Raafat Fares

Marion Moreau

Pascal E Sanchez

Béatrice Georges

Colette Moulin

Chantal Bonnet

Anne Morales

Laurent Bezin
email: laurent.bezin@univ-lyon1.fr

Beneficial effects of environmental enrichment on structural and genomic brain plasticity are abolished by forced physical exercise

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The aim of the study was to determine whether the effect of forced physical training from weaning to adulthood is the same between rats raised either in impoverished or enriched environments. For environmental enrichment, we have engineered a cage allowing standardization of the procedures of enrichment (under patent examination). This cage allows increased social interactions (12 rats per cage), increased voluntary exercise (increased size of the cage, running wheels), "diverting" activities (ladder, toboggan slide, tunnel), and cognitive stimulations using labyrinths, the configuration of which is changed 3 times a week. The forced physical training consists of a daily swimming program, the time spent in the swimming pool being increased from 5 min the first week to 45 min the fifth week. S rats were housed in groups of 6 from weaning to adulthood. Four groups are examined: standard housing (S), enriched housing (En), S + forced swimming (SFS) and EN + forced swimming (EnFS). We found that the effect of FS on structural plasticity is dependent on the initial housing condition: 1) FS increases cortical thickness in S rats, as En does; however, the increased cortical thickness measured in En rats is not observed in EnFS rats. [Fig. 1] 2) En increases neurogenesis in the hippocampus when examined at the end of the protocol (compared to S rats). FS is not a sufficient condition to increase neurogenesis in S rats. However, in rats raised in enriched conditions, FS brings back neurogenesis to the value measured in S rats.

[Fig. 2] 3) In addition, while BDNF, TrkBFL, TrkBT1 and TrkBT2 transcript levels (determined using calibrated RT and real time PCR) are significantly increased in En group compared to S, FS decreased transcript levels in En rats below levels determined in S rats. [Fig. 3] 4) Levels of mRNA encoding glucocorticoid and mineralocorticoid receptors (GR and MR) are greater in En rats compared to S rats in the hippocampus, but FS decreased GR and MR mRNA levels in En rats below levels determined in S rats. By contrast, GR mRNA levels remained stable in SFS rats compared to S rats, while MR mRNA levels were significantly increased. [Fig. 4] 5) Surprisingly, basal levels of corticosterone (CORT) were significantly lower in SFS rats and En rats compared to S rats, but also in EnFS rats compared to S rats. [Fig.5] Our hypothesis is that FS is not interpreted the same way by rats raised in standard (S) or enriched (En) environments. Since CORT plasma levels in SFS rats is below S rats, we suggest that FS decreased stress in rats raised in S conditions. By contrast, FS may be considered as a stressful condition in rats raised in En conditions. Indeed, when the labyrinth is changed, En rats are used to find quickly the issue; in the swimming pool, the same rats try to find an "exit" by diving and exploration of the bottom of the tub in the first sessions, and may rapidly be stressed by the fact that they do not find a solution to escape the swimming pool. In EnFS rats, basal level of CORT are lower than that found in S rats and similar to that measured in En rats. We propose that CORT levels in EnFS rats greatly increase during FS, and return to baseline after a longer time than that measured in En rats that do not experience FS, due to a lower level of GR and MR in the hippocampus. We are currently measuring the response of plasma CORT to restraint stress, to determine whether basal level is recovered faster in En rats compared to S rats. If that is the case, we will repeat the same kind of experience to compare the responses of En and EnFS rats.

Figure legends:

Fig. 1: Effects of forced physical exercise on structural plasticity are dependant on environmental conditions. Forced swimming increases central cortical thickness in standard rats (SFS), but not in enriched rats (EnFS) after 6 weeks of swimming. Results are expressed in percent ± SEM of standard (S) group. ** P< 0.01 in comparison to respective control (S or En). § § § P< 0.001 in comparison to standard group (S). Fig. 2: Forced swimming abolishes the effects of environmental enrichment on neurogenesis. (A) Fluorescent detection of BrdU labeled cells in the dentate gyrus of standard (S) and enriched (En) groups. (B). Environmental enrichment increases the number of BrdU-labelled cells in the dentate gyrus of adult rats. This effect is reversed when enriched environment is associated to forced swimming (EnFS). Forced swimming in standard rats (SFS) is not sufficient to induce significant neurogenesis. Results are expressed in percent ± SEM of standard (S) group. * P< 0.05 in comparison to respective control (S or En). § § § P< 0.001 in comparison to standard (S). Enriched environment increases the expression of transcript encoding for GR and MR. While forced swimming induces the mRNA encoding MR only in standard rats (S), it has a complete opposite effect on MR and GR in enriched rats (EnFS). Results correspond to the mean ± SEM of the number of cDNA copies quantified by real time PCR following reverse transcription of mRNAs contained in 500 ng of total RNAs and are expressed in percent of standard (S)., ** p<0.01, *** p<0.001 in comparison to control, § P< 0.05, in comparison to standard (S).

Fig. 3 :

 3 Fig.3: Expression profile of trophic factors in different brain regions of adult rats.Enriched environment up-regulates the expression of transcript encoding for BDNF, TrkBFL, TrkBT1 and TrkBT2 in all brain regions. Forced swimming is able to induce the expression of some transcripts in several regions of standard rats (SFS). However, forced swimming reverses completely the effects of enriched in (EnFS) group. Results correspond to the mean ± SEM of the number of cDNA copies quantified by real time PCR following reverse transcription of mRNAs contained in 500 ng of total RNAs and are expressed in percent of standard (S). * p<0.05, ** p<0.01, *** p<0.001 in comparison to respective control (S or En), § P< 0.05, § § P< 0.01, § § § P< 0.001 in comparison to standard (S).

