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

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.

The critical components for effective adaptive radiotherapy in patients with unresectable non-small-cell lung cancer: who, when and how

Adaptive radiotherapy (ART) is a new radiotherapy technology based on image-guided radiation therapy technology, used to avoid radiation overexposure to residual tumors and the surrounding normal tissues. Tumors undergoing the same radiation doses and modes can occur unequal shrinkage due to the variation of response times to radiation doses in different patients. To perform ART effectively, eligible patients with a high probability of benefits from ART need to be identified. Confirming the precise timetable for ART in every patient is another urgent problem to be resolved. Moreover, the outcomes of ART are different depending on the various image guidance used. This review discusses 'who, when and how' as the three key factors involved in the most effective implementation for the management of ART.

WAM to SeeSaw model for cancer therapy - overcoming LQM difficulties

PURPOSE

The assessment of biological effects caused by radiation exposure has been currently carried out with the linear-quadratic (LQ) model as an extension of the linear non-threshold (LNT) model. In this study, we suggest a new mathematical model named as SeaSaw (SS) model, which describes proliferation and cell death effects by taking account of Bergonie-Tribondeau's law in terms of a differential equation in time. We show how this model overcomes the long-standing difficulties of the LQ model.

MATERIALS AND METHODS

We construct the SS model as an extended Wack-A-Mole (WAM) model by using a differential equation with respect to time in order to express the dynamics of the proliferation effect. A large number of accumulated data of such parameters as α and β in the LQ based models provide us with valuable pieces of information on the corresponding parameter b 1 and the maximum volume V m of the SS model. The dose rate b 1 and the notion of active cell can explain the present data without introduction of β, which is obtained by comparing the SS model with not only the cancer therapy data but also with in vitro experimental data. Numerical calculations are presented to grasp the global features of the SS model.

RESULTS

The SS model predicts the time dependence of the number of active- and inactive-cells. The SS model clarifies how the effect of radiation depends on the cancer stage at the starting time in the treatment. Further, the time dependence of the tumor volume is calculated by changing individual dose strength, which results in the change of the irradiation duration for the same effect. We can consider continuous irradiation in the SS model with interesting outcome on the time dependence of the tumor volume for various dose rates. Especially by choosing the value of the dose rate to be balanced with the total growth rate, the tumor volume is kept constant.

CONCLUSIONS

The SS model gives a simple equation to study the situation of clinical radiation therapy and risk estimation of radiation. The radiation parameter extracted from the cancer therapy is close to the value obtained from animal experiment in vitro and in vivo. We expect the SS model leads us to a unified description of radiation therapy and protection and provides a great development in cancer-therapy clinical-planning.

Opportunities for Quantitative Translational Modeling in Oncology

A 2-day meeting was held by members of the UK Quantitative Systems Pharmacology Network () in November 2018 on the topic of Translational Challenges in Oncology. Participants from a wide range of backgrounds were invited to discuss current and emerging modeling applications in nonclinical and clinical drug development, and to identify areas for improvement. This resulting perspective explores opportunities for impactful quantitative pharmacology approaches. Four key themes arose from the presentations and discussions that were held, leading to the following recommendations: Evaluate the predictivity and reproducibility of animal cancer models through precompetitive collaboration. Apply mechanism of action (MoA) based mechanistic models derived from nonclinical data to clinical trial data. Apply MoA reflective models across trial data sets to more robustly quantify the natural history of disease and response to differing interventions. Quantify more robustly the dose and concentration dependence of adverse events through mathematical modelling techniques and modified trial design.

A mathematical model for predicting the changes of non-small cell lung cancer based on tumor mass during radiotherapy

This study aims to build a feasible mathematical model to analyze the mass evolution of NSCLC during standard fractionated radiotherapy. Seventy-three cases of NSCLC who were received radiotherapy with prescription dose of 2 Gy  ×  30 fx were selected retrospectively and divided into adenocarcinoma (ADC) group and squamous cell carcinoma (SCC) group according to the pathological type. A total of six sets of CT/CBCT images were collected. The tumor masses were measured according to each set of images. We build a mathematical model (Linear Quadratic_Repopulation&Reoxygenation& Dissolution model, LQ_RRD model), which was used to fit the first five sets of measured mass into a smooth curve. By adjusting the model parameters (λ, ν and µ), the optimal fitting results can be obtained. In order to verify the accuracy of model prediction, we measured the mass of the review images (MV, measured values), and found out the estimate point of the corresponding time (EV, estimated value) on the fitting curve. The difference and correlation between MV and EV were compared. It was found that the model could substantially simulate the tumor mass changes during radiotherapy, and it had a good fit to the clinical data (%RMSE-Median  =  5.52, %RMSE-Range  =  [3.19, 10.73]). Comparing the differences of model parameters between ADC and SCC group, there was no significant difference in λ (t  =  1.622, p   =  0.109), but the difference was significant in ν and µ (z  =  -7.270, p   =  0.000 and t  =  -10.205, p   =  0.000). Moreover, linear correlation analysis showed that there was a linear correlation between MV and EV no matter mass or volume (r  =  0.960, p   =  0.000 versus r  =  0.926, p   =  0.000). Nevertheless, the deviation between MV and EV of volume was larger than that of mass (z  =  -1.897, p   =  0.058 versus z  =  -3.387, p   =  0.001), and the deviation was more pronounced in larger tumors. We suggest that this mathematical model is more suitable to predict the tumor mass than volume for NSCLC during radiotherapy.

