There is a significant difference between photon therapy and the new proton therapy. Proton therapy treatments are currently planned and delivered assuming a proton RBE of 1.1. This average value neglects any dependency of RBE on dose, endpoint or proton beam properties. We develop an IMPT optimization model that guarantees homogenous biological effectiveness instead of common homogenous dose distribution over the target volume while minimizing the damage to multiple organs at risk. By combining IMPT optimization into POMDP fractionation optimization, we provide a comprehensive decision-making policy that practically assures the highest quality of a personalized treatment in terms of both fractionation planning and radiation delivery. We improve our POMDP framework by using texture analysis of the CT images to derive partial observation of the tumor state.

1.       Introduction

Radiation therapy is one of the most common methods for cancer treatment which uses high-energy particles such as photon or proton to destroy cancer cells. Unlike chemotherapy that exposes the whole body to the harmful side effects of the treatment, radiation therapy is a local treatment and only affects the organs located in the radiation field, commonly called organs at risk (OAR). [1] The aim of treatment planning for external radiation therapy is to reduce the OAR damage while killing the tumor cells. Therefore, the radiation dose required for destroying the tumor must be divided into several treatment sessions to reduce the damage to OARs and provide enough time for them to recover from incurred damages. The conventional fractionation plan divides the prescribed dose equally in all treatment session. The effectiveness of this plan needs to be investigated because the biological response of each patient can be different.

There is a significant difference between the common radiation therapy that uses photon beam and the new proton therapy. The photon beams when entered human body deposit most of their energy in the beginning of their path, causing more destruction to organs close to surface while delivering less of their energy to the tumor. However, the dose deposition of proton beams increases as the beam penetrates deeper in the tissue with a sharp longitudinal dose fall-off at the end of the particle range called Bragg Peak [Kramer & Scholz 2006]. The linear energy transfer (LET) of proton is lower than that of photon, therefore the biological effectiveness of the proton dose is different than the same photon dose. Relative biological effectiveness (RBE) is a measure for converting a reference dose into photon dose to yield the same biological effect. Currently, a proton RBE of 1.1 is assumed in treatment planning of proton therapy . This average value was deduced (a) for the center of the target volume, (b) at 2 Gy, and (c) averaged over various endpoints and so neglects any dependency of RBE on dose, endpoint or proton beam properties (Paganetti, 2014). Marshall et al. (2016) investigated the effect of variable RBE in proton therapy fractionation by comparing the survival fraction on distal, central and proximal areas of the SOBP of different fractionation regimens with an interaction period of 24 hours. They outlined that the strong dependence of RBE on proton LET The energy distribution of proton beams and their deposition in tissue volumes can be controlled by the physician, providing the possibility of conforming the 3D dose shape to the tumor volume. The protons are energized to specific velocities that indicate their stopping points in patient’s body. As the protons move inside the body, their velocity decreases and their interaction with molecules increases which results in maximum energy deposition. The deposited energy, known as proton linear energy transfer (LET), can be controlled by the physician so that the tumor receives highest energy while the OARs receive the lowest energy. Therefore, the dose level can increase without more damage to OARs. Taking LET into account will emerge a new direction in the fractionated treatment planning in proton therapy. Several clinical studies have investigated that shortening clinical fractionation schedules through hypofractionation might be beneficial due to unique characteristics of protons (Habl et al., 2014, Newhauser et al., 2015, Wang et al., 2013). Grassberger el al. (2011) compared 3D and EDT planning techniques in IMPT and showed that they yield equivalent dose distribution while resulting in considerably different LET distributions. Giantsoudi et al. (2013) utilized calculated LET distribution in a dose-based multi criteria optimization to enhance treatment plan. Our purpose in this study is to develop a new model to incorporate radiobiological effect of proton energy into fractionated treatment planning.

