1.

Introduction

Over the past years, photon-counting spectral detectors for computed tomography (CT) have received considerable attention. Compared to conventional energy-integrating CT detectors, in which the energy of many photons is integrated over a certain time period, photon-counting spectral detectors count the individual photons and measure their energies. This enables higher signal-to-noise ratio (SNR), higher spatial resolution, and improved spectral imaging.^1–3

Although most research into photon-counting spectral CT detectors is centered around CdTe (cadmium telluride) and CZT (cadmium zinc telluride),^4,5 silicon is also a candidate detector material.^6,7 An advantage of silicon compared to CdTe and CZT is that it is possible to manufacture as almost perfect crystals with fast charge collection that is robust even at very high count rates. The low atomic number results in no cross-talk due to K-fluorescence, but on the other hand, there is a significant fraction of Compton interactions. In Compton interactions, only a part of the photon energy is deposited. This results in counts with little to no energy information. However, as secondary interactions can be largely avoided using tungsten shielding, the registered Compton counts still correspond to unique photons. For density imaging tasks, these counts are therefore vital and contribute to both dose efficiency and contrast in the image. In spectral imaging, we have previously shown that the contrast can be increased with 5% to 14% by including Compton counts.⁶

Figure 1 shows a typical spectrum of the deposited energies in a silicon detector. Compton interactions are visible as the peak between 0 and 35 keV and to obtain a high dose efficiency, it is desirable to have the lowest threshold as close to 0 keV as possible. However, if the threshold is set too low, electronic noise will give unwanted false counts. Having a low electronic noise is therefore vital for a high dose efficiency.

Fig. 1

Simulated spectrum of the deposited energies in a silicon detector from an x-ray source operated at 120 kVp, including 20 cm soft tissue filtration between the source and the detector.

The electronic noise level is affected by the filter component in the readout electronics. While the shaping time remains shorter than microseconds, a longer shaping time leads to decreased noise, but this comes at the expense of increased pulse length. If two photons are registered within a time frame shorter than the pulse length they will add up, resulting in pileup. This can cause count loss or spectral distortion. Increasing the pulse length therefore reduces the ability of the detector to measure at high count rates.

For silicon detectors count-rate performance is not a problem.^8–10 The relatively long absorption length of silicon compared to other detector materials enables segmentation in the direction of the incident x-rays. Each segment then consists of a row of readout channels that measure only a fraction of the incident x-ray flux. As the total number of absorbed photons becomes divided among the segments the demand on high count-rate capability is reduced in each readout channel. In situations where power consumption is not an issue, the optimal number of depth segments would ideally be chosen to minimize pileup without generating excessive charge sharing between the depth segments. We have previously presented systems with 9 and 16 depth segments that have shown good performance for diagnostically relevant count rates with linearities of detected count rates up to 90 and $200\mathrm{Mcps}/{\mathrm{mm}}^2$, respectively.^8–10

For CT detectors in a clinical environment, power consumption is one of the largest system constraints and active cooling has to be employed. There is no hard limit for power consumption for future photon-counting detectors (including silicon based) but it is expected to be in the range of today’s CT detectors to avoid expensive installations. Depending on the detector width this typically means $<5\mathrm{kW}$. To decrease the power consumption, it is possible to decrease the number of depth segments since the power consumption scales with the number of readout channels. Decreasing the number of depth segments reduces the count-rate capability of the system and makes it more prone to pileup. To counteract pileup, it is desirable to have fast readout electronics with a short shaping time. If the power in each channel is kept constant, decreasing the shaping time leads to shorter pulses but also increased noise.

In a typical CT scan, the photon flux varies greatly between different projection lines. The number of electronic noise counts is largely unaffected by the photon flux, and at high count rates the electronic noise counts only constitute a small fraction of the total counts meaning that the impact of electronic noise counts on image quality is limited. At low photon fluxes, on the other hand, the false counts from electronic noise are more detrimental as they constitute a larger fraction of the total number of counts. However, at low fluxes there is no demand for short pulses, as there is no problem with pileup. In a detector with few depth segments, it would therefore be desirable to measure different projections with different shaping times, e.g., a long shaping time at low photon fluxes to reduce the electronic noise and potentially decrease the lowest threshold and a short shaping time to mitigate pileup at high photon fluxes.