Fig 4 :

 4 Fig 4: Expression profile of glucocorticoids (GR) and mineralocorticoids (MR) receptors in the hippocampus of adult rats after 6 weeks of housing in either standard or environmental condition associated or not with forced swimming.Enriched environment increases the expression of transcript encoding for GR and MR. While forced swimming induces the mRNA encoding MR only in standard rats (S), it has a complete opposite effect on MR and GR in enriched rats (EnFS). Results correspond to the mean ± SEM of the number of cDNA copies quantified by real time PCR following reverse transcription of mRNAs contained in 500 ng of total RNAs and are expressed in percent of standard (S)., ** p<0.01, *** p<0.001 in comparison to control, § P< 0.05, in comparison to standard (S).

Fig 5 :

 5 Fig 5: Basal levels of plasma corticosterone (CORT) after 24 hours of the last swimming session.Forced swimming decreases the basal level of plasma CORT in standard rats (S). Enriched environment also decreases basal level of CORT, but forced swimming has no effect in this condition (EnFS). Results are expressed in percent ± SEM of standard (S) group. * p<0.05, in comparison to respective control (S or En), § § § P< 0.001 in comparison to standard (S).


Pierre Jop

Artyom Petrosyan

Sergio Ciliberto

Comment on "Measurement of Effective Temperatures in an Aging Colloidal Glass"

Keywords: numbers: 64.70.Pf, 05.40.-a, 05.70.Ln, 82.70.Dd

In a Letter [1], Greinert et al. reported an increase of effective temperature (T ef f ) during the aging process of a Laponite suspension measured from the fluctuations of position of optically trapped beads. We think their results are not conclusive since there could be some artifacts in the experiment.

We measure the fluctuations of the position of a silica bead trapped by an optical tweezers during the aging of the Laponite. The laser beam (λ=980 nm) is focused by a microscope objective (×63) 20 µm above the coverslip surface to create a harmonic potential well where a bead of 1 or 2 µm in diameter (d) is trapped. The Laponite mass concentration is varied from 1.2 to 3% wt for different ionic strengths.

Following the method used in [1], the stiffness of the optical trap is periodically switched between two different values (k 1 and k 2 ) every 61 seconds by changing the laser intensity. The position of the bead is recorded by a quadrant photodiode at the rate of 8192 Hz, then we compute the variance of the position over the whole signal, δx 2 = x 2 -x 2 , where the brackets stand for average over the time. To avoid transients, each record is started 20 seconds after the laser switch. Assuming the equipartition principle still holds in this out-ofequilibrium system, the T ef f is computed as in [1].

Our results are shown on the Fig. 1a. We find that T ef f is constant at the beginning and is very close to T bath = 294 K, then when the jamming occurs, it becomes noisy without any clear increase with t w , contrary to Ref. [1].

We now make three remarks. First, we point out that the uncertainty of their results are underestimated. The error bars in Fig. 1(a) are here evaluated from the standard deviation of the variance using Eq. 2 in Ref [1] at the time t w . Although they are small for short time t w , (∆T ef f /T ef f ≤ 10%), they increase for large t w . This is a consequence of the increase of variabilities of δx 2 i as the colloidal glass forms. This point is not discussed in in Ref. [1] and we think that the measurement errors are of the same order or larger than the observed effect. The results depend on the length of the analysing time window and the use of the principle of energy equipartition becomes questionable for the following reasons.

First, these analysing windows cannot be made too large because the viscoelastic properties of Laponite evolve as a function of time. Second, the corner frequency of the global trap (optical trap and gel), the ratio of the trap stiffness to viscosity, decreases continuously mainly because of the increase of viscosity. At the end of the experiment, the power spectrum density of the displace-ment of the bead shows that the corner frequency is lower than 0.1 Hz. We thus observe long lived fluctuations, which could not be taken into account with short measuring times. This problem is shown on Fig. 1b. We split our data into equal time duration ∆τ , compute the variance and average the results of all samples. The dotted line represents the duration 3.3 s chosen in [1]. At the beginning of the experiment, the variance of the displacement is constant for any reasonable durations of measurement. However, we clearly see that this method produces an underestimate of ∆x 2 for long aging times, specially when the viscoelasticity of the gel becomes important. Long lived fluctuations are then ignored.

) ∆τ for analyses ( Finally, a remark has to be made concerning the mean position of the bead during aging. When the stiffness of the gel becomes comparable to the optical one, the bead starts to move away from the center the optical trap. We observe a drift of the bead position at long time, which could lead to the escape of the bead. Moreover we have performed simultaneous measurements with a multiple trap using a fast camera showing that at very long t w the mean trajectories of beads separated by 7 µm are almost identical. This proves that one must pay attention when interpreting such measurements, specially on the duration of measurements. We also have seen that the way the sample is sealed can accelerate the formation of the gel and the drift of the bead by changing the chemical properties in the small sample. We have used different types of cell, Laponite concentrations, bead sizes, stiff-

FIG. 1 :

 1 FIG.1: a) Evolution of the effective temperature with aging time for k1 = 6.34 pN/µm, k2 = 14.4 pN/µm, 2.3% wt of Laponite and 1 µm glass bead. The error bars are computed from the statistical error of δx 2 i at the time tw. Inset: Evolution of δx 2 with tw for both stiffnesses (k1 • and k2 ). b) Evolution of δx 2 ∆τ for the low stiffness as a function of the duration ∆τ of the samples (see text) for different aging times : from top to bottom tw = 30, 57, 83, 110, 137, 163, 190, 217, 243 and 270 minutes.

nesses of the optical trap. In each case we do not find any increase of the effective temperature.

