Tumor-volume simulation during radiotherapy for head-and-neck cancer using a four-level cell population model

Predicting Radiotherapy Patient Outcomes with Real-Time Clinical Data Using Mathematical Modelling

Longitudinal tumour volume data from head-and-neck cancer patients show that tumours of comparable pre-treatment size and stage may respond very differently to the same radiotherapy fractionation protocol. Mathematical models are often proposed to predict treatment outcome in this context, and have the potential to guide clinical decision-making and inform personalised fractionation protocols. Hindering effective use of models in this context is the sparsity of clinical measurements juxtaposed with the model complexity required to produce the full range of possible patient responses. In this work, we present a compartment model of tumour volume and tumour composition, which, despite relative simplicity, is capable of producing a wide range of patient responses. We then develop novel statistical methodology and leverage a cohort of existing clinical data to produce a predictive model of both tumour volume progression and the associated level of uncertainty that evolves throughout a patient's course of treatment. To capture inter-patient variability, all model parameters are patient specific, with a bootstrap particle filter-like Bayesian approach developed to model a set of training data as prior knowledge. We validate our approach against a subset of unseen data, and demonstrate both the predictive ability of our trained model and its limitations.

Do We Preserve Tumor Control Probability (TCP) in FLASH Radiotherapy? A Model-Based Analysis

Reports of concurrent sparing of normal tissue and iso-effective treatment of tumors at ultra-high dose-rates (uHDR) have fueled the growing field of FLASH radiotherapy. However, iso-effectiveness in tumors is often deduced from the absence of a significant difference in their growth kinetics. In a model-based analysis, we investigate the meaningfulness of these indications for the clinical treatment outcome. The predictions of a previously benchmarked model of uHDR sparing in the "UNIfied and VERSatile bio response Engine" (UNIVERSE) are combined with existing models of tumor volume kinetics as well as tumor control probability (TCP) and compared to experimental data. The potential TCP of FLASH radiotherapy is investigated by varying the assumed dose-rate, fractionation schemes and oxygen concentration in the target. The developed framework describes the reported tumor growth kinetics appropriately, indicating that sparing effects could be present in the tumor but might be too small to be detected with the number of animals used. The TCP predictions show the possibility of substantial loss of treatment efficacy for FLASH radiotherapy depending on several variables, including the fractionation scheme, oxygen level, and DNA repair kinetics. The possible loss of TCP should be seriously considered when assessing the clinical viability of FLASH treatments.

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.

Simulation analysis for tumor radiotherapy based on three-component mathematical models

OBJECTIVE

To setup a three-component tumor growth mathematical model and discuss its basic application in tumor fractional radiotherapy with computer simulation.

METHOD

First, our three-component tumor growth model extended from the classical Gompertz tumor model was formulated and applied to a fractional radiotherapy with a series of proper parameters. With the computer simulation of our model, the impact of some parameters such as fractional dose, amount of quiescent tumor cells, and α/β value to the effect of radiotherapy was also analyzed, respectively.

RESULTS

With several optimal technologies, the model could run stably and output a series of convergent results. The simulation results showed that the fractional radiotherapy dose could impact the effect of radiotherapy significantly, while the amount of quiescent tumor cells and α/β value did that to a certain extent.

CONCLUSIONS

Supported with some proper parameters, our model can simulate and analyze the tumor radiotherapy program as well as give some theoretical instruction to radiotherapy personalized optimization.

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.

Modelling and Bayesian adaptive prediction of individual patients' tumour volume change during radiotherapy

The aim of this study is to develop a mathematical modelling method that can predict individual patients’ response to radiotherapy, in terms of tumour volume change during the treatment. The main concept is to start from a population-average model, which is subsequently updated from an individual’s tumour volume measurement. The model becomes increasingly personalized and so too does the prediction it produces. This idea of adaptive prediction was realised by using a Bayesian approach for updating the model parameters. The feasibility of the developed method was demonstrated on the data from 25 non-small cell lung cancer patients treated with helical tomotherapy, during which tumour volume was measured from daily imaging as part of the image-guided radiotherapy. The method could provide useful information for adaptive treatment planning and dose scheduling based on the patient’s personalised response.

