Feriel Khellaf

Nils Krah

Jean Michel Létang

Simon Rit

Projection deconvolution for proton CT using the spatially variant path uncertainty

Keywords: deconvolution, deblurring, spatial resolution, proton CT, direct reconstruction

Proton computed tomography (pCT) suffers from a lower spatial resolution compared to X-ray CT due to the stochastic non-linear proton paths. The most likely path (MLP) formalism provides an estimate for the proton path as well as the uncertainty around this estimate. Using the MLP estimate for the image reconstruction instead of straight integration lines has been shown to improve the spatial resolution of pCT images. In this work, the aim is to further increase the spatial resolution by also including the path uncertainty in the reconstruction algorithm. We proposed a projection-based deconvolution method, applied within the framework of a direct reconstruction algorithm based on distance-driven binning. We used an MLP formalism accounting for tracker resolution in addition to multiple Coulomb scattering. We investigated deconvolution artifacts and proposed a method to mitigate them via spatial regularisation.

Our method was tested on Monte Carlo simulated data, using a water cylinder with aluminium inserts and two slices of an anthropomorphic phantom. Our results showed an improvement of spatial resolution in all cases (up to 29% or 60% for the cylindrical phantom, depending on whether deconvolution artifacts were corrected for or not). Overshoot artifacts were observed in the case of the cylindrical phantom but were less prominent in the case of the anthropomorphic phantom. In conclusion, we have shown that including the path uncertainty in the reconstruction can notably improve the spatial resolution.

I. INTRODUCTION

Proton computed tomography (pCT) has been investigated to complement or replace X-ray CT for proton treatment planning. While it offers a superior relative stopping power (RSP) accuracy compared to X-ray CT, it suffers from a lower spatial resolution. The spatial resolution of pCT is mainly impacted by multiple Coulomb scattering (MCS) of the protons and by the tracker resolution [START_REF] Johnson | Review of medical radiography and tomography with proton beams[END_REF]. In list-mode pCT systems, front and rear trackers measure the position and direction of each proton upstream and downstream from the patient [START_REF] Schulte | Conceptual design of a proton computed tomography system for applications in proton radiation therapy[END_REF]. The most likely path (MLP) for each proton inside the patient, which can be computed using a compact formalism [START_REF] Schulte | A maximum likelihood proton path formalism for application in proton computed tomography[END_REF], is obtained from these measurements and has been shown to improve pCT reconstruction [START_REF] Li | Reconstruction for proton computed tomography by tracing proton trajectories: A Monte Carlo study[END_REF]. This work did not involve human subjects or animals in its research. F. Khellaf was with University of Lyon, INSA-Lyon, Université Claude Bernard Lyon 1, UJM-Saint Etienne, CNRS, Inserm, CREATIS UMR 5220, U1206, F-69373, Lyon, France. She is now with Univ. Grenoble Alpes, CNRS, Grenoble INP, TIMC-IMAG, 38000 Grenoble, France.

N. Krah, J. M. Létang and S. Rit are with University of Lyon, INSA-Lyon, Université Claude Bernard Lyon 1, UJM-Saint Etienne, CNRS, Inserm, CREATIS UMR 5220, U1206, F-69373, Lyon, France.

N. Krah is also with Institut de Physique des Deux Infinis de Lyon, UMR 5822, F-69622, Villeurbanne, France.

The MLP itself has been included in both direct [START_REF] Khellaf | A comparison of direct reconstruction algorithms in proton computed tomography[END_REF] and iterative reconstruction algorithms [START_REF] Censor | Developments in mathematical algorithms and computational tools for proton CT and particle therapy treatment planning[END_REF] to improve spatial resolution of pCT images. However, the uncertainty envelope around the MLP, which can also be calculated by the MLP formalism, has not yet been successfully used for reconstruction. Using the path uncertainty to improve spatial resolution has been mentioned in some works [START_REF] Schulte | A maximum likelihood proton path formalism for application in proton computed tomography[END_REF], [START_REF] Williams | The most likely path of an energetic charged particle through a uniform medium[END_REF]. Wang et al. [START_REF] Wang | On the use of a proton path probability map for proton computed tomography reconstruction a[END_REF] have tried to include this uncertainty in their projection matrix and performed reconstruction using an algebraic reconstruction technique (ART). However, their results did not show an improvement of spatial resolution compared to images reconstructed without using the path uncertainty. Unlike the method used in the aforementioned study, we propose to deconvolve pCT projections as an additional step in a direct reconstruction method, the distance-driven binning algorithm of Rit et al. [START_REF] Rit | Filtered backprojection proton CT reconstruction along most likely paths[END_REF]. The objective is to include the path uncertainty in the reconstruction to improve spatial resolution without compromising RSP accuracy. To estimate the path uncertainty in "realistic" conditions, we used an extended MLP formalism that takes into account the tracker's spatial and angular resolution (due to the strip pitch, scattering inside the tracker, etc.) [START_REF] Krah | A comprehensive theoretical comparison of proton imaging set-ups in terms of spatial resolution[END_REF].

Including the path uncertainty in the reconstruction is similar to resolution modeling in positron emission tomography (PET) imaging [START_REF] Rahmim | Resolution modeling in PET imaging: theory, practice, benefits, and pitfalls[END_REF] where the resolution model is typically included in the system matrix and which has been shown to improve spatial resolution and contrast. It is known, however, that resolution modeling in PET may also lead to deconvolution artifacts, i.e. overshoots and ringing near edges [START_REF] Tong | Properties and mitigation of edge artifacts in PSF-based PET reconstruction[END_REF]- [START_REF] Nuyts | Unconstrained image reconstruction with resolution modelling does not have a unique solution[END_REF], which can cause quantification errors [START_REF] Bai | The effect of edge artifacts on quantification of positron emission tomography[END_REF]. Therefore, we investigated deconvolution artifacts in the context of pCT and proposed methods to control them.

Our deconvolution method was validated via Monte Carlo simulations in two phantoms: a water cylinder with aluminium inserts and an anthropomorphic phantom.

II. MATERIALS AND METHODS

A. Distance-driven projection binning

The measured quantity in pCT is the proton's water equivalent path length (WEPL) which is the integral of the RSP along the proton path. In order to reconstruct pCT images, we first start by binning this list-mode data into projections. We apply the distance-driven binning procedure of Rit et al. [START_REF] Rit | Filtered backprojection proton CT reconstruction along most likely paths[END_REF] to take into account the non-linearity of the proton paths. The main idea is to bin the measured data along several virtual planes k ∈ {1, ..., K} situated between the entrance and exit detectors according to the estimated MLP (Figure 1).

