MR to CT synthesis with paired data 

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Energy transport and deposit

This section presents the mechanism of transport and interaction be-tween a photon beam and the medium. This explanation will help us to understand how the dose is deposited in the body.

Photons, the carriers

The photon has zero mass and zero electric charge. Contrary to charged particles, its displacement does not necessarily imply an in-teraction with the environment and can therefore just cross a body without any interaction. The probability p that a photon interacts through a linear section dx is given by p = dx (1.1.1)
where is the linear attenuation coe cient (cm 1) and depends on the energy of the photon and the electron density of the material. When a photon interacts with an atom, a transfer of energy towards an electron (e ) of this atom is made. This transfer of energy has the e ect of setting the electron in motion outside the electronic cloud (ionization). This interaction is called the photoelectric e ect (Figure 1.1). The second possible interaction is the Comptom scatter. The in-cident photon will set in motion an electron of the peripheral layers outside the electronic cloud (ionization). The incident photon is de-viated from its initial trajectory, it is scattered (Figure 1.2). It is the predominant e ect in radiotherapy. Last inelastic phenomenon is pair production occurring only at higher energy level. Near a nucleus, the incident photon is absorbed and a positron-electron pair is produced (Figure 1.3). Ultimately, the positron (e+) will meet an electron. The two particles will annihilate each other by emitting two annihilation photons of equal energy in opposite directions (concept used in PET scan). All these interactions imply a transfer of energy and not a deposit of energy. Photon beam is an indirectly ionizing radiation. It is the released electrons that will deposit their kinetic energy on their path by Coulombic interactions in the medium.

Electrons, the ionizers

Unlike photons, electrons are charged particles. They are therefore subject to many more interactions with the medium, which has the e ect of quickly slowing them down, until they stop completely. Electrons passing through the medium can interact with an elec-tronic cloud. The collision between the incident electron and one at equilibrium will cause sometimes just a excitation or the ejection of this one (ionization, Figure 1.4), at the cost of a loss of energy of the incident electron. This is called inelastic collision. A hole appears on the electronic cloud which is quickly lled by an electron from a neighboring layer generating a X-ray. The emitted energy is characteristic of the di er-ence in energy between two electronic level. An electron of su cient energy has to interact inelastically several hundred times before losing all its kinetic energy. The succession of ionization is the main source of biological damage due to localized radiation (ionization).
Less often, the electron is slowed down and it is deviated while passing in the vicinity of a nucleus. The energy lost by the electron is carried away by a photon, called secondary (Bremsstrahlung X-ray, Figure 1.5). Electrons tend to lose kinetic energy continuously as trey travel, the rate of energy lost is called the stopping power1 and is related to the two previous phenomenons. The energy lost E by the collision of electrons in an in nitesimal sample @V of density is related to the dose received in the sample D = E = E (1.1.2) @V m
where m is the mass of the in nitesimal sample. Other phenomenons of increasing complexity can alter the dose deposit, we just give a brief explanation. Inverse-square law The uence which is the number of crossing pho-tons over a de ned surface area is inversely proportional to the square of the distance from the source. Heterogeneity and interfaces When several materials compose the medium, the dose distribution takes a less regular form. The greater the electronic density of the medium, the more the electron interacts, and the shorter its range.

Dose distribution and Monte Carlo

Several algorithms calculate a dose distribution in the patient accord-ing to the established treatment plan, i.e. a unique set of beam parameters (number of beams, orientation, MLC). Among the current dose calculation algorithms, Monte Carlo (MC) algorithms are considered to be the most accurate, because tissue heterogeneity and other com-plex interactions are fully taken into account for the dose calculation.
MC algorithms are stochastic methods for solving numerical prob-lems for which no analytical formulation is de ned2. From the proba-bility distributions governing the interactions of electrons and photons, the transport of these particles in medium are simulated. The resulting physical quantities, such as the deposited dose, can be calculated by generating a very large number of simulated particles, called histories. The system uses the physical properties of the radiation in combina-tion with random number generators to determine when the particles will interact in the medium, and the type of interaction. Each history can be summarize in two successive steps Straight line motion the particle (photon or electron) moves in a straight line and without any interaction. The length of its path is ob-tained by a random draw on the probability distribution of the length of the path of a particle. We can therefore know the position where the particle interacts (or not). Interaction The position of the interaction having been simulated, its type and characteristics must be determined. The type of interaction is rst determined by random draw, from the probabilities of each in-teraction. Then the interaction itself is simulated, new random draws give the energy loss and the new direction of the incident particle, but also the characteristics of any secondary particle created. These sec-ondary particles then enter in the Straight line motion step.

