**2. Compton scattering**

To appreciate the role of scattered radiation for imaging purposes, it is useful to recall some of the key points on Compton scattering. When a thin pencil of X- or gamma radiation shines through a medium, its intensity weakens as its traverses matter. One of the cause for this attenuation process is Compton scattering: deflected rays by scattering will not reach a detector placed along the incident direction.

From the geometry and kinematics of this scattering, see *e.g.* [6], one may infer that:

• the scattered photon flux density in a given spatial direction is given by the differential Compton scattering cross-section *dσC*/*d*Ω = *π r*<sup>2</sup> *<sup>e</sup> <sup>P</sup>*(*ω*), where *re* <sup>=</sup> 2.82 <sup>×</sup> <sup>10</sup>−<sup>15</sup> m is the classical electron radius and *P*(*ω*), the Klein-Nishina scattering probability under a scattering angle *ω*.

• the scattered photon energy *E* is directly connected to the scattering angle *ω* by the so-called Compton relation

$$E = E(\omega) = E\_0 \frac{1}{1 - \epsilon \cos \omega'} \tag{1}$$

**Figure 1.** Compton scattering

− *SM* 0

beam spreading factor *mσ*(*r*) is of the form

and the beta-plus-ray activity density in PET.

**3. The earlier CST modalities**

**3.1. Point by point scanning CST**

<sup>A</sup>*in*(*SM*) = exp

where the attenuation factors on the traveled distances *SM* and *MD* are given by

*<sup>m</sup>σ*(*r*) = <sup>1</sup>

In equation 4, *μ*(**M**) is the matter linear attenuation coefficient at site **M** and the respective

*πσ*

with *σ* the linear size of scattering volume and *r* the traversed distance. Note that for *σ* → 0, *<sup>m</sup>σ*(*r*) <sup>∼</sup> 1/*r*2, a well known photometric factor, which shall be used later. Equation 3 is

When *E*<sup>0</sup> is chosen in such a way that competing events such as photoelectric absorption and pair creation are virtually absent, then Compton scattering becomes the main phenomena to be considered for imaging, and the relevant physical quantity is *n*(**M**), the electron density in matter at site **M**, from which other quantities such as chemical composition may be deduced. As *n*(**M**) appears also in the attenuation process, its determination becomes quite involved since it must be retrieved from a non-linear expression in *n*(**M**). It is clear that, in CST imaging, *n*(**M**) plays the role of the attenuation map in CT, the gamma-ray activity density in SPECT

Equation 3 has inspired many of the earlier CST modalities which shall be described below.

The simplest and earliest procedure is schematically described by Fig. 2. Both point source and detector are equipped with an axial collimator (lead cylindrical tube). Their axis lie in a

, and <sup>A</sup>*out*(*MD*) = exp

Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations 103

tan−<sup>1</sup> *<sup>σ</sup>* 2*r* <sup>2</sup> − *MD* 0

, (5)

*ds μ*(**M** + *s***D**)

 . (4)

*ds μ*(**S** + *s***M**)

fundamental to the image formation by scattered radiation.

where *E*<sup>0</sup> is the energy of incident photons and *�* the ratio of *E*<sup>0</sup> to the electron rest energy.

Then the number of particles *d*2*Nsc* scattered in a solid angle *d*Ω*sc* along a direction making an angle *ω* with the incident direction follows from the definition of the differential scattering cross section *dσC*/*d*Ω, if the following quantities are given: a) *φin*, the incident photon flux density, b) *n*(**M**), the electron density at the scattering site **M**, surrounded by volume element *d***M**. Thus, for a given incident energy, the angular distribution of scattered photons around the scattering site **M** is no longer isotropic. The final form of this number of scattered photons in the direction given by the angle *ω* is

$$d^2 \mathbf{N}\_{\rm sc} = \phi\_{\rm in} \ n(\mathbf{M}) d\mathbf{M} \,\pi \, r\_{\rm c}^2 P(\omega) \, d\Omega\_{\rm sc} \,. \tag{2}$$

