employed.
When we explore the implications of other models, we will see that it is not always necessary to hypothesize the existence of absolutely refractory cells. Nevertheless, the inclusion of the concept of absolutely resistant cells in the Skipper-Schabel-Wilcox model can account for many observations. According to this reasoning, once all the cells that are sensitive are eliminated by a certain length of treatment, continuing the same therapy for a longer duration will not give better results. The reason is that all the cells left after that initial course are drug resistant and therefore cannot be killed by further use of the same drugs. It is further assumed that such resistance is acquired during a cancer's growth history (by the occurrence of mutations, a phenomenon called tumor progression). If that were the case, the only way to guarantee the absence of resistant cells is to initiate therapy at so small a tumor size that no recalcitrant mutants are as yet present. In L1210, the transplantable mouse leukemia that was used to formulate the Skipper model, drug-resistant cells are rarely found in small aliquots, which would seem to support the above reasoning. If this same reasoning were to apply to human cancer, it would mean that such drug-resistant cells would have to arise spontaneously between the time of the carcinogenic event and the time of the appearance of diagnostically large amounts of tumor.265 This concept leads to the conclusion that the development of a curative strategy depends entirely on the answers to two questions: When in the course of growth does resistance develop? Can tumors be diagnosed early enough so that we can start treatment when the (small) tumor is still curable?266
To try to answer these two questions, theoreticians have turned to quantitative models of the emergence of drug resistance. Indeed, the development of such models was one of the first major activities in the field of growth curve analysis. That is because drug resistance, by then as yet unknown biochemical mechanisms, was recognized quite early to be important in cancer therapeutics.267 The original work, however, was not based on the study of cancer cells, but rather some pioneering experiments in bacteriology. In 1943, Luria and Delbruck found that different bacterial cultures developed resistance to bacteriophage infection at random (and hence different) times in their growth histories. In fact, resistance often developed long before exposure to the viruses.268 Later, when the cultures were exposed to the viruses, the survival of the resistant bacteria could be assessed, thereby measuring the percentage of cells that had randomly acquired resistance. Luria and Delbruck reasoned that those cultures that had experienced a mutation earlier in their histories had more time to develop a high percentage of resistant bacteria. If a bacterium mutates toward property X with probability x at each mitosis, the probability of the cell not developing property X in one mitosis is 1 - x. In y mitoses, the probability of no mutations occurring is (1 - x)y. If each mitosis produces 2 viable cells (no cell loss), it takes N - 1 mitoses (over log2N generation times) for 1 cell to grow into N cells, that is, one mitosis produces 2 cells, each of these 2 cells undergoes mitosis (for a cumulative total of three mitoses) to produce 4 cells, each of these 4 divides (for a cumulative total of seven mitoses) to produce 8 cells, and so on. Hence, the probability of not finding any bacteria with property X in N cells is exp[(N - 1) * ln(1 - x)], which is approximately exp[-x(N - 1)], since x is small. (A numeric example of the application of this formula is given below.)
Within a decade of Delbruck and Luria's original observation regarding bacteria, the same pattern was found by Law to apply to the emergence of methotrexate resistance in L1210 cells.269 Thus, antimetabolite resistance was reasoned to be a trait acquired spontaneously at random times in the pretreatment growth of this cancer.
The more modern view of cancer biology has not diminished enthusiasm for the concept of acquired mutations. Abnormalities of the process regulating the entry of G1 cells into S could disinhibit replication, producing aberrant levels of DNA per neoplastic cell at each cell division.270–273 By this mechanism, aneuploidy as well as drug resistance should be a function of the number of mitoses. Cell loss would actually increase the probability of mutations per given cell number, since more cell divisions would be required to produce that cell number than if no cell loss had occurred.
In a qualitative sense, the kinetic observations of Delbruck, Luria, Law, and others were highly influential in the genesis and development of the concept of combination chemotherapy.274 If tumor cells could acquire resistance to a drug prior to exposure to that drug, then the therapist could be faced with a disease heterogeneous in drug sensitivity even at the time of first diagnosis. Since it is numerically unlikely that any one cell could spontaneously become resistant to many different drugs (particularly if the drugs have different biochemical sites of action), only with combinations of drugs could one hope to eradicate all cells.275 Because of its influence on the development of combination chemotherapy this concept has been formative of modern medical oncology.
In a quantitative sense, the Delbruck-Luria model was reapplied to human cancer in 1979 by the work of Goldie and Coldman.276,277 These authors later refined their original model to include multiple sublines with double or higher orders of drug resistance and also the presence of cell loss.278 Their analysis contended that there is a high probability that mutations arise over a two-log (100-fold) increase in tumor size. This can be shown by the following calculation. Let us take a tenable mutation rate x of 10-6.279 Using the expression exp[-x(N - 1)] that we derived above, the probability of no mutants in 105 cells is exp[-10-6(105 - 1)], which equals 0.905. Similarly, the probability of no mutants in 107 cells is .000045, that is, it is unlikely that 105 cells have at least 1 drug-resistant mutant, but extremely likely that 107 cells have at least 1 such cell.
In this regard, it should be noted that although Goldie and Coldman focused on the property of drug resistance, an even clearer illustration of their concept might be found in the acquisition of metastatic ability. The capacity to metastasize is now established to be a reflection of genetic lability.280 The approximate volume of 107 packed cells is 0.01 cubic centimeter. If tumor cells are mixed with benign host tissue (including stromal cells, fibrosis, extracellular secretions, blood and lymphatic vessels, cellular infiltrate, and fluid-filled space) at a packing ratio of 1:10, 107 cancer cells will occupy a volume of 0.1 cubic centimeter. At a packing ratio of 1:100, which is often more realistic, 107 cancer cells would be found in a tumor volume of about 1.0 cubic centimeter. This example of scaling relates directly to clinical data. In primary breast cancer, the best predictor of axillary metastases is tumor size. Only 17% of invasive ductal lesions < 1 cm in diameter are metastatic to the axilla, contrasted with 41% of lesions of 2 cm in diameter and 68% of tumors of 5 to 10 cm.281 For primary breast cancer that does not involve axillary lymph nodes, the probability of eventual metastatic spread increases sharply when the mass in the breast is > 1 cm in diameter.282 Hence, metastatic ability is conspicuously more common in tumors larger than this critical size. A 1 cm spherical tumor contains a volume of slightly over 0.5 cubic centimeter, which is right in the middle of the range of 0.1 to 1.0 cubic centimeter described above as likely harboring between 105 and 107 cancer cells.
These calculations fit the model with reassuring precision. However, they cannot be regarded as proof of the model, since other explanations are possible (see Mitotoxicity Hypothesis, below). Moreover, some very specific predictions of the Goldie-Coldman model have not been confirmed in the clinic, as discussed immediately below. This illustrates the complexity of the biology underlying common tumor growth curves and illustrates that all models are useful, only in that they summarize empiric observations and motivate experiments. Models never explain phenomena: they merely describe phenomena in mathematic language.
