Can a differential equation cure cancer?

A fascinating article in Forbes suggests that using mathematics may be able to create drug combinations that are far more effective than the ones now in use.¹ "I have a suspicion that we are using almost all the cancer drugs in the wrong way," Larry Norton, Memorial Sloan-Kettering Cancer Center deputy physician-in-chief for breast cancer, says. "For all I know, we may be able to cure cancer with existing agents."²

Norton says that current researchers tend to focus on "identifying cancer-causing genes rather than writing differential equations to describe the rate of tumor spread," how fast tumors grow, and how quickly cancer cells develop resistance to therapy.

In the 1970s Norton worked with a statistician to come up with a new model for tumor growth based on the work of 19th-century mathematician Benjamin Gompertz. Gompertz's curve is an equation that models a time series where growth is slowest at the start and end of a period - an S curve. Researchers in the 1960s modified the equation to better fit what was known about tumor growth.

Microscopic tumors below a certain threshold barely grow at all. Small tumors grow exponentially, but the rate of growth slows dramatically as tumors get bigger, until it reaches a plateau. A corollary of this: The faster you shrink a tumor with chemo, the quicker it will grow back if you haven't killed it all. Based on these rates of growth, Norton argued that giving the same total dose of chemotherapy over a shorter period of time would boost the cure rate by limiting the time tumors could regrow between treatments.

Although it took him decades, Norton finally proved his theory was correct - "in 2002 a giant government trial showed that giving chemotherapy every two weeks instead of every three lowered the risk of breast cancer recurrence by 26% over three years, even though the two groups got the same cumulative dose."

Unfortunately, as Forbes points out, designing better treatment schedules doesn't get as much credit as the glamorous business of inventing drugs.

Norton and colleague Joan Massague are using Gompertz's curve to tie together tumor cell growth and metastases. Massague, a biologist, was studying metastasis and discovered that none of the genes implicated in the spread of cancer to distant organs had to do with excessive cell division - instead, they all related to the ability to infiltrate and adapt to new environments.

The finding seemed to contradict doctors' impression that the fastest-growing tumors are also the most likely to spread. Pondering how to reconcile the two ideas, Norton and Massagué theorized that tumor cells released into the bloodstream sometimes are attracted back to the original tumor and help it expand.

Self-seeding may explain why large tumors tend to grow (in percentage terms) more slowly than small tumors: It could be that growth is a function of surface area rather than volume. Tumors that are efficient seeders may kill people by promoting the seeding process, not because they have a higher exponential growth rate.

In a paper from the December 24 2009 issue of Cell, Massague and colleagues state that self-seeding of breast cancer, colon cancer, and melanoma tumors in mice is preferentially mediated by aggressive circulating tumor cells [CTCs], including those with bone, lung, or brain-metastatic tropism [growth in response to a stimulus]. They found that the tumor-derived cytokines IL-6 and IL-8 act as CTC attractants, whereas MMP1/collagenase-1 and the actin cytoskeleton component fascin-1 are mediators of CTC infiltration into mammary tumors. They also showed that self-seeding can accelerate tumor growth, angiogenesis, and stromal recruitment through seed-derived factors including the chemokine CXCL1. "Tumor self-seeding could explain the relationships between anaplasia [loss of differentiation of cells], tumor size, vascularity and prognosis, and local recurrence seeded by disseminated cells following ostensibly complete tumor excision." That last part refers to the surprising reappearance of a tumor after a surgeon has, ostensibly, removed the tumor. It's not for lack of skill of the surgeon, in other words, but perhaps a factor of the tumor itself.

Now for the hard part - coming up with drugs that block tumor seeding. Massagué and Norton have identified four genes involved in seeding and are testing for drugs to block them, Forbes says. "Convincing drug companies to go along could be difficult; it's easier to see whether a drug shrinks tumors than to see whether it stops evil cells from spreading. But Norton believes that doing this hard work may be the key to a cure."

In an Epub ahead of print on February 15 2010 from the Journal of the American Chemical Society, Massague and colleagues show that an analog of migrastatin, called migrastatin ether, exhibits a concentration-dependent inhibitory effect on migration of breast cancer cells both in vitro and in vivo.



Migrastatin was so named for its ability as a migration inhibitor. While searching for potent analogs of migrastatin, Massague and colleagues found that deletion of the entire glutarimide side chain, as well as the α,β-unsaturated lactone moieties, from the migrastatin-like structure did not abrogate inhibition of cell migration (see red and blue portions, above). Interestingly, the migrastatin ether only has its effects on migration, not cell proliferation or viability.

Mice treated with varying doses of ME showed significantly and drastically reduced metastases (88%-93%), and 40% survival (low dose) and 50% survival (high dose) in treated groups versus 0% survival in the control group. Mice with already-circulating metastatic cells were also treated with ME, and the outcome trended toward inhibition of metastatic outgrowth (high variability in the cohort affected statistical significance). Once metastasis had already occurred, ME did not suppress metastatic tumor development - this is a logical finding, as ME inhibits migration, not proliferation or viability.

The research is literally brand new, so there is much more to be done. But this is incredibly exciting, as metastatic cancers are so deadly, and if a drug can inhibit metastases the patient has a much better chance at survival.

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¹ For those mathematically inclined, here's a page full of great math quotes and comics (even Calvin&Hobbes, for Josh).
² This is not to say that you can generically cure cancer. There is no magic bullet that will cure every patient of every cancer.