The coordinate system (u, v, w) is such that (u, v) define the detector plane, and w is taken along the main beam direction. The steps to compute the distance-driven projections are (i) to compute each proton's MLP and (ii) assign each proton's WEPL in the corresponding bin. More specifically, the MLP of the i-th proton is sampled at given distances w k , with k ∈ {1, ..., K}, to have u i,k and v i,k , the most likely position of the i-th proton in the k-th plane (Figure 1). The binned value in the j-th pixel of the k-th plane of the projection acquired with the p-th source position is

g j,k,p = i∈Ip ζ j (u i,k , v i,k , w k )WEPL i i∈Ip ζ j (u i,k , v i,k , w k ) (1)
with I p the subset of protons with source position p, ζ j the indicator function for pixel j, and WEPL i the water equivalent path length of proton i. The distance-driven projection can be interpreted as the backprojection of a single view p of the measured WEPL along the MLP. The rest of the reconstruction consists in filtering the sinogram with the usual ramp filter and doing a backprojection [START_REF] Rit | Filtered backprojection proton CT reconstruction along most likely paths[END_REF].

B. Path uncertainty

The proton path and the associated uncertainty can be computed using the MLP formalism [START_REF] Schulte | A maximum likelihood proton path formalism for application in proton computed tomography[END_REF]. The derivation assumes perfectly known entrance and exit positions and angles per proton y in = (u in , θ in ) and y out = (u out , θ out ), measured at distances w in and w out (Figure 1). In practice, the spatial and angular resolution of the trackers will induce an uncertainty on the measured parameters. Krah et al. [START_REF] Krah | A comprehensive theoretical comparison of proton imaging set-ups in terms of spatial resolution[END_REF] have included the impact of detector resolution in the MLP formalism. We briefly recall their formalism and give the formulas used to compute the uncertainty envelope.

First, the joint probability that a proton passes through y 1 , an intermediary vector inside the object, and ỹout given ỹin (tilde distinguishes measured from true vectors) is redefined as

L(y 1 , ỹout |ỹ in ) = L scat (y 1 |y in )L meas (ỹ in |y in ) dy in × L scat (y out |y 1 )L meas (ỹ out |y out ) dy out (2)
where L scat (y 1 |y in ) and L scat (y out |y 1 ) represent the likelihoods that a proton is scattered from y in to y 1 , and from y 1 to y out , respectively; and L meas (ỹ in |y in ) and L meas (ỹ out |y out ) represent the likelihoods of the measured values ỹin and ỹout to be the true parameters y in and y out , respectively. The probabilities L meas are modeled with a Gaussian distribution as it describes well the statistical error in experiments. Solving the integrals in Equation 2 and reshaping into Gaussian functions gives an expression for the MLP and an uncertainty matrix that takes into account measurement uncertainties. The corresponding error matrix is given by [START_REF] Krah | A comprehensive theoretical comparison of proton imaging set-ups in terms of spatial resolution[END_REF]

Σ MLP (w) = C 1 (C 1 + C 2 ) -1 C 2 (3)
with

C 1 = R 0 S in Σ in S T in R T 0 + Σ 1 (4)
C 2 = R -1 1 S -1 out Σ out (S -1 out ) T (R -1 1 ) T + R -1 1 Σ 2 (R -1 1 ) T . (5)
The uncertainty on the path position relative to the MLP σ MLP (w) = Σ MLP (w) 1,1 is given by the element in the first row and first column of Σ MLP . The expressions for rotation matrices R 0 , R 1 and scattering matrices Σ 1 , Σ 2 are the ones used in the usual MLP formalism [START_REF] Schulte | A maximum likelihood proton path formalism for application in proton computed tomography[END_REF]. The matrices S in and S out propagate the measured positions from the detector to the object surface

S in = 1 d entry 0 1 , S out = 1 d exit 0 1 , (6)
with d entry the distance from the front tracker to the entrance position and d exit the distance from the rear tracker to the exit position. The matrices Σ in and Σ out describe uncertainties due to the trackers' finite spatial and angular resolution. They are given by the sum of the uncertainty due to the spatial resolution of the trackers σ t and the error due to scattering inside the trackers

Σ in = σ 2 t T in T T in + Σ sc,in (7)
Σ out = σ 2 t T out T T out + Σ sc,out (8)
where T in and T out are the matrices relating the positions measured by the two detectors in each pair to the position/direction parameters:

T in = 0 1 -1/d T 1/d T , T out = 1 0 -1/d T 1/d T , (9)
with d T the distance between the trackers. Finally, the matrices Σ sc,in and Σ sc,out represent the impact of the scattering inside the trackers on the angular uncertainty

Σ sc,in = 0 0 0 σ 2 sc,in , Σ sc,out = 0 0 0 σ 2 sc,out , (10)
with

σ 2 sc,in = E 2 0 p 2 (E in )v 2 (E in ) x X 0 1 + 0.038 ln x X 0 2 , ( 11
)
σ 2 sc,out = E 2 0 p 2 (E out )v 2 (E out ) x X 0 1 + 0.038 ln x X 0 2 , (12)
where E 0 = 13.6 MeV and 0.038 are empirical constants, x/X 0 represents the material budget, i.e. the ratio between the tracker thickness and the radiation length of the tracker material (silicon), and the term 1/p 2 (E)v 2 (E) is approximated using a 5 th -degree polynomial [START_REF] Schulte | A maximum likelihood proton path formalism for application in proton computed tomography[END_REF]. Equation 3 computes the path uncertainty envelope for each proton. In order to use this information to deblur the binned distance-driven projections, we perform a distancedriven binning of the path uncertainty similarly to Equation 1:

σ j,k,p = i∈Ip ζ j (u i,k , v i,k , w k )σ 2 MLP,i (w k ) i∈Ip ζ j (u i,k , v i,k , w k ) (13)
with σ 2 MLP,i (w k ) the variance of the path of proton i relative to its MLP at distance w k . The value σ j,k,p represents the average standard deviation of the paths of protons emitted from the pth source position passing in the j-th pixel and the k-th plane relative to their MLPs.

C. Spatially variant blurring model

As the blur occurs in the (u, v) plane of the binned projections g j,k,p , we consider each plane k of the distancedriven projections individually. Although the blur in a real pCT projection is the result of vertical and lateral components, we do not consider the blur in the v-direction in our approach. We work with one-line projections of a 3D image, which limits our problem to a set of 1D deconvolutions as modeled below, and leave 2D deconvolution of 2D projections to future work.