The specific, testable predictions of the Goldie-Coldman model regard drug sensitivity. Using the calculations above, the model predicts that a cancer arising from a single, drug-sensitive malignant cell has a 90% chance of being curable at 105 cells. Yet, if it has a 90% chance of being curable at that size, it will almost certainly become incurable by the time it grows to 107 cells. Thus, tumors larger than 0.1 to 1.0 cubic cm should always be incurable with any single agent. This line of reasoning led these authors to conclude that the best chemotherapeutic strategy is to treat as small a tumor as possible as early as possible. The earliest possible treatment might be perioperative or even preoperative. They also concluded that once treatment is started, as many effective drugs as possible should be applied as soon as possible. This strategy, according to the model, is needed to prevent cells that are already resistant to one drug from mutating to resist others.
These recommendations are intuitive, conforming to established empiric principles of combination chemotherapy.283 They differ from classic principles, only in that they concentrate on the emergence of resistance during treatment, as contrasted with the other possibility that resistance is already present at the start of treatment. Most uniquely, these recommendations imply that if several drugs cannot be used simultaneously at good therapeutic levels (because of overlapping toxicity or competitive interference), they should be used in a strict alternating sequence.
The recommendation for strict alternation is based on several assumptions. All are based on the principle of symmetry. Imagine that a tumor has two types of cells, type A and type B. The A-cells are sensitive only to therapy A, whereas the B-cells are sensitive only to therapy B. Symmetry means that A-cells are as sensitive to therapy A as B-cells are sensitive to therapy B. The second assumption is that the rate of mutation toward biochemical resistance is also symmetrical. This means that the A-cells mutate toward resistance to A at the same rate as the B-cells mutate to resistance to B. The third assumption is that the growth patterns and growth rates of the two types of cells are equivalent.284 Critical appraisal of the various assumptions and conclusions of the Goldie-Coldman model has raised several interesting points. It would be informative to examine these in some detail, not only to illuminate this one hypothesis but also to show the relevance of growth curve analysis to clinical problems.
The first, and most basic, assumption that we must question concerns the notion that all chemotherapeutic failure is rooted in absolute drug resistance. This is a widely held assumption, but it is not above criticism. In fact, evidence to the contrary is found easily in clinical experience. For example, lymphomas and acute leukemias frequently respond to chemotherapy when they relapse from a complete remission induced by the same chemotherapy. Patients with Hodgkin's disease who achieve complete remission with combination chemotherapy and who relapse ≥ 18 months later have an excellent chance of attaining complete remission again when the same chemotherapy is re-applied.285 A similar situation is found in the treatment of breast adenocarcinoma. Stage IV (recurrent, metastatic disease) frequently responds to chemotherapy that had worked previously but failed to work subsequently. Various illustrations of this phenomenon have been documented in the literature. For example, the Cancer and Leukemia Group B (CALGB) treated patients with advanced breast cancer with cyclophosphamide, Adriamycin (doxorubicin), and 5- fluorouracil (CAF), with or without tamoxifen.286 Although none of these patients had prior chemotherapy for their advanced disease, some had had prior adjuvant chemotherapy. All parameters of disease sensitivity to treatment—the response rate, response duration, and overall survival—were unaffected by patients' past histories of adjuvant chemotherapy. Similarly, patients on trials at the National Cancer Institute in Milan who developed stage IV breast cancer after adjuvant cyclophosphamide, methotrexate, and 5-fluorouracil (CMF) responded as well to CMF for advanced disease as those who previously had been randomized to be treated with radical mastectomy alone.287 From these observations we may safely conclude that breast cancers that regrow after exposure to adjuvant CMF are not universally resistant to CMF.288 We are now observing that patients experiencing recurrence of stage IV disease after the failure of high-dose chemotherapy (autologous bone marrow transplantation [ABMT]) can benefit in terms of tumor response to the re-application of conventional doses of chemotherapy drugs. Hence, all chemotherapeutic failure cannot be attributed to permanent drug resistance. It is possible that some cancers escape cure because a temporary absolute drug resistance develops that then reverses over time. It is also possible, however, that at least some cancers can escape cure by the use of drugs, even though some of their cells are not absolutely resistant to these drugs. This important possibility will be developed further as we consider growth models other than simple, symmetric exponential growth.
Another prediction from the Goldie-Coldman model that is interesting to examine is that tumors > 1.0 cubic centimeter (107 cells at a packing ratio of 1:100, 109 cells at maximum density packing) cannot be cured with single drugs. Two rapidly growing cancers, gestational choriocarcinoma and Burkitt lymphoma, both with dense packing of their cancer cells, have been cured with single drugs.289 Cures are achieved even when therapy is initiated at tumor sizes much larger than 1.0 cubic centimeter. Childhood ALLs, other pediatric cancers, adult lymphomas, and germ cell tumors of greater than 1010 cells are frequently cured with two-drug and three-drug regimens. Hence, contradicting the model, the size of 107 cells does not always mean incurability.
For the purposes of planning chemotherapy schedules, the Goldie-Coldman model speculates that mutations develop rapidly during the treatable portion of a cancer's growth history. This may seem tenable, since in our previous discussion of cell proliferation, we established that genetic lability is a key attribute of neoplasia. Yet, clinical observations hint at a deeper level of complexity.
As a starting example, let us examine metastatic ability as a measure of the rate of mutations. A primary breast cancer left untreated to grow in the breast, as was standard practice in the 19th century, always became metastatic.290 Yet at 30 years of follow-up after radical mastectomy (with no adjuvant chemotherapy), more than 30% of patients are alive and free of disease.291,292 The mortality rate drops gradually from about 10% per year in the first year to about 2% per year by year 25, but a plateau is reached after 30 years, with a rate of mortality indistinguishable from that of the general population.293–295 Hence, although most, if not all, breast cancers can become metastatic if left alone long enough, and many have already done so by the time of initial presentation, some have not. This speaks against the universal, rapid development of mutations.
Let us consider another situation, the case of a primary cancer that is diagnosed in the breast before it has developed metastatic ability. If the cancer cells in the breast are not completely removed or destroyed, will the residual cells mutate rapidly to produce metastatic clones? A protocol of the NSABP considered this question.296 Some patients with primary disease were treated by lumpectomy without radiotherapy. The local relapse rate was significant, indicating that residual tumor was left unchecked. Yet, such patients did not have a higher metastatic rate (measured by the survival rate) than patients treated adequately de novo by lumpectomy plus immediate radiotherapy or by mastectomy.
This latter result is surprising, since some metastases from residual cancer should be expected even if that residual disease did not progress in its ability to release metastatic clones. Longer follow-up of this trial might eventually reveal a higher rate of distant metastases. However, the absence at 12 years of a major negative survival impact of local recurrence indicates that tumor can remain in a breast, grow in the breast, and yet not develop metastatic cells at a very high rate. If metastases develop, therefore, the odds are high that they have already done so before the time of first clinical presentation.