A mathematical model of tumor growth and its response to single irradiation

BACKGROUND

Mathematical modeling of biological processes is widely used to enhance quantitative understanding of bio-medical phenomena. This quantitative knowledge can be applied in both clinical and experimental settings. Recently, many investigators began studying mathematical models of tumor response to radiation therapy. We developed a simple mathematical model to simulate the growth of tumor volume and its response to a single fraction of high dose irradiation. The modelling study may provide clinicians important insights on radiation therapy strategies through identification of biological factors significantly influencing the treatment effectiveness.

METHODS

We made several key assumptions of the model. Tumor volume is composed of proliferating (or dividing) cancer cells and non-dividing (or dead) cells. Tumor growth rate (or tumor volume doubling time) is proportional to the ratio of the volumes of tumor vasculature and the tumor. The vascular volume grows slower than the tumor by introducing the vascular growth retardation factor, θ. Upon irradiation, the proliferating cells gradually die over a fixed time period after irradiation. Dead cells are cleared away with cell clearance time. The model was applied to simulate pre-treatment growth and post-treatment radiation response of rat rhabdomyosarcoma tumors and metastatic brain tumors of five patients who were treated with Gamma Knife stereotactic radiosurgery (GKSRS).

RESULTS

By selecting appropriate model parameters, we showed the temporal variation of the tumors for both the rat experiment and the clinical GKSRS cases could be easily replicated by the simple model. Additionally, the application of our model to the GKSRS cases showed that the α-value, which is an indicator of radiation sensitivity in the LQ model, and the value of θ could be predictors of the post-treatment volume change.

CONCLUSIONS

The proposed model was successful in representing both the animal experimental data and the clinically observed tumor volume changes. We showed that the model can be used to find the potential biological parameters, which may be able to predict the treatment outcome. However, there is a large statistical uncertainty of the result due to the small sample size. Therefore, a future clinical study with a larger number of patients is needed to confirm the finding.

A note on modeling of tumor regression for estimation of radiobiological parameters

PURPOSE

Accurate calculation of radiobiological parameters is crucial to predicting radiation treatment response. Modeling differences may have a significant impact on derived parameters. In this study, the authors have integrated two existing models with kinetic differential equations to formulate a new tumor regression model for estimation of radiobiological parameters for individual patients.

METHODS

A system of differential equations that characterizes the birth-and-death process of tumor cells in radiation treatment was analytically solved. The solution of this system was used to construct an iterative model (Z-model). The model consists of three parameters: tumor doubling time Td, half-life of dead cells Tr, and cell survival fraction SFD under dose D. The Jacobian determinant of this model was proposed as a constraint to optimize the three parameters for six head and neck cancer patients. The derived parameters were compared with those generated from the two existing models: Chvetsov's model (C-model) and Lim's model (L-model). The C-model and L-model were optimized with the parameter Td fixed.

RESULTS

With the Jacobian-constrained Z-model, the mean of the optimized cell survival fractions is 0.43 ± 0.08, and the half-life of dead cells averaged over the six patients is 17.5 ± 3.2 days. The parameters Tr and SFD optimized with the Z-model differ by 1.2% and 20.3% from those optimized with the Td-fixed C-model, and by 32.1% and 112.3% from those optimized with the Td-fixed L-model, respectively.

CONCLUSIONS

The Z-model was analytically constructed from the differential equations of cell populations that describe changes in the number of different tumor cells during the course of radiation treatment. The Jacobian constraints were proposed to optimize the three radiobiological parameters. The generated model and its optimization method may help develop high-quality treatment regimens for individual patients.

Mathematical modelling of tumour volume dynamics in response to stereotactic ablative radiotherapy for non-small cell lung cancer

This paper reports a modelling study of tumour volume dynamics in response to stereotactic ablative radiotherapy (SABR). The main objective was to develop a model that is adequate to describe tumour volume change measured during SABR, and at the same time is not excessively complex as lacking support from clinical data. To this end, various modelling options were explored, and a rigorous statistical method, the Akaike information criterion, was used to help determine a trade-off between model accuracy and complexity. The models were calibrated to the data from 11 non-small cell lung cancer patients treated with SABR. The results showed that it is feasible to model the tumour volume dynamics during SABR, opening up the potential for using such models in a clinical environment in the future.