If the blur were spatially invariant, the measured projection would be the result of the convolution of an ideal projection g * j,k,p with a shift-invariant convolution kernel. As the blur depends on the position u, which is referred to by the 1D pixel index j, we model the measured projection as the result of a shift-variant operation

g j,k,p = m h j-m,m,k,p g * m,k,p (14)
where h j-m,m,k,p is computed from a Gaussian function due to the Gaussian model used for the path uncertainty

h j-m,m,k,p = 1 √ 2πσ m,k,p exp - (j -m) 2 τ 2 2σ 2 m,k,p , (15)
with τ the projection spacing and σ m,k,p the path uncertainty, both in millimeters. In matrix notation, Equation 14becomes

g k,p = H k,p g * k,p (16)
where g k,p is a vector containing one line of the distancedriven projection, H k,p is the shift-variant system matrix, and g * k,p is a vector containing one line of the projection without blurring. The formalism presented in Section II-B provides an estimate of the path uncertainty due to multiple Coulomb scattering and tracker resolution. Using a phantom with three inserts at different depths, it has been shown that the best resolution for each insert is obtained at the distance corresponding to the depth of the insert [START_REF] Rit | Filtered backprojection proton CT reconstruction along most likely paths[END_REF]. At other distances, the non-linearity of the proton paths causes further blurring. In the final image, the distances contributing to reconstructing the inserts are those corresponding to their depths. Therefore, we neglect this effect and consider that modeling the blurring in the projections using the path uncertainty is sufficient. This assumption will be verified using a phantom with inserts at different depths.

D. Deconvolution method

In practice, due to the presence of noise, the true solution of an inverse problem such as the one in Equation 16, even if well-posed, cannot be found exactly. Approximate solutions can be obtained using, for example, a least squares estimate ĝk,p = arg min

g * k,p ||H k,p g * k,p -g k,p || 2 2 . (17)
However, the unregularized deconvolution problem does not necessarily have a unique solution [START_REF] Nuyts | Unconstrained image reconstruction with resolution modelling does not have a unique solution[END_REF]. In practice, this means it is impossible to recover high frequency components that could not be measured by the system due to its spatial resolution. Any values assigned to these high frequency components produce a different possible solution. Specifically, the overshoot artifacts usually observed in deconvolution problems correspond to the sharp frequency cut-off between the restored frequencies and the high frequencies set to zero. A regularization term is therefore necessary to constrain the problem, which becomes ĝk,p = arg min

g * k,p ||H k,p g * k,p -g k,p || 2 2 + α 2 ||∇g * k,p || 2 2 ( 18
)
with α the regularization parameter. Regularization helps to attenuate overshoot artifacts by imposing a smoother cut-off between low and high frequencies, but it does not completely suppress them, as our results will confirm.

These artifacts causing overshoots near edges have been observed in resolution modeling for emission reconstruction [START_REF] Tong | Properties and mitigation of edge artifacts in PSF-based PET reconstruction[END_REF]- [START_REF] Nuyts | Unconstrained image reconstruction with resolution modelling does not have a unique solution[END_REF]. Note that they persist even with perfect knowledge of the system matrix. A relationship between the amplitude of those artifacts and the imaging matrix which maps the object to the projection space has been established [START_REF] Tong | Properties and mitigation of edge artifacts in PSF-based PET reconstruction[END_REF]. As the point spread function (PSF) gets wider, the matrix rank gets lower (i.e. the problem is more underdetermined) and its condition number higher, which produces worse artifacts. Different methods have been proposed to deal with this problem in the context of PET imaging, e.g. post-smoothing the image or using different regularization procedures. But they are not all directly applicable to our problem since the resolution model is included in the system matrix in PSF-based PET reconstruction, which is not the case in our framework. We have selected one method, tested in different papers [START_REF] Tong | Properties and mitigation of edge artifacts in PSF-based PET reconstruction[END_REF], [START_REF] Stute | Practical considerations for image-based PSF and blobs reconstruction in PET[END_REF], which consists in underestimating the PSF. As the artifacts appear with high uncertainty values, underestimating those values will reduce the overshoots observed on deblurred images, although this will limit the improvement of the spatial resolution. In practice, this uncertainty underestimation can be interpreted as the reconstruction of an image acquired with better detectors and lower MCS rather than ideal detectors and no MCS. We denote our underestimated system matrix by H β k,p corresponding to the following kernel

h j-m,m,k,p = 1 √ 2πβσ m,k,p exp - (j -m) 2 τ 2 2β 2 σ 2 m,k,p (19)
where β ∈ (0, 1] represents the uncertainty underestimation factor. Setting β to 1 corresponds to the use of the exact path uncertainty. Using the scaled system matrix, we take the regularized least squares solution ĝk,p = arg min

g * k,p ||H β k,p g * k,p -g k,p || 2 2 + α 2 ||∇g * k,p || 2 2 . ( 20
)
This problem was solved using a conjugate gradient method, for all planes k and all projections p before reconstruction. The scaling and regularization parameters α and β were chosen to have a good trade-off between spatial resolution and overshoot as described in the next section.

E. Simulations

The first simulations, performed using GATE [START_REF] Jan | Gate v6: a major enhancement of the gate simulation platform enabling modelling of CT and radiotherapy[END_REF] version 9.0 with the QGSP BIC physics list, were used to validate our blurring model. The simulations are similar to the one performed by Rit et al. [START_REF] Rit | Filtered backprojection proton CT reconstruction along most likely paths[END_REF]. In each simulation, we acquired a single projection of a 20 cm wide shell of water with a 5 mm wide bone insert (Figure 2). Five different simulations were acquired for five positions of the insert (one insert per simulation). Since a fan beam was used, we made the water shell and the inserts truncated hollow spheres with the same origin as the beam, such that the ideal projection (if no MCS was involved) would be a rectangular function. The internal and external radii were 100±10 cm for the water shell, and 92±0.25, 96±0.25, 100±0.25, 104±0.25, and 108±0.25 for the inserts. The inserts were truncated using an angular span of 2 degrees. The beam energy was 200 MeV and the fluence at the isocenter was about 2 × 10 5 protons•mm -2 . The spatial resolution in the projection, as a function of depth, was compared to the predicted resolution using the path uncertainty. The spatial resolution was measured by fitting an error function to the edges of the projection of the inserts, similarly to [START_REF] Khellaf | A comparison of direct reconstruction algorithms in proton computed tomography[END_REF]. More specifically, for each insert, and for each distance w in the projection of the insert, the following function

A 2 1 + erf x -µ σ √ 2 + B ( 21
)
was fitted to the projection data as illustrated in Figure 5a. The fitted parameter σ was used as a measure for the spatial resolution.

