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The Oncologist, Vol. 10, No. 6, 370-381, June 2005; doi:10.1634/theoncologist.10-6-370
© 2005 AlphaMed Press

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Conceptual and Practical Implications of Breast Tissue Geometry: Toward a More Effective, Less Toxic Therapy

Larry Norton

Memorial Sloan-Kettering Cancer Center, New York, New York, USA

Correspondence: Correspondence: Larry Norton, M.D., Memorial Sloan-Kettering Cancer Center, 1275 York Avenue, New York, New York 10021-6007, USA. Telephone: 212-639-5319; Fax: 212-717-3743; e-mail: nortonl{at}mskcc.org


   ABSTRACT
Top
 Abstract
Brinker Award for Scientific...
 Introduction
 Gompertzian Growth and the...
 Breast Tissue Geometry
 Addendum
 References
 Related Articles in The...
 
Mathematics provides greater understanding of the complex process of tumorigenesis.

Based on the Gompertzian phenomenon and the Norton-Simon hypothesis, enhanced cell kill can be obtained through a greater chemotherapy dose rate. Results from the 1995 Bonadonna et al. study and the CALGB/Intergroup C9741 study demonstrated that patients in the dose-dense arms had significantly longer disease-free survival and overall survival.

Because of the demonstrated applicability of Gompertzian kinetics, attention has been turned to the etiology of the Gompertzian curve. Breast tumor dimensions, as with all tissue dimensions in biology, can be calculated by fractals. A less cell-dense tissue usually has a lower fractal dimension than a tissue with more cells (i.e., a higher cell density is usually due to a higher fractal dimension). Density is the number of cells divided by the tissue volume. When allowed to grow, the density of a tissue with a lower fractal dimension drops quickly. However, a tumor, since it has a higher fractal mass dimension, maintains a high density as it grows bigger, resulting in a more rapid growth rate and a larger final size. Fractal dimensions of infiltrating ductal adenocarcinomas of the breast are high (i.e., 2.98), which results in a very dense tissue compared with normal breast tissue (with a fractal dimension of about 2.25). As expected, the higher fractal dimension results in a high rate of growth.

The reason for this high fractal dimension is that breast cancer can be considered as a conglomerate of many small Gompertzian tumors, each of which has a high cell density and hence ratio of mitosis to apoptosis. In mathematical terms, each component of the conglomerate can be considered a small metastasis in itself. Thus, the primary tumor is composed of multiple self-metastases that form around a seed from the tumor to itself. Conventional thinking is that cancers metastasize because they are large, but in fact it may be that they are large because they are self-metastatic.

Many genes are associated with the biology of metastasis; these include: A) obligatory cancer genes (most of which regulate mitosis and mitotic rate); B) genes relating to self-metastasis and growth of tumors at local sites, conferring the ability to invade and grow with high cell density; and C) genes that relate to the ability of the cancer to metastasize to distant areas. Additionally, fibroblasts may send out abnormal growth signals causing abnormal breast tissue growth. Consequently, we are not only dealing with abnormal cancer cells, but also with the tissue that surrounds them, or the microenvironment, that is, the "Smith-Bissell" model.

These new insights may lead us to change the thrust of our attack from genes involved in mitosis to those involved in metastasis, including metastasis to self, and to use and further improve dose-dense regimens.

Key Words. Breast cancer • Growth kinetics • Geometry • Mathematics • Gompertzian • Dose dense • Self-metastasis • Self-seeding • Mitosis • Apoptosis


   BRINKER AWARD FOR SCIENTIFIC DISTINCTION
Top
 Abstract
 Brinker Award for Scientific...
Introduction
 Gompertzian Growth and the...
 Breast Tissue Geometry
 Addendum
 References
 Related Articles in The...
 

Larry Norton, M.D.

The Brinker Award for Scientific Distinction was established by the Susan G. Komen Breast Cancer Foundation in 1992 to recognize leading scientists whose work significantly advances breast cancer research and clinical applications in research, screening, and treatment of the disease. In 2004, the Brinker Award was bestowed upon Dr. Larry Norton for his, "noteworthy advancements made in cancer research, detection, diagnosis, and treatment." The Komen Foundation’s Brinker Award for Scientific Distinction "is intended to recognize and thank researchers and scientists whose ground-breaking work allows us to understand more about breast cancer, but more important, enables many more women and men to survive the disease than ever before."