Photon-counting detectors with multiple shaping times are commonly used in positron emission tomography and single photon emission computed tomography systems. In these systems, two shaping filters are used in parallel: one filter is equipped with a long shaping time and is used for energy classification, while the other filter has a short shaping time, which enables extraction of timing information.^11,12 Detectors with dual shaping times have also been presented for correction of charge loss.^13,14 In such systems, a short shaping time is used to measure the position of the event while the filter with the long shaping time provides energy information.

An alternative to decreasing the number of depth segments is to decrease the power consumption in each readout channel by reducing the power to the preamplifier. The preamplifier integrates the induced currents that result from photon interactions and is one of the main contributors of noise in the readout channel.¹⁵ The noise is dependent on the power at which the preamplifier is operated: as the power is decreased, noise increases. Similar to the case with fewer depth segments, it is therefore desirable to adapt the shaping time to the incident flux. Increasing the shaping time can then mitigate the noise increase that comes with decreasing the preamplifier power. For both of the presented alternatives to reduce the power consumption, the trade-off between pileup with long shaping times and high noise with short shaping times must also be taken into consideration.

In order to develop a detector with low power consumption, we propose using an adjustable shaping time that can be adapted according to the incident x-ray flux and thereby mitigate the trade-off between dose efficiency and count-rate capability as previously shown in a preliminary study.¹⁶ To experimentally demonstrate the potential of varying the shaping time, we present noise and pulse shape measurements from a prototype detector for two shaping times. Furthermore, we develop a simulation model of the detector system, which we fit to the measured pulse shapes, and we compare the measured noise level to the predictions of the model. We also propose a potential future detector channel design that is specifically adapted to use with an adjustable shaping time and investigate the potential for reducing the power consumption with this channel design.

2.

Method

2.1.

Readout Electronics

The detector readout is performed by a custom application-specific integrated circuit (ASIC). The ASIC is implemented in a 180-nm CMOS process and is based on an architecture previously presented by Gustavsson et al.¹⁷ that has been measured and evaluated for photon-counting spectral CT by Xu et al.^10,18 It is a mixed-signal circuit capable of photon-counting and energy classification.

The ASIC has 160 channels in total and eight energy bins per channel. In Fig. 2, an overview of the analog interfacing circuits in each channel is presented. The first component is the preamplifier, which is a charge-sense amplifier (CSA) that integrates the current generated by incident photons and creates an output voltage. The CSA is followed by a pole zero cancellation (PZC) circuit and a shaper amplifier. A shaping filter is implemented after the shaper amplifier using two cascaded lossy transconductance-C ($G_m\text{-}C$) integrator stages. The two stages effectively form one pole each and combined act as a second-order filter with a shaping time $\tau$ that is set by the values of the transconductance $G_m$ and the capacitance $C$. For a hypothetical, ideal implementation of this circuit, the filter output, $V_{\mathrm{f}}(s)$, can be expressed as

Eq. (1)

$$V_{\mathrm{f}}(s)=\frac{1}{C_{\mathrm{f}}}\frac{C_{\mathrm{pz}}}{C_{\mathrm{s}}}\frac{{\tau }_{\mathrm{s}}}{1+s{\tau }_{\mathrm{s}}}{\left(\frac{2}{1+s{\tau }_0}\right)}^2{I}_{\text{in}}(s)$$

in the frequency domain, with $I_{\text{in}}(s)$ as the input from the detector. $C_{\mathrm{f}}$ is the feedback capacitance of the CSA, $C_{\mathrm{pz}}$ is the capacitance of the PZC circuit, $C_{\mathrm{s}}$ is the shaper amplifier capacitance, and $s=j\omega$ where $j$ is the imaginary unit and $\omega$ is the angular frequency. The expression also contains two time constants: ${\tau }_{\mathrm{s}}$, which is the time constant of the shaper amplifier, and ${\tau }_0=\frac{\tau }{2}$, which is the filter time constant. The time constant ${\tau }_{\mathrm{s}}$ is typically set to be in the order of the time constant of the filter: ${\tau }_{\mathrm{s}}\approx {\tau }_0$.

Fig. 2

Block diagram of the first stages of the readout electronics, i.e., the interface to the detector diode. The CSA integrates the charge, the shaper amplifier, PZC circuit, and $G_m\text{-}C$ filter shape the pulse. The comparators discretize the pulse.