In conclusion, our results show no increase of T ef f in Laponite and are in agreement with those of Jabbari-Farouji et al. [2], who measured fluctuations and responses of the bead displacement in Laponite over a wide range of frequency and found that T ef f is equal to the bath temperature.


Robert E Alvarez
email: ralvarez@aprendtech.com

Near optimal neural network estimator for spectral x-ray photon counting data with pileup

Keywords: spectral x-ray, photon counting, pileup, dual energy, Cramèr-Rao lower bound

Purpose: A neural network estimator to process x-ray spectral measurements from photon counting detectors with pileup. The estimator is used with an expansion of the attenuation coefficient as a linear combination of functions of energy multiplied by coefficients that depend on the material composition at points within the object [R.E. Alvarez and A. Macovski, Phys. Med. Biol., 1976, 733-744]. The estimator computes the line integrals of the coefficients from measurements with different spectra. Neural network estimators are trained with measurements of a calibration phantom with the clinical x-ray system. One estimator uses low noise training data and another network is trained with data computed by adding random noise to the low noise data. The performance of the estimators is compared to each other and to the Cramer-Rao lower bound (CRLB). Methods: The estimator performance is measured using a Monte Carlo simulation with an idealized model of a photon counting detector that includes only pileup and quantum noise. Transmitted x-ray spectra are computed for a calibration phantom. The transmitted spectra are used to compute random data for photon counting detectors with pileup. Detectors with small and large dead times are considered. Neural network training data with extremely low noise are computed by averaging the random detected data with pileup for a large numbers of exposures of the phantom. Each exposure is equivalent to a projection image or one projection of a computed tomography scan. Training data with high noise are computed by using data from one exposure. Finally, training data are computed by adding random data to the low noise data. The added random data are multivariate normal with zero mean and covariance equal to the sample covariance of data for an object with properly chosen attenuation. To test the estimators, random data are computed for different thicknesses of three test objects with different compositions. These are used as inputs to the neural network estimators. The mean squared errors (MSE), variance and square of the bias of the neural networks' outputs with the random object data are each compared to the CRLB. Results: The MSE for a network trained with low noise data and added noise is close to the CRLB for both the low and high pileup cases. Networks trained with very low noise data have low bias but large variance for both pileup cases.

INTRODUCTION

The estimator is used with the Alvarez-Macovski method 1 . In this method, the x-ray attenuation coefficient is approximated as a linear combination of basis functions of energy multiplied by coefficients that depend only on the material composition at each point within the object. The estimator computes the line integrals of the coefficients from transmitted x-ray measurements with different energy spectra. Recently, neural networks have been suggested for this application [START_REF] Lee | Material depth reconstruction method of multi-energy x-ray images using neural network[END_REF][START_REF] Zimmerman | Experimental comparison of empirical material decomposition methods for spectral ct[END_REF][START_REF] Touch | A neural network-based method for spectral distortion correction in photon counting x-ray ct[END_REF] by using the universal function approximation theorem [START_REF] Cybenko | Approximation by superpositions of a sigmoidal function[END_REF] to invert the nonlinear transformation from the the line integrals to the spectral measurements.

If the number of spectral measurements is equal to the number of basis functions then any estimator that inverts the deterministic transformation is the maximum likelihood estimator and it gives optimal performance 1, [START_REF] Alvarez | Estimator for photon counting energy selective x-ray imaging with multi-bin pulse height analysis[END_REF][START_REF] Alvarez | Efficient, non-iterative estimator for imaging contrast agents with spectral x-ray detectors[END_REF] . That is, in the limit of large photon counts it is unbiased and has covariance equal to the Cramer-Rao lower bound (CRLB) [START_REF] Kay | Fundamentals of Statistical Signal Processing[END_REF] .

If the number of spectral measurements is greater than the number of basis functions then simply inverting the transformation does not necessarily give optimal results.

For this case, the optimal estimator needs to use the probability distribution of the measurement noise to provide optimal performance [START_REF] Kay | Fundamentals of Statistical Signal Processing[END_REF] . Systems with more spectral measurements than the number of basis functions are becoming increasingly important because of the introduction of photon counting detectors into medical x-ray imaging systems [START_REF] Taguchi | Vision 20/20: Single photon counting x-ray detectors in medical imaging[END_REF] . With these detectors, we can use pulse height analysis (PHA) to categorize the energy of individual photons into separate energy bins. The counts in each bin constitute a separate spectral measurement and the number of bins depends on technical factors of the detector design [START_REF] Knoll | Radiation Detection and Measurement[END_REF] but is typically larger than the basis set dimension.