   INTRODUCTION
Top
 Abstract
 Brinker Award for Scientific...
 Introduction
Gompertzian Growth and the...
 Breast Tissue Geometry
 Addendum
 References
 Related Articles in The...
 
Along with other medical oncologists, I have spent most of my career developing antimitotic therapies, convinced that cancer was primarily a disease of aberrant mitosis or insufficient apoptosis. However, the phenomenon of abnormal cell division on which we have focused our energies, while common in cancer and clearly important in tumorigenesis, may not after all be its defining feature. I state this even though mitosis is a useful target in cancer therapeutics, and may be linked to a more basic abnormality from which further increases in cell division may arise. To properly develop this concept, we must turn to fundamental consideration regarding the nature of cancer.

Traditionally, we have seen carcinogenesis as a progression from abnormal cell morphology through increasing cell number and density (atypical hyperplasia) to the development of an invasive and metastatic phenotype [1]. More recently, biologists have been telling us that other variables play significant roles in the process [2–7]. Abnormal cells derive from precursors that often remain present in the tumor. There is a close relationship between cancer cells and their stroma, epitomized by the fibroblast. Other cells are important as part of the normal parenchyma of the breast, such as blood vessels; and the induction of blood vessel formation by stimulation of endothelial cell proliferation is a major factor in tumor progression.

The picture underlying tumorigenesis is therefore complex. However, mathematical sense can be made of it; and this is important because understanding the math can make a difference, as is illustrated by a brief account of the concept of dose density.


   GOMPERTZIAN GROWTH AND THE BENEFITS OF DOSE DENSITY
Top
 Abstract
 Brinker Award for Scientific...
 Introduction
 Gompertzian Growth and the...
Breast Tissue Geometry
 Addendum
 References
 Related Articles in The...
 
In 1825, Benjamin Gompertz developed the notion that biological growth, whether of normal organs or malignancies, follows a characteristic curve (see Addendum). Cell number increases with time, but the relative rate of increase (the rate divided by the size of tissue) falls exponentially as the mass tries to reach a "plateau phase" of very slow actual growth [8–11].

In the case of tumors, cytotoxic therapy can induce regression (often multiple regressions, with repeated cycles). However, there is always regrowth between cycles of treatment (Fig. 1Go). Through work in the clinic and the laboratory, Richard Simon and I derived the Norton-Simon hypothesis: Therapy results in a rate of regression in tumor volume that is proportional to the rate of growth that would be expected for an unperturbed tumor of that size [12]. Smaller tumors, because they are growing relatively more rapidly than larger tumors of the same kinetics, experience greater log kill (cell kill on a logarithmic scale) when chemotherapy is applied. However, because of more rapid regrowth (the essence of the Gompertzian phenomenon), the eventual outcome is the same (Fig. 1Go). The only escape from this is if all tumor cells can be eradicated (Fig. 2Go) [9,10].






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  		Figure 1. Repeated cytotoxic therapy (arrows) induces multiple regressions; however, due to the Gompertzian phenomenon, tumor cell regrowth occurs [9, 10, 12].

 


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  		Figure 2. Tumor cell regrowth can be prevented if all tumor cells are eradicated using a denser dose rate of cytotoxic therapy (arrows) [9, 10, 12].

 
This conceptual model led to the suggestion that we could make chemotherapy more efficient by giving it at a greater dose rate; for example, by giving pulses of chemotherapy more often, although not so often that we are trying "to kill cells that are already dead." The idea is that by minimizing the regrowth of cancer between cycles of treatment, we might enhance cumulative cell kill and thereby achieve greater therapeutic benefit (Fig. 2Go) [9,10]. This was readily demonstrated in experimental models [13,14]. However, it took an ingeniously simple study by Gianni Bonadonna and colleagues to demonstrate that the same held true for human disease [15,16]. The setting was the adjuvant treatment of female breast cancer patients with four or more positive axillary lymph nodes.