Linear-quadratic (known as LQ model) model and biological effective dose (BED) are commonly used to calculate the biological effectiveness of fractionated radiotherapy plan. LQ model was extended to include more biological factors representing the time dependent evolution of tumor in response to treatment. The linear-quadratic model assumes tumor response and growth are constant over time; however, there are experimental evidence suggesting the response of tumor cells to radiation and their growth rate are significantly affected by cell cycle regulation. In this study, we considers the extended LQ model and use a partially observable Markov decision process (POMDP) framework to deal with uncertainty of tumor response and growth rate based on cell cycle regulations. POMDP is a sequential decision making tool used for stochastic systems where full observation of the system is not possible, therefore it is a suitable tool for our problem.

Since CT images are obtained at the beginning of each week of treatment but tumor contour information is not updated, we use texture analysis of the CT images to derive partial observation of the tumor state. Several studies have investigated the potential of enhancing texture analysis using CT images for tumor prognostic and diagnostic purposes. Tumor sites including non-small lung cancer, head and neck, colorectal and esophageal cancer have been investigated in this regard. Different imaging technics including CE (contrast enhanced) and non-CE CT images and FDG PTE images (fluorodeoxyglucose positron emission Tomography) have been used for better textural features. Studies majorly focus on: Evaluating textural characteristics (majorly heterogeneity) as markers of tumor aggression (Ganeshan et al., 2012, Ganeshan et al., 2012, Ng et al. 2013). Using radiomics to build classifiers for tumor stage diagnosis (Hawkins et al., 2014). Identifying textural features useful in distinguishing tumor from normal tissue (Hawkins et al., 2014, Yu et al., 2009) Investigating correlation between tumor textural features and tumor response and survival (Cook et al., 2013). The objective of this study is to identify textural features useful in distinguishing tumor and employ a classifier to discriminate image of tumor and normal tissues.

2.       Materials and Methods

2.1   LET-based IMPT model

The biological response to radiation is commonly characterized by standard linear-quadratic model which provides the survival fraction of cells as a function of the dose with the two parameters

The constrained and indefinite time POMDP problems can be presented as following:

eq (13)

Where T is the time when the system enters the terminal state and q is the upper limit on constraint determined by the decision maker. For any possible state s, b(s) is the probability that the current state is s and evolves every time a decision is made and an observation is received. The set of all possible belief states is called the belief space and denoted by B. Therefore, if the current belief is b, and the action a is  elected and observation o is received, then the belief system can be updated using:

Eq (14)

A POMDP policy is a mapping from B to A. The following Bellman’s equation is used for iterative updating of POMDP value function:

Eq (15)

This study aims to develop personalized treatment planning policies for proton therapy in an indefinite horizon and constrained POMDP framework. The states of the system, st, are defined as the normalized number of tumor cells XtX1 and a healthy state is considered as the terminal state of the system. We are interested in developing personalized treatment policy that helps us decide whether the traditional planning scheme is sufficient or alterations are required. In this regard, the set of actions includes: (i) to deliver the equally divided prescribed dose, d, (ii) to increase the delivered dose with a specific amount, d + _1, and (iii) to decrease the delivered dose with a specific amount, d _2. Since in proton therapy the LET and tissue type are important in biological effectiveness of the treatment, we define the set of possible actions as the optimal weights of the following inverse IMPT optimization to achieve the prescribed doses d + _1, d and d + _1.

eq (16)

The aim of this problem is to maximize the BED of the tumor and minimizing the BED of organs at risk during fractionated treatment plan while maintaining the tumor survival less than OAR survival. We define the reward and cost functions as:

eq (17)

eq (18)

Where indices T and OAR refer to tumor and OAR respectively, and _ denotes the weight factor of OAR and the average BED over structure volume is calculated as

eq (19)

The transition probability function, T(s0 j s; a) and observation probability function, Z(o j s; a) can be obtained from an experts’ opinion.