There is no explicit way to incorporate the noise probability distribution with a neural network but in the past noise has been added to the training data to improve the generalizability of the network [START_REF] Sietsma | Neural net pruning-why and how[END_REF][START_REF] Bishop | Training with Noise is Equivalent[END_REF][START_REF] An | The Effects of Adding Noise During Backpropagation Training on a Generalization Performance[END_REF][START_REF] Zur | Noise injection for training artificial neural networks: A comparison with weight decay and early stopping[END_REF] . For our application, the noise variance varies exponentially with the variables being estimated and the off diagonal terms of the covariance also change with these variables [START_REF] Alvarez | Signal to noise ratio of energy selective x-ray photon counting systems with pileup[END_REF] . Therefore, it is not clear whether adding noise to the training data will result in optimal performance and what is the proper level of noise to provide this performance. These questions are examined in this paper.

The approach used is to train the neural network with measurements of a calibration phantom in the clinical x- Current state of the art photon counting detectors have defects including incomplete photon energy measurement due to K radiation and Compton scattered photon escape, charge sharing and trapping, polarization and other effects [START_REF] Taguchi | Vision 20/20: Single photon counting x-ray detectors in medical imaging[END_REF][START_REF] Overdick | Towards direct conversion detectors for medical imaging with X-rays[END_REF] . As the detector state of the art improves, we can expect these defects may be reduced to negligible levels. However, all photon counting detectors have a finite response time and, since the inter-arrival times of

x-ray photons on the detector sensor are exponentially distributed [START_REF] Barrett | Foundations of Image Science[END_REF] , there will always be a non-zero probability for two or more photons to enter the detector during its response time no matter how small. In

METHODS

A. The estimation problem

For biological materials and an externally administered high atomic number contrast agent we need three or more functions to accurately approximate the attenuation coefficient [START_REF] Alvarez | Dimensionality and noise in energy selective x-ray imaging[END_REF] ,

µ(r, E) = a 1 (r)f 1 (E) + a 2 (r)f 2 (E) + a 3 (r)f 3 (E). (1)
In this equation, a i (r) are the basis set coefficients, f i (E)

are the basis functions and the subscripts are i = 1 . . . 3.

As implied by the notation, the basis set coefficients a i (r)

are functions only of the position r within the object and the basis functions f i (E) are functions only of the x-ray energy E. Contrast agents with more than one high atomic number element may require additional functions of energy.

Neglecting scatter, the expected value of the number of transmitted photons n k for effective spectrum S k (E) is

λ k = n k = ˆSk (E)e -´µ(r,E)dr dE, k = 1 . . . n spect (2)
where nspect is the number of spectral measurements, denotes expected value and the integral in the exponent is on a line from the x-ray source to the detector. With pulse height analysis the effective spectrum for an energy bin measurement can be idealized to be

S k (E) = S incident (E) Π k (E) (3)
where S incident (E) is the x-ray spectrum incident on the detector and Π k (E) is a rectangle function equal to one inside the energy bin and zero outside.

Using Eq. 1, the line integral in Eq. 2 can be expressed

as ˆµ (r, E) dr = A 1 f 1 (E) + A 2 f 2 (E) + A 3 f 3 (E). ( 4
)
where A i = ´ai (r) dr, i = 1 . . . 3. The A i can be summarized as the components of the A-vector A and the measurements by a vector N whose components are the measurements with the effective spectra, n k , k = 1 . . . n spect . Since the x-ray transmission is exponential in A, we can approximately linearize the measurements by taking logarithms. The results is the log measurement vector

L = -log(N/N 0 ), (5)
where N 0 is the expected value of the measurements with no object in the beam and the division means that corresponding elements of the vectors are divided.

Eq. 2 defines a relationship between A and the expected value of the measurement vector, L(A) . In an x-ray system, the objective of the estimator is to invert the relationship with noisy data and to compute the best estimate of the A-vector taking into account the probability distribution of the noise.

B. The neural network estimator

The neural network shown in Fig. 1 was used as an estimator. The inputs to the network are the components of the L vector, Eq. 5. The network had one hidden layer with 12 processing elements and three output elements, which are the estimates of the components of the A-vector, A. This simple network was found to give near optimal performance due to the near linearity of the relationship between L and A. Other networks may also give good performance but they cannot have lower noise than the CRLB.

The hidden elements had a sigmoid response sigmoid (x) = 2 1 + e -2x -1 and the output elements had a linear response.

The network was trained with calibration data measured with the clinical x-ray system as discussed in Sec. During the training, random subsets of the data were used for training (70%), validation (15%), and final testing (15%).

The stopping criterion was that the error with the validation set not decrease for six iterations or the maximum epochs, 500, were reached.

The neural network was used with all input data including those that fall outside the convex hull of the training data.

C. The neural network training data

The neural network was trained using measurements of the calibration phantom shown in Fig. 2 with the clinical x-ray system. The figure shows a side-view of a three material phantom. The purpose of the calibration phantom is to provide values of the measurement vector L for a set of points in three dimensional A-space. If we use the attenuation coefficient functions of the materials of the calibrator as the basis set [START_REF] Alvarez | A comparison of noise and dose in conventional and energy selective computed tomography[END_REF] , then the A vectors for each step are simply the thicknesses of the materials along lines from the x-ray source to the detector. The phantom can be constructed from stable, machinable materials such as acrylic plastic and aluminum. The third material could be, as an example, a plastic resin with molecular linked iodine whose attenuation simulates iodine contrast agent in blood [START_REF]QRM-CTiodine[END_REF] .