Sequential or Alternating Doxorubicin and CMF Regimens: The 1985 NCI Milan Study
The trial, begun in 1985, compared an alternating regimen, which the Goldie-Coldman hypothesis suggested might decrease the appearance of mutations toward biochemical resistance to chemotherapy, with a sequential regimen, which was predicted by the application of the Norton-Simon Model to maximize cell kill [17]. In the alternating arm of the study, two 3-week cycles of cyclophosphamide, methotrexate, and fluorouracil (CMF, given simultaneously) were followed by one of doxorubicin (Adriamycin®; Bedford Laboratories, Bedford, OH, http://www.bedfordlabs.com), and this was repeated for four iterations. In the sequential arm, all four 3-week cycles of doxorubicin were given first, followed by eight cycles of CMF (Fig. 3Go) [15,16].



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  		Figure 3. NCI Milan study design of the sequential or alternating doxorubicin and CMF regimens [15,16].

 
The doses of the drugs, interval between cycles, and 36-week duration of total therapy were the same in both arms. However, administration of both doxorubicin and CMF was more "dense" in the sequential arm, since, for each drug, subsequent administrations followed quickly after prior administrations of that drug or combination. (I discuss below the question of whether the word "dense" was a wise choice.) In the alternating arm, both CMF and doxorubicin were spread out more thinly over the full 36 weeks (Fig. 4Go).




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  		Figure 4. NCI Milan study (see Fig. 3Go): both CMF and doxorubicin treatment were more dense in the sequential arm than in the alternating arm of the study [15,16].

 
Contrary to the expectations of many (who predicted no difference between arms or the superiority of alternating chemotherapy based on the Goldie-Coldman reasoning), but as predicted by the Gompertzian model, the outcome was superior with the sequential regimen: the long-term follow-up data recently published show both significantly longer disease-free survival and significantly better overall survival [16]. The sequential administration of four cycles of doxorubicin followed by eight cycles of CMF resulted in a significantly lower risk for disease relapse (p = .0017) and death (p = .018) compared with the alternating regimen [16]. The concept of using drugs sequentially to achieve greater density therefore seemed valid.

CALGB 9741: Concurrent Versus Sequential, and the Role of Dose Density
Encouraged by this ongoing work, and ultimately by these results, my colleagues and I continued a series of pilot studies that sought to maximize cell kill by maximizing the dose rate of drug delivery. This development occurred at a time when much attention in our field was instead directed toward increasing the dose level of chemotherapy (i.e., the number of mg/kg or mg/m2 with each administration), sometimes requiring the reinfusion of autologous hematopoietic progenitor cells. In contrast, our work concentrated on schedule rather than dose level per administration. A major breakthrough, conceptually and practically, occurred with the development of G-CSF (filgrastim, Neupogen®; Amgen Inc., Thousand Oaks, CA, http://www.amgen.com), which permitted the administration of marrow-toxic chemotherapy in cycles shorter than the conventional 3 weeks. This eventually led to the design, conduct, and publication of a four-treatment 2 x 2 study aimed at providing a definitive test of the model [18]. The starting point of the Cancer and Leukemia Group B (CALGB)/Intergroup C9741 adjuvant trial in node-positive patients was the proven Intergroup regimen of four cycles of concurrent doxorubicin plus cyclophosphamide (AC) followed by four cycles of paclitaxel (Taxol®; Bristol-Myers Squibb, Princeton, NJ, http://www.bms.com). This prior regimen (CALGB/Intergroup C9344) gave all treatments with 3-week intervals between cycles, for a total treatment period of 24 weeks. Sequential therapy, as mentioned previously, is already a form of dose density. For two treatments in the 2 x 2 design of the C9741 trial, the two most dose-dense variations, G-CSF was used to allow the interval between cycles to be reduced from 3 weeks to 2 weeks, giving a total treatment period of just 16 weeks for the shortest of the four regimens (Fig. 5Go). I emphasize on this occasion, as I have many times previously, that these clinical trials were the results of the vision and dedication of many investigators, advocates, administrative and oversight colleagues, and volunteer patients, to whom we all owe our deepest appreciation.