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Performance comparison indicators

In order to evaluate the accuracy of a sCT generation method, we rst have to see the di erent comparison metrics used to compare a CT with a sCT. There are several of them because none of them is uniformly better than the others in every aspects. Basically, independently they give us just few hints but their their combination brings us deeper insights into their relative qualities.

Voxelwise comparison

The mean absolute error (MAE) and the mean error (ME) are two global indicators of the correspondence between the HU of each voxel in two CT images, typically an sCT and the associated CT image of the patient. The latter is considered as the reference since the image is actually produced during a CT examination. The respective formulas10 for the mean absolute error and the mean error are
1 N MAE = jct(i) sct(i)j N =0 (1.4.1)

Table of contents :

0 Introduction 
0.1 General introduction to radiotherapy
0.2 Techniques of irradiation
0.3 Radiotherapy in clinical work ow
0.4 Magnetic Resonance in Radiotherapy
0.5 Description of the project
0.6 Problem statement and contributions
0.7 Outline of the manuscript
1 Tools and theoretical aspects 
1.1 Energy transport and deposit
1.1.1 Photons, the carriers
1.1.2 Electrons, the ionizers
1.1.3 Dose distribution and Monte Carlo
1.2 Magnetic resonance imaging
1.2.1 Nuclear magnetic resonance
1.2.2 Image reconstruction
1.3 Computed tomography
1.3.1 X-ray tomography
1.3.2 In radiotherapy planication
1.4 Performance comparison indicators
1.4.1 Voxelwise comparison
1.4.2 Dose volume histogram
1.4.3 Voxelwise dose dierence
1.4.4 Gamma dose distribution evaluation tool
1.5 Deep learning in computer vision
1.5.1 The mechanics of machine learning
1.5.2 Perceptron
1.5.3 Multiple layer perceptron
1.5.4 Convolutional neural net
1.5.5 Generative adversarial net
1.5.6 Feature scaling or normalization
2 Synthesizing CT from MR images 
2.1 Bulk density method
2.2 Atlas-based method
2.3 Voxelwise conversion
2.4 Machine Learning
2.4.1 Patch-based
2.4.2 Convolutional neural network
2.4.3 Generative adversarial network
2.5 Conclusion
3 MR to CT synthesis with paired data 
3.1 Introduction
3.2 A conditional GAN for MR-to-CT synthesis
3.2.1 cGAN baseline
3.2.2 The pix2pixHD network
3.2.3 Tailored architecture for sCT generation
3.3 Experimental material and implementation details
3.3.1 Patient data collection
3.3.2 Image pre-processing
3.3.3 Training of the network
3.3.4 sCT evaluation
3.4 Results
3.5 Discussion and Conclusion
Appendices
3.A Network details
4 MR to CT synthesis with unpaired data 
4.1 Introduction
4.2 From cycle GANs to an augmented cycle GAN for MR-to-CT synthesis
4.2.1 Unsupervised learning of one-to-many
4.3 Experimental material and implementation details
4.3.1 Patient data collection
4.3.2 Image pre-processing
4.3.3 Training of the networks
4.3.4 sCT evaluation
4.4 Results
4.4.1 Image comparison
4.4.2 DVH analysis
4.4.3 Dose dierence
4.4.4 CycleGAN comparison
4.5 Discussion
4.6 Conclusion
Appendices
4.A Fiducial markers
4.B Network details
Discussion and conclusion
Bibliography 

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