The calibrator data are acquired with the clinical xray imaging system and do not require additional physics instruments that may be unavailable in clinical institutions.

With a fan beam computed tomography system, the actual path lengths through different parts of phantom can be computed from its dimensions and the known geometry of the x-ray system by developing a method to locate the calibrator accurately with respect the the scanner gantry such as affixing pins to the phantom and scanning it. Fig. 2 shows a phantom with uniform steps but exponentially spaced thicknesses were used in the Monte Carlo simulations. They provide better results since they have closer spaced samples in the region near the origin where the gradient of L(A) is highest.

D. Training with noisy data

The Monte Carlo simulation described in Sec.

E. Model for photon counting data with pulse pileup

The idealized model for photon counting data with pileup is described in detail in a previous paper [START_REF] Alvarez | Signal to noise ratio of energy selective x-ray photon counting systems with pileup[END_REF] and is summarized here. The response of photon counting detectors with pileup to an incident photon is modeled with the dead time parameter, τ 11 , which is the minimum time between two photons that are recorded as separate events. Two models are commonly used to describe the recorded counts with pileup: paralyzable and non-paralyzable. The nonparalyzable model is used in this paper. Measurements by Taguchi et al. [START_REF] Taguchi | Modeling the performance of a photon counting x-ray detector for CT: energy response and pulse pileup effects[END_REF] indicate it is more accurate at higher count rates with their detectors and it also leads to simpler analytical results [START_REF] Yu | Mean and variance of single photon counting with deadtime[END_REF] .

A model is also needed for the recorded energies with pileup. One approach is to assume that the recorded energy is proportional to the integral of the sensor charge pulses during the dead time [START_REF] Taguchi | Modeling the performance of a photon counting x-ray detector for CT: energy response and pulse pileup effects[END_REF] . An idealization of this model was used that assumes that the recorded energy is the sum of the energies of the photons that arrive during the dead time regardless of how close the arrival time of a photon to the end of the period [START_REF] Wang | Pulse pileup statistics for energy discriminating photon counting x-ray detectors[END_REF] . The idealized model assumes that the photon energy is converted completely into charge carriers so there are no losses due to Compton or Rayleigh scattering and all K-fluorescence radiation is re-absorbed within the sensor. All of the carriers are assumed to be collected so that there is no charge trapping or charge sharing with nearby detectors.

F. Probability distribution of pulse height analysis data with pileup

Using the idealized pulse pileup model described in the previous Sec. 2.E, the probability distribution of PHA data with pileup is modeled as multivariate normal with the expected value and covariance in Table I [START_REF] Alvarez | Signal to noise ratio of energy selective x-ray photon counting systems with pileup[END_REF][START_REF] Wang | Pulse pileup statistics for energy discriminating photon counting x-ray detectors[END_REF] . The table has two columns, the first for data with no pileup and the second with pileup. In the table, the subscript 'rec' denotes recorded data with pileup. The multivariate normal model can be shown to be accurate for the number of x-ray photons required for the measurements in material selective imaging [START_REF] Alvarez | Signal to noise ratio of energy selective x-ray photon counting systems with pileup[END_REF] .

In the table,

λ = ˆSincident_det (E)dE (6)
is the expected value of the number of photons incident on the detector during the measurement time, T , and [START_REF] Alvarez | Signal to noise ratio of energy selective x-ray photon counting systems with pileup[END_REF] .

ρ = λ /T
If the probability of a zero recorded photon count value is negligible, as is the case with the large expected values of counts required for material selective imaging, the parameters of the normal distribution of the logarithm L data 27 can be computed from the formulas in Table I using:

log(N) = log( N ) var (log(N)) = var(N) N 2 cov (log(N 1 ), log(N 2 )) = cov(N1,N2) N1 N2 . (7)

G. The CRLB for A-vector noise with pileup

The CRLB is the minimum covariance for any unbiased estimator and is a fundamental limit from statistical estimator theory. It is the inverse of the Fisher information matrix F whose elements are 28

F ij = - ∂ 2 L ∂A i ∂A j ( 8
)
where L is the logarithm of the likelihood and the symbol denotes the expected value. Kay [START_REF] Kay | Fundamentals of Statistical Signal Processing[END_REF] shows that the Fisher information matrix for multivariate normal data with expected value L(A) and covariance C L has elements

F ij = ∂ L(A) ∂Ai T C -1 L ∂ L(A) ∂Aj + . . . 1 2 tr C -1 L ∂C L ∂Ai C -1 L ∂C L ∂Aj (9)
where the tr [] is the trace of a matrix.

The CRLB was computed numerically by approximating the derivatives in Eq. 9 from the first central difference.

For example, to compute ∆L we first compute the spectra through the object with attenuation A and then with A+∆A. The transmitted spectra are not affected by pileup since they occur before the measurement. These transmitted spectra are then used to compute the expected values of the measurements with pileup using the formulas in Sec. 2.F. We can similarly compute the difference of the covariance of the log data C L using the formulas in that section.

H. The test object for Monte Carlo simulation

The performance of the neural network estimator was tested with objects with A-vectors on three lines through the A-vector space shown in Fig. 3. These correspond to a set of thicknesses of three uniform objects with different compositions. Each material has different basis set coefficients in its attenuation coefficient expansion, Eq. 1. Summarizing the coefficients as the components of the a vector, the line integrals are therefore A = aW , where W is the object thickness with units corresponding to attenuation coefficient, for example g/cm 2 . The A-vectors for different thicknesses of a material therefore fall on a straight line through the origin in A-vector space. The end points of the lines used in the simulations, which also specify the ratios of the a vector coefficients, were [16, 1.2, 0.1], [5, 0.9, 0.1125], and [16, 0.375, 0.1] g/cm 2 . Fig. 3 shows a three dimension plot of the lines.

