Ill-posed problem and regularization in reconstruction of radiobiological parameters from serial tumor imaging data

Equivalent uniform aerobic dose in radiotherapy for hypoxic tumors

Objective.Equivalent uniform aerobic dose (EUAD) is proposed for comparison of integrated cell survival in tumors with different distributions of hypoxia and radiation dose.Approach.The EUAD assumes that for any non-uniform distributions of radiation dose and oxygen enhancement ratio (OER) within a tumor, there is a uniform distribution of radiation dose under hypothetical aerobic conditions with OER = 1 that produces equal integrated survival of clonogenic cells. This definition of EUAD has several advantages. First, the EUAD allows one to compare survival of clonogenic cells in tumors with intra-tumor and inter-tumor variation of radio sensitivity due to hypoxia because the cell survival is recomputed under the same benchmark oxygen level (OER = 1). Second, the EUAD for homogeneously oxygenated tumors is equal to the concept of equivalent uniform dose.Main results. We computed the EUAD using radiotherapy dose and the OER derived from the18F-Fluoromisonidazole PET (18F-FMISO PET) images of hypoxia in patients with glioblastoma, the most common and aggressive type of primary malignant brain tumor. The18F-FMISO PET images include a distribution of SUV (Standardized Uptake Value); therefore, the SUV is converted to partial oxygen pressure (pO2) and then to the OER. The prognostic value of EUAD in radiotherapy for hypoxic tumors is demonstrated using correlation between EUAD and overall survival (OS) in radiotherapy for glioblastoma. The correction to the EUAD for the absolute hypoxic volume that traceable to the tumor control probability improves the correlation with OS.Significance. While the analysis proposed in this research is based on the18F-FMISO PET images for glioblastoma, the EUAD is a universal radiobiological concept and is not associated with any specific cancer or any specific PET or MRI biomarker of hypoxia. Therefore, this research can be generalized to other cancers, for example stage III lung cancer, and to other hypoxia biomarkers.

Towards integration of time-resolved confocal microscopy of a 3D in vitro microfluidic platform with a hybrid multiscale model of tumor angiogenesis

The goal of this study is to calibrate a multiscale model of tumor angiogenesis with time-resolved data to allow for systematic testing of mathematical predictions of vascular sprouting. The multi-scale model consists of an agent-based description of tumor and endothelial cell dynamics coupled to a continuum model of vascular endothelial growth factor concentration. First, we calibrate ordinary differential equation models to time-resolved protein concentration data to estimate the rates of secretion and consumption of vascular endothelial growth factor by endothelial and tumor cells, respectively. These parameters are then input into the multiscale tumor angiogenesis model, and the remaining model parameters are then calibrated to time resolved confocal microscopy images obtained within a 3D vascularized microfluidic platform. The microfluidic platform mimics a functional blood vessel with a surrounding collagen matrix seeded with inflammatory breast cancer cells, which induce tumor angiogenesis. Once the multi-scale model is fully parameterized, we forecast the spatiotemporal distribution of vascular sprouts at future time points and directly compare the predictions to experimentally measured data. We assess the ability of our model to globally recapitulate angiogenic vasculature density, resulting in an average relative calibration error of 17.7% ± 6.3% and an average prediction error of 20.2% ± 4% and 21.7% ± 3.6% using one and four calibrated parameters, respectively. We then assess the model's ability to predict local vessel morphology (individualized vessel structure as opposed to global vascular density), initialized with the first time point and calibrated with two intermediate time points. In this study, we have rigorously calibrated a mechanism-based, multiscale, mathematical model of angiogenic sprouting to multimodal experimental data to make specific, testable predictions.

Forecasting Individual Patient Response to Radiation Therapy in Head and Neck Cancer With a Dynamic Carrying Capacity Model

Purpose:

To model and predict individual patient responses to radiation therapy.

