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

Radio-immune response modelling for spatially fractionated radiotherapy

Objective.Radiation-induced cell death is a complex process influenced by physical, chemical and biological phenomena. Although consensus on the nature and the mechanism of the bystander effect were not yet made, the immune process presumably plays an important role in many aspects of the radiotherapy including the bystander effect. A mathematical model of immune response during and after radiation therapy is presented.Approach.Immune response of host body and immune suppression of tumor cells are modelled with four compartments in this study; viable tumor cells, T cell lymphocytes, immune triggering cells, and doomed cells. The growth of tumor was analyzed in two distinctive modes of tumor status (immune limited and immune escape) and its bifurcation condition.Main results.Tumors in the immune limited mode can grow only up to a finite size, named as terminal tumor volume analytically calculated from the model. The dynamics of the tumor growth in the immune escape mode is much more complex than the tumors in the immune limited mode especially when the status of tumor is close to the bifurcation condition. Radiation can kill tumor cells not only by radiation damage but also by boosting immune reaction.Significance.The model demonstrated that the highly heterogeneous dose distribution in spatially fractionated radiotherapy (SFRT) can make a drastic difference in tumor cell killing compared to the homogeneous dose distribution. SFRT cannot only enhance but also moderate the cell killing depending on the immune response triggered by many factors such as dose prescription parameters, tumor volume at the time of treatment and tumor characteristics. The model was applied to the lifted data of 67NR tumors on mice and a sarcoma patient treated multiple times over 1200 days for the treatment of tumor recurrence as a demonstration.

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 of tumor regression incorporating the concept of cancer stem-like cells for patients with locally advanced lung cancer

Background

Personalized medicine for patients receiving radiation therapy remains an elusive goal due, in part, to the limits in our understanding of the underlying mechanisms governing tumor response to radiation. The purpose of this study was to develop a kinetic model, in the context of locally advanced lung cancer, connecting cancer cell subpopulations with tumor volumes measured during the course of radiation treatment for understanding treatment outcome for individual patients.

Methods

The kinetic model consists of three cell compartments: cancer stem-like cells (CSCs), non-stem tumor cells (TCs) and dead cells (DCs). A set of ordinary differential equations were developed to describe the time evolution of each compartment, and the analytic solution of these equations was iterated to be aligned with the day-to-day tumor volume changes during the course of radiation treatment. A least squares fitting method was used to estimate the parameters of the model that include the proportion of CSCs and their radio-sensitivities. This model was applied to five patients with stage III lung cancer, and tumor volumes were measured from 33 cone-beam computed tomography (CBCT) images for each of these patients. The analytical solution of these differential equations was compared with numerically simulated results.

Results

For the five patients with late stage lung cancer, the derived proportions of CSCs are 0.3 on average, the average probability of the symmetry division is 0.057 and the average surviving fractions of CSCs is 0.967, respectively. The derived parameters are comparable to the results from literature and our experiments. The preliminary results suggest that the CSC self-renewal rate is relatively small, compared to the proportion of CSCs for locally advanced lung cancers.

Conclusions

A novel mathematical model has been developed to connect the population of cancer stem-like cells with tumor volumes measured from a sequence of CBCT images. This model may help improve our understanding of tumor response to radiation therapy, and is valuable for development of new treatment regimens for patients with locally advanced lung cancer.

Utilization of a hybrid finite-element based registration method to quantify heterogeneous tumor response for adaptive treatment for lung cancer patients

Tumor response to radiation treatment (RT) can be evaluated from changes in metabolic activity between two positron emission tomography (PET) images. Activity changes at individual voxels in pre-treatment PET images (PET1), however, cannot be derived until their associated PET-CT (CT1) images are appropriately registered to during-treatment PET-CT (CT2) images. This study aimed to investigate the feasibility of using deformable image registration (DIR) techniques to quantify radiation-induced metabolic changes on PET images. Five patients with non-small-cell lung cancer (NSCLC) treated with adaptive radiotherapy were considered. PET-CTs were acquired two weeks before RT and 18 fractions after the start of RT. DIR was performed from CT1 to CT2 using B-Spline and diffeomorphic Demons algorithms. The resultant displacements in the tumor region were then corrected using a hybrid finite element method (FEM). Bitmap masks generated from gross tumor volumes (GTVs) in PET1 were deformed using the four different displacement vector fields (DVFs). The conservation of total lesion glycolysis (TLG) in GTVs was used as a criterion to evaluate the quality of these registrations. The deformed masks were united to form a large mask which was then partitioned into multiple layers from center to border. The averages of SUV changes over all the layers were 1.0  ±  1.3, 1.0  ±  1.2, 0.8  ±  1.3, 1.1  ±  1.5 for the B-Spline, B-Spline  +  FEM, Demons and Demons  +  FEM algorithms, respectively. TLG changes before and after mapping using B-Spline, Demons, hybrid-B-Spline, and hybrid-Demons registrations were 20.2%, 28.3%, 8.7%, and 2.2% on average, respectively. Compared to image intensity-based DIR algorithms, the hybrid FEM modeling technique is better in preserving TLG and could be useful for evaluation of tumor response for patients with regressing tumors.

Prediction of Treatment Response for Combined Chemo- and Radiation Therapy for Non-Small Cell Lung Cancer Patients Using a Bio-Mathematical Model

The goal of this work was to develop a mathematical model to predict Kaplan-Meier survival curves for chemotherapy combined with radiation in Non-Small Cell Lung Cancer patients for use in clinical trial design. The Gompertz model was used to describe tumor growth, radiation effect was simulated by the linear-quadratic model with an α/β-ratio of 10, and chemotherapy effect was based on the log-cell kill model. To account for repopulation during treatment, we considered two independent methods: 1) kickoff-repopulation using exponential growth with a decreased volume doubling time, or 2) Gompertz-repopulation using the gradually accelerating growth rate with tumor shrinkage. The input parameters were independently estimated by fitting to the SEER database for untreated tumors, RTOG-8808 for radiation only, and RTOG-9410 for sequential chemo-radiation. Applying the model, the benefit from concurrent chemo-radiation comparing to sequential for stage III patients was predicted to be a 6.6% and 6.2% improvement in overall survival for 3 and 5-years respectively, comparing well to the 5.3% and 4.5% observed in RTOG-9410. In summary, a mathematical model was developed to model tumor growth over extended periods of time, and can be used for the optimization of combined chemo-radiation scheduling and sequencing.

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.