3D Nonlinear Least Squares Position Estimation in a

3D Nonlinear Least Squares Position Estimation in a Monolithic Scintillator Block Zhi Li, Peter Bruyndonckx, Mateusz Wedrowski, Gerd Vandersteen Abstract–The 3D interaction position of a 511 kev gamma ray in a monolithic scintillator can be determined by nonlinear least squares fitting of the measured light distribution at the exit surface to an underlying model. This model is based on solid angles of direct detected light and reflected light, combined with a background due to diffuse reflection. The advantage is that this method does not rely on training data or detector parameter calibration and it estimates x, y and DOI simultaneously. Simulating a 20x20x10 mm block of LSO mounted onto two adjacent S8550 Hamamatsu APD matrix, we achieved an intrinsic resolution over whole block of 0.81 mm, 0.98 mm and 1.30 mm FWHM in x, y and DOI direction.

I. INTRODUCTION spatial resolution of current PET systems based on THE discrete scintillator detectors is determined by the size of the individual crystals. Making the crystals smaller improves the spatial resolution but also decreases the sensitivity and increases the cost and complexity. In recent years continuous detectors have shown to be a quite interesting alternative to simultaneously improve energy resolution and sensitivity without degrading the spatial resolution [11]. Numerous methods have been presented to obtain spatial information from a monolithic scintillator block. The two main techniques are the localization of the 3D interaction position using Maximum Likelihood methods [2] or the determination of the 2D surface entry position using Neural Networks, which does not require depth of interaction (DOI) information since the DOI coordinate is fixed at the surface [1]. Maximum Likelihood methods incorporate a model for the statistical properties of the signals. A disadvantage of this approach is that a lot of parameters need to be calibrated in the case of 3D position estimation [3]. A Neural Network is a black box for training algorithm weights, which is easy to implement on hardware. However one needs to train different Neural Network specifically for small ranges of photon incidence angles [4]. Both methods required the acquisition of reference data sets. This can cause a problem if the measurement conditions, such as temperature or APD high voltage, have changed since the time the reference data sets were collected. The present study aims to develop a nonlinear least square estimation method to predict the 3D 511 keV gamma ray Manuscript received November 13, 2009. Zhi Li, Gerd Vandersteen are with Dept. of ELEC, Vrije Universiteit Brussels, Pleinlaan 2, 1050 Brussels Belgium (telephone: (+32) 2 6293665, email: zhili @vub.ac.be). Peter Bruyndonckx, Mateusz Wedrowski, are with Dept. of Physics, Vrije Universiteit Brussels, Pleinlaan 2, 1050 Brussels Belgium (e-mail: [email protected]).

interaction position simultaneously by modeling the relation between the scintillation light source position and the measured photo detector pixel signals. This should lead to a high spatial resolution detector which only depends on the information embedded in the signals of individual events, and therefore does not need any prior training. II. METHOD AND DATA ACQUISITION To simplify the problem, we first analyzed the case of photoelectric absorption, where a 511 keV gamma-ray transfers all its energy into isotropically emitted optical photons in a single interaction. The influence of Compton scatterings is included later in the simulation results. A. Proposed Models Optical scintillation photons produced at the interaction site inside a perfectly polished LSO block, wrapped with Teflon on 5 sides, can travel to the photo detector pixels along different routes (Fig. 1).

Fig. 1. Schematic drawing of the direct detected scintillation light and the internally reflected light in a polished LSO crystal wrapped in Teflon on 5 sides. The latter is represented by virtual light sources mirrored around the edges.

Optical photons that are emitted directly towards the bottom exit surface of the block, within a cone with an opening angle smaller than the critical angle (55.5o), which is determined by the LSO-epoxy interface, can immediately leave the block and be detected if they hit an APD pixel. Scintillation light that hits the side of the crystal with an angle bigger than the critical angle for an LSO-air interface (33.3o) will refract into air, undergo diffuse reflection on the Teflon layer and reenter the crystal with a random direction. These optical photons contribute to the background signal, which only adds to the total energy measured (and hence to the energy resolution) but contains no spatial information anymore. The rest of the light will undergo internal reflections back in the crystal. These

mirror-like reflections behave like if there is a virtual source at a symmetric light source position for a perfectly polished surface (refer section III.C for discussion of grounded surface). What interest us is the relationship between the number of optical photons detected by the APD pixel, which is presented by the known coordinates (xi, yi) of the pixel center, and the unknown coordinates of the interaction point . We assume the relationship expression is composed of three parts: 1. A constant Cest which is mainly due to the reflection of the optical photons on a diffuse reflector (e.g. Teflon) surrounding the scintillator block. 2. A distribution f corresponding to the cone of optical photons directly exiting the scintillator block. 3. Additional virtual light sources (with an identical distribution as 2.) mirrored around the surfaces to take internal reflections into account. Thus for each event, we can write:

B. Light distribution f(x,y,z|αk) 1. Solid Angle based f(x,y,z|αk) One of the choices for the light distribution

in model (1)

is based on the assumption that the number of detected photons in one pixel is proportional to the solid angle subtended by that pixel seen from the source location. Considering the geometry in Fig. 2, the solid angle seen from a point source S1 located at a distance d straight up from one corner of a rectangular shaped detector with length a and width b, is given by [5] (3) When the projection of the source is located outside of the detector, as S2 in Fig. 2, the solid angle is composed by the sum of 4 sub-solid angles, each of them similar to Eq (3). The formula for the solid angle then becomes: (4)

(1) where sjx , sjy and sjz are the coordinates of the mirror source corresponding to the jth surface of the block, are the parameters characterizing the distribution f(x,y,z) (e,g. amplitude A ). When the photon interaction happens in the upper part of the scintillator block, the measured light distribution is not very peaked. This introduces a larger uncertainty on the estimated distribution parameters. When and are both obtained simultaneously by fitting the model, this often results in a negative value for Cest, which is in conflict with its physical meaning as a background. Therefore Cest and αk are computed separately. The background parameter Cest is first computed by sorting 64 APD pixel values, and taking the average of the Onum smallest values. The optimal value of Onum is chosen experimentally by comparing the achieved resolution as a function of different Onum values. After Onum has been chosen, it becomes a fixed parameter for that detector and does not need to be changed, i.e. it is independent of the interaction positions of the events. As Cest is a constrained parameter, for each event, the 3D interaction position and distribution parameters

Fig. 2. The relative position of point sources at position S1 and S2 at a distance d above a rectangular detector of length b and width a.

In our simulations one APD pixel is 1.6x1.6 mm , as shown in Fig. 3. A unitary function of the solid angle can now we written as:

(5)

are then estimated by fitting model (1) to the data using a Levenberg-Marquardt least squares optimization: (2) where mi are the measured APD pixel values,

are

the estimated pixel values by model Eq (1). This procedure is independent from event to event and therefore it does not need any reference data.

Since LSO is transparent to its own optical photons, no attenuation factor has been added in the model. Only an amplitude parameter is added to solid angle. Eq 5 will be the distribution in Eq 1. The mirror sources distribution in Eq 1 have a similar function, where the position of the mirror sources (sjx , sjy ,sjz ) replaces the position of the direct source

.

III. SIMULATION RESULTS

Fig. 3. Geometry of our setup consisting of one 20*20*10 mm LSO (red line), and two adjacent Hamamatsu S8550 APD arrays (blue line).

2. Simplified Solid Angle based f(x,y,z|αk) If we assume the distance from the source to the central of each pixel is big compared to the size of the pixel, then we can write a simplified solid angle function:

A Monte Carlo studies using GATE [6] were performed to simulate LSO based detectors, 20*20*10 mm crystal read out by two Hamamatsu S8550 APDs. The four side surfaces and the top surface were covered by Teflon (95% reflectivity). The light yield of LSO is set at 26000 Photons/Mev, According to [7], there is no optical absorption under wavelengths 400 nm. Most of the photon emission of LSO excited by γ-rays have wavelengths more than 400nm [8]. Therefore there is no attenuation of optical photons included in the simulation. Simulation data was generated on a rectangular grid of beam position in steps of 0.5 mm. Since the simulation data only included photon statistics, we added the influence of a 70% quantum efficiency, 1.75 excess noise factor, a APD gain of 50 and a preamplifier equivalent noise charge of 600 electrons, using the method described in [9]. A. Solid angle or simplify solid angle? TABLE I. SOLID ANGLE VS. SIMPLIFY SOLID ANGLE

(6)

Ω model Complete Simplified

x-direction

y-direction

DOI-direction

FWHM (mm)

FWTM (mm)

FWHM (mm)

FWTM (mm)

FWHM (mm)

FWTM (mm)

0.70 0.83

3.40 3.68

0.88 0.98

3.72 3.58

1.30 1.41

4.27 4.66

Using the simplified solid angle as a model for the light distributions is easy, costs less time and achieves still reasonable resolution. B. How many mirror sources are needed?