Since this view was first expounded in the first edition of this textbook, more recent evidence concerning radiotherapy to the chest wall after mastectomy has become available. These data may be regarded as confirmatory. Several papers have shown that for patients with a high probability of local recurrence the use of such radiotherapy decreases the chances of local and (the key point) distant recurrence.297–299 Yet the differences are small, and are not apparent for many years after primary therapy. At high rates of mutations the local residual disease would be expected to produce metastatic clones universally: hence, the rate of mutations cannot be very high. For further discussion of the practical and natural philosophical implications of these results, the reader is referred to a recent paper by Hellman.300
In a similar vein, the Goldie-Coldman model concludes that for chemotherapy to be effective it must be started as soon as possible after diagnosis. To restate the rationale: the hypothesized rapid mutation rates would otherwise produce cells that would be resistant to all treatments. Here as well, however, contradictory evidence is well known. For example, in a pioneering trial in the treatment of acute leukemia, the response to an antimetabolite (A) was the same whether that drug was used first or sequentially after the use of a different antimetabolite (B).301 Had mutations toward the drug A occurred rapidly during treatment with drug B, we should not have expected these results. Another two examples are found in the treatment of breast cancer. In a randomized trial, the International (Ludwig) Breast Cancer Study Group found that it was equally effective to give node-positive breast cancer patients either 7 months of chemotherapy starting within 36 hours of surgery or 6 months of chemotherapy starting about 4 weeks later.302 Again, rapid mutation rates should have impeded response after the delay, but this did not occur. A very important result, unfortunately as yet published only in abstract, was obtained in a trial of high-dose chemotherapy (with autologous hematopoietic stem cell rescue, ABMT) for metastatic breast cancer.303 Here patients in complete remission from conventionally-dosed chemotherapy were randomized to receive ABMT immediately or were followed in an unmaintained remission. Those 90% who relapsed from the unmaintained remission were treated with ABMT. The duration of disease control and survival from the time of ABMT was the same in both arms of the trial. (Incidentally, this translated to an overall improved median duration of disease control and survival for the patients receiving delayed ABMT. This is partially because the 10% of patients who remained disease free for more than 5 years from the conventional therapy without ABMT raised the whole curve. It is also because the delayed ABMT patients had an overall duration of disease control that was the sum of that which they experienced from the conventional therapy and from the ABMT.) Had mutations toward resistance to chemotherapy occurred rapidly during the unmaintained remission, the delayed-ABMT arm should have done worse. A trial in stage B nonseminomatous testicular cancer provides yet another example.304 This trial randomized patients after retroperitoneal lymph node dissection either to two cycles of cisplatin combination chemotherapy or to observation. At a median follow-up of 4 years, 6% of patients randomized to adjuvant chemotherapy relapsed, compared with 49% of patients randomized to observation. Yet, because the response of relapsing cases to subsequent chemotherapy was excellent, there was no significant survival difference between the two approaches. Hence, this is evidence that most testicular carcinomas retained their chemosensitivity in spite of a prolonged period of unperturbed growth. We may conclude, therefore, that leukemia cells, and breast and testicular cancer cells that are residual after surgery, can grow unperturbed and yet not develop drug-resistant mutants at a fast rate.
Other controversial implications challenge the validity of the Goldie-Coldman model. The model concludes that adjuvant treatment must be instituted as early as possible in the growth history of a cancer to be effective. Yet several pilot studies and one major multiinstitutional trial failed to find an advantage to preoperative chemotherapy for primary breast cancer.305,306 The model also concludes that if drugs are used postoperatively, they have to be used as soon as possible after surgery to be effective. Hence, Goldie-Coldman recommends that all drugs in an adjuvant regimen be introduced immediately, lest their biologic impact be dampened by mutations toward drug resistance. This hypothesis was questioned in a trial by the CALGB that gave node-positive primary breast cancer patients 8 months of an adjuvant CMF (plus vincristine and prednisone) regimen.307,308 The CMFVP was followed by either more CMFVP or by 6 months of vinblastine, Adriamycin, thiotepa, and halotestin (VATH). Patients receiving the cross-over therapy had a significantly improved disease-free survival, especially those with four or more involved axillary nodes. In a similar vein, it is of note that a trial in Milan found no advantage to Adriamycin following CMF for patients with one to three involved nodes, that is, the cross-over effect was not seen in patients with lower risk of relapse.309 This differs from the results of a pivotal trial using paclitaxel in the adjuvant setting, which is discussed below. The point is that for many patients, dominant resistance to VATH did not develop during the 8 months of CMFVP treatment in those cells escaping CMFVP. This result, therefore, does not confirm the Goldie-Coldman hypothesis. The implications of the CALGB's results in patients with higher degrees of nodal involvement, including the issues of simultaneous versus sequential therapies, dose scheduling, and optimal duration, are discussed in more detail below.
The assertion most singularly identified with the Goldie-Coldman model is the recommendation for alternating chemotherapy sequences. To repeat; they say that it is so important to give drugs as early as possible that if one cannot deliver a true simultaneous combination using all the drugs, one should alternate sequences rather than use the drugs in sequential blocks. Has this strategy demonstrated unequivocal advantages? Numerous attempts to improve the prognosis of patients with SCLC by alternating chemotherapy sequences have resulted in little or no benefit.310 Another relevant trial concerns the treatment of diffuse aggressive non-Hodgkin's lymphoma. The National Cancer Institute (NCI) found no advantage to a ProMACE-MOPP hybrid, which delivered eight drugs during each monthly cycle, over a treatment plan delivering a full course of ProMACE (prednisone, methotrexate, Adriamycin, cyclophosphamide, etoposide), which was then followed by MOPP (mechlorethamine, vincristine, procarbazine, prednisone).311
The Goldie-Coldman principle was also examined in the context of advanced Hodgkin's disease, where MOPP was compared with MOPP alternating with Adriamycin, bleomycin, vinblastine, and dacarbazine (ABVD). ABVD is an effective first-line therapy for Hodgkin's disease and is also an effective salvage regimen for patients who are refractory to MOPP.312,313 Among chemotherapy-naive patients, MOPP-ABVD was found to be superior to MOPP, with regard to complete remission rate, freedom from progression, and survival.314,315 These results suggested that there might be some advantage to the “all drugs early” idea. However, the CALGB found that the complete remission rate and failure-free survival with MOPP-ABVD, although better than with MOPP alone, was not different from that with ABVD alone.316 Indeed, the superiority of MOPP-ABVD and ABVD over MOPP may have been due to differences in dose received, since only about 40% of MOPP patients received full doses of the cytotoxic agents by the third cycle, whereas these percentages were greater than 70% on ABVD and on MOPP-ABVD. At comparable levels of received dose, there were no clear advantages to the alternation of MOPP and ABVD over ABVD alone. Similarly, the NCI found no advantage to MOPP alternating with lomustine, Adriamycin, bleomycin, and streptozocin over MOPP alone.317 An American intergroup trial has found that a hybrid of MOPP-ABVD was superior in complete remission duration, failure-free survival, and overall survival to MOPP followed by ABVD.318,319 As with MOPP-ABVD in the CALGB trial, however, it is possible that this result may be explained by dose differences, that is, patients treated with the hybrid regimen received higher doses because of the necessity to modify for toxicity the doses of MOPP in the regimen that delivered MOPP followed by ABVD. It is also possible that the earlier introduction of Adriamycin in the hybrid might have been advantageous because such an approach could diminish the adverse impact of the emergence of multi-drug resistance. These points are discussed below in the context of the Norton-Simon model.
