Measurement technical background of γ photon position sensitive and energy selective detection

Below, we discuss the position and energy-dependent detection of gamma radiation and its physical basis, which is also the basis of modern nuclear imaging. We present the process using the so-called linear one-dimensional model depicted in Figure 1. This model is based on the fundamental principles of scintillation gamma detectors, which encompass the following elements:

A NaI(Tl) scintillation crystal, which can be considered linear, a linear glass window and a light guide optically coupled to the crystal, and three photomultiplier tubes (PMTs) optically coupled to the light guide, arranged linearly along the detector: PMT-A, PMT-B and PMT-C.

A well-collimated gamma-ray source can be moved parallel to the surface of the detector along the line of the scintillation crystal. In our case the gamma beam, the direction of which is perpendicular to the surface of the detector, can be considered a line source. Let us examine how the position and the energy of the gamma source moved into different positions can be identified, using the simple model described above.

As a first step, let us review briefly how the detector works and what the physical process of signal conversion encompasses. The scintillation crystal interacts with the gamma ray through the photoelectric effect, which causes a flash of light – with a wavelength of around 415 nm and a colour of blue – in the NaI (Tl) crystal itself, at the location of the interaction. This light is called scintillation light, which travels isotropically with little attenuation in the scintillation crystal; the index of refraction for air is n_NaI(Tl) = 1,85. The intensity of the scintillation light induced in one interaction depends on the energy of the gamma photons (40 photons are created/keV), while the duration of the flash of light is 230 ns. Subsequently, a light detector is needed which converts the scintillation light induced in the crystal into electrical signals, into pulses for further electronic signal processing. The most suitable device for this purpose is the photomultiplier tube (PMT), which gives electrical pulses that are proportional to the intensity of the scintillation light at the output of the preamplifier connected to the PMT. It is the maximum of the pulse appearing at the output of the preamplifier that is proportional to the intensity of the scintillation light, which depends on the energy of the detected gamma radiation. Henceforth, when the signals that appear on the PMTs are referred to, we always mean the maximum value of the signals, since only the maximums carry real physical meaning. Let A, B, and C denote the output signals of the individual PMTs at a certain point of the gamma-ray source along the surface of the detector, as shown in Figure 1. The point of interaction between the gamma ray and the scintillation crystal has to be estimated (decoded) using the signals A, B, and C provided by the PMTs.

Let us examine how the value of the signals A, B and C change by moving the collimated gamma source along the detector with an equidistant step distance. This is depicted in Figure 2. for a 160 mm long detector with a step distance of 2 mm. 8000 events were collected and averaged in every single point so that the statistical error that originates from the Poisson distribution will be around 1%. The functions thus obtained for A, B and C are called mean detector response functions (MDRFs).

Figure 1. [12], [13], [14], [15]

It can be seen that if the mean detector response function of the system shown in Figure 1. is measured sufficiently precisely (calibration) and the system is time-invariant, then the location of the gamma event can be estimated (decoded) based on the MDRF using the measured signals A, B and C. Since the signal conversion processes are achieved through quantum effects and quantum electrical effects, and since the calibration itself follows a Poisson distribution, position estimation can only be performed with a certain probability. In order to support all this, the MDRF is depicted together with the error measured for the individual points (MDRF_i±SD_i) in Figure 3.

Figure 2.

Figure 3.

Figure 4. shows the distribution of the signals A, B and C (with a resolution of 6 bits) when the gamma source is in the middle of the detector – x=0 position. The location estimation of the gamma radiation as an event in case of a calibrated system – known MDRF – can be performed with statistical methods (Maximum Likelihood ML, Minimum Square Error MSE) or with analytical methods, out of which the centroid method is the most prevalently used. The statistical methods are applied in digital signal processing, but they are still subject to research. The centroid method is very widespread, which is due to the fact that it is relatively simple; it can even be performed with so-called hard-wired analogue electronic devices.

Figure 4.

The essence of the centroid method is that it makes use of the part of the MDRF where the slope of the function is sufficiently high (it takes noise into consideration and it does not use the so-called plateau – saturation – region of the function that appears at the detector ends) and it can be linearized well. Accordingly, it can be seen from Figures 2. and 3. that slightly more than half of the diameter of the applied PMTs at the edges cannot be used by this technique . That is how much dead space is created in the detector. When applying this method, the useful field of view will be the region denoted as UFOV in Figure 2. The rest of the crystal is worthless as far as position estimation is concerned, but it plays an important role in the determination of the full scintillation light, which is practically the estimation of the energy of the gamma photons detected by the scintillation crystal. Position determination with the centroid method is performed as follows. Let W₁ , W₂ , ..., W_n , be arbitrarily chosen different positive values, weighting factors for the n PMTs. Additionally, let A_i be the output signal of the i-th PMT. Thus, the position of a gamma event based on the measured signals A₁, A₂, ..., A_i, ..., A_n can be calculated using the centroid method:

