Basic x-ray imaging concepts SNR^^ is given by the incident x-ray fluence, 0 ^ In this case the frequency dependent

DQE is given by

This equation is important from the experimental point of view since it indicates that DQE can in principle be determined from experimentally measured quantities.

1.8.3. Multi-stage imaging systems

If the digital radiographic system involves several steps in the process of image formation then the signal and noise transfer characteristics through the system can be analysed in terms of elementary amplification and scattering processes.

Rabbani et al (1987) have developed a model that uses multivariate moment generating functions to express the input-output relationships of noise power spectra through a series of cascaded steps. The model considers the information carriers (e.g. x-ray or light photons) as a series of random impulses. Only three elementary processes are used to describe the propagation of signal and noise through the system: binary selection, amplification and scattering. If and ( / ) represent the mean quantum fluence and the noise Wiener spectrum at the input of the i-th stage of the system, and 0 , and W s ^ f) represent the corresponding output, then the propagation of signal and noise for each elementary process are as follows:

Binary selection. A binary selection process is characterised by a probability ( l - q ) that an input quantum will appear at the output, 0 . = (1 - q) 0 , where q represents the probability of interaction. The Wiener spectrum is given by

W5,(/) = (l- q )^ W5,_,(/) + q ( l- q ) 0 ,_ , (1.57)

Scattering. The scattering process is characterised by the modulation transfer function of the stage, M TF.{f). The photon fluence is preserved, that is 0 , =0,_i and the Wiener spectrum is given by

Ws, ( / ) = [ Wi,_, ( / ) - 0,_, ] MTF;^ ( / ) + <!>,,, (1.58) Amplification. An amplification process is described by the mean gain g.^ and the gain

Chapter 1 Basic x-ray imaging concepts variance . The output photon flux increases by a a factor g., 0 . = g, 0,_,, and the Wiener spectrum is given by

(1.59) Using these expressions Cunningham et al (1994) obtained a generalised expression for the spatial-frequency dependent DQE of a A^-stage system with a Poisson distributed input, which is given by

D Q E ,A f) = N 1=1 i + e j M r f ; ( / ) | P, ( f ) (1.60) where ;=i

and e„ is the Poisson excess factor, which is defined as

(1.61)

(1.62)

Equations 1.51 and 1.52 can be used to obtain an expression for the DQE of a system with an arbitrary sequence of stages that can be described in terms of amplification or scattering mechanisms, given only the average gain g., the Poisson excess term and the modulation transfer function MTF^{f) of each stage. The particular values that these parameters take according to the type of process are given in table 1.1. Notice that in this model binary processes are considered a particular case of amplification.

Table 1.1. Values of the gain variance (a^ ) and Poisson excess factor (e^ ) for amplification and scattering processes in a multi-stage system.

Process Gain MTF Type

<

Deterministic 0 -1

Amplification _Si 1 Poisson _Si 0

Binary _{&(l-&) - S i}

Chapter 1 ______________________ Basic x-ray imaging concepts

1.9. Monte Carlo methods

The development of new x-ray imaging detectors requires the optimise the parameters that are related to image quality while maintaining low radiation dose to the patient. Computer simulation has proved to be a very useful tool for examining the merits of various design strategies and for finding the optimal choices of detector parameters, such as composition and thickness (Bencivelli e ta l 1991, Boone 1992).

The Monte Carlo method is probably the most common technique employed for the study of radiation transport in medical radiation physics. An extensive review of the Monte Carlo techniques as applied to medical radiation physics problems has been recently made by Andreo (1991).

In this thesis all the simulations that involve radiation transport were developed using the Electron-Gamma Shower v.4 (EGS4) code system (Nelson et al 1985). EGS4 is a general purpose package for the Monte Carlo simulation of the coupled transport of electrons and photons in an arbitrary geometry. EGS4 is not a stand-alone Monte Carlo code, but a set of subroutines that have to be linked to a series of user-written routines, which are called the user code. The user code contains a description of the geometry of the problem and the scoring algorithms needed to extract the relevant quantities from the transport subroutines. In a simulation each particle is followed until it reaches a pre-determined energy limit (cut-off) or a geometrical boundary defined by the user.

All the results reported in this thesis are based on very simple slab or cylindrical geometries. Accurate determination of the x-ray energy deposition required the introduction of two important modifications into the Monte Carlo code. First, correct energy scoring was ensured by including the parameter reduced electron-step transport algorithm (PRESTA) developed by Bielajew and Rogers (1987) into the EGS4 code. This algorithm was developed to facilitate the selection of the appropriate electron- step sizes required to avoid energy scoring artefacts when working at low energies. The second modification was the introduction of a method for the correct K-edge sampling in compounds, according to the algorithm developed by Del Guerra et al (1991). In the standard version of the EGS4 system the equivalent K-edge of a compound is assumed to be the weighted average of the K-edge of each element in the

Chapter 1 ______________________________________Basic x-ray imaging concepts