I. Random data for Monte Carlo simulation

The Monte Carlo simulation compared the mean squared error, the variance, and the square of the bias of the A-vector estimates to the Cramèr-Rao lower bound.

The generation of the random data for the simulation started with the computation of a 120 kilovolt x-ray tube spectrum using the TASMIP algorithm of Boone and Seibert [START_REF] Boone | An accurate method for computer-generating tungsten anode x-ray spectra from 30 to 140 kV[END_REF] . The number of photons incident on the object for each detector element or pixel was set to 10 6 . The measurement time was assumed to be 10 milliseconds so the rate of photons incident on the detector with no object in the beam was 10 8 photons per second. Two cases of pileup were computed by setting the detector dead times to 10 and 1 nanoseconds. These result in zero object thickness pileup parameters, η 0 , of 1 and 0.1 respectively. no pileup with pileup

λ = ´Sincident_det (E)dE {0 < E 1 < . . . < E nbins } photon number spectrum S(E) S rec (E) = λp rec (E) normalized spectrum p(E) = S(E)/λ p rec (E) = ∞ k=0 η k k! e -η p (k) * p bin probabilities P k = ´Ek E k-1 p(E)dE P rec,k = ´Ek E k-1 p rec (E)dE expected total counts N = λ N rec = λ 1+η variance total counts var(N ) = λ var(N rec ) = λ (1+η) 3 non-Poisson factor D = 0 D = var(N rec ) -N rec expected bin counts N k = λP k N rec,k = N rec P rec,k variance bin counts var(N k ) = λP k var(N rec,k ) = N rec P rec,k + DP 2 rec,k covariance bin counts 0 cov(N j , N k ) j =k = P rec,j P rec,k D
As discussed in Sec. A calibration phantom with 30 steps for each material was implemented. The thicknesses were geometrically spaced from zero to 20, 1.5 and 0.125 g/cm 2 for each of the calibration materials respectively. These were chosen to be greater than the object values so all L measurements except for noise will fall within the calibration data convex hull.

For a single A-vector on one of the lines in Fig. 3, the TASMIP x-ray tube spectrum was used with Eq. 6 to compute the spectrum and the expected value of the total number of transmitted photons incident on the detector sensor during the measurement time. The sensor was assumed to be perfectly absorbing so the signal for each photon was proportional to the photon energy. The recorded energy with pulse pileup was computed using the pileup model described in Sec. 2.E. Five bin PHA was done on the recorded energies. The PHA energy response functions were computed with an algorithm that maximized the SNR with no pileup and were assumed to be perfect rectangles. The algorithm for the optimal bins is described in a previous paper [START_REF] Alvarez | Signal to noise ratio of energy selective x-ray photon counting systems with pileup[END_REF] .

The expected values and covariance of the recorded PHA bin counts were computed from the formulas in Table I. These were used to compute the parameters of the multivariate normal distributed log data L using Eq. 7. These data were used to test the neural network estimators.

J. Estimator performance

The estimator performance was computed for the three training data sets and for the two cases of the dead time parameter, 1 and 10 nanoseconds. The neural networks were trained as described in Sec. 2.D. The input data to the estimators were computed as described in Sec. 2.I.

The estimates of the networks were computed using the same random data as inputs so their output noise could be directly compared. The mean square error MSE of the estimates for 2000 trials was computed as

MSE = 1 N trials N trials i=1 Â -A actual 2 (10)
where N trails is the number of trials, Â is the estimate and A actual is the actual A-vector value. Notice that the MSE is a vector quantity with a value for each component of A.

The sample variance and the square of the sample bias of the estimates were also computed. These were plotted after normalizing by dividing by the CRLB variance.

RESULTS

A. Mean squared error

Fig. 4 shows the mean squared errors of the neural network estimators as a function of A-vector magnitude for the three lines in Fig. 3. Panel (a) has the results for the low pileup case with zero object thickness dead time parameter, η 0 = 0.1, and panel (b) for the high pileup case, η 0 = 1. The A-vector magnitude is proportional to object thickness as explained in Sec.

  ray system. Low noise training data are computed by averaging together a large number (1000) of exposures of the phantom. Another set of training data is computed by adding zero mean multivariate normal random data with a covariance equal to the values on a properly chosen interior step of the calibration phantom to the low noise data. Training data with high noise are computed by using only one exposure. The output noise of neural network estimators trained with the three data sets are compared to each other and to the CRLB. Other factors that affect the estimator output noise include the number of steps in the calibration phantom as well as the architecture of the network.

  2.C. The Levenberg-Marquardt algorithm was used for training. This algorithm selected the weights and offsets of the network to minimize the mean squared error of the estimates of the calibration phantom's known A-vectors using the L data of the calibrator measured with the clinical x-ray system.

3 Figure 1 .Figure 2 .

 312 Figure 1. Neural network estimator block diagram. One hidden layer with 12 processing elements and three output elements was found to give near optimal performance.

Figure 3 .

