Effect of Temperature Variation on the Energy Response ... - IEEE Xplore

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IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 60, NO. 2, APRIL 2013

Effect of Temperature Variation on the Energy Response of a Photon Counting Silicon CT Detector Hans Bornefalk, Mats Persson, Cheng Xu, Staffan Karlsson, Christer Svensson, and Mats Danielsson

Abstract—The effect of temperature variation on pulse height determination accuracy is determined for a photon counting multibin silicon detector developed for spectral CT. Theoretical predictions of the temperature coefficient of the gain and offset are similar to values derived from synchrotron radiation measurements in a temperature controlled environment. By means of statistical modeling, we conclude that temperature changes affect all channels equally and with separate effects on gain and threshold offset. The combined effect of a 1 C temperature increase is to decrease the detected energy by 0.1 keV for events depositing 30 keV. For the electronic noise, no statistically significant temperature effect was discernible in the data set, although theory predicts a weak dependence. The method is applicable to all x-ray detectors operating in pulse mode. Index Terms—Photon counting multibin detector, spectral CT, temperature effect.

I. INTRODUCTION

M

ULTIBIN photon counting computed tomography (CT) is an imaging technology employing semiconductor detectors in pulse mode to register the deposited energy of single x-ray quanta and increment separate bins depending on the detected pulse height. The clinical motivation for the application is that energy resolving multibin detectors, compared to energy integrating detectors, make better use of the available contrast information of each photon. This can be translated into lower dose at similar image quality, or increased image quality at similar dose. Theoretical investigations have reported that optimal weighting of the bin images can result in 15%–60% increase in the signal-difference-to-noise ratio at constant dose for typical imaging tasks [1], [2]. Other benefits of photon counting multibin CT include the prospect of performing material decomposition with three or more material bases, which would allow improved contrast agent visibility and concentration quantification (k-edge imaging) as well as beam hardening free CT images [3]. The technology is still in its early stages and bringing it to clinical practice presents several challenges. Two of the main challenges are the high incident fluxes and the inhomogeneity

Manuscript received July 06, 2012; revised September 03, 2012, November 12, 2012, and January 28, 2013; accepted January 29, 2013. Date of publication March 12, 2013; date of current version April 10, 2013. H. Bornefalk, M. Persson, C. Xu, S. Karlsson, and M. Danielsson are with the Department of Physics, Royal Institute of Technology, AlbaNova University Center, SE-106 91 Stockholm, Sweden (e-mail: [email protected]. kth.se). C. Svensson is with the Department of Electrical Engineering, Linköping University, SE-581 83 Linköping, Sweden. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNS.2013.2244909

of channel responses. High flux, relative to the charge collection time in the diode and the characteristic shaping time of the detector electronics, is associated with pileup which leads to both loss of counts and spectral distortion. Simulations have shown that under certain conditions, this might reduce the performance of multibin detectors to that of energy integrating systems applying a dual kVp technique [4]. For this reason, count-rate linearity measurements are important and routinely presented by groups involved in photon counting detector development [5], [6], [7]. It is also well known that differences among individual detector elements result in ring artifacts in the final image if not properly compensated for. For multibin systems, such compensation schemes tend to be more complex than for energy integrating systems, but have been shown to yield satisfactory results [8] under realistic assumptions on detector channel inhomogeneities. Since a multibin CT detector assembly has high power consumption (measured in kilowatts), there will be a substantial temperature increase during the image acquisition time if heat is not properly dissipated. Such temperature changes can also result in differences between detector elements [9], [10], with an accompanying risk of artifacts. Since ring artifacts are intolerable in clinical CT images, the normal approach to avoid temperature induced ring artifacts is to keep the detector temperature nearly constant (within 2 C) [9]. System design would however be much simplified if temperatures were allowed to drift more. This paper is concerned with the channel response homogeneity of photon counting spectral CT detectors; in particular with the temperature effect on the energy response of separate channels. While the method is developed and presented on data from a multibin silicon CT detector, including comparison with theoretical predictions, the statistical framework is applicable to all pulse mode operated photon counting detectors. Topics treated are 1) how the temperature affects the energy response, 2) the magnitude of the effect, and 3) whether the effect is channel dependent or similar across channels. All these topics are of great concern when methods to avoid ring artifacts are devised. For instance the issue of channel independent temperature effects boils down to the number of needed lookup tables for calibration, one global or one per channel. For our proposed system with some 5000 ASICs each containing 160 channels, this indicates a difference in lookup table volume by close a to a factor of 1 million. II. BACKGROUND An overview of this system has been presented elsewhere [11], [12], the ASIC has been characterized in [13] and initial performance evaluations in [7], [14], [15]. Only a brief overview is given here.

