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Researchers Suggest That Universal ‘Law’ Governs Tumor Growth

Mike Martin

Modeling the growth and development of tumors with specialized software, mathematics, and biologic data is a burgeoning area of cancer research. Computers also use mathematical models to account for blood vessel growth, metastasis, and tumor type to generate complex 3-dimensional images of rapidly growing cancers. Using such techniques, researchers have extensively modeled brain, vulvar, and mammary tumors.

Biological scientists seeking to model nature have increasingly turned to "fractals"—jagged geometrical arrays that never simplify or smooth out, no matter how close you look at them. Snowflakes, coastlines, falling rain, and disease transmission in large populations all exhibit fractal complexity.

Fractals and tumor modeling have recently merged in a simple and ingenious "universal growth law." Originally formulated for normal organisms, this growth law may also apply to benign and metastatic tumors.

Such a theorem could have "far-reaching implications concerning the mechanism of tumor metastasis and recurrence, cell turnover, angiogenesis, and invasion," said Harvard University researcher Thomas Deisboeck, M.D., and a team from the University of Turin in Italy—Caterina Guiot, Ph.D., Pier Giorgio Degiorgis, Ph.D., Pier Paolo Delsanto, Ph.D., and Pietro Gabriele, M.D.

The Elegant Universal

The growth law makes an elegant and plausible assumption: The fractal nature of angiogenesis inherently limits tissue growth.

"It is based on the premise that the tendency of natural selection to optimize energy transport has led to the evolution of fractal-like distribution networks," wrote physicist Geoffrey West, Ph.D, and biologists James Brown, Ph.D., and Brian Enquist, Ph.D., in a 2001 paper in Nature that described a general model for tissue growth.

Branching capillaries coursing through a comparatively smooth organ or limb illustrate the idea. Where the mass, m, of the limb grows in proportion to a whole number exponent, m1, the capillary mass grows at a fraction of the rate—m3/4, for instance.

"The 3/4 exponent is well supported by data on mammals, birds, fish, mollusks, and plants," West and his co-authors noted.

Inherently less than the whole number growth rate of supplied tissue, the fractional, or fractal growth rate, of the blood supply creates a natural brake on further development.

"This imbalance between supply and demand ultimately limits growth," West and his co-authors claimed.

Factoring metabolic rate, body mass, and the fractal nature of blood vessel growth for hundreds of organisms, West and his co-authors plotted a simple curve they termed "universal." (see figure).



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In 2001, Geoffrey B. West, Ph.D., James H. Brown, Ph.D., and Brian J. Enquist, Ph.D., published their universal growth curve, which predicts that growth curves for all organisms should fall on the same universal parameterless curve. (Source: Nature 2001;413:628–31. Access at www.nature.com.) (© 2001 Nature Publishing Group. Reprinted with permission.)

 
"All species, regardless of taxon, cellular metabolic rate, or mature body size, should fall on the same parameterless universal curve," West and his colleagues predicted.

Tumor Growth

Deisboeck—director of the Harvard-MIT Complex Biosystems Modeling Laboratory—and his research team have now applied the growth model to tumors.



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This year, Thomas Deisboeck, M.D., and colleagues applied the same principles of a universal tissue growth law to tumor growth. They generated the graph above from breast and prostate tumor samples from patients. (Source: http://www.arxiv.org/ftp/physics/papers/0303/0303050.pdf) (Reprinted with permission.)

 
"It is very natural that the ‘universal’ growth theory we presented should be extendable in some appropriate form to cancer," said West, of the Los Alamos National Laboratory in New Mexico.

That form includes a variable that describes "the tumor’s ability to metastasize or invade, or its affinity for nutrient uptake," Deisboeck explained.

"The results from a variety of in vitro and in vivo data support the notion that neoplasms also follow a universal growth law," Deisboeck and his colleagues concluded. "The data fit the universal growth curve very well."

Following the Curve

There is no reason why tumors should follow a fundamentally different growth pattern than healthy organisms, said Reinhard Ebner, Ph.D., principal scientist at Avalon Pharmaceuticals in Germantown, Md. "If anything, because they sometimes consist of relatively homogenous assemblies of clonal, often identical cells (e.g., early-stage colonic neoplasms), solid tumors should follow a universal growth curve even more closely, especially in the primary, pre-metastatic phase," he said.