Lessons learned in the treatment of the lymphomas have extended to the treatment of breast cancer, that is, alternating cycles that have not resulted in a dosage difference have not proved advantageous. For example, the VATH regimen is active against tumors relapsing from or failing to respond to CMF, and thereby meets the non-cross-resistance requirements of the Goldie-Coldman model.319 Yet, in patients with advanced disease, the CALGB found no advantage to CMFVP alternating with VATH over CAF or VATH alone.320 A direct comparison of alternating and sequential chemotherapy in the adjuvant chemotherapy of breast cancer was conducted in Milan. This group had previously generated historically controlled data that suggested a benefit from a sequential approach, the rationale for which is discussed below.321,322 In the more recent study, female patients with stage II breast cancer involving four or more axillary lymph nodes were randomized between two arms.323 Arm I prescribed four 3-week courses of Adriamycin (A), followed by eight 3-week courses of intravenous CMF (C), symbolized as AAAACCCCCCCC. Arm II stipulated the use of two courses of intravenous CMF alternated with one course of Adriamycin four times for a total of 12 courses, symbolized as CCACCACCACCA. The total amounts of Adriamycin and CMF in both arms were equal. Yet the patients who received arm I had a higher diseasefree survival and a higher overall survival than those on arm II. With total dose controlled, alternating courses of chemotherapy were found to be inferior to a cross-over therapy plan. These preliminary results have been confirmed by long-term follow-up analysis.324
The sequential application of drugs has proved to be a useful strategy in the treatment of leukemias. In adult acute myelogenous leukemia, a high rate of complete remission is obtained with cytarabine plus anthracyclines, but the duration of the responses is short. Postremission maintenance therapy has been shown by the CALGB to be relatively ineffective when given at low doses.325 Moreover, a trial showed that 32 months of postremission therapy were not superior to 8 months of the same therapy, similar to the failure of longer courses of adjuvant chemotherapy to improve results achieved by 4 to 6 months of such treatment in breast cancer.264, 326 A randomized trial was recently reported that studied 596 patients of 1,088 who had achieved complete remission with induction chemotherapy.327 This trial was designed to question the effectiveness of intensive postremission chemotherapy, exploiting the steep dose-response curve for cytarabine.328 The study found that the high-dose regimen was the best of three different dose schedules of cytarabine. Indeed, the best results were comparable with those reported in similar patients undergoing allogeneic BMT during first remission.327,329 The Children's Cancer Group (CCG) has reported that intensive induction, followed sequentially by intensive consolidation and later intensification, was superior to other strategies in the treatment of childhood acute lymphoblastic leukemia.330 These observations have major practical and theoretic implications, as they suggest that strategies other than those advocated by the Goldie-Coldman hypothesis may have significant clinical impact.
The foregoing detailed examination of the Goldie-Coldman model was provided not only to provide discussion points regarding this particular concept but also to illustrate the relevance of growth curve analysis to treatment design. The Goldie-Coldman model is mathematically sensible and may well be applicable to some aspects of cancer biology. The model is also of major historic importance, in that its publication rekindled interest in the quantitative development of drug resistance. These two points are valid even though several of the model's major predictions have not been sustained by clinical data. Yet, we are left with an enigma: how can a model that is so reasonable and seemingly so well grounded in kinetic dogma (log-kill) fail to generate a successful clinical strategy? One reason for a discrepancy between tenable theory and empiric results is the invalidity of underlying assumptions. An assumption of particular consequence in this regard, one that merits reevaluation in face of the negative data reviewed above, concerns the concept of absolute drug resistance.
The Goldie-Coldman model is very concerned with absolute drug resistance. Yet, it is now well established that much drug resistance is relative rather than absolute.331 A cell that is absolutely resistant cannot be killed with any pharmacologic dose level of the agent. Relative drug resistance, on the other hand, depends on the dose level employed. In terms of the SkipperSchabel-Wilcox model, one tumor may experience a log kill of two (99% reduction in cell number) when it is exposed to a certain dose and duration of treatment. Another, more resistant, tumor may experience a log kill of one (90% shrinkage) when it is treated with exactly the same therapy. However, if the dose intensity of chemotherapy against the relatively resistant tumor is increased, the log kill can increase as well.332,333
Clinically, even two-fold increases in dose level can have profound effects on the curative impact of chemotherapy.331 Yet, this is not always seen with all drugs, nor in all diseases.334 In retrospective analyses of the adjuvant chemotherapy of operable breast cancer and of the chemotherapy of advanced lymphoma, a high dose seems to be a key beneficial variable.335–337 Yet, even here, the validity of conclusions based on retrospective data has been questioned.338,339 In randomized trials in childhood ALL, adult germ cell tumors, advanced breast cancer, and breast cancer in the adjuvant setting, the higher-dose regimen has proven superior.340–344 Yet, results do not indicate a strictly rising dose-response relationship. For example, doses of cyclophosphamide over 600 mg/m2 do not improve results in the adjuvant chemotherapy of breast cancer,344 nor do doses of doxorubicin over 60 mg/m2.345
How do we explain this complicated relationship between dose and effect? From a kinetic viewpoint, the importance of dose is defensible. In many animal experiments, the log kill will be greater for the regimen with a higher dose intensity.346 One problem is that the concept of dose intensity requires definition. It is not just the total amount of drug received, nor is it just the amount of drug received per unit of time; rather, it is a mathematic combination of both. Dose intensity is actually a combination of dose escalation (raising the dose level) and dose density (increasing the amount of drug per unit of time, usually by shortening the total duration of treatment, while keeping the total amount of drug constant). If regimen I gives X amount of drug over Y days, and if regimen II gives 2X amount of drug over Y days, then regimen II is clearly more dose intensive. Regimen III, giving X amount of drug over Y/2 days, is also more intensive than regimen I. Although the dose rate of drug delivery of regimen III (2X/Y drug per day) is equivalent to regimen II, regimen II delivers more total drug and thus may be superior to regimen III in clinical efficacy. Hence, dose intensity alone may not account for clinical superiority. Yet, sometimes, once a certain minimal total dose is achieved, further increases in total dose are unimportant. For example, a number of trials have shown that durations of adjuvant chemotherapy longer than 4 to 6 months do not improve clinical results in operable breast cancer.347–350 Therefore, once the minimal total dose is determined empirically and adhered to, dose intensity should be an important determinant of cell kill.
The shape of the relationship between cell-killing capacity and dose is not totally clear for any drug, but for some agents, some data suggest a strictly proportional relationship up to a point. We may use as an example the randomized trial by the CALGB that treated node-positive patients by one of three plans of CAF adjuvant treatment (cyclophosphamide, doxorubicin, 5-fluorouracil).343 (Further studies on dose levels greater than those employed in this trial are discussed in the next section.) Let Z equal a certain total cumulative dose of chemotherapy: the three regimens gave either 2Z over 4 months (plan I), 2Z over 6 months (plan II), or Z over 4 months (plan III). Plan I was superior to plan III in reducing the rate of recurrence, but no difference between plan I and plan II has as yet been reported, except for a subset of patients.351 Hence, the total anticancer influence of each of these regimens seems to be strictly proportional to the total dose administered. For plan I, it was 2Z, the sum of 2Z over the first 4 months plus zero for the 2 additional months. Plan II also gave 2Z but over the entire 6 months. Plan III delivered half as much total anticancer influence, the sum of Z over the first 4 months, then zero for the remaining 2 months. A proportional dose-response relationship would predict that plan III should be inferior to both plan I and plan II, which was observed. One qualifier in this argument is that if CAF chemotherapy cures some patients, then plan I might eventually prove to be superior to plan II. This is because the cancer cell killing accomplished at 4 months from 2Z given over 4 months should be greater than the cell killing measured at 4 or at 6 months from 2Z given over 6 months. For some patients given 2Z over 4 months, the log kill might be enough to preclude disease regrowth. This might explain the superiority of plan I in patients with HER-2-overexpressing tumors.351 That such tumors may be especially sensitive to higher dose levels of doxorubicin is now suggested by the results of several corroborating studies.352,353
The global conclusion of this analysis is that clinical treatment failure may be the consequence of insufficient dose intensity (ie, 2Z over 6 months when it could have been given over 4 months). A tumor may relapse because some of its cells, relatively but not absolutely insensitive to the agents applied, are not exposed to enough drug to be eradicated. This is analogous to a bacterial infection relapsing because an insufficient dose intensity of an antibiotic is applied, even though the microorganisms are sensitive in vitro. In both infection and neoplasia, however, prolonged or repeated episodes of low-dose therapy can give rise to absolute resistance by the selection of biochemically resistant cells.
