Optimization of Dose Fractionation for Radiotherapy of a Solid Tumor with Account of Oxygen Effect and Proliferative Heterogeneity

Rethinking the potential role of dose painting in personalized ultra-fractionated stereotactic adaptive radiotherapy

Fractionated radiotherapy was established in the 1920s based upon two principles: (1) delivering daily treatments of equal quantity, unless the clinical situation requires adjustment, and (2) defining a specific treatment period to deliver a total dosage. Modern fractionated radiotherapy continues to adhere to these century-old principles, despite significant advancements in our understanding of radiobiology. At UT Southwestern, we are exploring a novel treatment approach called PULSAR (Personalized Ultra-Fractionated Stereotactic Adaptive Radiotherapy). This method involves administering tumoricidal doses in a pulse mode with extended intervals, typically spanning weeks or even a month. Extended intervals permit substantial recovery of normal tissues and afford the tumor and tumor microenvironment ample time to undergo significant changes, enabling more meaningful adaptation in response to the evolving characteristics of the tumor. The notion of dose painting in the realm of radiation therapy has long been a subject of contention. The debate primarily revolves around its clinical effectiveness and optimal methods of implementation. In this perspective, we discuss two facets concerning the potential integration of dose painting with PULSAR, along with several practical considerations. If successful, the combination of the two may not only provide another level of personal adaptation ("adaptive dose painting"), but also contribute to the establishment of a timely feedback loop throughout the treatment process. To substantiate our perspective, we conducted a fundamental modeling study focusing on PET-guided dose painting, incorporating tumor heterogeneity and tumor control probability (TCP).

Optimization of Size of Nanosensitizers for Antitumor Radiotherapy Using Mathematical Modeling

The efficacy of antitumor radiotherapy can be enhanced by utilizing nonradioactive nanoparticles that emit secondary radiation when activated by a primary beam. They consist of small volumes of a radiosensitizing substance embedded within a polymer layer, which is coated with tumor-specific antibodies. The efficiency of nanosensitizers relies on their successful delivery to the tumor, which depends on their size. Increasing their size leads to a higher concentration of active substance; however, it hinders the penetration of nanosensitizers through tumor capillaries, slows down their movement through the tissue, and accelerates their clearance. In this study, we present a mathematical model of tumor growth and radiotherapy with the use of intravenously administered tumor-specific nanosensitizers. Our findings indicate that their optimal size for achieving maximum tumor radiosensitization following a single injection of their fixed total volume depends on the permeability of the tumor capillaries. Considering physiologically plausible spectra of capillary pore radii, with a nanoparticle polymer layer width of 7 nm, the optimal radius of nanoparticles falls within the range of 13-17 nm. The upper value is attained when considering an extreme spectrum of capillary pores.

Optimization of antitumor radiotherapy fractionation via mathematical modeling with account of 4 R's of radiobiology

A spatially-distributed continuous mathematical model of solid tumor growth and treatment by fractionated radiotherapy is presented. The model explicitly accounts for the factors, widely referred to as 4 R's of radiobiology, which influence the efficacy of radiotherapy fractionation protocols: tumor cell repopulation, their redistribution in cell cycle, reoxygenation and repair of sublethal damage of both tumor and normal tissues. With the use of special algorithm the fractionation protocols that provide increased tumor control probability, compared to standard clinical protocol, are found for various physiologically-based values of model parameters under the constraints of fixed overall normal tissue damage and maximum admissible fractional dose. In particular, it is shown that significant gain in treatment efficacy can be achieved for tumors of low malignancy by the use of protracted hyperfractionated protocols. The optimized non-uniform protocols are characterized by gradual escalation of fractional doses in their last parts, which start after the levels of oxygen and nutrients significantly elevate throughout the tumor and accelerated tumor proliferation manifests itself, which is a well-known experimental phenomenon.

Nonlocal Reaction-Diffusion Equations in Biomedical Applications

Nonlocal reaction-diffusion equations describe various biological and biomedical applications. Their mathematical properties are essentially different in comparison with the local equations, and this difference can lead to important biological implications. This review will present the state of the art in the investigation of nonlocal reaction-diffusion models in biomedical applications. We will consider various models arising in mathematical immunology, neuroscience, cancer modelling, and we will discuss their mathematical properties, nonlinear dynamics, resulting spatiotemporal patterns and biological significance.

