# Personalized Radiotherapy Planning Based on a Computational Tumor Growth Model

TL;DR: Two methods to derive the radiotherapy prescription dose distribution are introduced, which are based on minimizing integral tumor cell survival using the maximum a posteriori or the expected tumor cell density and it is shown how the method allows the user to compute a patient specific radiotherapy planning conformal to the tumor infiltration.

Abstract: In this article, we propose a proof of concept for the automatic planning of personalized radiotherapy for brain tumors. A computational model of glioblastoma growth is combined with an exponential cell survival model to describe the effect of radiotherapy. The model is personalized to the magnetic resonance images (MRIs) of a given patient. It takes into account the uncertainty in the model parameters, together with the uncertainty in the MRI segmentations. The computed probability distribution over tumor cell densities, together with the cell survival model, is used to define the prescription dose distribution, which is the basis for subsequent Intensity Modulated Radiation Therapy (IMRT) planning. Depending on the clinical data available, we compare three different scenarios to personalize the model. First, we consider a single MRI acquisition before therapy, as it would usually be the case in clinical routine. Second, we use two MRI acquisitions at two distinct time points in order to personalize the model and plan radiotherapy. Third, we include the uncertainty in the segmentation process. We present the application of our approach on two patients diagnosed with high grade glioma. We introduce two methods to derive the radiotherapy prescription dose distribution, which are based on minimizing integral tumor cell survival using the maximum a posteriori or the expected tumor cell density. We show how our method allows the user to compute a patient specific radiotherapy planning conformal to the tumor infiltration. We further present extensions of the method in order to spare adjacent organs at risk by re-distributing the dose. The presented approach and its proof of concept may help in the future to better target the tumor and spare organs at risk.

## Summary (3 min read)

### Introduction

- Previous works on computational growth models for gliomas have focused on reaction-diffusion equations to model cell proliferation and infiltration into surrounding brain tissue [1].
- In order to account for the infiltrative nature of the tumor, several studies recently proposed to personalize radiotherapy planning based on a computational growth model.
- They showed that personalizing the delivered dose could improve therapy in terms of days gained by the patients.
- In the third scenario, the authors include the uncertainty in the segmentation of the abnormality visible on the different MRIs to the personalization strategy.

### II. SEGMENTATION SAMPLES

- The T1Gd abnormality, which is the active part of the tumor, and the larger T2-FLAIR abnormality, which is usually called the edema, were segmented by a clinician.
- In order to take into account the uncertainty in the segmentation, the authors propose to randomly modify the original clinician segmentations.
- The samples are efficiently produced on the regular grid using the separability and stationary properties of the squared exponential covariance function (see [18] for details).
- Segmentation samples for the T1Gd and T2-FLAIR abnormalities at the first and second time points are generated.
- One could compare the output of the tumor growth model with probabilistic segmentation approaches which have been proposed for glioblastoma [20].

### III. TUMOR GROWTH MODEL

- Below, the authors identify the scalar parameter dw with D. The solution of the reaction-diffusion equation (1) is a tumor cell density u computed over the whole brain domain.
- The method is based on the assumption that the solution of equation (1) at the first time point has converged to its asymptotic, traveling wave type solution.
- Thereby, the tumor cell density is propagated outward (and inward), starting from the T1Gd segmentation, and drops approximately exponentially with distance.
- The reaction-diffusion equation is solved using the Lattice Boltzmann Method [21], [23], [24] which allows for easy parallelization and fast computations.
- The common approach taken in these works is to add a death term to the reaction-diffusion equation, which allows to model the shrinkage of the tumor due to the therapy.

### IV. PERSONALIZATION

- The personalization of the tumor growth model is combined with a dose response model in order to define the radiotherapy planning.
- First the authors only use a single time point (the second acquisition) to personalize the model such that the radiotherapy plan will be defined using a single acquisition, similarly to what is being done in clinic.
- Second the authors use two time points in order to personalize the model.
- The radiotherapy plan will then be defined on the latest acquisition.
- Third, the authors use two time points and include the uncertainty in the segmentation.

### A. Scenario 1: One time point only

- The authors further model the prior as log-uniform and independent between the parameters, P (θ) = P (D)P (ρ) (4) The authors sample from the posterior distribution using a Metropolis-Hasting algorithm.
- Note that this section only uses the initialization algorithm (see Section III) which only depends on the invisibility index λ = √ D/ρ.
- Note that this section can be related to the method described in [16], where a single time point is used to propose a dose planning.
- Moreover, the Bayesian methodology allows to take into account the uncertainty in the personalization.

### V. RADIOTHERAPY PLANNING

- The authors detail how they use the personalization of the tumor growth model in order to define the best radiotherapy plan at the time of the second acquisition.
- The authors start by coupling the growth model with a cell survival model (Section V-A).
- The authors then detail how to compute the prescription doses in Section V-B, and how to compute the delivered dose in Section V-C.