Other simulations were made to evaluate the capability of the proposed deconvolution method to improve spatial resolution. We used the same simulation for the spiral phantom as the one described in a previous study [START_REF] Rit | Filtered backprojection proton CT reconstruction along most likely paths[END_REF], except that the radius of the aluminium beads was increased from 5 to 10 mm. A 200 MeV fan beam with a fluence of 225 protons•mm -2 •projection -1 at the isocenter was simulated. The spiral phantom was also used to choose the scaling and regularization parameters α and β. In addition, we simulated an acquisition using the ICRP female phantom [START_REF]ICRP, Adult Reference Computational Phantoms, ser[END_REF]. A uniform 250 MeV beam energy was used to traverse the phantom volume everywhere (note that a variable energy could have been used depending on the scanned slice), with a flux of 190 protons•mm -2 •projection -1 . In all our simulations, a high number of protons was simulated in order to have low noise and thus a good measure of spatial resolution. A total of 720 projections over 360°were simulated. Two slices, one through the posterior fossa and the other through the heart, were reconstructed (Figure 3). To compute the MLP and the path uncertainty, the convex hull of the spiral phantom was known and that of the head and lung slices of the ICRP phantom was approximated by an ellipse for fast detection of the intersection between the patient and the proton path before and after the patient. The sampling for the distance-driven projections was 0.625 × 0.625 mm 2 for all phantoms. The spiral phantom was reconstructed on a grid of 0.5 × 0.5 mm 2 pixels, and the ICRP phantoms on a grid of 1.775×1.775 mm 2 pixels -the same sampling as the reference phantoms.

The GATE simulation was performed using an idealized setup, i.e. with error-less position and angle measurements. Tracker uncertainties were included by postprocessing the output of the simulation. Specifically, for each proton, a Gaussian error with a covariance given by Equations 7 and 8 is added to the ideal entrance/exit parameters. To compute the covariance matrices, we used a spatial resolution of σ t = 0.066 mm corresponding to a strip pitch of 228 µm, a material budget of x/X 0 = 5 × 10 -3 and a distance d T = 10 cm between the trackers in each pair. These values meet or exceed the minimum specifications for pCT scanner prototypes as detailed in [START_REF] Bashkirov | Development of proton computed tomography detectors for applications in hadron therapy[END_REF]. The closest trackers of each pair were placed at a distance of 40 cm from the isocenter, which corresponds to distances used in commercial X-ray CT scanners [START_REF] Krah | A comprehensive theoretical comparison of proton imaging set-ups in terms of spatial resolution[END_REF]. Images were reconstructed from data without and with tracker uncertainties using the standard MLP formalism [START_REF] Schulte | A maximum likelihood proton path formalism for application in proton computed tomography[END_REF] and the extended one [START_REF] Krah | A comprehensive theoretical comparison of proton imaging set-ups in terms of spatial resolution[END_REF], respectively, to compute the proton path and the uncertainty maps. Spatial resolution was calculated in the spiral phantom corresponding to an MTF value of 10% using the approximate method described in [START_REF] Khellaf | A comparison of direct reconstruction algorithms in proton computed tomography[END_REF] based on [START_REF] Richard | Towards task-based assessment of CT performance: system and object MTF across different reconstruction algorithms[END_REF]. The function given in Equation 21 was fitted to each bead's radial profile (see Figure 4 for an example of a radial profile), and the MTF10% was derived as

MTF10% = ln 10 2 1 πσ . (22)
In order to choose the scaling and regularization parameters, the spiral phantom was reconstructed using values of β ranging from 0.5 to 1 at intervals of 0.1, and values of α ranging from 0.05 to 0.9 at intervals of 0.05. The average spatial resolution and average overshoot over all beads were measured for each reconstruction, and the pair of parameters giving the best resolution/overshoot trade-off was chosen and applied to the anthropomorphic phantoms. The overshoot in each bead was measured by taking the radial profile (as for the spatial resolution), and computing the relative difference between the maximum value and the true RSP value of the beads (RSP = 2.1), after post-smoothing the profile with a Gaussian kernel of standard deviation 0.1 mm and width 8 mm (equal to the size of the profile) to reduce noise (Figure 4). Negative overshoot values occurred when there were no oscillations and the edges remained blurred due to loss of spatial resolution.

III. RESULTS

A. Resolution modeling and uncertainty maps

Figure 5b shows the spatial resolution measured in the five distance-driven projections of the inserts positioned as depicted in Figure 2, acquired using either ideal or realistic trackers. The spatial resolution was measured as explained in Section II-E, i.e. by fitting Equation 21 to the edge spread function of each insert and for each distance w. The theoretical path uncertainty shows a good agreement with the spatial resolution measured in the projections at the depths corresponding to the insert positions. At other depths, the path uncertainty does not reflect the spatial resolution due to the non-linearity of the proton path, as discussed in Section II-C. The dashed line shows the effect of pixel size (0.5 mm) on the overall uncertainty. The added uncertainty was taken into account by computing the quadratic sum σ 2 total = σ 2 MLP + σ 2 pixel , with σ pixel = 0.5/ √ 12 and σ MLP as defined in Equation 3. It is overall non negligible close to the entrance and exit of the phantom in the case of ideal trackers, while it is negligible in the case of realistic trackers because the path uncertainty dominates the loss of spatial resolution.

Figure 6 shows path uncertainty maps for the spiral phantom, with and without tracker uncertainties. For the case without tracker uncertainty, there is no variability as a function of the projection angle due to the fact that the object is circular. The uncertainty due to MCS reaches a maximum of about 0.5 mm approximately at the center of the object. The maximum is not exactly at the central depth due to the energy loss term in the computation of the scattering matrices Σ 1 , Σ 2 . It is slightly shifted towards the exit of the proton path. Adding tracker uncertainties increases the maximum uncertainty value to about 1.4 mm and shifts it towards the exit. We further observe high uncertainty streaks near the exit of the path of the protons, corresponding to protons traversing aluminium inserts and losing more energy than those traversing fewer or no inserts. Those lower energy protons scatter more in the rear tracker (see energy term in Equation 12), hence a higher uncertainty.

