Towards a new concept for high sensitivity Compton scatter emission imaging

A new efficient scheme for imaging gamma-emitting objects is advocated in this work. It is elaborated on the recent idea of collecting data, using a detector equipped with a parallel-hole collimator, from Compton scattered photons to reconstruct an object in three-dimensions. This paper examines a working mode without collimation, which should increase its sensitivity and field of view. To simplify the otherwise complex mathematical formulation, we choose to discuss the image formation process in two-dimensions, which can be implemented with a slit collimator. Comparison with the standard collimated case, via the analysis of the shapes of the respective point spread functions (PSF), shows marked improvements and numerical simulation results, obtained using a brain phantom, support the viability and attractiveness of this new imaging modality.


INTRODUCTION
Emission imaging with gamma rays is widely used in numerous fields such as medical imaging, non-destructive testing, gamma astronomy and environmental survey. In conventional nuclear imaging, a collimated gamma camera rotates in space to collect primary radiation emitted by an object under investigation. In this case Compton scatter radiation behaves generally as noise hindering image quality and consequently correction to scatter should be applied.
However recently an interesting new imaging concept, which precisely uses, as imaging agent, scattered radiation by the object medium (instead of primary radiation), has been proposed. A spatially fixed collimated gamma camera records now images labeled by the energy of scattered radiation (or equivalently its scattering angle). It is then shown that the reconstruction of a three-dimensional object is feasible using this data [1,2,3]. However in this situation, the image sensitivity is considerably affected due to the presence of the collimator. Only about one out of 10 4 scattered photons reaches the detector. Therefore in order to record a much larger amount of scattered radiation, we propose to extend the working principle of this gamma-ray imaging to a functioning modality without collimator, as depicted by Fig. 1. However, as can be seen, this procedure differs from other proposals on scattered radiation imaging processes [4,5,6,7], in particular: -Compton tomography [8], which reconstructs the electron density of the object (instead of its activity density), and uses a moving point-like detector collecting scattered radiation from an external radiation source, -Compton camera [9], which reconstructs the activity density of an object from scattered radiation using coincidence measurements between a site on a scatter layer-detector and another site on an absorption-detector.
As the true three-dimensional problem implies involved space geometry of scattered rays, we shall first study its twodimensional counterpart to test the viability of this idea. To this end, we give a careful analysis of the image formation. The corresponding PSF is derived and shall be compared with the previous case where a collimator is present. Next we perform numerical reconstruction of the object radiation activity from simulated scattered radiation (recorded by a camera without collimator) to illustrate the working of this concept.

IMAGE FORMATION
To understand image formation by scattered radiation, we follow radiation propagation in a two-dimensional (2D) slice (or thin section) of a scattering medium with an electronic density n e assumed to be approximately constant. This can be in principle implemented with a slit placed above a detector without collimator, see Fig. 2. Moreover, to concentrate chiefly on scattering effects radiation, attenuation will be left out as working hypothesis.  (1)).
The energy of scattered photon is related to the scattering angle ω by the Compton relation: where E 0 is the emitted photon energy, ε = E 0 /mc 2 and mc 2 the rest energy of the electron.
From Fig. 3, one can see that the photon flux density reaching a site M is the number of photons emitted into the angular fan dΩ per unit length and per unit time: where |SM | is the distance between sites S and M and dS S the area element around S. The fraction of photons scattered in the direction making an angle ω with the incident direction depends on the Compton differential cross section σ C S (ω) (which has the dimension of a length in 2D) and on the number of electrons n e (M) dS M at site M, dS M being the integration area element around M. Hence the scattering photon flux density received at the detection site D is given by : where θ is the angle between the outgoing photon unit vector with the detector normal unit vector, |M D| the distance from scattering site M to detection site D. In fact, for a given point source S, there will be two scattering sites M 1 et M 2 located on two arcs of circle subtending a scattering angle (π − ω), as shown in Fig. 4.
The total photon flux density at a site D is g (D, ω), the integral over all source sites and all scattering sites such that the scattering angle is ω. This last constraint is expressed by a δ-function in the integration as where SM D is the angle at vertex M of the triangle SM D (see Fig. 4). Eq. (4) describes the basic image formation process from scattered radiation measured on a detector without collimator.