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BORNEFALK et al.: EFFECT OF TEMPERATURE VARIATION ON THE ENERGY RESPONSE OF A PHOTON COUNTING SILICON CT DETECTOR

Fig. 1. Photograph of a detector module. X rays enter on the top in the plane of the paper and move downward, passing 16 individual depth segments.

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particular ASIC has eight thresholds processing the data from each channel. If the gain and offset vary between channels, this must be compensated for to avoid ring artifacts. The pulse heights generated by x rays depositing keV will vary around the expected value . We denote the standard deviation of this variation as . The main source of variation is the electronic noise (in mV). Other, less dominant, sources for the variation are the Poisson statistics and the pileup of signals from neighboring detector elements [12]. Charge sharing, whereby generated charge carriers diffuse to neighboring detector elements during collection, will also result in spectrum trailing, i.e. a skewness towards lower energies. The detector module was designed to counteract temperature drifts. Offset drift, for example, depends mainly on the difference between the drift of the two transistors in the differential pair used in the offset calibration circuit and analog gain is mainly controlled by capacitor ratios. Due to the detector module complexity accurate analytical modeling of the temperature effects is intractable but effort is put into high level physical modeling of the origin, sign and magnitude of the expected temperature effects of , the gain and the offset separately. While this increases the physical understanding of the detector, real measurements must be obtained and statistical methods applied to infer the actual mechanisms by which the temperature affects the detector and whether they are channel dependent or not. III. THEORETICAL MODEL A. Electronic Noise

Fig. 2. Schematic of the analog part of detector channel from [13].

Fig. 1 depicts the silicon wafer diode with slots for 5 ASICs. The detector is designed for edge-on geometry; i.e. x rays enter from the top. Along the path of the x ray, the diode is segmented in 16 depth segments to decrease the count rate of each channel. To obtain similar conversion rates, the depth segments have increasing length as depicted in the figure. The wafer is 0.5 mm thick and the pitch is 0.4 mm. The active part of the wafer is 20 mm wide which indicates that the x rays see 50 detector elements of size 0.5 mm 0.4 mm. Each ASIC chip contains 160 channels, each one of which is connected to an individual collection electrode defining one detector segment (acting as a small diode). The analog part of each ASIC is shown schematically in Fig. 2 and contains a charge sensitive amplifier, a pole-zero cancellation, a pulse shaper and an offset calibration circuit. The pulse shaper converts incoming pulses to a common shape with pulse amplitude proportional to the charge collected by the electrodes. There will also be a small offset; the expected value for the pulse height amplitude (mV) generated by an x ray depositing keV can be written as (1) where is the gain in mV/keV and (in mV) an offset. If falls between the thresholds and (user defined reference voltages), the counter of the th bin is incremented. This is how spectral information is extracted from multibin systems. This

First make the plausible assumption that the total electronic noise is the sum (in quadrature) of the electronic noise from the diode, (d for diode) and the ASIC electronics, including (t for the effect of the added capacitance from the diode, thermal): (2) The problem is that the noise level of the ASIC with added capacitance, , is not accessible in isolation. If the sensor diode is disconnected from the ASIC, the capacitive load will decrease which affects the noise level and would render measurements unrepresentative of the intended use. Before estimating the temperature effects, we first show that the pure diode contribution to (2) is negligible, i.e. that (3) is known from x-ray measurements, cf. Eqs. (16) and (21) and the contribution from the diode can be estimated by leakage current measurements. The current that flows into one ASIC channel from the corresponding diode segment is the sum of the leakage current and the x-ray generated charge carriers . The offset calibration described above seeks to constantly remove the DC level of the signal, including the leakage current, but the contribution of the leakage current to the total variation of the signal amplitude cannot be removed (the counting statistics variance for the number of charge carriers in the signal is accounted for by (21)). If the leakage current of a