If insufficient dose intensity is a major cause of failure to cure, then it is possible that increased dose intensity itself can improve clinical results.354,355 This statement is phrased as a possibility rather than as a certainty because it is highly dependent on the host tolerance to the chemotherapy and also on the shape (degree of steepness and nonlinearity) of the dose-response curve for each agent for each disease. It also depends on the shape of the curve of tumor volume regression, which is considered in the next section.
The log-kill model originated from, and is expressed in terms of, exponential growth. How realistic is this pattern of growth for human cancer? Only some tumors in some special situations seem to follow this pattern. Nodular pulmonary metastases and, much less commonly, measurable lesions in other sites do seem to follow exponential growth during periods of observation that are short in relation to the total life histories of the tumors.356–358 Doubling times, ranging from 1 week to 1 year, with a median of 1 to 3 months, correlate with histologic type, growth fraction, and cell loss fraction. Yet, many, if not all, human cancers do not grow exponentially because they do not have constant doubling times.359–361
Gompertzian model of breast cancer growth.
illustrates a typical breast cancer (the specifics of this tumor's Gompertzian growth are detailed below). Between 102 cells and clinical appreciation at 1010 cells, the shape of the growth curve on the semilogarithmic plot deflects downward. An exponential curve would appear as a straight line. The progressive respective slowing of Gompertzian growth may be more the result of decreased cell production than of increased cell loss in larger tumors, that is, the fraction of dividing cells seems to decrease as the tumor gets larger, possibly in proportion to the ratio of tumor cells over total tumor volume, which decreases as volume increases as a consequence of fractal geometry.110,111,363 Failure to appreciate the existence of Gompertzian growth can lead to certain errors. For example, the false assumption of exponential growth would suggest that the tumor's doubling time when below the level of clinical appreciation is the same as that which is clinically observed. That doubling time would be unrealistically slow.364 The assumption of exponentiality has led to some unrealistic estimates of the time from carcinogenesis to the appearance of clinical disease, that is, estimates that are too long.
The biologic basis for Gompertzian growth is still unclear. An old, now unpopular, concept is that a solid tumor “outgrows” its supply of nutrients and so cannot sustain unimpeded exponential growth. This has been challenged by evidence that large tumors, with relatively slow growth rates, usually have adequate vascularity. Indeed, that may be why they are large tumors: they can induce the blood vessels (neovascularization) that allow them to grow to large size.365 A new concept concerns the relation between the cancer cell and its local environment, which includes other like cancer cells.366 Most cancers are composed of repeating elements—such as branching tree patterns or multiple nodules—that are self-similar over various scales of size. Such patterns are called fractals. The dimension of a fractal is called its mass dimension: a mass dimension of 3 means that the structure is solid and regular (like densely-packed sand); a mass dimension of 2 means that the cells are arranged in a sheet. The average mean mass dimension for a normal tissue is about 2.7, which means that the number of cells is proportional to the length of the tissue raised to the 2.7 power.367–372 The volume of the tissue is proportional to the length raised to the power of 3.0. Therefore, a fractal geometric pattern means that the number of cells is proportional to the tumor volume raised to a power ≤ 1, that power being a function of the mass dimension, say 2.7/3.0 in the above example. Cancers tend to have mass dimensions of greater than 2.7, but they vary widely, with the more malignant tumors having mass dimensions closer to 3.0. Smaller mass dimensions produce lower power constants and, therefore, low ratios of number of cells per volume of tumor. Such tumors, with relatively few cells per microscopic field, tend to be more benign, whereas cancers with higher mass dimensions (more cancer cells and little intervening stroma) tend to be more malignant. It has been shown that masses growing in a manner that preserves the power relationship between cell number and volume follow a Gompertzian curve. The rate of deviation from exponentiality is functionally related to the power constant: values close to 1 give more aggressive growth, and smaller values give Gompertzian curves that plateau at a benign size, as in ductal carcinoma in situ of the breast.366 An interesting aspect of this thesis is that a precancerous mass can suddenly become recognizable as malignant with just a small additional increment in the power constant over a certain threshold. Since the power constant reflects the mass dimension, tissues with widely varying mass dimensions can be benign, but once the power constant is close to a critical degree (about 2.85/3.00), a further small change toward increased mass dimension could result in malignant transformation. The molecular bases of the power constants that define Gompertzian growth are an active topic of study, but current hypotheses concern autocrine and paracrine growth factor loops, which might also determine invasion and metastases.366 That is, if the cells are responding to a concentration of growth factors and if that concentration is proportional to the number of cells divided by their total volume, this would be enough to explain Gompertzian growth. Clearly, rates of apoptosis, which, like rates of mitosis, vary as a function of tissue size, relate to these phenomena as well, including sensitivity to ambient soluble factors. That so many of the genes implicated in malignant transformation affect mitosis, apoptosis, and microanatomy (measured by the fractal dimension), such as adhesion molecules, provides a potential bridge between modern cytokinetics and the molecular biology of cancer.
Some of the important characteristics and implications of Gompertzian growth will be illustrated below.373
is the median curve from the family of simple Gompertzian curves mentioned above.373 Note that it takes just 3 months for the tumor to increase by two logs from 102 to 104. Yet, it takes 5 months for 109 cells to grow just one log to 1010. This is a relevant example of increasing doubling time with increasing tumor volume.The Skipper-Schabel-Wilcox model is so meaningful because it conceptualizes both tumor growth (exponential) and tumor regression (log-kill) in response to chemotherapy. We have already discussed the profound implications of the positive association between the rate of tumor regression and the dose intensity of chemotherapy. Experimental and clinical data also indicate that the rate of tumor regression is positively related to the growth rate of the unperturbed tumor just prior to treatment.378,379 This important observation is corroborated by experimental data: the logarithm of the surviving fraction of an experimental neoplasm is negatively correlated with the logarithm of the tumor size at the time of treatment.380
This log-log relationship extends the Skipper-Schabel-Wilcox model. In exponential growth, the growth rate is always proportional to tumor size. If a tumor at size X is growing at rate Y, the same tumor at size 2X would grow at rate 2Y. On a logarithmic scale, these growth rates would appear to be the same since the rate of growth per tumor size (Y/X) is the same in both cases. A rate of regression proportional to growth rate is, therefore, also proportional to tumor size, which results in a constant proportional (or “log”) kill, that is, imagine that a tumor at size X shrinks at rate Z to achieve a size X/2 in 1 week (a change in size by the proportion of one half). The same tumor at size 2X, if treated with the same chemotherapy, would shrink at rate 2Z to achieve size X in 1 week (also a change by the proportion of one half). The absolute volume shrinkage would be X/2 in the first case and X in the second case, but the proportional change would be one half in both cases (X to X/2; 2X to X). The distinction between the Skipper-Schabel-Wilcox model and the Norton-Simon model is that in Gompertzian growth, unlike exponential growth, the growth rate of the unperturbed tumor is always changing, that is, if a tumor at size X grows at rate Y, the same tumor at size 2X would grow at a rate less than 2Y.