Methods and Materials:

We modeled tumor dynamics as logistic growth and the effect of radiation as a reduction in the tumor carrying capacity, motivated by the effect of radiation on the tumor microenvironment. The model was assessed on weekly tumor volume data collected for 2 independent cohorts of patients with head and neck cancer from the H. Lee Moffitt Cancer Center (MCC) and the MD Anderson Cancer Center (MDACC) who received 66 to 70 Gy in standard daily fractions or with accelerated fractionation. To predict response to radiation therapy for individual patients, we developed a new forecasting framework that combined the learned tumor growth rate and carrying capacity reduction fraction (δ) distribution with weekly measurements of tumor volume reduction for a given test patient to estimate δ, which was used to predict patient-specific outcomes.

Results:

The model fit data from MCC with high accuracy with patient-specific δ and a fixed tumor growth rate across all patients. The model fit data from an independent cohort from MDACC with comparable accuracy using the tumor growth rate learned from the MCC cohort, showing transferability of the growth rate. The forecasting framework predicted patient-specific outcomes with 76% sensitivity and 83% specificity for locoregional control and 68% sensitivity and 85% specificity for disease-free survival with the inclusion of 4 on-treatment tumor volume measurements.

Conclusions:

These results demonstrate that our simple mathematical model can describe a variety of tumor volume dynamics. Furthermore, combining historically observed patient responses with a few patient-specific tumor volume measurements allowed for the accurate prediction of patient outcomes, which may inform treatment adaptation and personalization.

Spatially fractionated radiotherapy: tumor response modelling including immunomodulation

A mathematical tumor response model has been developed, encompassing the interplay between immune cells and cancer cells initiated by either partial or full tumor irradiation. The iterative four-compartment model employs the linear-quadratic radiation response theory for four cell types: active and inactive cytotoxic T lymphocytes (immune cells, CD8⁺T cells in particular), viable cancer cells (undamaged and reparable cells) and doomed cells (irreparably damaged cells). The cell compartment interactions are calculated per day, with total tumor volume (TV) as the main quantity of interest. The model was fitted to previously published data on syngeneic xenografts (67NR breast carcinoma and Lewis lung carcinoma; (Markovskyet al2019Int. J. Radiat. Oncol. Biol. Phys.103697-708)) subjected to single doses of 10 or 15 Gy by 50% (partial) or 100% (full) TV irradiation. The experimental data included effects from anti-CD8⁺antibodies and immunosuppressive drugs. Using a new optimization method, promising fits were obtained where the lowest and highest root-mean-squared error values were observed for anti-CD8⁺treatment and unirradiated control data, respectively, for both cell types. Additionally, predictive capabilities of the model were tested by using the estimated model parameters to predict scenarios for higher doses and different TV irradiation fractions. Here, mean relative deviations in the range of 19%-34% from experimental data were found. However, more validation data is needed to conclude on the model's predictive capabilities. In conclusion, the model was found useful in evaluating the impact from partial and full TV irradiation on the immune response and subsequent tumor growth. The model shows potential to support and guide spatially fractionated radiotherapy in future pre-clinical and clinical studies.

Proliferation saturation index in an adaptive Bayesian approach to predict patient-specific radiotherapy responses

Purpose: Radiotherapy prescription dose and dose fractionation protocols vary little between individual patients having the same tumor grade and stage. To personalize radiotherapy a predictive model is needed to simulate radiation response. Previous modeling attempts with multiple variables and parameters have been shown to yield excellent data fits at the cost of non-identifiability and clinically unrealistic results. Materials and methods: We develop a mathematical model based on a proliferation saturation index (PSI) that is a measurement of pre-treatment tumor volume-to-carrying capacity ratio that modulates intrinsic tumor growth and radiation response rates. In an adaptive Bayesian approach, we utilize an increasing number of data points for individual patients to predict patient-specific responses to subsequent radiation doses. Results: Model analysis shows that using PSI as the only patient-specific parameter, model simulations can fit longitudinal clinical data with high accuracy (R²=0.84). By analyzing tumor response to radiation using daily CT scans early in the treatment, response to the remaining treatment fractions can be predicted after two weeks with high accuracy (c-index = 0.89). Conclusion: The PSI model may be suited to forecast treatment response for individual patients and offers actionable decision points for mid-treatment protocol adaptation. The presented work provides an actionable image-derived biomarker prior to and during therapy to personalize and adapt radiotherapy.