Assessment of interpatient heterogeneity in tumor radiosensitivity for nonsmall cell lung cancer using tumor-volume variation data

PURPOSE

In our previous work, the authors showed that a distribution of cell surviving fractions S2 in a heterogeneous group of patients could be derived from tumor-volume variation curves during radiotherapy for head and neck cancer. In this research study, the authors show that this algorithm can be applied to other tumors, specifically in nonsmall cell lung cancer. This new application includes larger patient volumes and includes comparison of data sets obtained at independent institutions.

METHODS

Our analysis was based on two data sets of tumor-volume variation curves for heterogeneous groups of 17 patients treated for nonsmall cell lung cancer with conventional dose fractionation. The data sets were obtained previously at two independent institutions by using megavoltage computed tomography. Statistical distributions of cell surviving fractions S2 and clearance half-lives of lethally damaged cells T(1/2) have been reconstructed in each patient group by using a version of the two-level cell population model of tumor response and a simulated annealing algorithm. The reconstructed statistical distributions of the cell surviving fractions have been compared to the distributions measured using predictive assays in vitro.

RESULTS

Nonsmall cell lung cancer presents certain difficulties for modeling surviving fractions using tumor-volume variation curves because of relatively large fractional hypoxic volume, low gradient of tumor-volume response, and possible uncertainties due to breathing motion. Despite these difficulties, cell surviving fractions S2 for nonsmall cell lung cancer derived from tumor-volume variation measured at different institutions have similar probability density functions (PDFs) with mean values of 0.30 and 0.43 and standard deviations of 0.13 and 0.18, respectively. The PDFs for cell surviving fractions S2 reconstructed from tumor volume variation agree with the PDF measured in vitro.

CONCLUSIONS

The data obtained in this work, when taken together with the data obtained previously for head and neck cancer, suggests that the cell surviving fractions S2 can be reconstructed from the tumor volume variation curves measured during radiotherapy with conventional fractionation. The proposed method can be used for treatment evaluation and adaptation.

Predicting oropharyngeal tumor volume throughout the course of radiation therapy from pretreatment computed tomography data using general linear models

PURPOSE

The purpose of this work was to develop and evaluate the accuracy of several predictive models of variation in tumor volume throughout the course of radiation therapy.

METHODS

Nineteen patients with oropharyngeal cancers were imaged daily with CT-on-rails for image-guided alignment per an institutional protocol. The daily volumes of 35 tumors in these 19 patients were determined and used to generate (1) a linear model in which tumor volume changed at a constant rate, (2) a general linear model that utilized the power fit relationship between the daily and initial tumor volumes, and (3) a functional general linear model that identified and exploited the primary modes of variation between time series describing the changing tumor volumes. Primary and nodal tumor volumes were examined separately. The accuracy of these models in predicting daily tumor volumes were compared with those of static and linear reference models using leave-one-out cross-validation.

RESULTS

In predicting the daily volume of primary tumors, the general linear model and the functional general linear model were more accurate than the static reference model by 9.9% (range: -11.6%-23.8%) and 14.6% (range: -7.3%-27.5%), respectively, and were more accurate than the linear reference model by 14.2% (range: -6.8%-40.3%) and 13.1% (range: -1.5%-52.5%), respectively. In predicting the daily volume of nodal tumors, only the 14.4% (range: -11.1%-20.5%) improvement in accuracy of the functional general linear model compared to the static reference model was statistically significant.

CONCLUSIONS

A general linear model and a functional general linear model trained on data from a small population of patients can predict the primary tumor volume throughout the course of radiation therapy with greater accuracy than standard reference models. These more accurate models may increase the prognostic value of information about the tumor garnered from pretreatment computed tomography images and facilitate improved treatment management.

A mathematical model of tumor volume changes during radiotherapy

PURPOSE

To develop a clinically viable mathematical model that quantitatively predicts tumor volume change during radiotherapy in order to provide treatment response assessment for prognosis, treatment plan optimization, and adaptation.

METHOD AND MATERIALS

The correction factors containing hypoxia, DNA single strand breaks, potentially lethal damage, and other factors were used to develop an improved cell survival model based on the popular linear-quadratic model of cell survival in radiotherapy. The four-level cell population model proposed by Chvetsov et al. was further simplified by removing the initial hypoxic fraction and reoxygenation parameter, which are hard to obtain in routine clinics, such that an easy-to-use model can be developed for clinical applications. The new model was validated with data of nine lung and cervical cancer patients.