Fig. 4. Two example sources A, B and two pixels

where dx and dy are width and length of one pixel (both 1.6 mm in our geometry). However the above condition is not satisfied when the interaction point is very close to the crystal bottom. To check the validity, the solid angle subtended by two neighboring pixels at positions (0,0) and (2.3,0) was computed using eq (5) and eq (6) for two different light source positions A and B (Fig. 4). Light source B at position (10,0,5) is far away from the pixels. In this case the numerical results from equations (5) and (6) were identical. On the other hand, light source A at position (0,0,1) is very close to the pixels. The numerical results for the solid angle subtended by pixel (0,0) differed by 59.6%. The difference between the results obtained from eq. (5) and (6) for the neighboring pixel at (2.3,0) is already significantly reduced to 9.9 %. Hence for interactions occurring very close to the bottom of the crystal, the solid angle of only 1 pixel will be estimated incorrectly while the approximated solid angles of the 63 remaining pixels are close to the true solid angle.

From Fig. 1, we expect that the top mirror source would not have much influence due to the critical angle. Also the contribution of the virtual light sources diminished when the gamma interaction position shifts closer to the center of the monolithic crystal. Based on simulation data, adding detector noise as mentioned above, and applying an energy threshold of 350 kev, we have evaluated several virtual light source geometries. The results are shown as Table 2. It shows that adding a virtual corner source (2nd table entry), or four virtual sources (3rd table entry) or a virtual source on top of the crystal (4th table entry) will not improve the resolution. The model with only two close-by virtual sources has the best performance. Therefore function (1) can be rewritten as follows in our crystal geometry: (7)

TABLE II. INFLUENCE OF MIRRORS SOURCES ON 3D SPATIAL RESOLUTION x-direction Mirror Sources

y-direction

IV. CONCLUSIONS

DOI-direction

FWHM (mm)

FWTM (mm)

FWHM (mm)

FWTM (mm)

FWHM (mm)

FWTM (mm)

0.81

3.76

0.98

3.64

1.30

4.37

0.93

3.46

1.10

3.35

1.28

4.36

0.83

3.68

0.98

3.58

1.41

4.66

0.84

3.78

0.95

3.71

1.38

4.95

C. Grounded side surfaces We use the UNIFIED model in GATE [6] to model grounded surfaces, which consist of small micro-facets. The normals of the micro-facets are slightly different from the average surface normal, whose mean is 0 and standard deviation is α. This is show in Fig. 5 where α=0 means a perfect polished surface. When the surface is rough, the virtual source position will influenced by each micro-facet. A cluster of virtual sources will be formed instead of a single one, as would be the case for a perfectly polished surface. Since a cluster of virtual sources is difficult to model, we will still use one single virtual source to present a cluster. Table 3 shows the spatial resolution in case of two levels of surface roughness α, using the simplified solid angle model with 2 mirror sources. We can see that polished (0.1o) or unpolished (6o) surfaces have similar spatial resolution.

The purpose of this work is to obtain the interaction position of a gamma in a monolithic scintillator block without having to acquire training data sets or calibrate block parameter. We have described a nonlinear least square method to fit a model, based on virtual mirror sources and solid angle estimation, to light profiles generated in a monolithic block. From the fitted model we obtain the x,y and DOI coordinate of the interaction point . Two basic functions to form the model have been studied: solid angle and simplified solid angle. Using the simplified solid angle as a model for the light distributions is easy, cost less time to evaluate and achieves still good resolution. Adding only the two closest virtual sources in x and y direction is the best choice. Polishing the crystal surfaces does not seem to have any influence of the resolution. We are now in progress to acquire data on an experimental set-up identical to the detector described in this paper. This will allow us to verify our model on realistic data. REFERENCES [1]

[2] [3]

[4] [5] [6] [7] [8]

VOL.55. No3, 3, June 2008 P. Bruyndonckx, et al “Impact of instrumentation parameters on the performance of neural network based positioning algorithms for monolithic scintillator blocks”, 2007 IEEE Nuclear Science Symposium Conference Record. [10] P. Bruyndonckx, et al, “Study of Spatial Resolution and DOI of APD Based PET Detector Modules Using Light Sharing Schemes”, 2002 IEEE Nuclear Science Symposium Conference Record. [11] S. Tavernier. et al“A high-resolution PET detector based on continuous scintillators”, Nuclear Instruments and Methods in Physics Research A 537 (2005) 321-325 [9]

Fig. 5. Grounded surface Table III. Influence of surface roughness on the 3D spatial resolution

α (o) 0.1 6

x-direction

y-direction

DOI-direction

FWHM (mm)

FWTM (mm)

FWHM (mm)

FWTM (mm)

FWHM (mm)

FWTM (mm)

0.81 0.82

3.76 3.97

0.98 0.92

3.64 3.94

1.30 1.30

4.37 4.25

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