Detector Modeling Techniquesfor Pre-Clinical 3D PET Reconstruction on the GPU

We present different approaches to the efficient handling of gamma interaction within the detector crystal matrix for ML-EM-based 3D reconstruction algorithms for Positron Emission Tomography (PET). Geometric calculations and the approximation of inter-crystal scattering are executed on the GPU. The main characteristics of the scattering effect are captured by factorized models incorporating both on-the-fly simulation and the use of pre-computed data.

“Effective Radius” ModelOur first approximate detector response model is the introduction of an effective detector ring radius. The model is very easy to implement: the detector surfaces are shifted radially outwards by a constant displacement with the average scattering depth, but we still work with the geometric model. The difference of the real and effective radii, i.e. the average depth of gamma interaction is calculated off-line with a Monte Carlo simulation of the detector module.

LOR shiftingScattering and absorption in the detector can be modeled by a transport function as p(z, ω→d) that gives the probability of a hit reported in crystal d provided that a photon enters the detector module at point z from a direction ω. In the LOR shifting scheme, we assume that a finite element representation of the transfer function is available, which can be determined off-line using MC simulation. Thanks to spatial invariance, it is enough to compute this function only for a single detector d and dense enough directional samples ω and positional samples z. These discrete data are stored in tables and we linearly interpolate in between the sample values. This transfer function is incorporated into the expected value formula. The expected number of hits in detector pairs is a five-dimensional integral, where 2 × 2 dimensions are associated with the two detector surfaces, and one dimension is with the line between the two hit points. We propose to evaluate this high dimensional quadrature with quasi-Monte Carlo methods.

Parameterized ModelThe method of the parameterized model fits a combination of simple continuous functions to the pre-computed transport probabilities, which is appropriate for on-line importance sampling. Exploiting the radial symmetries of the transport function with respect to ω, it can be approximated as a separable extension of a function along the flight path and a function specifying the distance from the linear path.

Full MC ModelWe also implemented a full GPU-based MC method, which uses no pre-computation but traces the photon until it is absorbed or leaves the system. The registered detection position is the energy-weighted centroid of the position of energy loss events.

ResultsWe have modeled the geometry and crystal matrix of the NanoPET™/CT pre-clinical imaging system. In case of this PET scanner, 12 detector modules are organized into a ring, with each detector module consisting of 81×39 tightly packed LYSO crystals (1.12 mm × 1.12 mm × 13 mm).

The reconstruction algorithm that we apply is an ML-EM method in which the forward and back projectors are implemented on the GPU in two different ways: by using Monte Carlo particle transport simulation and by using adjoint Monte Carlo approximation.

The verification of the 3D reconstruction implementations was performed by using mathematical phantoms, such as the simulated micro Derenzo phantom (1.0 - 1.5 mm) as well as small animal measurements.

M. Magdics1, B. Tóth1, L. Szécsi1, B. Csébfalvi1, L. Szirmay-Kalos1, Á. Szlavecz1, G. Hesz1, B. Benyó1, Á. Cserkaszky2, D. Légrády2, Sz. Czifrus2, A. Wirth3, B. Kári3,J. Lantos4, G. Patay4, D. Völgyes4, P. Major4, G. Németh4, T. Bükki4, and B. Domonkos4

1Department of Control Engineering and Information Technology, 2Institute of Nuclear Techniques, Budapest University of Technology and Economics.3Faculty of Medicine Department of Diagnostic Radiology and Oncotherapy, Semmelweis University.

4Mediso Medical Imaging Systems, Hungary

Reconstructions of the simulated micro Derenzo phantom (1.0 - 1.5 mm) after 100 iteration steps, using different detector models.

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Geometry-only with MCEffective radius with MC

LOR shiftingParametric model

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Correlation-based error curves of the simulated Derenzo phantom in the function of the number of iterations for the different methods.

Mouse 18F bone PET study taken by NanoPET™/CT reconstructed by using the “effective radius” (left) and the LOR shifting model (right). Data courtesy of P. Blower, G. Mullen, and P. Mardsden, Rayne Institute, King’s College, London.

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