Figure 7 shows uncertainty maps for the lung and head phantoms for two different source positions, with and without tracker resolution. There is a larger variability of the uncertainty values between different projections compared to the maps of the spiral phantom. This is due to the elliptical shapes of the convex hulls used to approximate the phantom surface. This inter-projection variability is more pronounced for the lung phantom, with a maximum path uncertainty in the first projection (traversing the left-right axis of the patient) of 1.7/1.8 mm with/without tracker resolution, and a maximum for the second projection (traversing the anteroposterior axis of the patient) of 0.8/1 mm. Spatial variability due to RSP heterogeneities is visible in both phantoms when using realistic trackers. For example, in the lung phantom, trajectories traversing the lung tissues are associated with a lower uncertainty (visible near the exit of the phantom).

B. Spiral phantom and choice of scaling and regularization parameters

Figure 8 shows the quantification of the overshoot (see Section II-E) as a function of spatial resolution for different scaling and regularization parameters. Each data point represents the average overshoot and spatial resolution (averaged over all inserts) in the spiral phantom reconstructed with a different pair of (α, β) parameters. The goal was to find the pair (α, β) which yields about zero overshoot and the highest possible resolution. In terms of the figures, we seek the rightmost data point on the dotted line corresponding to an overshoot of zero. With ideal trackers, such a point can be found with β = 1, i.e. without underestimation of the uncertainty. With realistic trackers, setting β = 0.7 gave the highest resolution for an overshoot very close to zero (given the tested β values, the actual optimal parameter is somewhere between 0.7 and 0.8).

Representative reconstructions of the spiral phantom, used to plot Figure 8, are shown in Figure 9. The deblurred results with (β = 0.7) and without (β = 1) uncertainty underestimation are shown for a regularization of α = 0.2. Overshoots appear as ring-like structures, as in the lower central panel. The reconstructions are consistent with the measured overshoots in Figure 8: no artifact is visible in the images acquired with ideal trackers, while the reconstruction from realistic trackers without underestimation shows an overshoot which is corrected when using β = 0.7. The corresponding spatial resolutions are plotted in Figure 10 as a function of distance from the phantom center. For ideal trackers, there is an improvement of the spatial resolution for the central beads, up to 28% without overshoot correction and up to 11% with overshoot correction. For beads that are far away from the center, the spatial resolution is reduced due to the regularization. For realistic trackers, the spatial resolution is improved for all beads. The gain of resolution is up to 29% with overshoot correction and up to 60% without correction.

C. Anthropomorphic phantoms

The reconstructions for the head and lung phantoms using realistic trackers are shown in Figure 11. The difference maps indicate the improvement in spatial resolution along anatomical edges. Overall, the deblurred reconstructions show a better spatial resolution. For the head phantom, the error is more reduced without underestimation of the uncertainty (β = 1); while for the lung phantom, the error near anatomical edges is smaller with a scaling factor β = 0.7. This is reflected by the root mean square error (RMSE) measured inside the zoomed regions-of-interest. For reconstructions using ideal trackers, no improvement was seen.

IV. DISCUSSION

The purpose of this work was to investigate the use of the path uncertainty to improve spatial resolution in proton CT images. First, we verified that the path uncertainty is an accurate measure for the spatial resolution in the projections at the distance corresponding to the position of the edge in the object (Figure 5). Second, we generated distance-driven uncertainty maps corresponding to the pCT projections. As the estimated path uncertainty depends on the shape of the object to be imaged and on the heterogeneities inside the object (in the case of realistic trackers), the values in the maps vary as a function of the projection angle, the depth and the transverse position. Specifically, the less circular the object, the more the uncertainty values vary between different projections. For example, there was about a factor 2 between the maximum uncertainty for protons traversing the large and short axes of the lung phantom (Figure 7). Including tracker resolution in the MLP model increased the path uncertainty compared with ideal trackers, where only MCS in the object is relevant. Although both the standard and the extended MLP formalism assume a homogeneous medium, the impact of heterogeneities inside the phantom on the path uncertainty was partially taken into account through the energy term used for the computation of the scattering inside the rear detector (p 2 (E out )v 2 (E out ) term in Equation 12). For example, protons that traversed aluminium inserts in the spiral phantom had lower exit energies and scattered more in the rear tracker than protons that traversed no inserts, and therefore led to higher MLP uncertainties along projection lines crossing the inserts (Figure 6). Conversely, protons that traversed air in the lung phantom led to lower MLP uncertainties.

Using the computed uncertainty maps to deconvolve the pCT projections, we observed overshoots near edges in the deblurred reconstructions. Those are common side effects to deconvolution [START_REF] Tong | Properties and mitigation of edge artifacts in PSF-based PET reconstruction[END_REF]- [START_REF] Nuyts | Unconstrained image reconstruction with resolution modelling does not have a unique solution[END_REF]. The RSP accuracy of the deblurred image can be compromised by such artifacts: in order to limit this effect to an acceptable level, stringent requirements must be set (our objective is to have an average overshoot close to zero). We studied the effect of overshoots in the spiral phantom (Figure 8). In the ideal case, as the uncertainty values are smaller, the average overshoot was small (maximum measured average overshoot of less than 0.5% in the spiral phantom). However, with realistic trackers, the average overshoot could exceed 3% and was not effectively suppressed by the regularization term alone. Indeed, using either the regularization or the β parameter alone was not sufficient to reduce the average overshoot to zero: in Figure 8, the line corresponding to β = 1 is well above the zero average overshoot line regardless of the regularization; similarly, very small regularizations (α = 0.05) led to high average overshoots regardless of the β parameter. In addition, using both strategies allows for a better tradeoff between overshoot and spatial resolution. Increasing the regularization factor α alone managed to reduce the average overshoot, but at the cost of a considerable loss of spatial resolution: for example, without underestimation, reducing the average overshoot below 2% required a regularization of at least α = 0.7 which led to a resolution of 0.37 lp/mm. In contrast, underestimating the uncertainty by 10% (β = 0.9) managed to reduce the average overshoot well below 2% with α = 0.1 and a resolution of 0.58 lp/mm. Therefore, combining both methods was necessary to produce an overshoot-free image with a high spatial resolution. In this work, we chose the β value which corresponded to an average overshoot of zero and a high resolution, as our objective was to improve spatial resolution without compromising the image quality by introducing overshoot artifacts. However, depending on the specific application, if a certain amount of artifacts is acceptable in the image, further improvement of the spatial resolution is possible by using a larger β parameter. Indeed, overshoots are object dependent: regularly shaped objects such as cylinders lead to characteristic artifacts, e.g. rings; while in less regularly structured objects, overshoot artifacts might appear more like noise.