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detector element is (varying between channels as the strip detector elements have varying area) and a typical pulse length at the comparator is , the expected value of the number of electrons collected during the integration time is (4) where is the elemental charge. Since electron transfer is a stochastic process, is a random variable. In the limiting case of uncorrelated electron transfer, will follow a Poisson distribution. On the other hand, if the diode is ideally transmitting, there will be no variation at all around the expected value. The truth will lie somewhere between these limiting cases, resulting in a variance in the electron number . is a Fano factor for electron transmission through the reverse biased diode and, although unknown, obeys according to the limiting cases described. If is the variance of the collected number of electrons, the corresponding variance in detected pulse height, in mV , will be (5) where the ionization energy is expressed in keV. A typical leakage current under relevant conditions has been measured to be 150 pA which is used as an estimate for . This is expected to double every 7 degrees [16]. However, using the estimated average value 1.64 mV/keV for the gain (cf. Section VII-B), evaluates to mV which, when taken as an estimate for , is negligible compared to the estimated average value 3.66 mV for . One can thus safely assume that the entire contribution to the electronic noise stems from the ASIC, i.e. that (3) holds. For the ASIC electronics, the thermal noise is proportional to the square root of the temperature, i.e. , which directly yields the expected temperature coefficient: (6) With C K, which we assume to be a typical (future) operating temperature, the temperature coefficient of electronic noise should be 0.0016 K . B. Gain The gain is controlled by the transfer function of the analog channel, cf. Fig. 2 from [13]. For our circuit we can derive a transfer function as: (7) is the output voltage, is the input current, and , where and are the transconductance and the capacitance respectively of the Gm-C filters. by design which results in a peak output voltage

where equation 8 of [20] has been applied with and is the charge in the detected pulse. The gain, by definition, is now (9) where, as above, is the elemental charge and the ionization energy. Note that this formula is not expected to show any temperature dependence at all as the capacitances are not temperature dependent. However, and are not exactly equal, and they have different temperature dependence. An approximate correction of (8) taking the difference in time constants into consideration is (10) leading to a possible temperature dependence of the gain given ( is a constant). From this, the temperby ature coefficient can be expressed as (11) is implemented as a high resistivity poly resistor with a temperature coefficient of about K (according to the ASIC vendor). is proportional to the transistor transconductance of the filter which in turn is controlled by the bias circuitry of the chip. The bias circuitry strives to keep all transistor drain currents constant by relating them to an externally derived reference current. We may assume that the effective gate voltage of each transistor, adjusts to a value which keeps the drain current constant: (12) is the gate capacitance, the channel length, the charge carrier mobility, the gate voltage and the threshold voltage of the transistor [19]. Inserting the value of from (12) into the expression for , i.e. , gives (13) The only temperature dependent part is the charge carrier mobility . For holes in silicon, the mobility depends on temperature like [21]. We therefore expect , and thus , to be proportional to , i.e. to have a temperature dependence. It follows that K

By insertion in (11), the theoretically estimated temperature coefficient of the gain is obtained:

where

(8)

(14)

K

(15)

C. Offset The inherent offset can be expected to have a temperature dependence of the order of 3 mV/K (typical temperature dependence of transistor threshold voltage [19]). However, we expect


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a much smaller temperature effect for the offset, since a continuous offset calibration mechanism based on a continuous time nonlinear filter is applied [15] and this circuit strives to keep the offset constant. The offset calibration circuit utilizes an offset calibration amplifier with a differential input stage. Only a small residual of this 3 mV/K is expected for , possibly due to a mismatch between the temperature coefficients of the two input transistors in the offset calibration amplifier.