In our current ASIC prototype, there is a parasitic load from the comparators at the filter output. We expect this to result in a time constant of the second $G_m\text{-}C$ stage that is twice as large as the time constant of the first stage. Furthermore, the current ASIC prototype supports two shaping time settings that are achieved by altering the capacitor values in the $G_m\text{-}C$ stages. The shaping time settings correspond to a long shaping time, which we expect to be $\sim 40\mathrm{ns}$, and a short shaping time, which is dependent on the bandwidth of the amplifiers as well as parasitics varying with ASIC samples but expected to be in the range between 20 and 40 ns.

A general representation of the filter output from the ASIC, both prototype and ideal, is given as

Eq. (2)

$$V_{\mathrm{f}}(s)=\frac{1}{C_{\mathrm{f}}}\frac{C_{\mathrm{pz}}}{C_{\mathrm{s}}}\frac{{\tau }_{\mathrm{s}}}{1+s{\tau }_{\mathrm{s}}}\frac{2}{(1+s{\tau }_1)}\frac{2}{(1+s{\tau }_2)}{I}_{\mathrm{in}}(s),$$

in which the time constant of the shaper amplifier, ${\tau }_{\mathrm{s}}$, and the time constants of the two $G_m\text{-}C$ stages, ${\tau }_1$ and ${\tau }_2$, are defined by Table 1 for each ASIC type. In the prototype ASIC, we represent the impact of the shaping time setting on the time constants of the $G_m\text{-}C$ stages with a parameter $B$. There are two selectable values of $B$ corresponding to the two shaping time settings: $B_{\text{long}}=1$ and $B_{\text{short}}$. With the shaping time defined as $\tau =2B{\tau }_0$, the two shaping time settings result in ${\tau }_{\text{long}}=2{\tau }_0$ and ${\tau }_{\text{short}}=2B_{\text{short}}{\tau }_0$. The value of $B_{\text{short}}$ is not known due to the above-mentioned comment but is assumed to be in the interval $0.5\le B_{\text{short}}<1$ in accordance with the expected shaping time range of 20 to 40 ns. In the current ASIC prototype, we also expect the time constant ${\tau }_{\mathrm{s}}$ to be nominally 14 ns.

Table 1

Time constants in the current ASIC prototype and the ideal ASIC.

	τs	τ1	τ2
Current ASIC prototype	14 ns	$B{\tau }_0$	$2B{\tau }_0$
Ideal ASIC	${\tau }_0$	${\tau }_0$	${\tau }_0$

In the time domain, $v_{\mathrm{f}}(t)$ is the filter output voltage and each voltage peak corresponds to the energy deposited by a photon. To register the peak voltage of each pulse, the filter output is digitized by eight comparators. The voltage $v_{\mathrm{f}}(t)$ is discretized into eight different levels using a set of controllable thresholds (energy bins), one in each comparator. The comparator outputs are sampled at a fixed time period, the clock cycle ${\tau }_{\mathrm{c}}$, and whenever the sampled voltage exceeds the lowest set comparator threshold, a count is registered. The voltage pulses are typically longer than the clock cycle and each count is therefore followed by a deadtime to avoid multiple counts from a single photon event. The deadtime is a multiple of the clock cycle during which the height of the pulse is determined and the channel is unable to register new counts. The pulse is registered in the energy bin of the highest comparator threshold that has been triggered by the pulse.

2.1.1.

Shaping time

The pulse shaping is semi-Gaussian, and it is realized by lowpass filters with cut-off frequency determined by the shaping time. For shaping times that are shorter than a few microseconds, an increase of the shaping time results in a lower cut-off frequency.¹⁵ In the case of white noise, a lower cut-off frequency decreases the total noise power, but on the other hand, the cut-off frequency also affects the pulse height of the photons and thereby the gain in mV/keV. The SNR is here defined as the ratio between the gain and the standard deviation of the noise, $\sigma$. Therefore, in order for the SNR to increase when the shaping time is increased, the reduction of noise must be greater than the reduction of gain.

2.2.