The reconstructed images of the spiral phantom using realistic trackers showed an increase of spatial resolution between 22% and 29% when an underestimation of the uncertainty was included and between 49% and 60% without underestimation (Figure 10). The gain of spatial resolution was largest at the center of the object where spatial resolution is the lowest. The results with ideal trackers for the spiral phantom showed an improvement of spatial resolution only at the center of the object. At the surface, the resolution was not enhanced as the path uncertainty was already close to zero. In fact, the spatial resolution was even slightly reduced, depending on the value of the regularization parameter α. In this work, α was empirically chosen based on average measurements over different locations in the phantom. In principle, α could be spatially variant as less regularization is needed at the surface. In practice, i.e. with realistic trackers, path uncertainty within the object is more homogeneous than with ideal trackers, and we deem global parameters sufficient in that case.

For the anthropomorphic phantoms, differences between the reconstructions and the reference were observed mainly along edges (Figure 11), and were caused by a lack of resolution or the presence of overshoot artifacts. For a rising edge, we have an overestimation followed by an underestimation if the spatial resolution is not optimal, and the opposite (underestimation followed by an overestimation) in the case of an overshoot due to deconvolution (Figure 4). A small RMSE represents an adequate trade-off between spatial resolution and overshoot.

The reconstructions of the ICRP phantom took into account tracker resolution to provide a realistic scenario. Our results showed a visible improvement of spatial resolution, with a reduced RMSE near the edges in all cases. In the case of the lung phantom, when no underestimation of the uncertainty was applied, the gain in spatial resolution was countered by an error due to overshoot artifacts (e.g. near the spine in Figure 11). With the underestimated convolution kernel, no overshoot was visible and the error due to spatial resolution was reduced everywhere. In the case of the head phantom, the RMSE was reduced for both deconvolutions, i.e. with β = 1 and β = 0.7, compared to the non-deblurred image, but the results were better for the case without underestimation (β = 1). We note that the scaling parameter β was chosen based on the results for the spiral phantom and applied to both anthropomorphic phantoms. Given that the path uncertainty values in the head were smaller than in the spiral phantom (maximum uncertainty of 0.8 mm in the head against 1.5 mm in the spiral), and that overshoot artifacts appear for high uncertainty values, it is likely that indeed no overshoot correction was needed for the head phantom. The maximum uncertainty for the lung phantom was 1.8 mm, which is closer to that of the spiral phantom. On the other hand, given that the path uncertainty was computed assuming homogeneous water, and that protons traversed a certain amount of air in the lung phantom, the uncertainty in the lung might be overestimated. In this case, underestimating the deconvolution kernel might correct for this overestimation, and explain why the results for the lung phantom with β = 0.7 were better. A different MLP formalism taking into account heterogeneities would help to compute more accurate uncertainty maps. The methods proposed to compute a heterogeneous MLP [START_REF] Collins-Fekete | Extension of the Fermi-Eyges most-likely path in heterogeneous medium with prior knowledge information[END_REF], [START_REF] Brooke | An inhomogeneous most likely path formalism for proton computed tomography[END_REF] require knowledge about the tissue composition which can be obtained by iteratively updating the tissue composition from the current image in the case of an iterative reconstruction method. Alternatively, using an energy-adaptive parametrization of the MLP [START_REF] Krah | Energyadaptive calculation of the most likely path in proton CT[END_REF] might improve the path uncertainty estimation by providing a better estimate of the 1/p 2 (E)v 2 (E) term, which is currently approximated with a polynomial fitted to energy loss simulations in water. Still, in both slices and for all tested pairs (α, β), the measured RMSE indicates that the potential bias induced by deconvolution is still below the error due to low spatial resolution. As mentioned earlier, overshoot artifacts are object-dependent. Ultimately, in human-like geometries, they are expected to be less structured and therefore more benign than in the spiral phantom, as our results seem to indicate.

In this work, we have chosen to perform the deconvolution in the projection space to minutely take into account the spatial variation of the path uncertainty. However, it could be possible to do the deconvolution in the image space after reconstruction, especially in the case of realistic trackers for which the spatial variation of the uncertainty values is not as high as for ideal trackers. In an image domain deconvolution, we would need to approximate the uncertainty value in each pixel of the reconstructed image via a backprojection of the uncertainty maps, and then solve the same type of system as in Equation 20, only in two dimensions for a single slice. For one reconstructed image slice, one would perform one two-dimensional deconvolution as opposed to separate one-dimensional deconvolutions per projection, as in the current work. We expect that, in objects with little variation of the path uncertainty among projections, image-based deconvolution might provide similar results as projection-based deconvolution.

This work was conducted using a distance-driven projection binning algorithm, however, it can also be used with other direct methods [START_REF] Poludniowski | Proton computed tomography reconstruction using a backprojection-then-filtering approach[END_REF]- [START_REF] Khellaf | 2D directional ramp filter[END_REF] by deconvolving projections or partial backprojections. While other factors such as the reconstruction algorithm might affect spatial resolution, their impact is limited compared to MCS and tracker resolution [START_REF] Khellaf | A comparison of direct reconstruction algorithms in proton computed tomography[END_REF]. Direct application of our deconvolution method to iterative reconstruction methods is not straightforward because it is projection-based. The authors of a previous study [START_REF] Wang | On the use of a proton path probability map for proton computed tomography reconstruction a[END_REF] included path uncertainty in their projection model to iteratively reconstruct a pCT image. Surprisingly, their results did not show any improvement of spatial resolution. The projection deconvolution method described in this work could also be used to reconstruct a better starting iterate for an iterative reconstruction algorithm including path uncertainty. Alternatively, an image-based deconvolution as discussed above could be done independently of the reconstruction method.

V. CONCLUSION

The purpose of this work was to investigate whether including path uncertainty in a deconvolution framework might improve spatial resolution in direct pCT reconstruction. It was shown that including path uncertainty to perform deconvolution of blurred distance-driven projections can significantly increase spatial resolution of pCT images when using realistic trackers. Particularly, the resolution in a water cylinder of 20 cm diameter with aluminium inserts was increased by up to 29% without introducing overshoot artifacts. Reconstructions of anthropomorphic phantoms also showed an enhancement of spatial resolution when tracker resolution was included in the MLP model. Deconvolution artifacts were less prominent for anthropomorphic images, and our results indicate that their impact on RSP accuracy might be lower than that of a poor spatial resolution.

Fig. 1 .

 1 Fig. 1. Drawing of the coordinate system used for the distance-driven binning, for details see text.

Fig. 2 .