IV. EXPERIMENTAL SETUP AND RAW DATA Raw response data was collected at the SYRMEP beam line of the ELETTRA Synchrotron Light Laboratory (Trieste, Italy) by means of threshold scanning. The initial data analysis and treatment is identical to that of [15]: for each of the synchrotron x-ray energies keV one detector threshold was scanned and the resulting number of events depositing energies above this threshold (counts) in each channel were stored. The entire detector (diode and ASIC) was held at constant temperatures C during the scan by insertion in a small oven. (For the 18 keV beam, no measurements were obtained for 36.5 C.) The temperature range was selected to be representative of the modest elevations from room temperature that can be expected in clinical practise. The relatively low energy of the synchrotron radiation resulted in only the top part of the segmented detector diode wafer yielding any appreciable count numbers and only those channels were selected for further analysis. In total channels generated valuable data. S-curves for two selected channels are shown in Fig. 3. For most combinations of energy and temperature, the measurements are consistent: higher temperature measurements result in lower measured energies (s-curves shifted to the left). For channel 19, this effect in not readily apparent for the second leftmost curve family (corresponding to 22 keV). The same holds for the rightmost curve families (34.9 keV) for both channels; the effect is difficult to judge by visual inspection. One explanation could be that the s-curves do not level out at the same number of counts making the comparison difficult. Another explanation could be that the temperature behavior actually differs between channels. As mentioned, the purpose of this investigation is to find out whether any visible, and possibly invisible, discrepancies fall within the statistical uncertainties (given by the sample variation) and whether temperature effects can be modeled as channel independent or not. The threshold scans also illustrate that the magnitude of the s-curve shift increases with energy: the higher the x-ray energy (curve families to the right) the larger is the apparent temperature effect. This indicates that there is a separate temperature effect on the gain. Finally, the data illustrates no clear effect of temperature on electronic noise: while shifted horizontally by temperature the curves corresponding to a particular x-ray energy seem to move in parallel which would not be the case if electronic noise was greatly influenced by temperature.

Fig. 3. Registered counts as a function of threshold value for five different energies and three different temperatures. The five curve families from left to right correspond to increasing synchrotron x-ray energies from 18 to 34.9 keV.

V. INITIAL DATA PROCESSING A modified complementary error function [17]

(16) was fitted by means of weighted nonlinear least squares to the counts in each channel [18]. is the detector threshold value (in mV) and and the parameters of the underlying normal probability distribution function in the same units (subscript for the normal distribution). and phenomenologically capture charge sharing; the lower the threshold, the more charge packets that leak from neighboring pixels will result in counts and thus the upper part of the s-curve will not be flat, see Fig. 4. In the fitting procedure, the weights are the inverse of the observed count number at each detector threshold (due to the Poisson statistics). The inflection point is taken as the threshold value that corresponds to the incident x-ray energy [17]. It should be noted that the inflection point of an s-curve corresponds to the most likely energy (the mode of the energy probability density function) that would flow into the ASIC from the detector and that it therefore includes an average energy loss in the detector diode due to charge sharing.

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Preferably, the electronic noise should be the only contribution to . However, in practise it is impossible to separate the widening of the s-curve by electronic noise from that due to signal pileup and other variance sources. Such signal pile up could be with the signals from primary interaction locations in the same detector element or with charge packets that leak from neighboring pixels [12]. In this study we ensured a low flux (10–100 kcps) to minimize the effect of pileup on the widening of the s-curve. The typical contribution to the spreading of the s-curve due to statistical noise (Poisson statistics) is mV where is the gain, the Fano factor, the deposited energy and the ionization energy for silicon. Since all other sources of noise are small, (21) Fig. 4. Registered counts as a function of threshold value and complementary error function fitted according to (16).

The inflection point is not given by as sometimes assumed, but rather by setting and solving for :

(17) The difference might appear subtle, but actually lacks a clear physical interpretation after the inclusion of the factor in (16); it can no longer be simply interpreted as the midpoint of the curve since if there is a slope in the upper left part of the s-curve there is no well defined 50% level of the curve at all. In this work, since we find that the temperature effects arise in the ASIC rather than the diode, is an appropriate estimate for the threshold voltage. Gain and offset values (1) thus pertain to the ASIC rather than the entire detector system. Since charge sharing is limited in the detector diode [11], this distinction is of little practical importance. The error in the estimate of , where is the channel index, is given by error propagation (18) is the Jacobian of with respect to the fitting parameter and is the covariance matrix of . The channel gain and offset , for any given temperature , is now obtained as a result of a weighted least squares regression (19) where the weights are . The hat throughout this paper indicates that an entity has been estimated. Subscript indicates that and the error have been determined for channel at energy , . of (16) is closely related to the total uncertainty in the pulse height determination of the detector assembly and used as our estimate for : (20)