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  		Figure 5. CALGB 9741 study design of the sequential or concurrent doxorubicin/cyclophosphamide and paclitaxel regimens [18].

 
The two axes of the 2 x 2 design of the C9741 trial were "pure sequential" A->T->C versus "concurrent" AC->T and "dose dense" administration every 2 weeks versus "conventional" administration every 3 weeks. The longest, of course, was the every-3-week A->T->C arm, in which four cycles of doxorubicin were followed by four cycles of paclitaxel and then four cycles of cyclophosphamide for a total treatment period of 36 weeks. In the dose-dense variant, the same drugs were delivered in the same amount and sequence but with only 2 weeks between cycles (treatment period of 24 weeks) (Fig. 5Go).

The prediction was that the more dose-dense concurrent regimen would be superior to the less dose-dense concurrent regimen, and that the same difference would be evident between the two sequential regimens. That is, the efficacies of both AC->T and A->C->T would be improved by giving treatment every 2 weeks. It was also predicted that, for a given dose density, there would be no difference between the concurrent and sequential approaches. That is, however one gives the treatments every 3 weeks, the results would be approximately the same, since the dose densities are approximately equal. (The minor qualifier here is that AC->T is a bit more dose dense than A->C->T since the cyclophosphamide is given several weeks sooner.)

Both predictions have so far been upheld. Longer follow-up will be necessary to assess if there is a slight advantage to giving AC followed by paclitaxel rather than the purely sequential regimen of doxorubicin followed by paclitaxel followed by cyclophosphamide [18]. Additionally, we expect that dose density will be more advantageous in hormone receptor (HR)-negative disease because the concept is based on cell kill from chemotherapy, which is lower in HR-positive breast cancer. (Also, any survival benefit from dose density in HR-positive disease may be masked for several years by the activity of adjuvant hormonal therapies, like tamoxifen [Nolvadex®; AstraZeneca Pharmaceuticals, Wilmington, DE, http://www.astrazeneca-us.com] in this example, and the efficacy of hormonal treatment of recurrent metastatic cancer in those cases that do develop stage IV disease.) These issues will require further, long-term analysis. In that regard, analysis of the data from C9741 at a median follow-up of about 5 years is under way.

However, at the first protocol-specified analysis, the clear result, as predicted, was the superiority of the more dose-dense approach: the every-2-weeks regimen reduced the annual odds of disease recurrence by 26% and the annual odds of death by 31%. Both effects were statistically significant at p = .01.

Lessons Learned
The development of the dose-dense approach has marked a recent step in the progressive improvement of prospects for women with node-positive primary breast cancer, especially HR-negative cases. Other stages on the way have been the benefits achieved by increasing the doses of cyclophosphamide, doxorubicin, and 5-fluorouracil used in CAF, and the advent of the taxanes. Further improvements may stem from current research aimed at: A) reducing the interval between cycles from 14 days to 10 or 11 days; B) extending the period for which anthracyclines and taxanes can be given; C) adding noncytotoxic agents such as the humanized anti-HER2 antibody trastuzumab (Herceptin®; Genentech, Inc., South San Francisco, CA, http://www.gene.com) to chemotherapy in HER2-positive cases [7, 19–24]; and D) adding antiangio-genesis agents, e.g. bevacizumab (Avastin®; Genentech, Inc.).

In reviewing the lessons already learned, we should note that attention to Gompertzian growth kinetics has led, at least in breast cancer, to better chemotherapy. "Better," in this context, is not just "greater cell kill." It is now clear that the best dose level is not necessarily the highest dose level that can be administered [25–28]. The C9741 study used dose levels that were determined by much prior research to be optimal, and in all cases these dose levels were below the maximum-tolerated dose levels. Indeed, increasing the dose level per se can be detrimental to patients, adding to toxicity but not to efficacy. Furthermore, the "best" regimens in the C9741 trial were the shorter ones, allowing patients to get back to their normal lives more quickly. Hence, "best" can be defined in terms of quality as well as length of life.