Measurement Setup

To determine the pulse shape, noise, and gain for each of the two shaping time settings, measurements were performed at Diamond Light Source (United Kingdom), beamlines B16 and I15. These beamlines are able to produce monochromatic x-rays of 4 to 45 keV and 20 to 80 keV, respectively, which make them suitable for detector characterization at diagnostic x-ray energies. The measurement setup included a silicon strip detector with wire-bonded ASICs. Figure 3 shows one such detector prototype with nine depth segments. The detector was mounted in a light-tight box and aligned to the beam in edge-on geometry. The incident x-ray beam was narrowed by collimators to illuminate the center of one pixel, i.e., the center of the projection of the strip as seen from the source, ensuring complete charge collection and eliminating edge effects such as charge-sharing. A bias voltage of 400 V was applied as described by Xu et al.¹⁰ and measurements were performed on four ASICs in separate detector modules. To keep the noise level approximately equal, only one depth segment in each strip (the one closest to the x-ray source) was used for the measurements.

Fig. 3

Photograph of a segmented silicon strip detector module. The module is oriented with its upper edge toward the x-ray source. Three wire-bonded ASICs are visible as the black rectangular shapes to the right.

For each detector module and shaping time setting, the gain was measured by first scanning the lowest digital-to-analog (DAC) threshold voltage across the whole DAC range with monochromatic x-ray beams with a subset of the energies [0, 40, 60, 70, 80] keV where 0 keV represents a noise-only-measurement with no incident x-rays. Modified complementary error functions of the form $f(x,\mu ,\sigma ,A_1,A_2,A_3,A_4)=\frac{1}{2}\mathrm{erfc}(\frac{x-\mu }{\sqrt{2}\sigma })(A_1(x-\mu )+A_2)+A_3(x-\mu )+A_4$ were then fitted to the measured data with $x$ as the DAC voltage and $\theta =[\mu \sigma A_1{A}_2{A}_3{A}_3]$ as the fitting parameters. The gain was obtained by fitting a linear relation between the $\mu$ values obtained by the fitting and the incident x-ray energies.¹⁰ The noise level $\sigma$ in keV was calculated from the fitting to the 0 keV measurement by dividing the obtained value of $\sigma$ with the calculated gain.

In the ASIC implementation, there is no direct access to measure the shape of each pulse due to a high sensitivity toward additional load and the bandwidth required to capture the pulses reliably. Therefore, in this paper, the pulse shapes were measured using an indirect method in which the lowest DAC threshold was scanned across the entire DAC range with a monochromatic beam for different deadtime settings. Measurements that are performed with a deadtime that is shorter than the pulse length result in double counting [see Fig. 4(a)]. As the pulse length depends on the threshold energy at which it is measured, a certain deadtime might result in double counts for a certain threshold, while at a higher threshold the pulses are single counted. By scanning the DAC threshold across the entire DAC range, it is possible to observe when pulses transition from being double counted to single counted. The threshold energy at which this occurs, $E_{1/2}$, then represents the energy at which the pulse length is equal to the deadtime. This is depicted in Figs. 4(b) and 4(c) where DAC scans have been simulated for two different deadtimes ${\tau }_{\mathrm{d},1}$ and ${\tau }_{\mathrm{d},2}$.

Fig. 4

(a) Deadtime behavior of the ASIC. (b) Simulated DAC scans of 60 keV pulses for two different deadtimes ${\tau }_{\mathrm{d},1}$ and ${\tau }_{\mathrm{d},2}$. The curves have been normalized according to the number of simulated pulses. Each dashed line shows the threshold energy $E_{1/2}$ where the pulse length is equal to the used deadtime. (c) Simulated 60 keV pulse. The energy coordinate of each dashed line corresponds to the identified energy threshold in the left figure and the pulse length at this energy is equal to the deadtime. (b) and (c) Apply to a case where the ASIC is designed to trigger immediately when the pulse rises above the threshold. For the detector study here, the pulses are sampled with a 10-ns sampling interval, which means that the actual pulse length is on average 5 ns longer than the deadtime used in the measurement.

In the limit of an infinitely short clock cycle, the threshold at which the pulse length equals the deadtime, $E_{1/2}$, is found where half of the pulses are counted twice and the rest are not [Fig. 4(b)]. In our ASIC, pulses are sampled at intervals given by the clock cycle ${\tau }_{\mathrm{c}}=10\mathrm{ns}$, which means that each count is triggered with a random delay between 0 and ${\tau }_{\mathrm{c}}$ from the moment when the signal crosses over the lowest threshold. The pulse length at $E_{1/2}$ is therefore equal to ${\tau }_{\mathrm{d}}+{\tau }_{\mathrm{c}}/2$, where ${\tau }_{\mathrm{d}}$ is the deadtime.