is a decent approximation for the electronic noise level (2) and is the entity for which we desire to evaluate the temperature behavior. VI. STATISTICAL MODEL The goal of this section is two-fold. The first is to present regression models that estimate the temperature coefficients that were derived theoretically above. The second goal is to determine whether the temperature coefficients for gain and offset differ across channels. We do not expect the electronic noise level to depend on the energy of the incident x rays. Therefore the following two nested regression models are set up (22) (23) where is the deviation from the average temperature. Model 1 has fitted parameters whereas model 2 has . After running a weighted linear regression (with weights given by the square root of the inverse of the second diagonal element of the corresponding covariance matrix from the curve fitting procedure) the residual sum of squares for model 1 (similar for model 2 but without ) is obtained as (24) be that model 2 describes the Let the null hypothesis noisy data set as good as model 1. This would correspond to there not being a statistically significant temperature effect on the electronic noise level in the data set at hand. Under , the statistic (25) follows an -distribution with degrees of freedom. If , the null hypothesis that model 2 describes the data equally well as the extended model 1, is rejected at the -level.


Now consider the regression model (26) designed to determine the temperature coefficient of the gain, , and the temperature coefficient of the offset, : (26) The parameter estimates and are the values that minimize the weighted sum of squared residuals where the weights are . The regression (26) is non-linear in the parameters due to the cross terms and but can be solved iteratively [18]. Excellent start values are obtained by setting and solving the resulting linear system analytically for and . To determine whether and are significantly different from zero the weighted residual sum of squares,

(27) is then by successive -tests compared to the sum of squares obtained by similar fittings of models 4 and 5, where the temperature effect on gain and on the offset have been omitted successively: (28) (29) The number of estimated parameters for models 3–5 are and . Under the successive null hypotheses and , assuming that the nested models 4 and 5 describe the temperature effect equally well as the full model 3 (i.e. that there is no temperature effect on the gain and the offset), the test statistic (30) follow -distributions with degrees of freedom. Again, if , the null hypothesis that model describes the data equally well as the extended model 3, is rejected at the -level. Finally, we want to determine whether making the temperature effects on the gain and offset channel dependent adds any significant explanatory power to the model, since this is important information when developing channel inhomogeneity adjustment schemes. This is achieved by running the regression

and then determine the weighted sum of residuals evaluate

(31) and

(32) (with ) against a suitable cutoff value of the corresponding -distribution. If , the null hypothesis that model 3 describes the data equally well as the extended model 6, is rejected at the -level.

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VII. RESULTS A.

-Statistics

of (25) evaluates to 1.16 which should be compared to the cutoff value of the corresponding distribution which is 6.66 at the -level. This indicates that there is no statistically significant temperature effect on the electronic noise level, i.e. the null hypothesis cannot be rejected. and of (30) evaluates to 81.8 and 37.4 which should be compared to the cutoff value 6.67 at the -level. This is strong support against the null hypotheses that model 4 and 5 explain the data equally well as model 3. We therefore conclude, from a pure statistical data processing point of view, that the temperature affect both the gain and the offset , i.e. favors model 3. is 0.72 which should be compared to the 1%-cutoff value 1.37. This provides no support for rejecting that the temperature effect on gain and offset is channel independent. The overall conclusion of the statistical modeling is that there is no statistically significant effect of temperature on the electronic noise level and that the gain and offset of all channels are affected similarly by temperature changes, i.e. behave according to model 3 rather than model 6. B. Regression Coefficients Model 3 (the most appropriate model among those describing the temperature effect on the measured energy) yields the following estimates for the gain and offset effects, K and K . The estimators have standard deviations K and K . The average value of in model 3 over the measured channels is . The average standard error due to the regression (i.e. the estimated uncertainty in the parameter estimates ) is . The sample standard deviation of the estimates is with (33) (The value for is only included for the sake of completeness and reference, the variability in gain and offset across channels have been characterized elsewhere [7], [15] but under slightly different circumstances. It should be noted though that the differences the gain between channels is appreciably larger than the uncertainty in gain of individual channels.) For the corresponding values are mV and mV (and mV). Also for completeness, model 2 which was shown more appropriate than model 1, yields an average value of the regression constant of 3.66 mV, which corresponds to an average electronic noise of 2.2 keV for all channels. C. Normality Assumption of Residuals The standard method of comparing nested regression models by means of an -test is dependent on the weighted residuals of (26) being independent and identically normally

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Fig. 7. Temperature bias on energy determination as a function of deposited x-ray energy . Dashed lines indicate plus/minus one standard deviation. Measurements were made in the interval 18–35 keV. Fig. 5. Histogram of the weighted residuals (unitless) of model 3 (26), showing a modest right skew.