 3 Figure 3. Three lines in A-vector space used in the Monte Carlo simulation. Each line is the A-vectors of different thicknesses of a material with a specific a vector of coefficients in its attenuation coefficient expansion.

  2.C, a basis set consisting of the attenuation coefficients of acrylic plastic, aluminum, and an iodine contrast agent simulant consisting of 20% fraction by weight iodine in paraffin (C 31 H 64 molecular composition) was used as the basis functions of energy in Eq. 1. With this choice, the A-vectors were the thicknesses of each of the materials in the calibration phantom 22 . The attenuation coefficients of the materials were computed as the fraction by weight of each element in its chemical formula multiplied by the attenuation coefficient of that element. The elements' attenuation coefficients were computed by piece-wise continuous Hermite polynomial interpolation of the standard Hubbell-Seltzer tables 31 .

  Fig. 6 shows the square of the bias for the two pileup cases. The bias squared data are normalized by dividing by the CRLB variance and a dashed line is drawn at a ratio of one.

Fig. 7 Figure 4 .

 74 Figure 4. Mean squared error vs. A-vector magnitude for the (a) low zero object thickness pileup case, η 0 = 0.1, and (b) high pileup case, η 0 = 1.The errors are plotted as a function of the A-vector magnitude for the three lines in Fig.3. The A-vector magnitude is proportional to object thickness as explained in Sec. 2.H. The results for each of the lines are in separate rows with the A-vector components in separate columns. The two sets of training data noise were plotted using different symbols: 1000 trials averaged,♦ (low noise), and added random noise * .

Figure 5 .Figure 6 .

 56 Figure 5. Variance for the (a) low zero object thickness pileup case, η 0 = 0.1, and (b) high pileup case, η 0 = 1, plotted as a function of A-vector magnitude. The symbols are the same as in Fig.4. The data are normalized by dividing by the CRLB variance and a dashed line is drawn at a ratio of one. Notice that the neural network trained with low noise data has variance substantially larger than the networks trained with added noise.

Figure 7 .

 7 Figure 7. Bias (a) and variance (b) of network trained with high noise data with large pileup parameter. The data are normalized by dividing by the CRLB variance and a dashed line is drawn at a ratio of one.

  Lee et al.[START_REF] Lee | Material depth reconstruction method of multi-energy x-ray images using neural network[END_REF] used a neural network estimator but did not examine noise in the output. Zimmerman and Schmidt[START_REF] Zimmerman | Experimental comparison of empirical material decomposition methods for spectral ct[END_REF][START_REF] Zimmerman | Experimental study of two material decomposition methods using multi-bin photon counting detectors[END_REF] studied noise but used noisy training data from a single exposure of the calibration phantom. Touch et al.

addition, x-ray quantum noise is, of course, universal. In order to focus on these fundamental issues an idealized model that includes only quantum noise and pileup is used with a Monte Carlo simulation to test the neural network estimators' performance.

  is the average rate of photon arrivals. The pileup parameter, η = ρτ , is the expected number of photons arriving during a dead time period, τ . The total number of photons recorded by the detector in all PHA bins is N rec and the number recorded in PHA bin k is N rec,k . Notice that with non-zero pileup parameter the non-Poisson factor D is not zero so the counts are not Poisson distributed. The formulas in the table were validated by a Monte Carlo simulation

Table I PHA

 I PROBABILITY DISTRIBUTION. λ IS THE EXPECTED NUMBER OF PHOTONS INCIDENT ON THE DETECTOR DURING THE MEASUREMENT TIME, η IS THE EXPECTED NUMBER OF PHOTONS ARRIVING DURING A DEAD TIME PERIOD, Nrec IS THE TOTAL NUMBER OF PHOTONS RECORDED BY THE DETECTOR IN ALL PHA BINS AND N rec,k IS THE NUMBER RECORDED IN PHA BIN k.


Jérôme Rivière
email: jerome.riviere@univ-reunion.fr

Alexandre Douyere

Jean-Daniel Lan

Sun Luk

Design of a CPW fed circular slot loop antenna for RF/DC rectifier at low power level

This work deals with the design and implementation of a planar and compact antenna realized on Arlon AD1000 substrate, εr=10.35. The antenna consists of a CPW fed circular slot loop antenna with two stubs as impedance matching circuitry. The antenna was designed for 2.45 GHz in order to be implemented with a RF/DC rectifier optimized for low power level in CPW technology. Thus, allowing easy and fast fabrication of a rectenna array with a high reproducibility between one rectenna conception to another one.

I. INTRODUCTION

Recently, energy harvesting has been the center of attention with the growth of wireless communication and development of ideas such as smart cities and the Internet of Things [START_REF] Perera | Context aware computing for the internet of things: A survey[END_REF]. Those inevitable wireless data communication development and growth tend to provide reliable and constant wireless energy. Because electromagnetic waves convey information, they are also an energy vector. The majority of the communicative devices spread their signal in all direction in space. Hence, most of the signal is not used for is communication property, and so energy are wasted in the air. Rectenna are a special kind of antenna that converts electromagnetic waves to direct current. They can be used to harvest this and powered small devices and are generally build using microstrip waveguide technology. But one of the challenges relies on the repeatability of vias needed to connect surface mountain component (SMC) in this technology; as the ground and the conductive plane are on different plane in space separate by a substrate. Robustness and repeatability in rectenna should be as perfect as possible to limit differences in one conception from another. Moreover, many applications require power characteristics that may not be achievable by a single rectenna element. It may, however, be possible that an aggregate of radiating elements in an electrical and geometrical arrangement (an array) will result in the desired power characteristics. But, as the complexity of antenna and frequency number increases, phase shifting and complex matching impedance circuitry are needed. Another solution relies on rectenna array in which the combining of the DC voltage of several rectenna is accomplished to power a device. Then this second solution requires reliable and accurate rectenna conception. This work is dedicated to rectenna array solution. In order to suppress vias in rectenna conception we uses coplanar waveguide technology (CPW). In this technology, the conductive line and the ground plane are etched on the same plane.