 2 Fig. 2. Setup of the five simulations with a bone insert in water. Only one insert was modeled per simulation. The position of the insert for simulations 1 to 5 is shown. The inserts and the water shell have the same curvature.

Fig. 3 .

 3 Fig. 3. Reference RSP maps of the reconstructed slices. The dotted yellow lines represent the hulls used to compute the MLP.

Fig. 4 .

 4 Fig. 4. Example profiles of inserts in the spiral phantom used to measure the overshoot. The top profile is from a deblurred image without underestimation, and the bottom one from a non-deblurred image, hence the negative "overshoot".

Fig. 5 .

 5 Fig. 5. (a) Distance-driven projection for insert position 3 (see Figure 2). One line of the projection (indicated by the blue vertical line) is plotted against the absolute position |u|, and the erf fit is used to compute the spatial resolution σ. The fit parameters of Equation 21 are given in the text box. (b) Spatial resolution as a function of depth measured on the projections corresponding to each insert position, compared with the path uncertainty, for ideal trackers (top) and realistic trackers (bottom). The vertical bars indicate the depth of the insert in colors corresponding to the curves.

Fig. 6 .

 6 Fig. 6. Uncertainty maps of the spiral phantom introduced by an ideal tracker (top) or a realistic tracker (bottom) for two different source positions (the beam directions are indicated by red arrows). The phantom geometry is depicted with low opacity in a different color map for better visualization.

Fig. 7 .

 7 Fig. 7. Uncertainty maps without tracker resolution (first and third rows) and with tracker resolution (second and fourth rows) for two different source positions (the beams directions are indicated by red arrows). The ellipse used to approximate the phantom surface is shown with a yellow dashed line.

Fig. 8 .

 8 Fig. 8. Overshoot vs spatial resolution in the spiral phantom for ideal trackers (top) and realistic trackers (bottom). Each curve is computed for a different β parameter. The regularization parameter α, corresponding to the different data points for each curve, ranges between 0.05 and 0.9 in steps of 0.05 (low values of α lead to high resolution and vice versa as shown by the two circled data points). The arrows point to the tested (α, β) parameters giving the highest spatial resolution for an overshoot close to zero.

Fig. 9 .

 9 Fig. 9. pCT reconstruction of the spiral phantom: without deconvolution (left), with deconvolution without uncertainty underestimation (middle), and with underestimation (right), for ideal trackers (top) and realistic trackers (bottom). The zoomed inset represents the central bead. The range of the grayscale of the image is [0 -2.2], and that of the inset is [0.9 -2.2].

Fig. 10 .

 10 Fig. 10. Spatial resolution in the spiral phantom as a function of the distance from the center. The results without overshoot correction correspond to β = 1 and α = 0.2, and those with overshoot correction to β = 0.7 and α = 0.2.

Fig. 11 .

 11 Fig. 11. Reconstruction of the head and lung slices using realistic trackers, and difference maps with the reference RSP images. The root mean square error (RMSE) is calculated inside the zoomed regions-of-interest.

ACKNOWLEDGMENTS

The work of Feriel Khellaf and Nils Krah was partially supported by the project DIC20161236452 of the Fondation pour la Recherche Médicale (FRM). This work was performed within the framework of the SIRIC LYriCAN

INCa INSERM DGOS 12563 and the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). All authors declare that they have no known conflicts of interest in terms of competing financial interests or personal relationships that could have an influence or are relevant to the work reported in this paper.


Karthika Sankar

R Baer

§,^,ǁ, ∇ Chloé Grazon

Robert C Sabatelle

Sébastien Lecommandoux

Cathe- Rine M Klapperich

James E Galagan

Mark W Grinstaff

An Allosteric Transcription Factor-DNA Binding Electrochemical Biosensor for Progesterone

Keywords: Transcription factor, progesterone, impedance, DNA, electrochemical biosensor, ferrocyanide/ferricyanide, detection

We describe an electrochemical strategy to transduce allosteric transcription factor (aTF) binding affinity to sense steroid hormones. Our approach utilizes square wave voltammetry (SWV) to monitor changes in current output as a progesterone (PRG) specific aTF (SRTF1) unbinds from the cognate DNA sequence in the presence of PRG. The sensor detects PRG in artificial urine samples with sufficient sensitivity suitable for clinical applications. Our results highlight the capability of using aTFs as the biorecognition elements to develop electrochemical point-of-care biosensors for detection of small molecule biomarkers and analytes.