Cancerous cells, for example, share universal traits and metabolic requirements.

"All cells in the body require oxygen, and the distance that oxygen can diffuse out of a red blood cell within a vessel is universally limited," said Isaiah Fidler, D.V.M., Ph.D., of the University of Texas M. D. Anderson Cancer Center, Houston. In 2002, Fidler published data showing that in human patients with lung cancer and brain metastasis, all dividing viable cells are within 100 microns from the nearest capillary and all apoptotic cancer cells are located at distances exceeding 150 microns from the nearest capillary.

Debating the Merits

Although a universal growth law may govern both normal and cancerous tissue, experts debate the role of the vascular system in such a hypothesis.

"The generic growth law can equally be applied to a symbiotic situation such as tumor or fetal growth," West said. "In all cases, the vasculature is hierarchical and fractal-like, though there may be interesting and important differences."

Ebner disagreed. "To conclude from the ‘fractal look’ of a capillary network that capillaries always grow at a fraction of the rate of tissue is misleading and ignores the possibility that some components within the tissue architecture also bear fractal growth characteristics," Ebner said. "As for tumors, the general view of cancer geneticists is that the complexity of a tumor genome does not lend itself well to applying a universal law that rescales tumor masses and growth times as directly proportional to rates of neovascularization and angiogenesis."

Growth factors, too, can supersede supply constraints in growth regulation, Ebner added.

"This can be observed in the chicken chorioallantoic membrane (CAM) assay, where capillaries form at a rate much faster than the tissue they supply if treated with angiogenic agents," Ebner said.

Apples and Oranges?

Some experts also worry that the old "apples and oranges" comparison may apply to cancerous and normal cells, hampering attempts to universalize their basic features.

"Cancer is a heterogeneous disease," Fidler said. Unlike normal tissue, "each cancer consists of heterogeneous subpopulations of genetically unstable cells."

Deisboeck concurred that "the metabolic rate of a single cell may vary between tumor cell subpopulations, e.g., according to the cells’ location within the tumor." To account for this difference, "we would have to incorporate a ‘spatial dimension’ into our model," he said.

Normal cell growth is also "much more tightly regulated, and the cells respond to external signals," said oncologist Clay Anderson, M.D., director of clinical services for the Ellis Fischel Cancer Center in Columbia, Mo. Cancer cells, on the other hand, "have multiple genetic and phenotypic abnormalities that allow unregulated growth," Anderson explained.

Regulated or not, however, "our model should apply to tumor growth," said Brown, of the University of New Mexico.

The growth hypothesis presents two possible ways to influence or predict a cancer’s course: calculation of both a therapeutic agent’s lethal dose and an approximate time of tumor metastasis.

"The extension of West’s model to tumors suggests the rate of cell turnover at any developmental stage can be predicted," Deisboeck explained. Clinicians can therefore use the growth theory, he said, to determine the "lethal dose of a given therapeutic agent at any time the dose for which the cell turnover rate vanishes."

However, using the growth law to calculate lethal doses in patients with multiple tumors may be problematic. "Would it require measuring cell turnover rates in all or most of the tumors?" Ebner asked. He noted that Deisboeck and his colleagues have not addressed how this could be achieved.

The proposed growth law also predicts "the critical tumor volume at the onset of metastasis and invasion," Deisboeck said. Coupled with clinical data, he added, the growth law hypothesis may be used to predict when a neoplasm would reach such a volume.

"In principle, metastasis can be linked to a critical tumor volume, as this volume can be a reflection of how many mitoses the tumor cell population has completed," Ebner said. "However, dissemination of primary tumor cells can also be achieved by many other means, including mechanical spread following surgical removal of the primary tumor."

Proposed Amendments

Deisboeck and his team "show that the growth of various cancers fit our growth curve, and I like what they have done," said West, who nevertheless wants the research to answer such questions as, "What is the tumor’s dependence on the mass of the host?"

Although Anderson noted that he "probably cannot be convinced" that a universal law governs tumor growth, he said he could be further persuaded by data from animal studies showing that the "therapeutic implications of the universal growth pattern are truly independent of all other phenotypic characteristics that tend to create bias."

More research doubtless lies ahead, but Deisboeck’s work on a universal law of tumor growth "represents a pioneering first step," Brown said.



             
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