But this is still not realistic. For radiation emitted from a point source and incident on site **M**, see Fig. 1, attenuation effects in matter before and after scattering are to be taken into account, as well as beam spreading due to straight line propagation, in the evaluation of the detected photon flux density. Equation 2 reads now for a point source emitting isotropically *I*<sup>0</sup> photons per second and per steradian

$$d^2 \mathcal{N}\_{\mathbf{sc}} = \frac{I\_0}{4\pi} \, m\_{\sigma}(SM) \, \mathcal{A}\_{\text{in}}(SM) \, n(\mathbf{M}) d\mathbf{M} \, \pi \, r\_{\varepsilon}^2 P(\omega) \, \mathcal{A}\_{\text{out}}(MD) \, m\_{\sigma'}(MD) \, d\Omega\_{\text{sc}} \tag{3}$$

**Figure 1.** Compton scattering

2 Will-be-set-by-IN-TECH

gone through revolutionary conceptual steps which have led to what is nowadays known as Compton scatter tomography (CST). Yet it has stirred continuous interest in numerous applications, see *e.g.* [2], [4], [11], [18], [5], [22], [29], [1], [23]. The aim of this chapter is to recount the past episodes of research, to give a comprehensive account of what has been accomplished in CST and to describe some new ideas which have arisen recently, see [42, 54]. The emphasis of the discussion shall be placed at the theoretical level. Negative effects on imaging such as beam attenuation and multiple scattering, which complicate enormously the analytic treatment, shall be dealt with conventional retrieval or compensation methods. The crux of the matter is to see how fertile ideas evolve in time and generate new fruitful concepts.

To appreciate the role of scattered radiation for imaging purposes, it is useful to recall some of the key points on Compton scattering. When a thin pencil of X- or gamma radiation shines through a medium, its intensity weakens as its traverses matter. One of the cause for this attenuation process is Compton scattering: deflected rays by scattering will not reach

• the scattered photon flux density in a given spatial direction is given by the differential

classical electron radius and *P*(*ω*), the Klein-Nishina scattering probability under a scattering

• the scattered photon energy *E* is directly connected to the scattering angle *ω* by the so-called

where *E*<sup>0</sup> is the energy of incident photons and *�* the ratio of *E*<sup>0</sup> to the electron rest energy. Then the number of particles *d*2*Nsc* scattered in a solid angle *d*Ω*sc* along a direction making an angle *ω* with the incident direction follows from the definition of the differential scattering cross section *dσC*/*d*Ω, if the following quantities are given: a) *φin*, the incident photon flux density, b) *n*(**M**), the electron density at the scattering site **M**, surrounded by volume element *d***M**. Thus, for a given incident energy, the angular distribution of scattered photons around the scattering site **M** is no longer isotropic. The final form of this number of scattered photons

1

2

2

*<sup>e</sup> <sup>P</sup>*(*ω*), where *re* <sup>=</sup> 2.82 <sup>×</sup> <sup>10</sup>−<sup>15</sup> m is the

<sup>1</sup> <sup>−</sup> *�* cos *<sup>ω</sup>* , (1)

*<sup>e</sup> P*(*ω*) *d*Ω*sc*. (2)

*<sup>e</sup> P*(*ω*) A*out*(*MD*) *mσ*�(*MD*) *d*Ω*sc*, (3)

From the geometry and kinematics of this scattering, see *e.g.* [6], one may infer that:

*E* = *E*(*ω*) = *E*<sup>0</sup>

*d*2*Nsc* = *φin n*(**M**)*d***M** *π r*

*mσ*(*SM*) A*in*(*SM*) *n*(**M**)*d***M** *π r*

But this is still not realistic. For radiation emitted from a point source and incident on site **M**, see Fig. 1, attenuation effects in matter before and after scattering are to be taken into account, as well as beam spreading due to straight line propagation, in the evaluation of the detected photon flux density. Equation 2 reads now for a point source emitting isotropically *I*<sup>0</sup> photons

**2. Compton scattering**

angle *ω*.