, a realistic numeric example illustrates the implications of Gompertzian regression. In Figure 43-2A the tumor is observed to grow to clinical diagnosis at 1010 cells (about 10 cubic centimeters of packed tumor cells or about 100 cubic centimeters at a packing ratio of 1:10). Let us assume for the purposes of this illustration that 90% of the tumor is in the breast and axillary lymph nodes, and about 10% of the cells are scattered in various micrometastatic sites. The mass in the breast itself would be about 5 cm in diameter. If this mass and the axillary contents are removed completely (or destroyed completely with radiotherapy), the total body's burden of tumor is reduced to the 109 metastatic cells. Since the 109 cancer cells are spread throughout the body, they are invisible to our diagnostic tests. No adjuvant therapy is given. The tumor grows for 13.5 months until it reaches about 1011 cells in total number, which is large enough for detection as metastases. At this time, chemotherapy is employed, reducing the total cell number to about 109 (a two-log kill). A period of remission is experienced but the tumor eventually relapses, leading to death at 1012 cells.
graphs the same tumor, but here the same chemotherapy is applied in the adjuvant setting at a total tumor size of 109 cells. The relative rate of growth (the slope of the curve on this semilogarithmic plot) is faster for the tumor at 109 cells than it would be for the same tumor at 1011 cells. This is clear from inspection of Figure 43-2A. According to the Norton-Simon model, the relative rate of regression of the 109-cell tumor will be faster as well, even though the dose and schedule of chemotherapy are identical. Figure 43-2B shows that the chemotherapy that had caused a two-log kill of 1011 cells causes instead a five-log kill of 109 cells. The 104 cells that result regrow to relapse as stage IV disease at 1011 cells and to kill the patient at 1012 cells. Comparison of Figures 43-2A and 43-2B demonstrates a remarkable result. The time from surgery to stage IV is clearly longer when adjuvant chemotherapy is applied. However, the time from surgery to death is identical! The greater fractional kill in the adjuvant setting is counterbalanced by a faster fractional regrowth. This may explain why the adjuvant chemotherapy of breast cancer has less impact on overall survival (a function of eventual tumor body burden) than on disease-free survival. It may also explain why the survival duration of patients with stage IV breast cancer has remained fairly stable in recent decades in spite of more aggressive approaches to management.381–383
. If 102 cells are left instead of 104, it will take only 3.5 months longer for the tumor to reach 1012, since the growth from 102 cells to 104 cells is very rapid. Hence, adjuvant therapies can differ greatly in log kill, with only a slight impact on eventual clinical results measured years later. This slight impact could easily be lost in the “noise” caused by random fluctuations, especially in clinical data sets of small size.
The pessimistic side of this observation is that much more aggressive chemotherapy may produce little real clinical benefit. Short-course very-high-dose chemotherapy with hematopoietic stem cell rescue has been employed in an effort to eradicate all breast cancer cells in the adjuvant setting. The results of randomized trials in this regard are largely negative. At a landmark annual meeting of the American Society of Clinical Oncology in 1999, several studies of the short-course high-dose approach were presented, all negative except for one (Bezwoda) which was later retracted over issues of data veracity.384–387 The largest American trial—coordinated for the Intergroup by the CALGB—was not positive at that time, and the final paper, showing no impact on survival after adequate follow-up, is soon to be published.385 A confirmatory trial by the ECOG is expected to be reported in mid-2003.
The optimistic side of this analysis is that if the model holds, current adjuvant chemotherapies for breast cancer are actually bringing us much closer to total cellular eradication than we might otherwise be led to suspect. It is possible that the use of multiple cycles of effective therapy might be enough to bring some patients beyond the threshold of disease eradication.388 Even with drug-sensitive diseases, such as bacterial infections responsive to antibiotics, more than one cycle is almost always necessary for cure. Further studies of multi-cycle high-dose regimens have completed accrual, and some preliminary results are expected soon. Of course, it is not known if dose escalation is obligatory for the success of adequately planned multicycle regimens since other considerations—such as dose density—might prove to be more potent as a therapeutic manipulation.
The basic concept is that patient survival can be improved to a significant degree only when tumor cell populations are actually eradicated or when their regrowth is otherwise meaningfully impeded. In our previous discussion of cellular proliferation, we concluded that heterogeneity in drug sensitivity is a characteristic of neoplasia. How can tumor cell eradication be accomplished in a heterogeneous cancer? The answer may lie in the application of kinetic principles. Gompertzian regression means that slower-growing collections of tumor cells will tend to regress more slowly in response to a given therapy than will the faster-growing tumor cells treated at the same time.389 In a heterogeneous cancer, therefore, the slower-growing clones are also the most kinetically resistant. These slower-growing cells should be in the minority by the time of diagnosis because, by then, they should have been overgrown by the faster-growing cells. The existence of a population of slow-growing cells may also be the consequence of the hypothetic ability of chemotherapy to differentiate cells that are not killed.390
The best way to treat a heterogeneous population is to treat the dominant, faster-growing populations as efficiently as possible and then to treat the numerically inferior, slower-growing populations as efficiently as possible.322 As in the Skipper-Schabel-Wilcox model, the most efficient therapy is the most dose-dense therapy, giving as much drug as possible over as short a period as possible.387 This pattern of therapy is accomplished much better by sequential treatment than by strict alternation. For example, in the adjuvant breast cancer trial from Milan described above, the alternating plan, CCACCACCACCA, gave eight cycles of CMF over 30 weeks and four cycles of Adriamycin over 33 weeks.324 The cross-over, sequential plan, AAAACCCCCCCC, gave eight cycles of CMF over 33 weeks and four cycles of Adriamycin over 9 weeks. The dose density of the CMF was almost the same, but for Adriamycin the cross-over significantly improved the density. This could, by itself, account for the superiority of the AAAACCCCCCCC treatment. A similar result has also been seen in the adjuvant chemotherapy of resected osteosarcoma: Adriamycin alone was superior to Adriamycin alternating with high-dose methotrexate, presumably because the dose density of the superior agent (Adriamycin) was diluted by the alternation.391 The results of trials in acute leukemia in adults and children330 described above are also consistent with the concept of dose-dense sequential treatment plus dose escalation as a means of increasing dose intensity and thereby inproving clinical benefit.
In the breast cancer trial from Milan,324 the use of Adriamycin initially might have caused greater cell kill by avoiding the expression of the multi-drug resistance gene, which tends to progress over time, independent of treatment.392,393 Conversely, the delayed use of Adriamycin might have compromised the efficacy of two other regimens described previously: ABVD following prolonged MOPP for advanced Hodgkin's disease and Adriamycin following 6 months of CMF for primary breast cancer with low degrees of nodal involvement.309,318
Although the invention and interpretation of clinical trials intended to test cytokinetic principles are fraught with subtleties and complexities, dose-dense sequential therapy has been successful in the laboratory. The only way to cure 108 L1210 cells is by induction with cytosine arabinoside plus 6-thioguanine for two or three courses, followed by one course of high doses of cyclophosphamide and carmustine (BCNU) given simultaneously.394 In the treatment of BDF1 mice bearing the M5076 tumor, the addition of one dose of l-phenylalanine mustard (l-PAM) (a drug that by itself is only weakly active) after four doses of methyl-lomustine (CCNU) doubles the complete remission rate and the median survival.395 The presumed mechanism for this latter effect is that the few cells left after methyl-CCNU induction are l-PAM sensitive, whereas in the untreated situation, most cells are methyl-CCNU sensitive, and l-PAM resistant. In general, alkylating agents seem particularly helpful as the cross-over therapy.