RESULTS

Out of the nine cases, the new model can predict tumor volume change in six cases with a correlation index R (2) greater than 0.9 and the rest of three with R (2) greater than 0.85.

CONCLUSION

Based on a four-level cell population model, a more practical and simplified cell survival curve was proposed to model the tumor volume changes during radiotherapy. Validation study with patient data demonstrated feasibility and clinical usefulness of the new model in predicting tumor volume change in radiotherapy.

Modelling the interplay between hypoxia and proliferation in radiotherapy tumour response

A tumour control probability computational model for fractionated radiotherapy was developed, with the goal of incorporating the fundamental interplay between hypoxia and proliferation, including reoxygenation over a course of radiotherapy. The fundamental idea is that the local delivery of oxygen and glucose limits the amount of proliferation and metabolically-supported cell survival a tumour sub-volume can support. The model has three compartments: a proliferating compartment of cells receiving oxygen and glucose; an intermediate, metabolically-active compartment receiving glucose; and a highly hypoxic compartment of starving cells. Following the post-mitotic cell death of proliferating cells, intermediate cells move into the proliferative compartment and hypoxic cells move into the intermediate compartment. A key advantage of the proposed model is that the initial compartmental cell distribution is uniquely determined from the assumed local growth fraction (GF) and volume doubling time (TD) values. Varying initial cell state distributions, based on the local (voxel) GF and TD, were simulated. Tumour response was simulated for head and neck squamous cell carcinoma using relevant parameter values based on published sources. The tumour dose required to achieve a 50% local control rate (TCD50) was found for various GFs and TD's, and the effect of fraction size on TCD50 was also evaluated. Due to the advantage of reoxygenation over a course of radiotherapy, conventional fraction sizes (2-2.4 Gy fx(-1)) were predicted to result in smaller TCD50's than larger fraction sizes (4-5 Gy fx(-1)) for a 10 cc tumour with GFs of around 0.15. The time to eliminate hypoxic cells (the reoxygenation time) was estimated for a given GF and decreased as GF increased. The extra dose required to overcome accelerated stem cell accumulation in longer treatment schedules was estimated to be 0.68 Gy/day (in EQD26.6), similar to published values derived from clinical data. The model predicts, for a 2 Gy/weekday fractionation, that increased initial proliferation (high GF) should, surprisingly, lead to moderately higher local control values. Tumour hypoxia is predicted to increase the required dose for local control by approximately 30%. Predicted tumour regression patterns are consistent with clinical observations. This simple yet flexible model shows how the local competition for chemical resources might impact local control rates under varying fractionation conditions.

Image-based modeling of tumor shrinkage in head and neck radiation therapy

PURPOSE

Understanding the kinetics of tumor growth/shrinkage represents a critical step in quantitative assessment of therapeutics and realization of adaptive radiation therapy. This article presents a novel framework for image-based modeling of tumor change and demonstrates its performance with synthetic images and clinical cases.

METHODS

Due to significant tumor tissue content changes, similarity-based models are not suitable for describing the process of tumor volume changes. Under the hypothesis that tissue features in a tumor volume or at the boundary region are partially preserved, the kinetic change was modeled in two steps: (1) Autodetection of homologous tissue features shared by two input images using the scale invariance feature transformation (SIFT) method; and (2) establishment of a voxel-to-voxel correspondence between the images for the remaining spatial points by interpolation. The correctness of the tissue feature correspondence was assured by a bidirectional association procedure, where SIFT features were mapped from template to target images and reversely. A series of digital phantom experiments and five head and neck clinical cases were used to assess the performance of the proposed technique.

RESULTS

The proposed technique can faithfully identify the known changes introduced when constructing the digital phantoms. The subsequent feature-guided thin plate spline calculation reproduced the "ground truth" with accuracy better than 1.5 mm. For the clinical cases, the new algorithm worked reliably for a volume change as large as 30%.

CONCLUSIONS

An image-based tumor kinetic algorithm was developed to model the tumor response to radiation therapy. The technique provides a practical framework for future application in adaptive radiation therapy.