Compton relation

a detector placed along the incident direction.

in the direction given by the angle *ω* is

per second and per steradian

*<sup>d</sup>*2*Nsc* <sup>=</sup> *<sup>I</sup>*<sup>0</sup>

4*π*

Compton scattering cross-section *dσC*/*d*Ω = *π r*<sup>2</sup>

where the attenuation factors on the traveled distances *SM* and *MD* are given by

$$\mathcal{A}\_{\rm in}(SM) = \exp\left(-\int\_0^{SM} ds \,\mu(\mathbf{S} + s\mathbf{M})\right), \quad \text{and} \quad \mathcal{A}\_{\rm out}(MD) = \exp\left(-\int\_0^{MD} ds \,\mu(\mathbf{M} + s\mathbf{D})\right). \tag{4}$$

In equation 4, *μ*(**M**) is the matter linear attenuation coefficient at site **M** and the respective beam spreading factor *mσ*(*r*) is of the form

$$m\_{\sigma}(r) = \left(\frac{1}{\pi\sigma}\tan^{-1}\frac{\sigma}{2r}\right)^{2},\tag{5}$$

with *σ* the linear size of scattering volume and *r* the traversed distance. Note that for *σ* → 0, *<sup>m</sup>σ*(*r*) <sup>∼</sup> 1/*r*2, a well known photometric factor, which shall be used later. Equation 3 is fundamental to the image formation by scattered radiation.

When *E*<sup>0</sup> is chosen in such a way that competing events such as photoelectric absorption and pair creation are virtually absent, then Compton scattering becomes the main phenomena to be considered for imaging, and the relevant physical quantity is *n*(**M**), the electron density in matter at site **M**, from which other quantities such as chemical composition may be deduced. As *n*(**M**) appears also in the attenuation process, its determination becomes quite involved since it must be retrieved from a non-linear expression in *n*(**M**). It is clear that, in CST imaging, *n*(**M**) plays the role of the attenuation map in CT, the gamma-ray activity density in SPECT and the beta-plus-ray activity density in PET.

## **3. The earlier CST modalities**

Equation 3 has inspired many of the earlier CST modalities which shall be described below.

#### **3.1. Point by point scanning CST**

The simplest and earliest procedure is schematically described by Fig. 2. Both point source and detector are equipped with an axial collimator (lead cylindrical tube). Their axis lie in a

#### 4 Will-be-set-by-IN-TECH 104 Numerical Simulation – From Theory to Industry Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations <sup>5</sup>

plane and the detector is connected to a multichannel analyzer. When the axis of a collimated point-like detector is made to intersect the incident pencil beam, the intersection site is actually a scattering site of the Compton effect. So by measuring the scattered photon flux density at a given energy, by estimating the strength of the attenuation factors, and by evaluating the beam spreading factors one can obtain *n*(**M**). The process is then repeated for all sites in a trans-axial slice. It is interesting to note that equation 3 has been recast in the framework of an inverse problem by E. M. A. Hussein et al.[33], who have set up a discretization scheme to solve it.

a moving platform which allows a raster scanning motion. The detected number of photons at each point (in fact a small "sensitive volume") is directly mapped onto the screen of an

Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations 105

Later on, S. R. Gautam et al. [21] presented an imaging system called Compton Interaction Tomography, which consists of a pencil incident beam falling on a large object, the backscattered pencil is registered by a collimated detector situated on a line parallel to the large object. This is also a point by point determination of the electron density, but with the idea that the object is on one side of the line source-detector. Numerous works have been done by the NDE community in the 80's, see *e.g.* [10], [27], [30], including some reviews such