Goldie and Coldman's prediction of the superiority of alternating chemotherapy assumed stringent conditions of symmetrical tumor cell numbers, growth rates, and mutation rates. Day has performed computer simulations of mutation to drug resistance under asymmetrical conditions.396 He came to a conclusion similar to the Norton-Simon model regarding the expected superiority of a cross-over, sequential plan.397 By his worst drug rule, in a coordinated two-regimen plan, the therapy with a lower cell kill per treatment (the worst drug) should be used either first or, if it is used second, for a longer duration. However, the Norton-Simon model qualifies this to specify that the induction therapy must be sufficiently cytoreductive for the residual tumor cell burden to be low. This is another possible reason for the inferiority of ABVD following dose-reduced (and, hence, less cytotoxic) MOPP, compared with a hybrid MOPP/ABV, which could be delivered at fuller dosages.318 Theory therefore supports an efficient induction followed in sequence by one or more aggressive chemotherapeutic cross-overs. Indeed, in the treatment of ALL in children, a classic trial demonstrated that induction by vincristine plus prednisone facilitates the anticancer activity of sequential methotrexate.398 The Children's Cancer Study Group (CCSG) trial in childhood leukemia that gave intensive induction, consolidation, and intensification also demonstrated the importance of an efficient initial cytoreduction.330
The concept of dose-dense sequential therapy has been applied in several important clinical trials. A pilot study in breast cancer used Adriamycin following just 16 weeks of CMFVP for patients with node-positive primary disease.399 C9344, a major adjuvant trial in node-positive breast cancer, gave doxorubicin plus cyclophosphamide with one of three dose levels of cyclophosphamide, followed by four cycles of paclitaxel or not. The justification of cross over to paclitaxel (rather than forcing a simultaneous combination that would certainly have increased toxicity) was the concept of dose density. Although escalating the dose of doxorubicin did not improve results, the use of paclitaxel in patients with estrogen-receptor negative tumors reduced the rate of recurrence by and of death to a degree comparable with the effects of CMF adjuvant chemotherapy over no therapy in the Worldwide (Oxford) Overview.263,344 This trial, coordinated by the CALGB for the Intergroup, led to the US Food and Drug Administration (FDA) approval of paclitaxel for adjuvant use. The NSABP has completed accrual to a trial of similar design. Although results are anticipated soon interpretation of this trial will be complicated by its simultaneous use of tamoxifen with chemotherapy (which could impair the chemotherapy effect). The Eastern Cooperative Oncology Group (ECOG) is now coordinating an Intergroup trial of doxorubicin plus cyclophosphamide followed either by paclitaxel or docetaxel each four doses for 3 weeks, or 12 weekly administrations of each of these two taxanes. (The weekly administration is another method of increasing dose density, although the ultimate impact of this manipulation depends upon the balance between inhibition of regrowth via frequent administration and reduced cell kill because of lower doses of drug per weekly administration.) The North-Central Cancer Treatment Group is coordinating a trial of doxorubicin plus cyclophosphamide followed by weekly paclitaxel with or without trastuzumab, the anti-HER-2 monoclonal antibody.400 The CALGB is conducting a trial of similar design in locally-advanced disease. Other treatment plans exploit the ability of hematopoietic growth factors, such as granulocyte colony-stimulating factor (G-CSF) and granulocyte-macrophage (GM)-CSF401 to increase dose density and other means of hematopoietic reconstitution to permit dose escalation.402,403 In the adjuvant chemotherapy of low-risk breast cancer, the Intergroup has recently reported preliminary results of a Southwest Oncology Group (SWOG)-coordinated study of doxorubicin followed by G-CSF-supported high-dose cyclophosphamide versus a more conventional, simultaneous doxorubicin plus cyclophosphamide combination.404 This trial showed no advantage to the dose-dense, dose-escalated sequential regimen, but interpretation is complicated by more recent observations suggesting that the efficacy of doxorubicin is capped at 60 mg/m2 and cyclophosphamide at 600 mg/m2. Investigators at the Memorial SloanKettering Cancer Center have published data about a regimen called ATC that gives dose-dense doxorubicin followed by dose-dense paclitaxel followed by dose-dense cyclophosphamide.405 On the basis of results that hint at considerable efficacy, this regimen was being compared with dose-escalated ABMT-supported treatment of women with stage II breast cancer and four or more involved axillary lymph nodes, but the study was terminated prematurely because of poor accrual following multiple negative reports on the efficacy of ABMT. The Intergroup has recently reported the results of a CALGBcoordinated 2 by 2 factorial trial that applied several types of dose density (C9741).136 All patients received postoperative doxorubicin (A), cyclophosphamide (C), and paclitaxel (T). One axis was simultaneous AC followed by T vs sequential A followed by T followed by C. The other axis was treatment every 3 weeks (conventional scheduling) vs every 2 weeks (dose dense scheduling) by the use of G-CSF. The dose-dense schedule resulted in a major decrease in the rates of recurrence and death with no increase in toxicity. Sequential scheduling was not inferior to simultaneous combination chemotherapy, noting that both delivered equivalent dose density since dose level and schedule were identical in both approaches. These results are entirely consistent with the modeling considerations discussed above. 406
For diffuse large cell lymphoma, an induction regimen with Adriamycin, vincristine, and prednisone has been followed by sequential high-dose cyclophosphamide, then methotrexate (plus vincristine), then etoposide, then l-PAM (plus total body irradiation), all with GM-CSF support. In a randomized comparison against a standard aggressive combination, the induction intensification plan proved superior in complete remission rate, failure from relapse, failure from progression, and event-free survival.407 At least one trial of dose density (chemotherapy every 2 weeks via the use of G-CSF) is underway in the treatment of malignant lymphoma.
These cytokinetic considerations may be as applicable to radiation therapy as to chemotherapy. The Gompertzian phenomenon of rapid repopulation of clonogenic cells after cytoreductive treatment is well documented in radiobiology. Moreover, clinical data suggest an acceleration of growth of the remaining viable tumor during the second part of protracted “split-course” radiation therapy.408,409 In the treatment of head and neck cancer, split-course treatment has been used to allow normal tissues to recuperate from radiation damage. In this treatment plan, it has been observed that an additional radiation dose is needed to overcome tumor regrowth during the rest interval between the split courses. The alternative hypothesis, that the higher radiation dose in split-course treatment could be needed because of increased radio-resistance of the tumor following the first part of the split course, is felt to be implausible.410 In fact, the tumor is actually more completely oxygenated during the second course of treatment, which should render it more radiosensitive.411 Hence, we are left with the likelihood of rapid regrowth between courses, more rapid than could be explained by exponential growth. The mechanism of such rapid regrowth relates to the three parameters that determine Gompertzian growth: mitotic cycle time, growth fraction, and cell loss (apoptotic) fraction. Cell cycle times of 2 to 4 days are commonly measured in head and neck cancers in the unperturbed state and after radiation therapy.412 Nevertheless, the doubling time can decrease from 60 to 4 days because of a persistence of the clonogenic cells (ie, high growth fraction) resulting from a decrease in their tendency to differentiate or die by apoptosis (ie, low cell loss fraction). It is important to note that this increase in proliferative parameters is occurring at a time of volume regression induced by the radiotherapy, that is, the observer may see cancer shrinkage, although the cells may be experiencing growth acceleration as a means of compensating for the effects of therapy. The same kinetic principles applicable to chemotherapy may be needed to overcome this potential cause of treatment failure. In this regard, a review of studies of head and neck radiotherapy has calculated the dose of irradiation needed to achieve local control in half the cases.411 This standard benchmark is consistently greater when the treatment is given over a 6-week interval than over a 4-week period,411 which is entirely consistent with the principle of dose density.