2.3   Texture analysis

In supervised learning is a machine learning technique in which the inputs and outputs are given to train a machine that has the ability of classifying given examples (inputs). During training, the machine is shown an input and using internal parameters (weights) produces an output in the form of a vector of scores which. The objective is to minimize the error (or distance) between the output scores and the actual scores. Therefore, the machine repeats this process on several examples during several iterations to find the optimal weights that minimize the error. In a deep learning system, that uses multilayer neural networks to train a classification model and extract features.

Assuming (x,z) as a pair of input and output, the aim of deep learning is to estimate the function f such that z=f(x). A multilayer neural network consists of an input layer, several hidden layers and an output layer. The input layer has several nodes each taking one of the features of the scores (features) from the input data and encoding them into hidden layer using weighted links. Denoting xi, yj and zk as node in input, hidden and output layer respectively, the input layer nodes are encoded into hidden layer by

eq (20)

Where wij is the weight of the links between nodes of input and hidden layer and F is a nonlinear activation function that is commonly sigmoid. The machine then decodes the hidden layer into output layer using derivative of the activation function.

eq (21)

A machine learning system takes batches of input data with n samples and estimates the outputs.

Figure 1: Multilayer neural network

But the estimated outputs are not close to actual outputs before training is completed. An algorithm is required to improve the weights w in a way that improves the performance of the model. One way of doing so is to minimize the mean squared error of the model on each data batch.

Eq (22)

Because of nonlinear nature of the model, the machine reduces MSE using Newton based algorithm. In this study, we use quantitative information of tumor and OAR structures obtained from contoured pre-treatment CT as input of a 3-layer neural network to train a model that classifies the tissue structure. The trained model then can be applied to weekly CT images during the treatment horizon to distinguish the tumor volume without contouring and obtain partial observations from the tumor condition.

2 Future work

In future work of this study, we will first implement the LET-based IMPT model to derive the

optimal weights and thus the set of actions and immediate reward values for the POMDP model.

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In the next step we will perform data mining on the voxel features obtained from CT images to

train a model classifying the cancerous and non-cancerous cells. We will use the result of this

work to derive practical observations for the POMDP model. Also, the possibility of control

limits will be discussed due to complexity of proton therapy model and mechanism.

The milestones of this work is presented in Figure 2.

Figure 2: Estimated milestone of the study: Personalized and LET-based proton therapy treatment

planning considering tumor biological response uncertainty

References and Notes

1. T. I. Marshall, et al., International Journal of Radiation Oncology* Biology* Physics 95,

70 (2016).

2. G. Habl, et al., BMC cancer 14, 202 (2014).

3. W. D. Newhauser, et al., Cancers 7, 688 (2015).

4. Y.Wang, J. A. Efstathiou, H.-M. Lu, G. C. Sharp, A. Trofimov, Medical physics 40 (2013).

5. C. Grassberger, A. Trofimov, A. Lomax, H. Paganetti, International Journal of Radiation

Oncology*Biology*Physics 80, 1559 (2011).

6. D. Giantsoudi, et al., International Journal of Radiation Oncology* Biology* Physics 87,

216 (2013).

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7. B. Ganeshan, S. Abaleke, R. C. Young, C. R. Chatwin, K. A. Miles, Cancer imaging 10,

137 (2010).

8. B. Ganeshan, K. Skogen, I. Pressney, D. Coutroubis, K. Miles, Clinical radiology 67, 157

(2012).

9. F. Ng, B. Ganeshan, R. Kozarski, K. A. Miles, V. Goh, Radiology 266, 177 (2013).

10. S. H. Hawkins, et al., IEEE Access 2, 1418 (2014).

11. H. Yu, C. Caldwell, K. Mah, D. Mozeg, IEEE transactions on medical imaging 28, 374

(2009).

12. G. J. Cook, et al., Journal of nuclear medicine 54, 19 (2013).

13. B. Jones, R. G. Dale, International Journal of Radiation Oncology* Biology* Physics 48,

1549 (2000).

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