COMPUTATION OF THE IMAGE OF A POINT SOURCE
We now compute and study the essential object in this new imaging concept: the Point Spread Function (PSF). It is, by definition, the image of a single point source at site S. Eq. (4) can be rewritten in terms of the PSF as follows : As shown before for a camera without collimator, the scattering sites due to a single point source S are located on two circular arcs subtending an angle (π − ω) (Eq. (4)). The photon flux density received at site D is then given by an integration over these two circular arcs (see Fig.4). The circular arcs have polar equations: where γ is the angle between −→ DS and − − → DM . The distance |SM | can be extracted from a simple identity in the triangle DSM : And the integration area dS M is now reduced to the arc element: Hence the PSF is given by the sum of the two integrations on γ: sin ω sin γ arctan σ 2d sin ω sin(ω ± γ) , (9) where K(ω) = 4n e σ C S (ω)f 0 /2π, σ C S (ω) the differential Compton cross section at scattering angle ω, f 0 the intensity of the single point source and cos θ = sin(α − γ) if the detector lies along the 0x axis and l is the distance between the line detector and the linear lower boundary of the medium L. The integration is carried out over the points inside the scattering medium. Therefore when the medium is of finite extent, the limit of the integration γ l (ω), which corresponds to the intersection of the arcs of circle with the scattering medium, should be calculated beforehand, see Fig. 4. Now if the collimator is mounted on the detector, then only one scattering site M, located on the perpendicular to the detector at site D, will contribute to detection site D, (see Fig. 5). Thus the integration on γ is restricted by a delta function which picks out only the corresponding value of γ, i.e.: The resulting PSF expression for a collimated detector is: sin ω cos α arctan σ 2d sin ω ± cos(α ∓ ω) .
Now at a fixed scattering angle ω, the PSF curve as function of the detector position with collimator has a Mexican hat shape (see Fig. 6) while the PSF curve without collimator has a wide Lorentzian shape (see Fig 7).  Fig. 8 shows that the PSF without collimator is about 10 times stronger than the PSF with collimator.

NUMERICAL RECONSTRUCTION RESULTS
As an illustration of this new imaging concept, we carried out numerical reconstructions of a two-dimensional Shepp-Logan medical phantom from data computed with our model. The 2D original object (see Fig. 9) is placed at the center of the scattering medium and a unit distance above the detector. A line detector of 55 pixels of 1 unit length, placed on  A series of 55 images of the object corresponding to 55 different scattering angles (12 • < ω < 132 • ) have been simulated. We construct the 3025 × 3025 weight matrix by computing, for each mesh point source, the PSF at the different scattering angles for each site on the detector. The reconstruction is carried out by inverting this weight matrix using the Singular Value Decomposition method, which is less time consuming, compared to other reconstruction methods.  Fig. 11. Shepp-Logan phantom reconstruction without collimator Fig. 10 and Fig. 11 show the reconstruction results with and without collimator. One observes a better agreement with the original object when the collimator is removed. In Fig.  10, we can see that the part of the object near the detector is better reconstructed than the upper part of the object. The three small structures are invisible. The reconstructive relative error is about 9.13 %. But without collimator (see Fig.  11), the whole object is correctly reconstructed. All structures are visible and the relative error is about 1.17 × 10 −4 %.
These results have been subsequently validated by Montecarlo simulations [10].

CONCLUSION
The feasibility of image reconstruction using Compton scattered rays detected by a gamma-camera without collimator, operating in a fixed position, is demonstrated in this study. This is the essence of a new concept of high sensitivity imaging, which takes advantage of scattering rays instead of rejecting them as done usually. The main point in this imaging process by Compton scattered radiation is the fact that data acquisition is performed without the usual motion of the detector. This is a major advantage compared to existing imaging systems which require a heavy, bulky and costly mechanical rotation mechanism to move the detector around in space. Work towards an extension to three-dimensional imaging is in progress. The modeling and simulations of multiple scattered radiation in this context will be also subjects of future investigations. These promising results may open the way to new high sensitivity imaging devices which will have applications in nuclear medicine, non-destructive industrial control, high energy astrophysics, environmental survey, etc.