Both the Skipper-Schabel-Wilcox and Norton-Simon models are based on the observation that the rate of tumor regression is positively related to the rate of unperturbed growth. The most obvious explanation for this observation is the mitotoxicity hypothesis: tumors regress most rapidly when they are growing most rapidly because more of their cells are then synthesizing DNA and other macromolecules in preparation for mitosis. Such metabolically active cells are thereby at particular risk for cytotoxicity by drugs that interfere with such synthetic processes.413 The intuitive notion is that poisoning the S-phase renders cells incapable of progressing successfully through the M-phase. This is a dominant idea in cytokinetic thinking, and it undoubtedly has considerable merit. Growth-stimulating substances (ie, estradiol, epidermal growth factor) increase both cell proliferation and cell kill from Adriamycin in MCF-7 cells in vitro.414 Pharmacologic concentrations of estradiol enhance the cytotoxicity of the chemotherapeutic agent melphalan in hormone-responsive cell lines.415 These observations have been applied clinically, and hormone recruitment schemes have indeed resulted in high local response rates in locally advanced breast cancer.416,417 However, such treatments have proved only slightly better or no better than chemotherapy alone in metastatic breast cancer, except in data-driven subsets.418,419 Even when benefits were seen, methodologic issues have cast doubts on the analyzability of results.420
to result in a five-log kill. It may, therefore, be implausible that such significant cytoreductions are due to mitotoxicity alone. Indeed, cytokinetic analysis of MCF-7 cells exposed to low levels of Adriamycin does not show an immediate S-phase reduction.421 There is an accumulation of cells in late S, G2, and M, but also a block of the G1 to S transition starting 2 days after treatment.
Another puzzle concerns the effect of chemotherapy on normal host tissues. Chemotherapy is certainly toxic to rapidly dividing bone marrow, alimentary mucosa, and hair follicles. Yet, these tissues usually recover from the impact of chemotherapy. Some cancers, however, that are growing no more rapidly than these normal tissues may experience cytoreductions from which they never recover, that is, acute leukemias, malignant lymphomas, choriocarcinomas, and germ cell cancers may be cured by chemotherapy regimens that do not eradicate the patient's normal tissues that have comparable growth kinetics.
There is, at present, no established alternative to the mitotoxicity hypothesis that successfully relates cytokinetics to therapeutic cytotoxicity. One possibility is that chemotherapy could damage G0 cells that later exhibit their lethal injuries as they are recruited into cycle. Another, perhaps related, possibility is suggested by the thought that the hormonal therapy of responsive cancers works by growth factor perturbation, not by mitotoxicity.422 Could chemotherapy share with hormone therapy this mode of action? Long-term data on the probability of breast cancer relapse after adjuvant tamoxifen and CMF350 show similar qualitative changes.423 Breast cancer is a particularly relevant example because it is modulated by endogenous growth factors secreted by a subset of tumor cells in an individual cancer.424 The concept, however, may be generalizable since growth factors are important in many cancers. In the very genesis of cancer, malignant transformation frequently alters gene expression for growth factors, their receptors, and intracellular signal transduction proteins.425 Leukemogenic drugs, such as alkylating agents, are known to cause cytogenetic abnormalities, frequently at loci coding for products related to growth factors.426 It is even possible that the relation between tumor size and metastatic behavior, described in the context of the Goldie-Coldman model, is a consequence of the dependence of tumor cells on growth factors produced by the supporting stroma or the cells themselves.427
This discussion raises the possibility that chemotherapy, in addition to a gross mitotoxicity, might share with hormonal therapy an influence on growth factor loops.428 When hematopoietic cells are deprived of essential growth factors, they die by apoptosis.429,430 It has been well established that almost all chemotherapeutic drugs, as well as other lethal cytotoxins, also cause apoptosis.431 The existence of chemotherapy-induced apoptosis by growth factor disruption could clarify several mysteries. It could explain why the histologic analysis of breast cancers regressing after chemotherapy does not always reveal a high degree of necrosis.432 It could explain why the TLI of breast cancer appeared not to predict chemosensitivity in locally advanced disease and in the adjuvant setting.433 By implicating host-tumor paracrine interactions, the growth factor hypothesis might explain how tumor resistance to alkylating agents could be operant in vivo but not in vitro.434 The theory would not, moreover, be incompatible with mitotoxicity itself: rapidly growing cells that are dependent on growth factors would be expected to regress most rapidly when their growth support system is perturbed.
In the laboratory, chemotherapy can influence growth factor pathways. Doxorubicin, for example, may upregulate EGFR in HeLa and 3T3 cells.435 Activation of protein kinase C (an intracellular signal of growth factor ligand-receptor interaction) enhances the cytotoxicity of cisplatin without increasing drug uptake.436 In the treatment of human cancer xenografts, antibodies to EGFR, which can by themselves inhibit growth, synergize with cisplatin.437,438 Such antibodies also synergize with Adriamycin in the treatment of A431 cells in athymic mice.439 A major multi-institutional clinical trial has established that trastuzumab, which inactivates HER-2, increases response rate, duration, and survival in combination with doxorubicin plus cyclophosphamide or in combination with paclitaxel.400
A consideration of the impact of anticancer therapy on growth factor mechanisms must eventually encompass the diversity of cytokinetic features present in most clinical cancers. For example, clonogenic cells are those cells capable of inexhaustible proliferation. These are understood to have cytokinetic parameters that are markedly different from other cancer cells with more limited proliferative capacity. Although the clonogenic stem cells are overshadowed numerically by the majority of cells in the tumor, these minority cells are the most important to eliminate in order to prevent tumor recurrence from unstable remission. Malignant clonogenic cells may cycle more quickly than nonclonogenic cells, but this is usually mitigated by a high cell loss fraction. Cell loss from the clonogenic pool is accomplished by multiple mechanisms: differentiation, apoptosis, necrosis, exfoliation, and transportation away from the tumor in blood and lymph. Clearly, these cells differ biologically and cytokinetically from other cancer cells, as determined by genotypic differences that must be exploited to effect a cancer cure. It is, therefore, encouraging that the cytotoxic effects of chemotherapy might extend well beyond crude mitotoxicity. In this regard, cytokinetic analysis may play a key role in unraveling the relationships between cytotoxicity and molecular growth control. It is worthwhile to note that both aspects of cytokinetics, the study of cell proliferation, and the analysis of growth curves, are relevant to this field of inquiry.