Adaptive individualized radiotherapy for cancer: how OR can help

A few years ago, I attended a seminar where I learned that in external beam radiotherapy, radiation beams that are used to kill tumor cells also pass through and damage healthy tissue nearby. Thus the goal is to maximize tumor cell-kill while minimizing toxic effects of radiation on healthy tissue. As shown in the schematic below, this is achieved in part by optimizing the radiation intensity profile so that it conforms, as much as possible, to the tumor’s shape. Operations Research (OR) has played an important role in this spatial aspect of radiotherapy. Numerous formulations of the above optimization problem have been developed over the last fifteen years and efficient algorithms for their solution have been incorporated into commercial treatment planning systems.

Figure 1. A schematic of intensity modulation in radiotherapy. A cashew nut shaped tumor and a round healthy tissue are depicted. Three radiation beams are shown. Their intensity profiles are tuned so that high radiation is delivered to the tumor whereas low radiation is delivered to the healthy tissue. Long arrows=high intensity radiation; short arrows=low intensity radiation.

As I got more interested in this application of OR, and started working with my former doctoral student Minsun Kim who is a medical physicist, I found that radiotherapy is typically administered daily over several weeks. This gives the healthy tissue, which has better damage-repair capability than tumors, sufficient time to recover between consecutive treatment sessions. It also allows tumor cells to reorganize into more radiosensitive phases of the cell cycle. More generally, it is believed that the efficacy of radiotherapy depends, among many other factors, on patient- and tumor-specific complex biological processes that occur in the cancerous region over several weeks of treatment.

However, I observed, to my surprise, that optimization formulations used in treatment planning were “static”. That is, they did not explicitly model changes in the tumor’s condition over time. At that time, I thought, “but surely, there must be some health benefit in dynamically optimizing treatment plans over time based on biological information acquired over the treatment course and on the patient’s actual response to radiation.” However, I was a bit disappointed when I soon realized that several hurdles would need to be overcome before this idea can be implemented in practice. For instance, it has traditionally been very difficult to model tumor-response and to accurately image spatiotemporal evolution of intra-tumor biological processes. I thus abandoned the idea of working on dynamic optimization in radiotherapy. Well, almost – until I learned from Minsun that functional images can now provide quantitative biological information such as the density of active tumor cells, their radiosensitivity and rate of proliferation. Such biological information could be further utilized to develop patient-specific tumor-response models. In fact, some medical physicists and doctors now believe that spatiotemporally adaptive individualized treatment plans that utilize such response models and functional imaging techniques will be feasible in the not-so-distant future. But we noticed that all existing research in this area was clinical and a mathematical framework for this futuristic treatment paradigm was missing.

Some of our work in applied OR now focuses on building a rigorous mathematical foundation for adaptive individualized radiotherapy. Our goal is to build optimization models and algorithms that deliver the right radiation to the right location at the right time. In Minsun’s doctoral dissertation, we took an initial step[1] toward this objective. We formulated dynamic optimization models that account for uncertainty in tumor-response and let treatment planners tune beam intensities depending on the tumor’s condition, as observed in functional images acquired prior to each treatment session. Exact solution of these optimization models was intractable. We therefore designed approximate solution techniques, and demonstrated, using computer generated medium-scale test cases, that this stochastic dynamic approach to treatment planning can reduce the number of tumor cells by as much as 98% compared to the traditional static approach. Minsun won the Bonder award and the Dantzig dissertation prize for this work and my NSF CAREER grant award further builds upon these ideas. But this is just the tip of the iceberg.

I believe that OR can help make adaptive individualized radiotherapy a reality. This approach to treatment planning will generate a rich array of challenging stochastic control problems[2]. There will be much room for creativity in coming up with sufficiently realistic and yet tractable formulations. For truly patient-specific treatment, these will need to be integrated with methods to estimate the parameters of tumor-response models. The stochastic control problems in our aforementioned preliminary work were solved using approximate control schemes called open loop control, certainty equivalent control, and open loop feedback control. These methods were computationally efficient for our medium-scale test problems because our original model was formulated such that the resulting nonlinear programming sub-problems were convex. In the future, as we attempt to tackle much larger-scale treatment planning problems, we will need to develop specialized nonlinear programming algorithms and approximate control schemes, and thus there will great opportunities for methodological work. In the end, these dynamic optimization models and algorithms will have a real impact on radiotherapy practice only if they can achieve a demonstrable improvement in health outcomes and if they are embedded into commercial treatment planning systems. Thus the pursuit of mathematically rigorous, radiobiologically and clinically sound, and cost-effective adaptive individualized radiotherapy will call for a close collaboration and knowledge-exchange among doctors, medical physicists, radiobiologists, industry, and optimization experts. The 2008 World Cancer Report predicts that by 2030 there could be 17 million cancer deaths annually worldwide. Thus, if clinically successful, this interdisciplinary application of OR has the potential for significant societal impact.

Research supported in part by NSF Grant CMMI-1054026

-Post by Prof Archis Ghate, University of Washington, Seattle

archis at u dot washington dot edu

[1] M Kim, A Ghate, and M H Phillips, A stochastic control formalism for dynamic biologically conformal radiation therapy, forthcoming in European Journal of Operational Research, 2011.

[2] A Ghate, Dynamic optimization in radiotherapy, forthcoming in TutORials in Operations Research, INFORMS 2011.