$$ X=\frac{\left( {{W}_{1}}{{A}_{1}}+{{W}_{2}}{{A}_{2}}+...+{{W}_{n}}{{A}_{n}} \right)}{\left( {{W}_{1}}+{{W}_{2}}+...+{{W}_{n}} \right)\left( {{A}_{1}}+{{A}_{2}}+...+{{A}_{n}} \right)}$

that is, a so-called resultant weighting factor is calculated, which is proportional to the location of the gamma event.
In our example of three PMTs, if W_A=1, W_B=2, W_C=3, and the output signals of the individual PMTs are A, B and C, this will be:

$$ X=\frac{A+2B+3C}{\left( 1+2+3 \right)\left( A+B+C \right)}=\frac{A+2B+3C}{6\left( A+B+C \right)}$

In a case when our coordinate system is placed in the middle of the detector and the values of the weighting factors are chosen to be W_A=1, W_B=0, W_C=1 , then we obtain the following formula:

$$ X=\frac{W}{A+B+C}=\frac{{{W}^{+}}-{{W}^{-}}}{A+B+C}$

where

$$ {{W}^{+}}=\frac{{{W}_{C}}C+{{W}_{B}}B}{{{W}_{C}}+{{W}_{B}}}$

$$ {{W}^{-}}=\frac{{{W}_{A}}A+{{W}_{B}}B}{{{W}_{A}}+{{W}_{B}}}$

Substituting the weighting factors and equations F5 and F4 into equation F3 we get:

$$ X=\frac{C-A}{A+B+C}$

It can be seen that PMT-B (in the middle) only plays a role in normalization during the position estimation.
The expression in the denominator:

is the full scintillation light detected by the scintillation crystal, which is proportional to the energy of the gamma photon. The curve denoted by E in Figure 2. is the same that equation F7 gives, i.e. the position dependence of energy along the detector. It can be read from the figure that function E(x) does not change much, it is a good approximation to consider it constant at a given gamma energy, thus the system can be triggered with the energy signal by an appropriate choice of window. This means that if a detected energy signal E is within a pre-chosen window, then the position estimation is carried out in the signal processing unit that follows the detector, otherwise it is not. In this case only gamma photons with the desired energy are identified and only their position estimation will be performed, all other events will be filtered as noise.

In the model described so far we made use of the assumption that the gamma source is pointlike and it emits collimated radiation that is perpendicular to the surface of the detector, which is transferred to the desired position by a scanner moving along a straight line. That is what needs to be performed during calibration.

However, it is known that in reality a free pointlike gamma source emits its radiation isotropically. This needs to be imaged on the position sensitive surface of the detector. Imaging is performed by a special imaging unit called a COLLIMATOR, which is placed in front of the surface of the detector and can be considered a “lens” for the gamma photons. The collimator completes imaging based on radiation absorption. Only those gamma rays reach the surface of the detector that arrive parallel to the wall of the collimator. All other rays are absorbed by the wall of the collimator.

Figure 5.

Thus, the system outlined in Figure 5. selects and detects the activity distribution of an arbitrary linear gamma radiation source as a function of the position − A $\gamma$ (x) − according to the energy of the gamma photons. The following question arises: with what probability can the position sensitive detector identify the position of the gamma-ray source moved into a known position, if the MDRF is also known. This can be obtained from the statistical analysis of the point spread function (PSF) W (x|A,B,C).

Figure 6. shows the results of such an analysis, where the scattering, the full width at half maximum (FWHM), the full width at tenth maximum (FWTM) and the distortion was depicted as a function of the measurements performed at the individual positions with respect to the probability of the position recognition of the gamma source moved in the known points. At least 8000 events were registered in every point to ensure that the error that originates from the Poisson distribution will be around 1%, and that there is a sufficiently large number of events so that the so-called Gaussian “bell curve” can be applied with a very good approximation for the PSF analysis. (The statistical meaning of the FWHM and the FWTM in a given point of the detector – in our case in the middle – is shown in Figure 7. with a resolution of 7 bits.)

Figure 6.

Figure 7.

It is possible to see from Figure 6. the exact definition of “Useful Field of View - UFOV /Figure 2., Figure 3./ to be estimated from the MDRF - based on criteria of stochastical analysis. The MDRF based UFOV definition provide a good picture of what can be expected from the position sensitive detector. The UFOV can also be determined via the quantitative PSF based method.
The curve shapes of deviation (σ), FWHM and FWTM provide information about how well the PSF follows the “bell shape” curve, i.e. the Gaussian distribution (Gauss process).
Ideally, the three above parameters can be calculated from each other by a constant multiplication factor (only, if the process is Gaussian).

$$ \[FWHM=\sigma 2 \sqrt{2 ln (2)}$

$$ \[FWHM=\sigma 2 \sqrt{2 ln (10)}$

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