Mathematical Oncology

Cancer is complex and dynamic

Ergo, dose optimization calls for mathematical modeling

Written by Sandy Anderson, Robert Gillies, Robert Gatenby - May 18, 2022

This article originally appeared in "The Cancer Letter," found here.

We wish to applaud the sentiments expressed in the recent article in The Cancer Letter titled “Oncologists, advocates, FDA call for an end to MTD and ‘more is better’ era in cancer drug dosing,” but also raise several concerns to be addressed as this initiative moves forward.

In particular, we are concerned that proposed methodologies for determining “optimal dosing”—as presented at the April 21 meeting of the FDA Oncologic Drugs Advisory Committee—fail to account for a cancer’s evolving complexity during therapy (The Cancer Letter, April 29, 2022).

As noted in The Cancer Letter, the guiding principle for dosing cancer agents during the past five decades has typically been continuous administration at Maximum Tolerated Dose until progression (The Cancer Letter, April 29, 2022). For many reasons, there is growing recognition that this is an over-simplified approach, leading to proposals for “optimal” dosing.

We agree and have noted in multiple prior publications that the MTD approach is evolutionarily unwise.1, 2 However, we also wish to express concern that the proposed alternative—“optimal dosing”—incurs challenges that cannot be overstated.

Moving away from MTD dosing requires the treating physician to directly confront the reality of malignant tumors as complex, dynamic, adaptive systems: complex because they consist of multiple cellular and microenvironmental components; dynamic because the components interact with each other through a complex network of interactions that change in space and time; and adaptive because critical elements of the network as well as the network itself can change and adapt to perturbations.9 Each treatment represents a perturbation of this complex system that elicits equally complex responses—such that optimizing subsequent treatments requires knowledge of the patient, disease, and drug dynamics, as well as the eco-evolutionary responses to the therapy.

Fortunately, physicists have studied complex dynamic systems (the weather, for example) for centuries through the language of mathematics, and oncology can build upon this experience. An important lesson at this early stage is that such systems are dominated by nonlinearities and are thus inaccessible to the linear thinking that constitutes human intuition. In fact, a complete understanding of the dynamics of a complex system is impossible without framing the dynamics mathematically.

Thus, early weather forecasting—based on the “intuitively obvious” but incorrect assumption that storms were carried by the wind—has long been replaced by modern methodologies which combine selective data acquisition, atmospheric first principles encompassed in the Navier-Stokes equation, and powerful computational methods.

Articles showing improved clinical outcomes when the MTD dose is reduced are a simple example of non-linear dynamics in action. It is likely these results have been ignored because they do not fit the intuitive model that, in optimal dosing, more is always better. A humbling but fundamental truth must be acknowledged: in complex, dynamic, adaptive cancer systems, an “optimal” treatment strategy can simultaneously be intuitively obvious and completely wrong.

The figures in the FDA ODAC presentation demonstrate the trade-off between toxicity and efficacy, but how is the latter determined or measured? We propose that efficacy first requires clearly establishing the goal of treatment. If there is a reasonable likelihood of cure, then the goal of treatment must be maximal killing of cancer cells within acceptable limits of toxicity.

However, it is important to recognize that curative therapy represents an anthropogenic extinction event and lessons from extinction during the Anthropocene era have clearly demonstrated that these are typically the result a sequence of different, self-reinforcing perturbations.3, 4 These dynamics are quite different from the oncologic practice of continuous MTD dose until progression.

Furthermore, if the clinical goal is palliation, killing the maximum number of cancer cells may accelerate the emergence of resistance by applying intense evolutionary selection for resistance while also eliminating all potentially competing treatment-sensitive cells, a well-recognized evolutionary dynamic termed “competitive release.”5 Here, optimal dosing is fundamentally linked to suppression of resistance cancer cells and maintenance of high quality of life, which can be achieved by strategic application of therapies that exploits underlying Darwinian dynamics that use treatment-sensitive cells to suppress proliferation of the resistant population.6-8

We wish to express concern about the implication that dose optimization cannot necessarily be simply obtained from randomized clinical trials and pre-treatment molecular screening. The history of cancer research has been dominated by an implied assumption that acquisition of more and more molecular data will result in a spontaneous organization into a state of complete knowledge of the system dynamics.

Lessons from centuries of physical science investigations, from planetary motion in the 16th century to the “particle zoo” in the 1950s, clearly demonstrate the transition from data to knowledge in complex systems cannot and will not happen without an integrative approach combining data-driven investigations with fundamental theory.

Similarly, arrival at “optimal” dosing strategy cannot be achieved by a few preclinical experiments and intuitive thinking. Rather, as in weather forecasting, cancer therapy must become an integrated and multidisciplinary pursuit.

Data collection from clinical trials must go beyond simply determining the statistical superiority of, for example, treatment A over B. While such trials may be sufficient for regulatory approval, they fail to answer the most critical question: why is A better than B? And, more generally, why did both A and B ultimately fail and allow tumor progression?