3.

Results

3.1.

Measurements

The measured gain in mV/keV is presented in Table 2 for each shaping time setting and detector module. The table also shows the standard deviation $\sigma$ of the noise. These values were obtained by fitting modified complementary error functions to the measured data.

Table 2

Measured gain and noise for each shaping time setting and detector module.

Detector module	Gain for τlong (mV/keV)	Gain for τshort (mV/keV)	σlong (mV)	σshort (mV)
1	1.45	1.89	3.99	5.50
2	1.54	2.00	3.48	4.92
3	1.51	1.89	3.40	4.58
4	1.63	2.10	3.73	5.01

To enable analysis and comparison between the two shaping time settings, the determined gain was used to convert the measured noise from mV to keV. The calibrated noise values for each detector module and shaping time are presented in Table 3. In Fig. 5, the measured noise curves from one detector module are presented along with fitted modified complementary error functions.

Table 3

Measured noise in keV for the long and short shaping time settings.

Detector module	σlong,keV (keV)	σshort,keV (keV)
1	2.76	2.90
2	2.26	2.46
3	2.25	2.43
4	2.29	2.38

Fig. 5

Measured noise in one detector module using different shaping time settings along with fitted modified complementary error functions $\frac{1}{2}\mathrm{erfc}(\frac{x-\mu }{\sqrt{2}\sigma })(A_1(x-\mu )+A_2)+A_3(x-\mu )+A_4$ with $x$ as the lowest threshold in keV.

Threshold scans to obtain the pulse shape were performed for the energies and deadtimes presented in Table 4. In Fig. 6, the data from one such threshold scan with the long shaping time and a beam energy of 60 keV are shown. The curves have been normalized by the number of unique counts, measured as the height of the plateau between 40 and 55 keV in Fig. 6. Analogously to the calibrated noise, all threshold scans have been converted from mV to keV using Table 2.

Table 4

Energy and deadtime sets used in the threshold scans to obtain the pulse shapes.

	τlong		τshort
Detector module	Measured energies (keV)	Used deadtimes (ns)	Measured energies (keV)	Used deadtimes (ns)
1	40, 60, 70	70, 80, 90a, 100a	60, 70	70a, 80a
2	40, 60, 80	70, 80, 90, 100a	40, 60	70, 80, 90a
3	40	40, 50, 60, 70, 80, 90, 100	40	40, 50, 60, 70
4	40	40, 50, 60, 70, 80, 90, 100	40	40, 50, 60, 70

^aDeadtimes used only for energies 60, 70, and 80 keV.

Fig. 6

Threshold scan to obtain the pulse shape at 60 keV using four different deadtimes and the long shaping time setting.

The resulting simulated pulse shapes for one detector module are shown in Fig. 7 together with measured data points for the positive and negative flank of the pulse. The energy coordinate of each pair of data points corresponds to the threshold energy and the horizontal distance between the points in the pair is the used deadtime. The fitted shaping times are presented in Table 5.

Fig. 7

Measured pulse lengths with different shaping time settings for one detector module along with simulated pulse shapes. The parameters of the ASIC model used to simulate the pulse shapes have been fitted such that the pulse amplitudes agree with the nominal energies and the pulse shapes agree with the measured pulse lengths.

Table 5

Determined shaping times from the curve fit to the measured pulse lengths.

Detector module	τlong (ns)	τshort (ns)
1	39.2	27.8
2	41.4	28.6
3	38.4	27.6
4	38.6	28.2

In Table 6, the measured noise ratios and gain ratios between the long and the short shaping times are presented along with the corresponding simulated values. The noise ratio for each detector module, ${\sigma }_{\text{long},\mathrm{keV}}/{\sigma }_{\text{short},\mathrm{keV}}$, is the ratio between the values in Table 3 and the gain ratio, $G$, is the ratio between the gain values for the long and the short shaping time settings presented in Table 2. The corresponding simulated values were obtained from the simulated ASIC model using the determined shaping times in Table 5.

Table 6

Measured noise and gain ratios, G, for the two shaping times along with the corresponding simulated values.

Detector module	σlong,keV/σshort,keV	Simulated σlong,keV/σshort,keV	G	GSimulated
1	0.95	0.95	0.76	0.77
2	0.92	0.94	0.77	0.75
3	0.92	0.95	0.80	0.78
4	0.96	0.95	0.78	0.79

3.2.