Dose-Dense Does Not Mean More Toxic
I acknowledge, in retrospect, that the choice of the term "dose dense" was not a wise one, even though it describes the principles accurately from a mathematical point of view. The reason that it is a poor term is that, to many minds, it erroneously implies "greater toxicity." This misinterpretation is most unfortunate, since it may well be just the opposite. In fact, the C9741 trial showed that, compared with conventional regimens, more dose-dense (and shorter-term) administration was associated with less neutropenia and less need for hospitalization due to neutropenic fever. With the possible exception of anemia, which was eminently treatable with erythropoietins, the dose-dense approach was also associated with a small but notably lower incidence of all other common toxicities. There is, even to the present day, no greater incidence of myelodysplasia or leukemia associated with the every-2-week administrations. There is no evidence of greater cardiotoxicity. Indeed, even if disease-free or overall survival was not better, the dose-dense approach would still be preferable because of less toxicity, which also ultimately reduces costs and leads to a faster return to a normal life. When I am sometimes asked "Does this case require dose-dense chemotherapy?" I often answer "Why would you choose to treat this patient with a more toxic regimen, one without dose density?"

What Underlies Gompertzian Kinetics?
In addition to leading us toward more effective, less toxic treatment, Gompertzian analysis may be suggesting a radically different concept of cancer, which may—in the long run—be a more important development. In the early portion of the Gompertz curve, in which cell numbers rapidly increase, mitosis seems to outweigh apoptosis. Toward the plateau phase, they begin to balance each other. However, it is an open question as to whether this is because of a reduction in mitotic rate or an increase in apoptosis, or some other phenomenon. Findings relating to one process or the other, and the genetic mechanisms that regulate them, are nowhere near as consistent as the basic phenomenon itself. My thesis in the rest of this lecture is that the explanation may lie in an unexpected place, in the very geometry of the tumor.


   BREAST TISSUE GEOMETRY
Top
 Abstract
 Brinker Award for Scientific...
 Introduction
 Gompertzian Growth and the...
 Breast Tissue Geometry
Addendum
 References
 Related Articles in The...
 
Geometry is the mathematical art of the measurement of objects, one aspect of which is the calculation of dimensions. According to Euclid, who studied idealized objects, a point has a dimension of zero; a line has a dimension of one, its length; a plane (or sheet) has a dimension of two, its length and height; and all solid objects have three dimensions, length, height, and depth. However, how does one quantify the dimensions of biologic entities, which are more than sheets, in that they have depth, but don’t fill up all the three-dimensional space they occupy, as do solids? These are objects whose dimensionalities lie somewhere between a plane and a solid object. Their dimensionality can be calculated in fractals [29–34].

As the length of an object that exists in three-dimensional space increases, its volume increases as the cube of the length. If that object is solid, the ratio of its mass to its volume remains one as the object increases in length. However, cell mass (or cell number) in biologic entities increases as a function of the length raised to a fractal dimension that is less than three and usually greater than two. As illustrated in Figure 6Go, we see a graph of the volume of a biologic entity (with a cell mass dimension of 2.5) as a function of length, and the corresponding relationship between time and cell number. As a consequence of the mass fractal dimension being less than three, as the length increases, the ratio of cell mass to volume, that is, the cell density, decreases. That is, the cell density is the ratio of the length raised to a power less than three divided by the length raised to the power of three. If the fractal dimension were higher, say 2.8, the cell mass would be closer to the volume, and the cell density would decrease less quickly with time (Fig. 6Go).




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  		Figure 6. As length increases, volume (dimension [D] = 3) increases faster than cell mass (D = 2.5). At a fractal dimension of 2.8, cell mass is closer to the volume (D = 3) and cell density decreases slower with time.

 
This theoretical relationship is confirmed when one analyzes actual biologic data. For example, Pierce and colleagues in the laboratory of Hal Moses have published whole-mount photographs of mouse breast ductal trees in the wild-type state and in transforming growth factor beta (TGF-ß) hyperexpressing transgenics. My analysis of these data is that the normal breast has a fractal dimension of 2.5 while the breast in the transgenic mouse model has a fractal dimension of 2.1. What is striking about these data is that the wild-type breast is larger. That is, higher fractal dimension results in larger organ size, which is logical considering that the density of the tissue is preserved longer so that more cells per unit of volume are contributing to the growth. The cell density in the TGF-ß transgenic mouse model, with the lower fractal mass dimension, drops off quickly, leaving few cells to divide to contribute to increasing breast volume [35].