Asking these questions requires integration of predictive mathematical models into trial design, acquisition of longitudinal data during the trial to update model parameters, and predictions.9 This will facilitate the simulation of every subject in the trial, defining the underlying dynamics of an individual patient’s disease and allowing us to understand why treatment ultimately failed and how the outcome could have been improved.

Therefore, “optimal” dosing should instead be a patient-specific treatment schedule that evolves with the patient response dynamics, driven by predictive mathematical models that integrate relevant biological processes and eco-evolutionary dynamics.10

Ultimately, we propose that optimal cancer therapy will require well-integrated multidisciplinary teams that, as in weather predictions, mathematically frame the underlying evolutionary first principles and use computer simulation to predict the results of available therapeutic perturbations.11, 12 To this end, the Moffitt Cancer Center has established an Evolutionary Tumor Board that is directed by oncologists, but also includes faculty mathematicians and evolutionary biologists who are actively engaged in providing the theoretical support.


  1. Gatenby RA. A change of strategy in the war on cancer. Nature. 2009;459(7246):508-9. Epub 2009/05/30. doi: 10.1038/459508a. PubMed PMID: 19478766.
  2. Stankova K, Brown JS, Dalton WS, Gatenby RA. Optimizing Cancer Treatment Using Game Theory: A Review. JAMA Oncol. 2019;5(1):96-103. Epub 2018/08/12. doi: 10.1001/jamaoncol.2018.3395. PubMed PMID: 30098166; PMCID: PMC6947530.
  3. Gatenby RA, Artzy-Randrup Y, Epstein T, Reed DR, Brown JS. Eradicating Metastatic Cancer and the Eco-Evolutionary Dynamics of Anthropocene Extinctions. Cancer Res. 2020;80(3):613-23. Epub 2019/11/28. doi: 10.1158/0008-5472. CAN-19-1941. PubMed PMID: 31772037.
  4. Gatenby R, Zhang, J., Brown, J. First strike – second strike strategies in metastatic cancer: Lessons from the evolutionary dynamics of extinction. Cancer Research. 2019; In print.
  5. Gatenby RA, Brown JS. Integrating evolutionary dynamics into cancer therapy. Nat Rev Clin Oncol. 2020;17(11):675-86. Epub 2020/07/24. doi: 10.1038/s41571-020-0411-1. PubMed PMID: 32699310.
  6. Zhang J, Cunningham JJ, Brown JS, Gatenby RA. Integrating evolutionary dynamics into treatment of metastatic castrate-resistant prostate cancer. Nat Commun. 2017;8(1):1816. Epub 2017/11/29. doi: 10.1038/s41467-017-01968-5. PubMed PMID: 29180633; PMCID: PMC5703947.
  7. Zhang J C, J, Brown J, Gatenby R. Evolution-based mathematical models significantly prolong response to Abiraterone in metastatic castrate resistant prostate cancer and identify strategies to further improve outcomes Elife.2022;inpress.
  8. Gatenby RA, Silva AS, Gillies RJ, Frieden BR. Adaptive therapy. Cancer Res. 2009;69(11):4894-903. Epub 2009/06/03. doi: 10.1158/0008-5472. can-08-3658. PubMed PMID: 19487300; PMCID: 3728826.
  9. Strobl MAR, West J, Viossat Y, Damaghi M, Robertson-Tessi M, Brown JS, Gatenby RA, Maini PK, Anderson ARA. Turnover Modulates the Need for a Cost of Resistance in Adaptive Therapy. Cancer Res. 2021;81(4):1135-47. Epub 2020/11/12. doi: 10.1158/0008-5472.CAN-20-0806. PubMed PMID: 33172930; PMCID: PMC8455086.
  10. Rockne RC, Scott JG. Introduction to Mathematical Oncology. JCO Clin Cancer Inform. 2019;3:1-4. Epub 2019/04/27. doi: 10.1200/CCI.19.00010. PubMed PMID: 31026176; PMCID: PMC6752950.
  11. Kim E, Brown JS, Eroglu Z, Anderson ARA. Adaptive Therapy for Metastatic Melanoma: Predictions from Patient Calibrated Mathematical Models. Cancers (Basel). 2021;13(4). Epub 2021/03/07. doi: 10.3390/cancers13040823. PubMed PMID: 33669315; PMCID: PMC7920057.
  12. West JB, Dinh MN, Brown JS, Zhang J, Anderson AR, Gatenby RA. Multidrug Cancer Therapy in Metastatic Castrate-Resistant Prostate Cancer: An Evolution-Based Strategy. Clin Cancer Res. 2019;25(14):4413-21. Epub 2019/04/18. doi: 10.1158/1078-0432.CCR-19-0006. PubMed PMID: 30992299; PMCID: PMC6665681.
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