The Evolution of Tumour Composition During Fractionated Radiotherapy: Implications for Outcome

Current protocols for delivering radiotherapy are based primarily on tumour stage and nodal and metastases status, even though it is well known that tumours and their microenvironments are highly heterogeneous. It is well established that the local oxygen tension plays an important role in radiation-induced cell death, with hypoxic tumour regions responding poorly to irradiation. Therefore, to improve radiation response, it is important to understand more fully the spatiotemporal distribution of oxygen within a growing tumour before and during fractionated radiation. To this end, we have extended a spatially resolved mathematical model of tumour growth, first proposed by Greenspan (Stud Appl Math 51:317-340, 1972), to investigate the effects of oxygen heterogeneity on radiation-induced cell death. In more detail, cell death due to radiation at each location in the tumour, as determined by the well-known linear-quadratic model, is assumed also to depend on the local oxygen concentration. The oxygen concentration is governed by a reaction-diffusion equation that is coupled to an integro-differential equation that determines the size of the assumed spherically symmetric tumour. We combine numerical and analytical techniques to investigate radiation response of tumours with different intratumoral oxygen distribution profiles. Model simulations reveal a rapid transient increase in hypoxia upon regrowth of the tumour spheroid post-irradiation. We investigate the response to different radiation fractionation schedules and identify a tumour-specific relationship between inter-fraction time and dose per fraction to achieve cure. The rich dynamics exhibited by the model suggest that spatial heterogeneity may be important for predicting tumour response to radiotherapy for clinical applications.

Theoretical effectiveness of cell survival in fractionated radiotherapy with hypoxia-targeted dose escalation

PURPOSE

The goal of this article is to compute the cell survival during fractionated radiotherapy with non-uniform hypoxia-targeted dose distribution relative to cell survival for a uniform dose distribution with equal integral tumor dose. The analysis is performed for different parameters of radiotherapy with conventional and hypofractionated dose regimens.

METHODS

Our analysis is done using a parsimonious tumor response model that describes the major components of tumor response to radiotherapy such as radiosensitivity, cell proliferation, and hypoxia using as few variables as possible. Two levels of oxygenated and hypoxic cells with the survival curves described by the linear quadratic (LQ) model are implemented in the model. The model allows for analytical solutions for relative cell survival in a single fraction simulation which can be used for additional validation of our numerical simulations. The relative cell survival was computed for conventional and hypofractionated dose regimens in a model problem with radiobiological parameters for the non-small-cell lung cancer. Sensitivity of cell survival to different parameters of radiotherapy such as the relative volume of hypoxic fraction, boost dose ratio, re-oxygenation rate, and delivery errors was investigated.

RESULTS

Our computational and analytical results show that relative cell survival for non-uniform and uniform dose distributions is larger than 1.0 during the first few fractions of radiotherapy with conventional fractionation. This indicates that the uniform dose distribution produces a higher cell killing effect for the equal integral dose. This may stem from domination of linear contribution to the cell killing effect seen in the dose range of 1-2 Gy and a steeper slope of survival curve in the oxygenated tumor region. This effect can only happen if the distribution of clonogens is nearly uniform; therefore, after the first few fractions, the non-uniform dose distributions outperform the uniform dose distribution and relative cell survival becomes less than 1.0. However, re-oxygenation and delivery errors can also reduce the effectiveness of hypoxia-targeted non-uniform dose distributions toward the end of treatment. For hypofractionated radiotherapy with fractional dose >3 Gy, the relative cell survival was always below 1.0, which means the non-uniform dose distributions produced higher cell killing effect than the uniform dose distribution during the entire treatment.

CONCLUSION

The data obtained in this work suggest that non-uniform hypoxia-targeted dose distributions are less effective at cell killing than uniform dose distributions at the beginning of radiotherapy with conventional fractionation. However; non-uniform dose distributions can outperform uniform dose distribution by the end of the treatment if the effects of re-oxygenation and delivery errors are reduced. In hypofractionated radiotherapy, non-uniform hypoxia-targeted dose distributions are more efficient than uniform dose distributions during the entire treatment.