The same phenomenon is mirrored in data from human breast cancers. With Carlos Cordon-Cardo and his colleague Gloria Juan at Memorial Sloan-Kettering Cancer Center, we have shown that the fractal dimension in infiltrating ductal adenocarcinoma is greater than that in normal breast tissue. Read off an ordinary hematoxylin and eosin-stained microscope slide, an adenocarcinoma might have a high fractal dimension, say 2.98. This is very dense compared with the dimension of 2.25 of normal breast tissue. Density is better preserved in the tumor than in the normal cell line, giving it a higher rate of growth and a much, much larger potential tissue size.

Staining studies have show that this is true both of mitotic and apoptotic cells. There is a common expectation that apoptosis is deficient in cancer. In fact, this is generally not the case: apoptosis is normal or actually enhanced [36–39]. However, the fractal dimensionality of the mitotic compartment is increased to a greater extent than that of the apoptotic compartment. This is the explanation for why the tumor grows at the expense of normal tissue.

There is another aspect of the fractal biology of growth that it especially fascinating. Regardless of the magnitude of the fractal dimension, smaller tumors have a higher cell mass to volume ratio (or density) than larger tumors. This actually might explain why cancers have higher total fractal dimensions than normal tissues. A normal breast is a cohesive organ, organized and complete as a whole. However, a breast adenocarcinoma does not have the appearance of a cohesive organ, but rather seems to be a conglomerate of small, contiguous tumors. The elements of the conglomerate, being small, have high cell densities, explaining the overall high cell density of the malignant lump. Furthermore, if each component of the conglomerate follows a Gompertzian growth curve, the relative growth rate of each will be rapid, so the relative growth rate of the conglomerate would be proportionately more rapid. In other words, the geometry of a tumor as a conglomerate rather than an organized mass could explain both increased cell density and increased growth rate, two of the principle characteristics of cancer.

How do these small components arise? Perhaps they are metastases from the primary tumor mass not to a distant site (as we usually think of metastases), but rather to the mass itself!

The Concept of Self-Metastasis
According to this model, the primary tumor itself is composed of multiple self-metastases, each of which may be started by a stem cell (using the term loosely, these "seeds" may sometimes be committed progenitor cells incapable of self-renewal) that has traveled a short distance from its site of origin (Fig. 7Go). Because the progeny of a traveling seed cell forms a small mass (only a tiny part of the conglomerate), and because each small mass—being small—has a high cell density, the total conglomerate mass that sums all the small masses must have a high cell density. Additionally, because each component small mass follows its own Gompertzian curve (perhaps a consequence of its high cell density), each component mass and the total conglomerate has a high ratio of mitosis to apoptosis. This explains the rapid growth of the total tumor. Taking this one more step, the genes that allow self-metastasis may be part of the collection of genes that allows a stem cell to metastasize to distant sites. That is, the ability to self-seed may be far along the way toward the ability to distant seed, that is, metastasize distantly. Once in a distant metastatic site, the process can start all over again, with the tumor in that site growing by a process of metastasis (seeding) to the self-same mass. For this reason, the mass in the distant metastatic site would also grow rapidly to a large size, that size being a conglomeration of many Gompertzian curves.






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  		Figure 7. The self-metastasis model.

 
The bottom line is this: We have tended to think that cancers metastasize because they are large; the reality may be that they are large because they are self-metastatic (Fig. 8Go).



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  		Figure 8. The concept of self-metastasis.

 
This concept can explain an increased growth rate even when the growth fraction is not very large: ten subtumors each of size x, each growing at a certain rate, y, will, as a conglomerate of size 10x, grow faster, at rate 10y, than one tumor of size x growing at the same rate y. The concept explains cell density, because what we have is not one tumor but a collection of tiny tumors. And it explains the appearance of invasion, because self-metastasis into surrounding tissue contiguous with the bulk of the tumor will look the same as a direct extension into that space.

