Quantum entanglement for identifying true coincidences in a CZT-based PET system

INTRODUCTION

Recently, quantum-based techniques utilizing entanglement of positron annihilation photons in positron emission tomography (PET) have become an alternative approach for scatter and random correction to optimize the signal to noise ratio (SNR). [1–2]

Gammas originating from the same electron-positron annihilation are generated exclusively in an entangled Bell state.

Unentangled gamma rays:

• Do not share an annihilation origin event, such as randoms

• A gamma which undergoes an internal scatter becomes decoherent (unentangled) from its coincidence pair.

Entangled gamma rays:

• True coincidence gammas contain scattering probabilities unique from randoms and scatters.

  • Entangled gammas have a high probability of large differences in azimuthal scattering angles Δφ compared to nonentangled gammas.

In this work, we implement quantum entanglement analysis in the annihilation photon data of cadmium zinc telluride (CZT) to identify random coincidences and in-patient scatter events.

Quantum Entanglement

The distribution of the relative azimuthal angle (Δφ) between the two photons can be used to predict the Δφ ranges that result in high fractions and low fractions of true events. Based on the result and according to the Bell state of the two photons’ entangled wave function [7], a range of Δφ for correlated events can be determined. The polarization dependence of the Compton scattering, is the result of the Kleins-Nishina formula for the differential cross section as described by Kuncic et al [6].

Kleins-Nishina formula:

$$\frac{d\sigma }{d\Omega }=\frac{1}{2}{r}_0^2{\left(\frac{v^{\prime }}{v_0}\right)}^2\left(\frac{v_0}{v^{\prime }}+\frac{v^{\prime }}{v_0}-2{\text{sin}}^2\theta {\text{cos}}^2\varphi \right)$$

METHODS

Detector and System Design

• Dual panel system using 80 CZT semiconductor crystals in each panel

• Each CZT uses cross-strip configuration consisting of 39 anode strips and 8 cathode strips. [3–5]

Dataset Acquisition

• Experimental data was obtained by positioning a point source within the dual panel scanner

• Data is acquired in list-mode (event-by-event) format; each line of input data file provides information regarding:

  • Level of deposited energy in Analog to Digital Converter (ADC) units (calibrated for conversion to keV)
  • Location of the electrode where the event took place
  • Timestamp that is necessary to group events together

Data Processing

1. Events resulting from a pair of photons are grouped together using a coarse 1 us time window.

2. Groups of four are further processed to select only interactions resulting from double Compton scatters (DCSc),

3. Interactions are classified as Compton or Photoelectric.

4. We compute the relative azimuthal angle of each DCSc using the coordinates of the classified interactions.

5. We generate a histogram of these Δφ to identify regions of true counts

Compton Scattering Angle:

$${\theta }_E={\text{cos}}^{-1}\left(1-m{c}^2\left(\frac{1}{E_s}-\frac{1}{E_i}\right)\right)$$

Line of response (LOR) angle:

$${\theta }_p={\text{cos}}^{-1}\left(\frac{A\cdot B}{\left|A\cdot B\right|}\right)$$

RESULTS

We compare the experimental data to a computationally generated LOR dataset of the same subsystem, shown below.

While the azimuthal angle histogram shows an increased count rate centered around the Δφ at 0 and 180, we expect a distribution around 90°according to results achieved by [1].

Next, we compare the computationally generated results of the four crystal subset to the dual panel system. Since the computationally generated data do not incorporate quantum physics, the dual panel results better reflects the distribution of random Δφ.

These results show a strong dependence on the geometry of the system, which must be further investigated. The histograms will be regenerated using data from the scaled up system, seen in Figure 2.

Figure 2.

Dual Panel setup using CZT detectors in a cross-strip configuration.

SUMMARY

• We used the quantum entanglement effect to filter random and scatter events from true coincidences

• We calculated the Δφ of the DCSc’s for a subset and for the full dual panel system

• We observed a strong geometric dependence on the Δφ distribution, as validated by our computational results

• Greater inspection of systems geometry effects is needed to better understand these results

Figure 1.

Double Compton scattering of two entangled photons from an annihilation event. Direction vectors (and planes) depicted following the Compton events within the CZT detectors with polar and azimuthal scattering angles of θ and ϕ, respectively. Adapted from [1].

Figure 3.

(Left) Energy resolution from the experimental data from the dual panel system. (Right, top) Scatter plot with original registered coordinate. (Right, bottom) Scatter plot following the estimation of the z coordinate after applying depth of interaction using the energy deposited in the anode and cathode.

Figure 4.

Compton scattering angle and line of response (LOR) angle are calculated from energy and position of events to classify anode-cathode pairs corresponding to first position of interaction. [3] E_s is the scattered photon energy, E_i is the incident photon energy.

Figure 5.

(Left top) Azimuthal angle in the DCs event in experimental data, captured by a subset of the dual panel system using four crystals. (Right top) Computational data for a four crystal subset of the dual panel (Left bottom) Computational data for the full dual panel shown in figure 2 [3–5].