VIII. DISCUSSION If two measurements are obtained at and , the typical change in the detected energy is given by the parameter estimates of model 3 (26):

With

(34) and denoting the th component of , error propagation yields the standard deviation

of (35)

Fig. 6. Scatter plot of the weighted residuals of model 3 vs.

.

distributed. The histogram plot in Fig. 5 of the weighted residuals (unitless) of model 3 casts doubt on this assumption, indicating that the error distribution is slightly skewed to the right. The Bowman-Shenton test for normality of the residuals [22], more popularly known as the Jarque-Bera test [23], also indicates that the null hypothesis of normally distributed residuals should be rejected with . However, the deviation from normality is not large and there is no clear pattern of the residuals when examined in a scatter plot against the dependent variable of the regression, Fig. 6. For that reason, and because of the similarity of the theoretically predicted values with the ones obtained from the regression, we trust the model and its results.

and are used as typical errors for and . where Equation (34) is expressed in units of mV and is a comparator reference voltage against which pulse amplitudes are compared. and have units mV/K. Thus it is difficult to get an intuitive feeling for the magnitude of the temperature effect. By division with the average gain , the overall temperature effect on the detected pulse heights (i.e. the bias in determining the energy) in keV is obtained. In Fig. 7 in keV/K is plotted for different x-ray energies. The interpretation of Fig. 7 is that a 10 C temperature increase would result in a downward shift of the detected energies by 1 keV for events depositing 30 keV. Whether this energy shift, smaller than the electronic noise, results in artifacts or not has not yet been established but in general even small biases are discernible in noisy data sets if the number of measurements is high enough. In CT there are in the range of 2000 projections needed to reconstruct one image slice so artifacts from a 10 C temperature increase seem likely (and are known to appear in current systems)[9]. Even if channels are affected homogenously by the temperature increase, the calibration (flatfielding) might be invalidated by


large temperature drifts; this will not only destroy the quantitative use of CT-numbers, but also likely translate to ring artifacts if the weighting scheme is not taking the effect on gain into consideration (the measured x-ray spectrum would be more downward shifted for higher energies than for lower energies). It should be mentioned that in a real implementation, temperature drift would never be allowed to result in artifacts; then more cooling or more lookup tables would have to be implemented and this would come at the cost of higher system complexity. IX. CONCLUSION We conclude that the measurements give strong support that temperature affects the ASIC according to model 3, with independent gain and offset effects and that these are channel independent. For the electronic noise level, no statistically significant effect of temperature was observed. The expected temperature coefficient of the gain is K . This has the same sign and is of the same order of magnitude as the observed temperature coefficient K . The difference may be due to a slight temperature dependence of the external bias current , as it is generated through a resistor in series with an MOS diode. The observed offset temperature drift is mV/K. This should be compared to a typical expected value of 3 mV/K in the absence of offset calibration. We conclude that the offset calibration mechanism is very efficient and reduces the temperature coefficient to 1/45 of its original value. The statistical method presented in this paper can help developers of multibin detectors to assess the temperature dependence of their detector. This in turn is important input when devising strategies for handling temperature variation; if temperature increases and their corresponding effects on pulse height determination are within what compensation schemes can handle, real-time temperature measurements inside the detector cradle might be enough and developers need not resort to mechanically complex cooling and heat dissipation solutions. For the system under investigation, the final strategy has not yet been determined but a combination of some heat dissipation mechanism and temperature measurements and adjustments does not seem unlikely. ACKNOWLEDGMENT The authors would like to thank the staff of the SYRMEP beam line at the Trieste Synchrotron facilities for kind assistance and also Stefan Rydström for assembling the heating device for maintaining constant detector temperature. REFERENCES [1] P. M. Shikhaliev, “Computed tomography with energy resolved detection: A feasibility study,” Phys. Med. Biol., vol. 53, pp. 1475–1495, 2008. [2] T. G. Schmidt, “Optimal ‘image-based’ weighting for energy-resolved CT,” Med. Phys., vol. 36, no. 7, pp. 3018–3027, 2009.

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