This hypothesis has a molecular aspect. For example, Laura van ’t Veer and colleagues in Amsterdam, as well as many others, have identified genes that are associated with a poor prognosis in breast cancer [40]. Many of these genes are associated with the biology of metastasis. They include metalloproteases, signaling molecules that are involved in cell adhesion as well as cell division, genes relating to angiogenesis, and other genes related to the interaction between the cancer and its environment. That is, many of the genes that are expressed in poor-prognosis cancers are not directly related to the regulation of mitosis and apoptosis.

It is logical to assume that these same genes are responsible for the biologic behavior of the cancer that leads to a poor prognosis, including and especially the ability to metastasize to distant sites. Yet work by Kang and colleagues in the laboratory of Joan Massagué leads to another conclusion [41]. Human breast cancers have been expanded in culture and passed sequentially in mice to develop a subset of cancers with a very high potential for metastasis to bone. Microarray profiling demonstrates that many of these tumors express genes found in van ’t Veer’s poor-prognosis set. However, the genes that best distinguish the parental lines from those that are highly metastatic to bone (and indeed other sites also) are not part of the poor-prognosis profile! These additional genes are also almost exclusively ones that concern the relationship between the cancer cell and its microenvironment. But if the microenvironmental genes in the poor-prognosis set are not responsible for the distant metastases, what is their function in the primary tumor? Perhaps their function is to permit metastases of tumor cells not to distant sites but back to the tumor itself.

Hence, as illustrated in Figure 9Go, the full hypothesis is as follows. Cancer is the manifestation of a multigenetic picture. First, there are obligatory cancer genes, most of which regulate mitosis and apoptosis and are often expressed in pre-neoplastic states—such as ductal or lobular carcinoma in situ—as well as in frank malignancy. Second, there are genes (such as many of the ones included in various poor-prognosis sets) that relate to self-metastasis, promoting the growth of a tumor at its local site, including its ability to invade, achieve high cell density, and exhibit a high mitosis/apoptosis ratio. Third, there are additional genes that relate to the cancer’s ability to metastasize to a distant site such as bone, lung, or liver. This multigenetic, stepwise, evolutionary concept of breast cancer is reminiscent of one first proposed by Helene Smith and colleagues several years ago.






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  		Figure 9. Understanding tumor development through the Smith-Bissell model [42–46]. Abbreviations: HGF = hepatocyte growth factor; TGF-ß1 = transforming growth factor beta one.

 
Concerning the relationship between the cancer per se and its environment, recent data presented by Kuperwasser et al. from Bob Weinberg’s laboratory in Boston are quite intriguing [42]. In that work, normal breast tissue from reduction mammoplasty was placed in cleared mammary mouse pads. If normal fibroblasts were admixed, normal breast tissue growth ensued. However, if the fibroblasts were engineered to overexpress hepatocyte growth factor or TGF-ß1, what resulted were true breast cancers, histologically identical to clinical cases, including blood vessels and cellular heterotypia (Fig. 9Go).

This experiment suggests that some normal breast samples may contain foci of partially transformed tissue that are quiescent only as long as they are in contact with fibroblasts sending out normal growth signals. The full display of biologic behavior depends upon the interplay between the parenchymal epithelial cells of the breast and stromal cells that are intrinsic to their milieu. It emphasizes that, when we talk of cancer, we are dealing not just with abnormalities of the cancer cell but also with abnormalities of the parenchymal–stromal interface [43–46]. Mina Bissell and colleagues have been developing such concepts for many years. Indeed, it is important that we include elements of Mina’s ideas in any formulation of a molecular basis for a self-and-distant-metastasis model of cancer. Hence, the model of Figure 9Go should, in historic context, properly be labeled the Smith-Bissell model.

Implications
Just as Gompertzian mathematics formerly led us toward the concept of dose density, which seemed counterintuitive to many at that time, they are now leading us toward a new concept in which abnormal cell mobility—loss of anchoring, invasion, migration, readhesion, and proliferation appropriate for a small mass—dominates mitotic and apoptotic irregularities as a prime defect in cancer. How the idea of self-metastasis relates to advances in cancer stem cell biology remains to be determined. Yet these new insights may lead us to change the thrust of our attack from the gene products involved in mitosis to those involved in self and distant metastasis. After all, if the fractal and Gompertzian theses are correct, higher mitotic rates may merely be the passive result of the active change of increased cancer cell mobility. That is, the real problem may be self-metastasis masquerading as increased mitosis.

