Adapted from Y. Iwasa, et al., Proc R soc Lond B Sci, 270: 2573–8, 2003.
In this example, genes can mutate at the rate u per gene, per cell division. Gross chromosomal changes occur at the rate p. For this tumor suppressor gene, TSP, Cancer cells have two functioning alleles prior to chemotherapy. Cell reproductive rates Ro-R4 are assumed equal, and a = R/(1-R). Therefore, the critical population size before escape from the therapeutic could happen, N*, can be represented by the equation N* = 1/[a(1 + 2a)u(u + 2p)]. Supposing a given u and a given R, cancers with chromosomal instability, defined by increased rate p, have a much lower critical population size for escape from chemotherapy.
Natural selection remains modern biology's leitmotif 145 years after Darwin published
CHANGES UNDER PRESSURE
Tumors can adapt while under selective pressure from anticancer drugs. James Goldie, clinical professor emeritus at the University of British Columbia, recalls treating a man with liposarcoma of the throat in 1969. Initially, the tumor responded to standard methotrexate doses of between 5 and 10 mg per day. After the disease recurred, 500 mg of methotrexate daily produced a "spectacular improvement." After a later recurrence, doses as large as 20,000 mg failed to stop tumor growth, Goldie says. "This terrible sequence of events was, and continues to be, the all too frequent outcome."
Developing mathematical representations of tumor resistance involves modeling systems researchers describe as both complicated and complex. Victorian engineers built complicated steam engines with thousands of precisely engineered, intricately linked working parts. But a steam engine does not adapt, evolve, or become more than the sum of its parts. In contrast, a tumor is complex. Its complicated molecular biology adapts, allowing the malignancy to escape pharmacological or natural regulation.
Intracellular and extracellular fluxes make mathematical analysis all the more difficult. "Malignant cells become genomically unstable as DNA replication runs amok," says Yves Moreau, University of Leuven, Belgium. "This instability means that there is not one model of the cell; the model itself is dynamic." Researchers have yet to develop a single coherent mathematical model of tumorigenesis, but they are beginning to model several key aspects of tumor development and the evolution of resistance.
Harvard University's Martin Nowak, for example, recently published a mathematical analysis that determines the probability that a cancer will produce mutations, enabling it to escape selection pressure.1 With colleagues at Kyushu University in Fukuoka, Japan, Nowak used a technique called multitype branching to analyze the impact of events that destabilize the genome. According to the model, cancers with unstable genomes (through nucleotide polymorphisms, gross chromosomal changes, or recombination, for example) show a lower critical population before resistance emerges, compared to wild type. The model allows researchers to estimate the probability of a treatment's success on rapidly evolving cells.
Other models examine particular aspects of resistance. For example, redundancy protects cells against mutations: several genes may contribute to the same phenotype; the genome may contain several copies of the same gene; and redundancy in the genetic code (where multiple codons encode the same amino acid) minimizes the impact of mutation. These and other mechanisms allow biological systems to absorb and conceal genetic damage.
David Krakauer of the Santa Fe Institute argues that in rapidly dividing cells showing a "hypersensitivity to mutation," another strategy, converse to redundancy, might be just as important in ensuring tissue stability.2 Apoptosis offers an example of one of these so-called antire-dundancy mechanisms that remove rather than hide mutants.
Krakauer uses a mathematical quasi-species model to study antiredundancy.2 In it, cancer cells are modeled as if they behaved like a multitude of genetically heterogeneous variants, or so-called quasispecies, with copious mutations that could confer a selection advantage. "We need to be aware that at any one time a cancerous tissue is a highly heterogeneous mix of genetic strains, rather than a single mutant, and this has serious consequences for chemotherapy," he explains.
For example, apoptosis contributes to cancer pathogenesis, chemotherapeutic effectiveness, and tumor resistance, making it a crucial target for researchers. "Interfering with apoptosis is one of the key steps in transforming a cell into the malignant state," Goldie notes. Most cancer chemotherapeutics either interfere with DNA synthesis or damage the mitotic spindle, both of which can trigger apoptosis. Furthermore, blocking the transmission of proapoptotic signals might be, he says, "a very common" resistance mechanism.
Researchers targeting angiogenesis may also find their mark through mathematical modeling, though it's far from being an easy shot. "Modeling angiogenesis involves describing a spatial population of tumor cells and their interactions," Moreau comments. "This type of mathematical modeling is notoriously difficult."
Wodarz and Krakauer used mathematics to show that genetic instability, which they define as an increased mutation rate among somatic cells, selects cell lines that promote angiogenesis. The mutated cells overcome the factors produced by nonmalignant tissue and tumor cells that inhibit blood vessel formation.3
To improve the models, Wodarz is currently examining the intricacies of cancer cells' dependence on the blood supply. "An intriguing example is breast cancer," Wodarz says. "Even if the tumor can initially be removed or destroyed, it can come back after 10 or 20 years. It seems that the cells are in some kind of dormancy, possibly because of the lack of blood supply, that prevents them from growing, but which might also prevent them from being eradicated by treatment or immunity."
Wodarz collaborates with other researchers to validate his models and predictions using in vitro, animal, and clinical studies. He says he's setting up a yeast lab to complement the theoretical work.
MATH BY THE BEDSIDE
Ultimately, of course, the models need to improve outcomes for cancer patients. This means, according to Moreau, uniting medical informatics and bioinformatics into a discipline he calls computational biomedicine.4 For example, microarray tests predicting prognosis or treatment response may include a few dozen molecular markers. This makes statistical models essential. "While physicians will never accept a computer program that decides which therapy should be applied to the patient, they will accept a diagnostic protocol that establishes that their patients' tumor is likely to be sensitive to this or that chemotherapy regimen," Moreau says.
Callaghan agrees that future mathematical models may help clinicians decide which drug combination to give a particular patient. Currently, he remarks, clinicians make many of these decisions empirically, by combining drugs with different resistance mechanisms, for example. Mathematical models could help design a drug combination that limits the risk of resistance and ensures the optimal pharmacokinetic profile. "Combining two drugs is complex enough, but the pharmacokinetics of three or four together is just a nightmare."
It'll be several years before these models move to the bedside. In the meantime, tumor resistance offers a poignant reminder of the potency and applicability of biology's leitmotif. Human interlopers can select some organisms' characteristics, such as a dog's shape or temperament, through selective breeding. But Darwin noted that natural selection is "immeasurably superior to man's feeble efforts." On the other hand, Darwin remarked that natural selection ensures "that the vigorous, the healthy, and the happy survive and multiply." For the host, at least, thriving cancer cells are hardly healthy. But the growing sophistication of mathematical models could eventually allow clinicians to predict when resistance is likely to occur and enable them to develop chemotherapeutic and other regimens that overcome this perennial problem.