A Century of Fractionated Radiotherapy: How Mathematical Oncology Can Break the Rules

Radiotherapy is involved in 50% of all cancer treatments and 40% of cancer cures. Most of these treatments are delivered in fractions of equal doses of radiation (Fractional Equivalent Dosing (FED)) in days to weeks. This treatment paradigm has remained unchanged in the past century and does not account for the development of radioresistance during treatment. Even if under-optimized, deviating from a century of successful therapy delivered in FED can be difficult. One way of exploring the infinite space of fraction size and scheduling to identify optimal fractionation schedules is through mathematical oncology simulations that allow for in silico evaluation. This review article explores the evidence that current fractionation promotes the development of radioresistance, summarizes mathematical solutions to account for radioresistance, both in the curative and non-curative setting, and reviews current clinical data investigating non-FED fractionated radiotherapy.

Improving cancer treatments via dynamical biophysical models

Despite significant advances in oncological research, cancer nowadays remains one of the main causes of mortality and morbidity worldwide. New treatment techniques, as a rule, have limited efficacy, target only a narrow range of oncological diseases, and have limited availability to the general public due their high cost. An important goal in oncology is thus the modification of the types of antitumor therapy and their combinations, that are already introduced into clinical practice, with the goal of increasing the overall treatment efficacy. One option to achieve this goal is optimization of the schedules of drugs administration or performing other medical actions. Several factors complicate such tasks: the adverse effects of treatments on healthy cell populations, which must be kept tolerable; the emergence of drug resistance due to the intrinsic plasticity of heterogeneous cancer cell populations; the interplay between different types of therapies administered simultaneously. Mathematical modeling, in which a tumor and its microenvironment are considered as a single complex system, can address this complexity and can indicate potentially effective protocols, that would require experimental verification. In this review, we consider classical methods, current trends and future prospects in the field of mathematical modeling of tumor growth and treatment. In particular, methods of treatment optimization are discussed with several examples of specific problems related to different types of treatment.

Combined Influence of Nutrient Supply Level and Tissue Mechanical Properties on Benign Tumor Growth as Revealed by Mathematical Modeling

A continuous mathematical model of non-invasive avascular tumor growth in tissue is presented. The model considers tissue as a biphasic material, comprised of a solid matrix and interstitial fluid. The convective motion of tissue elements happens due to the gradients of stress, which change as a result of tumor cells proliferation and death. The model accounts for glucose as the crucial nutrient, supplied from the normal tissue, and can reproduce both diffusion-limited and stress-limited tumor growth. Approximate tumor growth curves are obtained semi-analytically in the limit of infinite tissue hydraulic conductivity, which implies instantaneous equalization of arising stress gradients. These growth curves correspond well to the numerical solutions and represent classical sigmoidal curves with a short initial exponential phase, subsequent almost linear growth phase and a phase with growth deceleration, in which tumor tends to reach its maximum volume. The influence of two model parameters on tumor growth curves is investigated: tissue hydraulic conductivity, which links the values of stress gradient and convective velocity of tissue phases, and tumor nutrient supply level, which corresponds to different permeability and surface area density of capillaries in the normal tissue that surrounds the tumor. In particular, it is demonstrated, that sufficiently low tissue hydraulic conductivity (intrinsic, e.g., to tumors arising from connective tissue) and sufficiently high nutrient supply can lead to formation of giant benign tumors, reaching tens of centimeters in diameter, which are indeed observed clinically.

A fast numerical method for oxygen supply in tissue with complex blood vessel network

Angiogenesis plays an essential role in many pathological processes such as tumor growth, wound healing, and keloid development. Low oxygen level is the main driving stimulus for angiogenesis. In an animal tissue, the oxygen level is mainly determined by three effects—the oxygen delivery through blood flow in a refined vessel network, the oxygen diffusion from blood to tissue, and the oxygen consumption in cells. Evaluation of the oxygen field is usually the bottleneck in large scale modeling and simulation of angiogenesis and related physiological processes. In this work, a fast numerical method is developed for the simulation of oxygen supply in tissue with a large-scale complex vessel network. This method employs an implicit finite-difference scheme to compute the oxygen field. By virtue of an oxygen source distribution technique from vessel center lines to mesh points and a corresponding post-processing technique that eliminate the local numerical error induced by source distribution, square mesh with relatively large mesh sizes can be applied while sufficient numerical accuracy is maintained. The new method has computational complexity which is slightly higher than linear with respect to the number of mesh points and has a convergence order which is slightly lower than second order with respect to the mesh size. With this new method, accurate evaluation of the oxygen field in a fully vascularized tissue on the scale of centimeter becomes possible.