The two ideas derived from Gompertzian kinetics may not be so divergent after all. The seeds of self-metastasis must divide to form the small Gompertzian masses that in conglomerate populate the whole tumor mass. Perhaps dose density is the best way to perturb these cells with their ability to repopulate rapidly, especially were we to have agents that selectively disrupt the biology of migratory, self-seeding "stem" cancer cells. We need to apply all of our weapons against cancer, not forgetting dose scheduling while we turn so enthusiastically to targeted molecular therapeutics, not ignoring the lessons of the past as we charge headlong into an exciting—if sometimes confusing and unpredictable—future of breast cancer management.


   ADDENDUM
Top
 Abstract
 Brinker Award for Scientific...
 Introduction
 Gompertzian Growth and the...
 Breast Tissue Geometry
 Addendum
References
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Benjamin Gompertz

The Gompertz family were merchants who—like many others in the late 1700s—left Holland to try their luck in a London booming with trade. Three brothers were born in the new family home, in Spitalfields, with Benjamin arriving in 1779. The young Gompertz showed a prodigious ability in mathematics, but there was a major obstacle. The family was Jewish, meaning Benjamin was denied entrance to university. He set about educating himself, learning mathematics by reading Newton and Maclaurin. He was greatly helped in his mathematical education by the Spitalfields Mathematical Society which was later to become the London Mathematical Society.

Gompertz married Abigail Montefiore who came from a wealthy Jewish family with strong links with the stock exchange. Gompertz himself joined the stock exchange in 1810 and he became a Fellow of the Royal Society in 1819. The following year he read a paper to the Society which applied the differential calculus to the calculation of life expectancy. Gompertz was now wedding his mathematical brilliance to the cold science of insurance, and in 1824, the year his 10-year-old son died, he was appointed as actuary and head clerk of the Alliance Assurance Company. And in 1825, he observed that after the age of 20, there was a doubling of the "force of mortality" every 7 years. The Gompertz equation reflected the cellular and molecular deterioration that pushes us towards disease and death.

The Spitalfields Mathematical Society had been founded by Joseph Middleton, a marine surveyor, in 1717. Unlike grand institutions such as the Royal Society, there was no difficulty becoming a member. The Spitalfields Mathematical Society met in pubs around the East End and anyone could join in. There was just one rule. Anyone receiving tuition from the Society had to make himself available in turn as a tutor. If a fellow member asked a question, their colleague had to endeavor to find an answer – or pay a fine of one penny. This system of peppercorn fines established a marvelous cooperative spirit, and the free flow of ideas back and forward created a fertile breeding ground for mutual education and the development of new ideas and theories. In short Gompertz was working out, for every age of a person, how likely they were to die. The seeds of Gompertz’s Law of Mortality had been sown.

This geometric progression led to the plotting of a straight line, known as the Gompertz Function. Gompertz had plotted the actuarial principles which would underpin life insurance to this day. It is the reason premiums go up and dividends go down as one gets older. It is a stark fact of ageing: the older we get, the greater our risk of contracting serious diseases and dying. This may seem commonsense, but the unwelcome truth was first cast into mathematical form by the amateur London East End mathematician Benjamin Gompertz. Ironically, Gompertz himself went off the scale. He lived to the ripe old age of 86, dying in 1865.


   ACKNOWLEDGMENT
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I gratefully acknowledge the Susan Komen Foundation for the opportunity to present this Brinker Award Lecture and the many basic science and clinical colleagues and others who have contributed over the years to what has been a genuinely communal, collaborative, and cumulative research effort. Above all, I acknowledge the courage and vision of the patients who have participated in the many clinical studies needed for us to reach our present state of understanding of breast cancer. Along with the funding foundations and the breast cancer advocacy movement, they are responsible for all of our advances in breast cancer biology, therapeutics, and prevention.

DISCLOSURE OF POTENTIAL CONFLICTS OF INTEREST
The author indicated no potential conflicts of interest.


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Received April 4, 2005; accepted for publication April 29, 2005.




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