Personalized Radiotherapy Design for Glioblastoma: Integrating Mathematical Tumor Models, Multimodal Scans, and Bayesian Inference

Glioblastoma (GBM) is a highly invasive brain tumor, whose cells infiltrate surrounding normal brain tissue beyond the lesion outlines visible in the current medical scans. These infiltrative cells are treated mainly by radiotherapy. Existing radiotherapy plans for brain tumors derive from population studies and scarcely account for patient-specific conditions. Here, we provide a Bayesian machine learning framework for the rational design of improved, personalized radiotherapy plans using mathematical modeling and patient multimodal medical scans. Our method, for the first time, integrates complementary information from high-resolution MRI scans and highly specific FET-PET metabolic maps to infer tumor cell density in GBM patients. The Bayesian framework quantifies imaging and modeling uncertainties and predicts patient-specific tumor cell density with credible intervals. The proposed methodology relies only on data acquired at a single time point and, thus, is applicable to standard clinical settings. An initial clinical population study shows that the radiotherapy plans generated from the inferred tumor cell infiltration maps spare more healthy tissue thereby reducing radiation toxicity while yielding comparable accuracy with standard radiotherapy protocols. Moreover, the inferred regions of high tumor cell densities coincide with the tumor radioresistant areas, providing guidance for personalized dose-escalation. The proposed integration of multimodal scans and mathematical modeling provides a robust, non-invasive tool to assist personalized radiotherapy design.

Hybrid modeling frameworks of tumor development and treatment

Tumors are complex multicellular heterogeneous systems comprised of components that interact with and modify one another. Tumor development depends on multiple factors: intrinsic, such as genetic mutations, altered signaling pathways, or variable receptor expression; and extrinsic, such as differences in nutrient supply, crosstalk with stromal or immune cells, or variable composition of the surrounding extracellular matrix. Tumors are also characterized by high cellular heterogeneity and dynamically changing tumor microenvironments. The complexity increases when this multiscale, multicomponent system is perturbed by anticancer treatments. Modeling such complex systems and predicting how tumors will respond to therapies require mathematical models that can handle various types of information and combine diverse theoretical methods on multiple temporal and spatial scales, that is, hybrid models. In this update, we discuss the progress that has been achieved during the last 10 years in the area of the hybrid modeling of tumors. The classical definition of hybrid models refers to the coupling of discrete descriptions of cells with continuous descriptions of microenvironmental factors. To reflect on the direction that the modeling field has taken, we propose extending the definition of hybrid models to include of coupling two or more different mathematical frameworks. Thus, in addition to discussing recent advances in discrete/continuous modeling, we also discuss how these two mathematical descriptions can be coupled with theoretical frameworks of optimal control, optimization, fluid dynamics, game theory, and machine learning. All these methods will be illustrated with applications to tumor development and various anticancer treatments.

This article is characterized under:

Analytical and Computational Methods > Computational Methods

Translational, Genomic, and Systems Medicine > Therapeutic Methods

Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models

Effect of Changing Phantom Thickness on Helical Radiotherapy Plan: Dosimetric Analysis

Purpose: The aim of this study is to investigate the effect of changing phantom thickness on high dose region of interest (HD_ROI) and low dose ROI’s (LW_ROI’s) doses during helical radiotherapy (RT) by utilizing Adaptive RT (ART) technique.

Materials and Methods: The cylindrical phantom (CP) is wrapped with different thickness boluses and scanned in the kilovoltage computed tomography (KVCT). HD_ROI and LW_ROI’s were created in contouring system and nine same plans (1.8 Gy/Fr) were made with images of different thicknesses CP. The point dose measurements were performed using ionization chamber in Helical Tomotherapy (HT) treatment machine. For detecting thickness reduction effect, CP was irradiated using bolus-designed plans and it was irradiated using without bolus plan. The opposite of this scenario was applied to determine the thickness increase. KVCT and megavoltage CT (MVCT) images were used for dose comparison. The HT Planned Adaptive Software was used to see the differences in the planning and verification doses at dose volume histograms (DVH).

Results: Point dose measurements showed a 4.480% dose increase in 0.5 cm depth reduction for HD_ROI. These differences reached 8.508% in 2 cm depth and 15,279% in 5 cm depth. At the same time, a dose reduction of 0.665% was determined for a 0.5cm depth increase, a dose reduction of 1.771% was determined for a 2 cm depth increase, a dose reduction of 5.202% was determined for a 5 cm depth increase for the HD_ROI. The ART plan results show that the dose changes in the HD_ROI was greater than the LW_ROI’s.

Conclusion: Phantom thicknesses change can lead to a serious dose increase or decrease in the HD_ROI and LW_ROI’s.

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.