Simulated Performance of Proposed Future System

3.2.1.

Dose efficiency

The dose efficiency based on the lowest threshold was calculated for the three spectrums in Fig. 8(a), which shows the deposited energies in a silicon detector with 20, 30, and 40 cm soft tissue filtration between the x-ray source and the detector. The dose efficiency as a function of the lowest threshold is shown in Fig. 8(b). Assuming a lowest threshold of $4\sigma$, the mapping from lowest threshold to $\sigma$ is displayed as an additional axis in the figure.

Fig. 8

(a) Simulated spectrums of the deposited energies in a silicon detector from an x-ray source operated at 120 kVp including 20, 30, and 40 cm soft tissue filtration between the source and the detector. (b) Dose efficiency as a function of lowest threshold for the three spectrum shapes normalized to 100% at a 0 keV threshold. The dose efficiency is the fraction of interactions registered above the lowest threshold.

In order to evaluate the dose efficiency related to high count rates, the lowest threshold and deadtime were determined for each shaping time and are presented in Table 7 along with the dose efficiency resulting from the lowest threshold. The count characteristics are shown in Fig. 9 and were obtained by simulating the registered number of counts at different input count rates for each shaping time with the corresponding threshold and deadtime.

Table 7

Lowest threshold and deadtime values for each shaping time. The lowest threshold was set in relation to the noise level σ as 4σ. The deadtime corresponds to the length of a 100-keV pulse at the lowest threshold. The dose efficiency is the fraction of interactions registered above the lowest threshold.

Shaping time (ns)	Lowest threshold (keV)	Deadtime (ns)	Dose efficiency (%)
40	8	140	68.8
100	5.2	370	78.5
400	1.9	890	92.1

Fig. 9

Count characteristics for different shaping time values on different input count rate intervals. The vertical lines show (from left to right) the input count rate for a detector with 16 and 9 segments, respectively, assuming an incident count rate of $300\mathrm{Mcps}/{\mathrm{mm}}^2$ and a detection efficiency of 80%.^8,10 The spectrum of input pulses was obtained from a 120-kVp x-ray tube spectrum assuming 20 cm soft tissue filtration together with the response function of the silicon detector.

3.2.2.

Power consumption

Noise as a function of power is presented in Fig. 10(a) for the different shaping time values. The power values are given as multiples of $P_0$, which is the power level that gives a noise level of $\sigma =2\mathrm{keV}$ at a 40-ns shaping time. In Fig. 10(b), the noise levels for three different power settings are shown as functions of shaping time. To show the potential for decreasing the power consumption by increasing the shaping time while keeping the noise constant, the resulting relation between power consumption and shaping time is shown in Fig. 11.

Fig. 10

(a) Noise as a function of power for different fixed shaping times. (b) Noise as a function of shaping time for different fixed power levels.

Fig. 11

Power consumption and required shaping time to maintain a constant noise level of $\sigma =2\mathrm{keV}$.

4.

Discussion

As seen in Table 2, the gain is lower for the long shaping time than for the short shaping time. This is expected as the low-pass filtration decreases the pulse height. The measured noise values in Table 2 also show a reduction with the long shaping time. The small variations that can be seen between the measured gain and noise values for different detector modules could be caused by fabrication imperfections in the ASICs or statistical errors in the measurements. Based on the calibrated noise presented in Table 3 and Fig. 5, the long shaping time results in an average noise decrease of 6.1% compared to the short shaping time. This corresponds to an SNR increase of 6.5%, and it is clear that even though the gain decreases when the shaping time is increased, the noise decreases more, and the resulting SNR becomes higher.

The measured counts as a function of the threshold for different deadtime settings in Fig. 6 are similar to the simulated curves in Fig. 4. However, at $\sim 15\mathrm{keV}$, the normalized counts become larger than 2 resulting from triple counted pulses and noise counts. These data points have been excluded in the curve fitting.

The two $G_m\text{-}C$ stages that form the filter in our ASIC are expected to give a semi-Gaussian pulse shape as described by the system transfer function [Eq. (2)] used in the simulation model. The measured pulse lengths in Fig. 7 show good correspondence with this predicted pulse shape and similar fitting results were obtained for all ASICs. Note that as the pulse shape was measured by measuring the pulse length at different pulse heights, the relative time points of the measurements for different deadtime settings are not known, i.e., there is not sufficient information to determine if the pulse rises faster than it drops off. However, the measurements are consistent with the theoretical model.