The topic of this paper focuses on the design and conception of a planar and compact antenna to be used with a RF/DC rectifier in CPW technology. This antenna is also simple and fast to realize, as few milling tools process is required. It consists of a patch surrounded by a ground conductor on the same plane of the substrate, without background metallization to avoid connection transition with the ungrounded CPW rectifier.

II. DESIGN AND SIMULATION

A. Initial Design: Circular Monopole Antenna

The antenna for this rectenna conception is a modified CPW fed circular slot loop antenna. The study starts with a circular monopole antenna because it is one of the simplest patterns to study [START_REF] Xian | Ultra-Wideband Antenna and design[END_REF]. In this configuration, only two parameters are optimized: Patch radius (R) and distance (d) between patch and ground The antenna feed is performed with a quarter wavelength 50 Ω CPW line. The initial patch radius R was chosen using the equation [START_REF] Perera | Context aware computing for the internet of things: A survey[END_REF].

R= λ g 2h ( 1+ √ π * ( ϵ r +1 ) * λ g * [ 1,7726+ln ( 0,5 λ g * π h )] ) (1)

B. CPW-Fed Circular Slot Loop Antenna

Plotting the current distribution of the patch at 2,45 GHz (Fig. 1.b), we can notice that the current propagate along the edge of the circular patch and is "null" at the top of the circular monopole pattern. Hence, shielding the patch with a ground plane compels the magnetic field to propagate along the newly created slot (Fig. 2.a); thus increasing the current at the edges while maintaining similar resonant frequency.

Following this method and using a quarter wavelength matching feeding line to match the 50 Ω port, the simulated antenna shows resonant frequency at 2.2 GHz. This shift could be explained by the increase of effective dielectric of the circular patch. Indeed, with the surrounding ground the field path is changed and with it the value of the equivalent uniform dielectric, εeff. To correct this shift, a diminution of the patch radius was sufficient. The total antenna dimension was of 45.2*53 mm 2 (Fig. 2.a). The goal of this antenna is to be connected to a RF/DC rectifier in CPW technology. The rectifier is designed with an input impedance characteristic of the CPW line of 50 Ω. In order to connect this antenna to the rectifier, one could add a small line portion of 50 Ω and take in consideration the coupling between the quarter wavelength line and the 50 Ω one [START_REF] Nie | A compact 2,45 GHz broadband rectenna using grounded coplanar waveguide[END_REF].

C. Compact CPW-Fed circular Slot Loop Antenna

To decrease the size of the antenna we use another technique for matching impedance. Two stubs are inserted inside the radiating element to match the antenna impedance at 50 Ω (Fig. 2.b). This solution gives a more compact antenna with a shorter feed line. This method because of the nature of the CPW feeding line can also be seen as a moving of the feed-point closer to the center of the circular patch. Thus, the antenna size is reduced down to 45.2*50,57 mm 2 with gain of 2.34 dBi, directivity of 3.87 dBi and peak current of 29,8 A/m.

III. MEASUREMENT RESULTS

The proposed antenna is realized and measured (Fig. 3). Measurement presents a bandwidth B of 250 MHz and return loss of -23,7 dB at 2,45 GHz. The simulation and measurement are in good agreement with a small frequency drift ΔF of 10 MHz. IV. CONCLUSION In this work a compact CPW-fed circular slot loop antenna was designed and measured. We show that such antenna can be derived from a circular monopole antenna as a starting point with little tuning process. This design is interesting because it has several advantages over the well-known patch antenna; such as, bad impedance matching at first higher order harmonics, larger bandwidth than microstrip patch antenna, bidirectional radiation pattern in opposite direction of space and antenna size reduction.

Subsequently to this work, the simulation of a complete rectenna in CPW technology is performed with a more accurate diode model than our previous work [START_REF] Rivière | DC-pass filter design with notch filters supperposition for CPW rectenna at low power level[END_REF]. The rectifier is composed of a single series diode HSMS7630. The diode was modeled through I/V characteristics and S-parameter fitting between diode model and measurement at Pin=-20 dBm. It shows rectifying efficiency of approximately 9 % at 2.45 GHz for Pin=-20 dBm and a resistive load Rload=5,7 kΩ.

Fig. 1 .

 1 Fig. 1. (a) CPW-fed circular monopole and dimensions (mm) (b) Current propagation of the antenna at 2.45 GHz

Fig. 2 .

 2 Fig. 2. CPW-fed circular slot loop antenna with (a) 1/4 wavelength matching (b) stub matching and dimensions (mm)

Fig. 3 .

 3 Fig. 3. Reflection coefficient of the simulated and realized antenna

ACKNOWLEDGEMENT

This work received financial support from Reunion island Regional Council and Union European -European Regional Develoment Fund (FEDER)