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.

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.

Kinetic modeling and numerical simulations to predict patient-specific responses to radiotherapy

Recent research indicates that quiescent tumor cells are significantly less radiosensitive with a greater repair capacity than proliferative cells. In order to better predict patient-specific responses to radiotherapy, we develop a mathematical model with treatment terms to describe dynamical behaviors of tumor growth. The global stabilities of the tumor-free equilibrium, the tumor-present equilibrium and the corresponding sufficient criteria are obtained. In addition, we simulate volumetric imaging data from 12 head-and-neck cancer patients and estimate the patient-specific responses to radiotherapy. Results indicate that radiosensitivity of proliferative cells is a critical factor that determines a successful radiotherapy. By comparison with previous simulation results, we find that the model presented in this paper is more suitable to describe the radiotherapy procedure of head-and-neck cancer. Finally, we discuss the influences of different radiotherapy strategies on therapeutic effect. The results show that treatment strategies with large dose or short treatment cycle can obtain better treatment effect.

Accurate outcome prediction after neo-adjuvant radio-chemotherapy for rectal cancer based on a TCP-based early regression index

•

A TCP-based early regression index (ERI_TCP) was previously introduced.

•

ERI_TCP was associated to improved survival after neo-adjuvant therapy for rectal cancer.

•

Distant-metastasis-free survival was predicted by ERI_TCP and 5-FU dose.

•

The resulting AUC (0.86) was significantly higher than models not including T ERI_TCP.

•

ERI_TCP is a promising tool for therapy personalization.

Background and purpose

An early tumor regression index (ERI_TCP) was previously introduced and found to predict pathological response after neo-adjuvant radio-chemotherapy of rectal cancer. ERI_TCP was tested as a potential biomarker in predicting long-term disease-free survival.

Materials and methods

Data of 65 patients treated with an early regression-guided adaptive boosting technique (ART) were available. Overall, loco-regional relapse-free and distant metastasis-free survival (OS, LRFS, DMFS) were considered. Patients received 41.4 Gy in 18 fractions (2.3 Gy/fr), including ART concomitant boost on the residual GTV during the last 6 fractions (3 Gy/fr, D_mean: 45.6 Gy). Chemotherapy included oxaliplatin and 5-fluorouracil (5-FU). T2-weighted MRI taken before (MRI_pre) and at half therapy (MRI_half) were available and GTVs were contoured (V_pre, V_half). The parameter ERI_TCP = −ln[(1 − (V_half/V_pre))^Vpre] was calculated for all patients. Cox regression models were assessed considering several clinical and histological variables. Cox models not including/including ERI_TCP (CONV_model and REGR_model respectively) were assessed and their discriminative power compared.

Results

At a median follow-up of 47 months, OS, LRFS and DMFS were 94%, 95% and 78%. Due to too few events, multivariable analyses focused on DMFS: the resulting CONV_model included pathological complete remission or clinical complete remission followed by surgery refusal (HR: 0.15, p = 0.07) and 5-FU dose >90% (HR: 0.29, p = 0.03) as best predictors, with AUC = 0.75. REGR_model included ERI_TCP (HR: 1.019, p < 0.0001) and 5-FU dose >90% (HR: 0.18, p = 0.005); AUC was 0.86, significantly higher than CONV_model (p = 0.05). Stratifying patients according to the best cut-off value for ERI_TCP and to 5-FU dose (> vs <90%) resulted in 47-month DMFS equal to 100%/69%/0% for patients with two/one/zero positive factors respectively (p = 0.0002). ERI_TCP was also the only variable significantly associated to OS (p = 0.01) and LRFS (p = 0.03).

Conclusion

ERI_TCP predicts long-term DMFS after radio-chemotherapy for rectal cancer: an independent impact of the 5-FU dose was also found. This result represents a first step toward application of ERI_TCP in treatment personalization: additional confirmation on independent cohorts is warranted.

Studying the regression profiles of cervical tumours during radiotherapy treatment using a patient-specific multiscale model

Apart from offering insight into the biomechanisms involved in cancer, many recent mathematical modeling efforts aspire to the ultimate goal of clinical translation, wherein models are designed to be used in the future as clinical decision support systems in the patient-individualized context. Most significant challenges are the integration of multiscale biodata and the patient-specific model parameterization. A central aim of this study was the design of a clinically-relevant parameterization methodology for a patient-specific computational model of cervical cancer response to radiotherapy treatment with concomitant cisplatin, built around a tumour features-based search of the parameter space. Additionally, a methodological framework for the predictive use of the model was designed, including a scoring method to quantitatively reflect the similarity and bilateral predictive ability of any two tumours in terms of their regression profile. The methodology was applied to the datasets of eight patients. Tumour scenarios in accordance with the available longitudinal data have been determined. Predictive investigations identified three patient cases, anyone of which can be used to predict the volumetric evolution throughout therapy of the tumours of the other two with very good results. Our observations show that the presented approach is promising in quantifiably differentiating tumours with distinct regression profiles.

Principal component analysis modeling of Head-and-Neck anatomy using daily Cone Beam-CT images

PURPOSE

To model Head-and-Neck anatomy from daily Cone Beam-CT (CBCT) images over the course of fractionated radiotherapy using principal component analysis (PCA).

METHODS AND MATERIALS

Eighteen oropharyngeal Head-and-Neck cancer patients, treated with volumetric modulated arc therapy (VMAT), were included in this retrospective study. Normal organs, including the parotid and submandibular glands, mandible, pharyngeal constrictor muscles (PCMs), and spinal cord were contoured using daily CBCT image datasets. PCA models for each organ were developed for individual patients (IP) and the entire patient cohort/population (PP). The first 10 principal components (PCs) were extracted for all models. Analysis included cumulative and individual PCs for each organ and patient, as well as the aggregate organ/patient population; comparisons were made using the root-mean-square (RMS) of the percentage predicted spatial displacement for each PC.

RESULTS

Overall, spatial displacement prediction was achieved at the 95% confidence level (CL) for the first three to four PCs for all organs, based on IP models. For PP models, the first four PCs predicted spatial displacement at the 80%-89% CL. Differences in percentage predicted spatial displacement between mean IP models for each organ ranged from 2.8% ± 1.8% (1st PC) to 0.6% ± 0.4% (4th PC). Differences in percentage predicted spatial displacement between IP models vs the mean IP model for each organ based on the 1st PC were <12.9% ± 6.9% for all organs. Differences in percentage predicted spatial displacement between IP and PP models based on all organs and patients for the 1st and 2nd PC were <11.7% ± 2.2%.

CONCLUSION

Tissue changes during fractionated radiotherapy observed on daily CBCT in patients with Head-and-Neck cancers, were modeled using PCA. In general, spatial displacement for organs-at-risk was predicted for the first 4 principal components at the 95% confidence levels (CL), for individual patient (IP) models, and at the 80%-89% CL for population-based patient (PP) models. The IP and PP models were most predictive of changes in glandular organs and pharyngeal constrictor muscles, respectively.

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.

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.