In 1971, F. T. Farmer & M. P. Collins [19] advocated the use of a *wide angle* collimated detector limited by two plates parallel to the incident radiation pencil. Fig. 3 shows how schematically how this procedure works. The detector is coupled to a multichannel analyzer. This design allows to obtain rapidly the electron density profile along the probing incident radiation pencil. This is the reason why it is called line scanning technique. A global image of an object slice is realized when parallel lines are put together. This technique has been refined in

Variants of this design have been developed by the NDE community to combat the low efficiency of the single voxel technique. In [25], G. Harding described a Compton scatter imaging system for NDE with horizontal scanning and incident pencil beam perpendicular to the scanning direction. Scatter detectors are disposed on both sides of the incident beam. Fig. 4 shows a sketch of the COMSCAN (Compton Scatter Scanner), developed by the Philips

Then emerged several planar scatter imaging systems as extensions of linear scatter imaging systems in NDE [30], [53] as well as in medicine [24], which work at fixed scattering angle with relative success, since most of the reconstruction methods are still mathematically ill-defined and the technological problems not yet under control. Fig. 5 shows a plane by plane scanning

Research Laboratories in Hamburg, Germany which was quite successful (see [26]).

oscilloscope which yields a density image in a slice.

as [31],[32],[8], and the use of dual sources by [28].

**3.2. Line by line CST**

**Figure 3.** CST with line by line scanning

**3.3. Plane by plane CST**

[20].

**Figure 2.** CST point by point scanning

At the turn of the twentieth century, the idea of using scattered gamma rays for investigating hidden structures in tissues seems to have been proposed for the first time by F. W. Spiers [52]. In 1959, P. G. Lale [40], realizing that radiographic images of organs fail to reveal their inner structure, has suggested a technique using Compton scattered radiation from a thin pencil of X-rays. Using equation 1, it is possible to move a collimated detector in space to measure the scattered radiation flux density and deduce the electron density at a precise point on the incoming pencil of X-rays. When this measurement can be performed in a planar slice of an object, it said that an image of the slice is obtained by Compton scatter tomography (CST). Of course this is clearly a point by point procedure which is quite time consuming.

An analogous problem arises in nuclear industry. There heat transfer research in circulating two-phase fluids requires knowledge and measurement of their densities without disturbing or arresting their flow. It was realized that penetrating gamma-radiation would be the most suitable for this purpose. As matter density is responsible for traversing radiation attenuation (since most of the attenuation occurs as a result of Compton scattering) measurements of radiation attenuation along linear paths crossing the section of a pipe can be easily done. D. Kershaw [36] has shown how the matter density distribution can be reconstructed from these measurements. In fact this problem is practically identical to the medical computed tomography which was developed simultaneously at that time and Kershaw has actually performed the mathematical inversion of the classical Radon transform, but most probably without being aware of the seminal works of J. Radon [49] and A. M. Cormack [15]. But it was N. N. Kondic [37], who has first suggested the use of scattered radiation for obtaining the two-phase fluid density in a pipe.

In 1973, R. L. Clarke et al. [13] used both transmitted and scattered gamma-rays to measure bone mineral content. In [14], they have introduced the term gamma-tomography for their apparatus which is made of a fixed collimated pencil source and four focussing collimated detectors positioned so as the scattering angle is about 450. The investigated object is placed on a moving platform which allows a raster scanning motion. The detected number of photons at each point (in fact a small "sensitive volume") is directly mapped onto the screen of an oscilloscope which yields a density image in a slice.

Later on, S. R. Gautam et al. [21] presented an imaging system called Compton Interaction Tomography, which consists of a pencil incident beam falling on a large object, the backscattered pencil is registered by a collimated detector situated on a line parallel to the large object. This is also a point by point determination of the electron density, but with the idea that the object is on one side of the line source-detector. Numerous works have been done by the NDE community in the 80's, see *e.g.* [10], [27], [30], including some reviews such as [31],[32],[8], and the use of dual sources by [28].