The determined shaping times (Table 5) result in an average shaping time of 39.4 ns with the long shaping time setting and 28.1 ns with the short shaping time setting. When the determined shaping times are used to simulate the noise ratio, ${\sigma }_{\text{long}}/{\sigma }_{\text{short}}$, and the gain ratio, $G$, the resulting values agree well with the measured values as presented in Table 6. The discrepancy could be caused by small differences in the performed curve fittings due to statistical errors in the measurements. Overall, the ASIC model is able to describe the measurements well, and this indicates the potential of using the model in future investigations.

The deadtimes in this work were chosen in order to obtain double counts at each beam energy. The number of deadtimes was determined in relation to the available synchrotron beam time during each synchrotron session. To improve the capture of the pulse shape, the number of deadtimes could be increased as this would provide information about the pulse length at more energy levels. However, as the deadtime is a multiple of the clock cycle and the resulting double counts must arise in the threshold range between the noise floor and the energy of the pulse, the number of possible deadtimes is limited. This could be mitigated by utilizing a shorter clock cycle, but this would put much harder constraints on, e.g., the speed of the digital readout circuitry as well as the bandwidth of the analog circuitry.

The dose efficiency based on the lowest threshold presented in Fig. 8 shows that the dose efficiency is largely independent of the soft tissue thickness. It is also clear that there is a benefit in dose efficiency if the lowest threshold is decreased. Reducing the threshold from, e.g., 8 to 4 keV increases the dose efficiency from 68.8% to 85.3%.

The dose efficiency at high input count rates depends on the pulse length and the corresponding deadtime. In Fig. 12, the relation between shaping time and pulse length is presented. As increasing the shaping time results in longer pulses and thereby a longer deadtime, the count characteristics in Fig. 9(a) show that the short shaping time has the best count-rate performance at high input count rates. Saturation occurs when the curves become flat and represents the maximum count-rate capability of the channel. Due to the short pulse length, the 40-ns shaping time saturates at a higher input count rate. However, at low input count rates, the lower threshold levels enabled by the 100- and 200-ns shaping times provide better performance and give higher output count rates compared to the 40-ns shaping time. This can be seen in Fig. 9(b).

Fig. 12

Pulse length of a 100-keV pulse at a threshold of $4\sigma$ for different shaping times.

In Fig. 10(a), noise is described as a function of power for different shaping times. From the figure, it is clear that the noise level decreases as the input power increases. Correspondingly, Fig. 10(b) shows how the noise level decreases when the shaping time is increased. The resulting relation between power consumption and noise in Fig. 11 shows that it is possible to counteract the additional noise caused by decreasing the power consumption by simultaneously increasing the shaping time. The potential for decreasing the noise by increasing the shaping time is particularly high if the initial shaping time is low. If the starting point is 200 ns, a 50% decrease in power consumption can be counteracted by increasing the shaping time by a factor of 2.5 to 500 ns. At a starting point of 40 ns, the corresponding required shaping time increase is a factor of 1.875 to 75 ns.

For future work, there are a number of ways in which varying the shaping time could be used to increase the dose efficiency. One example is to adjust the shaping time for each projection line based on the flux. The pulse length would then be adapted to the count-rate demand. Another example is a dual or multiple shaper system in which two or more filters could be implemented in parallel to process the same signal. This could be, e.g., one filter of a short shaping time to enable a high count-rate capability and one filter of a longer shaping time to measure low-energy interactions. Such an implementation would increase the dose efficiency while maintaining a high count-rate capability for all projection lines.

5.

Conclusions

We have presented noise and pulse shape measurements for a photon-counting ASIC with two different shaping times. Our results show that increasing the shaping time from 28.1 to 39.4 ns decreases the noise and increases the SNR with 6.5% at low count rates. At the same time, the pulse length increases, which results in a lower count-rate capability. This shows that adjusting the shaping time can be an important tool to adapt the pulse length and noise level to the photon flux and thereby optimize the dose efficiency of photon-counting silicon detectors. With our proposed future ASIC design, we also show that it is possible to maintain a constant noise level while decreasing the power consumption by 50% by increasing the shaping time with a factor of 1.875.