### A. Cell survival

- Cell survival after irradiation is often modeled using the linear-quadratic model.
- The authors follow the derivations of [16], and consider the linear approximation of the linear-quadratic model.

### B. Prescription Dose Optimization

- A prescription dose can be defined as the dose minimizing the surviving fraction of tumor cells.
- The personalization of the tumor growth model provides samples {θl} from the posterior distribution P (θ|S).
- The probabilistic dose is defined as the dose minimizing the expectation of the surviving fraction of tumor cells.

### VI. RESULTS

- The authors first present the results for one high grade glioma patient.
- For scenarios 2 and 3, 2000 samples are drawn from the posterior distribution, leading to an acceptance rate of 60%.
- Including the second time point, and the uncertainty in the segmentation, increases the uncertainty in the invisibility index.
- Those two figures highlights the two sources of uncertainty for scenario 3: the segmentation, and the tumor infiltration.
- This can be more clearly observed by looking at the dose volume histograms on Figure 15.

### VII. DISCUSSION

- Extensions of the model could be considered in order to include the noise level σ, the thresholds τ1, and τ2, as free parameters of the model.
- This level of noise allows to explore the parameter space, and to focus on a region of interest which is in accordance with the lowest distances found.
- It was set manually after a few experiments.
- The authors did not include it in this study for several reasons.
- Second, this adds complexity and increases the computational cost of the method.

### VIII. CONCLUSION

- The authors presented the proof of concept for a method combining a computational model of tumor growth with a dose response model in order to optimize radiotherapy planning, which takes into account the uncertainty in the model parameters and the clinical segmentations.
- In the third one, the authors include uncertainty in the segmentation process.
- The MAP dose minimizes surviving tumor cells after irradiation of the most probable situation, while the probabilistic dose allows one to take into account the uncertainty by minimizing the expected surviving tumor cells.
- To that end, the model should be extended in order to take into account the complex therapy the patient is undergoing.
- Finally, it should be investigated if more conformal dose delivery techniques such as proton therapy lead to IMRT planning more conformal to the prescribed dose.

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##### Citations

84 citations

### Cites background from "Personalized Radiotherapy Planning ..."

...For example, multi-modality MR images have been collectively utilized to study the neuroanatomy of human brains for disease diagnosis [3] or therapy planning [4]....

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30 citations

### Cites methods from "Personalized Radiotherapy Planning ..."

...This model structure was also recently used to investigate the personalization of radiotherapy strategy for brain cancer patients.(66) Setting f (c (x,t))= ⋅c (x,t), a similar model structure was also used to simulate the growth of glioblastoma based on previous reported parameters estimated from patients and estimated the survival times of patients under different parameter settings....

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23 citations

### Cites background or methods from "Personalized Radiotherapy Planning ..."

...They have been used to (i) study tumor growth patterns in individual patients [45, 127, 135, 167, 189], (ii) extrapolate the physiological boundary of tumors [113, 148], (iii) or study the effects of clinical intervention [116, 117, 139, 164, 169, 191, 203]....

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...finite-difference approximations to the gradient [101], or tackle the parameter estimation problem within a Bayesian framework [48, 91, 116, 117, 122, 123, 138, 154] (see Rem....

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...A common assumption in many models is that the rate of migration depends on the tissue composition; typically, it is modeled to be faster in white than in gray matter [116, 117, 189]....

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...such example is to relate (manual) segmentations of imaging abnormalities to a detection thresholds for the computed population density of cancerous cells [70, 89, 116, 117, 127, 133, 191]....

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##### References

^{1}, ETH Zurich

^{2}, University of Bern

^{3}, Harvard University

^{4}, National Institutes of Health

^{5}, University of Debrecen

^{6}, University Hospital Heidelberg

^{7}, McGill University

^{8}, University of Pennsylvania

^{9}, French Institute for Research in Computer Science and Automation

^{10}, University at Buffalo

^{11}, Microsoft

^{12}, University of Cambridge

^{13}, Stanford University

^{14}, University of Virginia

^{15}, Imperial College London

^{16}, Massachusetts Institute of Technology

^{17}, Columbia University

^{18}, Sabancı University

^{19}, Old Dominion University

^{20}, RMIT University

^{21}, Purdue University

^{22}, General Electric

^{23}

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...The five presented samples per abnormality correspond to an average DICE of 87%, which is comparable to the inter-expert DICE measured in the BraTS Challenge for brain tumors delineation [19]....

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##### Frequently Asked Questions (2)

###### Q2. What have the authors stated for future works in "Personalized radiotherapy planning based on a computational tumor growth model" ?

The authors presented the proof of concept for a method combining a computational model of tumor growth with a dose response model in order to optimize radiotherapy planning, which takes into account the uncertainty in the model parameters and the clinical segmentations. In the second one, the authors use two time points in order to personalize the model and plan radiotherapy. In the future, the inclusion of the fractionation scheme of the delivered dose could be optimized. To that end, the model should be extended in order to take into account the complex therapy the patient is undergoing.