Biosensors play a pivotal role in medicine as well as the food, environmental, and agricultural industries, and their potential to positively impact society is increasing as our world becomes more digital. [START_REF] Hasan | Recent Advances in Application of Biosensors in Tissue Engineering[END_REF][START_REF] Rogers | Biosensors for Environmental Applications[END_REF][START_REF] Patel | Biosensors in Health Care: The Milestones Achieved in Their Development towards Lab-on-Chip-Analysis[END_REF] Analytes for detection widely vary based on application, and our particular interest is in hormones. Progesterone (PRG) is a steroid hormone secreted by the corpus luteum and is central to fertility planning, assisted reproductive technologies, and endocrine disorders. Among women, one million in the US suffer from infertility, approximately 7.5 million suffer from reduced fertility, and about 15 million miscarry, with similar numbers in the EU. In developing countries, the statistics are even higher and the rates of infertility approach 30%, with only half of the people in developed and developing countries seeking infertility care. [START_REF] Inhorn | Infertility around the Globe: New Thinking on Gender, Reproductive Technologies and Global Movements in the 21st Century[END_REF] As simple as tracking PRG levels during an ovulation cycle increases fertility odds. Furthermore, during pregnancy, PRG balance helps nurture and develop the fetus. Determining ovulation windows is challenging due to fluctuating menstrual cycles and inaccurate interpretations of hormone levels. Currently PRG is measured from serum via blood collection at the clinic/hospital or at home, followed by analysis at a commercial laboratory with results provided in a few days to a week. Today, at-home urine tests measure pregnanediol glucuronide (PDG), a metabolic product of PRG. However, PDG levels are more variable than PRG levels, and up to 12% of women do not metabolize PRG sufficiently to produce detectable PDG. [START_REF] Ecochard | Use of Urinary Pregnanediol 3-Glucuronide to Confirm Ovulation[END_REF] A rapid, facile at-home fertility planning and monitoring point-of-care (POC) device will alleviate the apprehensiveness associated with routine screenings and give couples the privacy desired during fertility planning. With a mindset towards this goal and recognizing that conventional antibody 6-10 -or aptamer [START_REF] Zhu | An Antibody-Aptamer Sandwich Cathodic Photoelectrochemical Biosensor for the Detection of Progesterone[END_REF][START_REF] Samie | Label-Free Electrochemical Aptasensor for Progesterone Detection in Biological Fluids[END_REF][START_REF] Alhadrami | High Affinity Truncated DNA Aptamers for the Development of Fluorescence Based Progesterone Biosensors[END_REF][START_REF] Velayudham | Hydrogel-Aptamer Nanocomposite Based Electrochemical Sensor for the Detection of Progesterone[END_REF][START_REF] Contreras Jiménez | Aptamer-Based Label-Free Impedimetric Biosensor for Detection of Progesterone[END_REF] -based sensors (summarized in Table S1) have not yet provided a solution tailored towards POC approach, we are evaluating transcription factors as the biorecognition elements for sensors. Recently our group success-fully mined the microbial microverse for a bacterial allosteric transcription factor (aTF), SRTF1, responsive to PRG. [START_REF] Grazon | A Progesterone Biosensor Derived from Microbial Screening[END_REF] Bacteria have evolved over 3 billion years to respond to virtually all classes of stimuli, including chemical analytes of interest and represent an almost limitless database to search for unique aTFs. Like antibodies, aTFs recognize analytes with high sensitivity and specificity. In contrast to antibodies, aTFs respond to analyte binding by changing their affinity for a cognate DNA sequence. This change in DNA binding affinity provides an intrinsic transduction mechanism that we capture, electrochemically, for the development of a novel aTF based PRG biosensor. Specifically, our biosensor comprises a gold electrode with a surface immobilized DNA-aTF complex (Figure 1). In the absence of analyte (e.g., progesterone), the overall negative charge of the DNA-aTF complex blocks the negatively charged mediator (Fe(CN)6 3-/4-) approaching the gold electrode and affords a low current output. Addition of the analyte binds the aTF, releasing it from the DNA strand on the surface and decreasing the barrier for the mediator to reach the electrode, thereby resulting in an amplified current output. As binding of analyte to the DNA-aTF complex is a dose-dependent reaction, the difference in the current output before and after addition of analyte correlates to analyte concentration. Herein, we demonstrate the above biosensor concept with a working prototype. We further explore the effect of increasing the number of aTF binding pockets on DNA to improve sensor output. First, we performed bio-layer interferometry (BLI) experiments using synthetic DNAs containing SRTF1 binding site(s), SRTF1, and PRG (Table S2). SRTF1 rapidly binds to the DNA, conjugated to the probe via streptavidin-biotin interactions, and we observe an increase in the BLI binding signal that corresponds to an increase in the thickness from the sensing tip. (Figure 2a). The DNA strand possessing two or three SRTF1 binding sites (DNA 2 and DNA [START_REF] Patel | Biosensors in Health Care: The Milestones Achieved in Their Development towards Lab-on-Chip-Analysis[END_REF] , respectively) affords a larger BLI signal in the presence of SRTF1, consistent with greater binding signal with no significant change in the binding constant (KD ~ 4 nM) between DNA and SRTF1 (Figure 2b, S1; Table S3). The scrambled DNA, lacking a binding site for SRTF1, shows no change in BLI signal and highlights the importance of aTF recognition and specificity. In the presence of PRG, SRTF1 rapidly dissociates from the DNA-SRTF1 complex in comparison to the buffer control where no significant change in BLI signal occurs. Varying the concentration of PRG yields a dose-dependent BLI signal (Figure 2c, S1, and S2). Together these data confirm: i) SRTF1 binds to the cognate DNA sequence in the absence of PRG; ii) SRTF1 allosterically unbinds from the DNA in presence of PRG; and, iii) adding an additional SRTF1 binding site to the DNA increases the amount of aTF bound. This is crucial for the electrochemical sensor design as an increase in amount of aTF will result in higher impedance changes. The electrochemical biosensor utilizes ferrocyanide/ferricyanide as the mobile electrochemically active mediator, and, thereby, avoids the requirement to covalent conjugate a redox probe to the protein or DNA. The biosensor consists of three electrodes: gold working (diameter 1.6 mm) and counter electrodes along with a Ag/AgCl reference electrode. Following the electrode cleaning by cyclic voltammetry in 0.5 M H2SO4 (Figure S3), we immobilized the 5'-thiol modified DNA and 1 mM mercapto-6-hexanol, to prevent non-specific interactions, on the working electrode following published procedures. [START_REF] Steel | Electrostatic Interactions of Redox Cations with Surface-Immobilized and Solution DNA[END_REF][START_REF] Steel | Electrochemical Quantitation of DNA Immobilized on Gold[END_REF] The DNA surface coverage is ≈3*10 -11 mol/cm 2 (Figure S4 and S5), as determined by cyclic voltammetry measurements with tris-ruthenium hexamine, and the value is in agreement with previous reports. [START_REF] Ecochard | Use of Urinary Pregnanediol 3-Glucuronide to Confirm Ovulation[END_REF] Electrochemical and AFM studies support the DNA immobilization and SRTF1 binding. Upon immobilization of the DNA and protein to the electrode, the resistance increases as does the height from the electrode surface from ~700 pm to 8 nm (Figure S6, S7, and S8). DNA derivatized electrodes are widely used in biosensors with elegant strategies employing DNA-mediated charge transport, [START_REF] Yu | Voltammetric Procedure for Examining DNA-Modified Surfaces: Quantitation, Cationic Binding Activity, and Electron-Transfer Kinetics[END_REF][START_REF] Genereux | DNA-Mediated Charge Transport in Redox Sensing and Signaling[END_REF] conformational changes [START_REF] Zhang | Applications and Recent Developments of Multi-Analyte Simultaneous Analysis by Enzyme-Linked Immunosorbent Assays[END_REF][START_REF] Baker | An Electronic, Aptamer-Based Small-Molecule Sensor for the Rapid, Label-Free Detection of Cocaine in Adulterated Samples and Biological Fluids[END_REF][START_REF] Immoos | DNA-PEG-DNA Triblock Macromolecules for Reagentless DNA Detection[END_REF][START_REF] Immoos | Conformationally Gated Electrochemical Gene Detection[END_REF] during binding events, and electrocatalysis events by covalent and non-covalent association of redox mediators [START_REF] Steel | Electrochemical Quantitation of DNA Immobilized on Gold[END_REF][START_REF] Kang | Comparing the Properties of Electrochemical-Based DNA Sensors Employing Different Redox Tags[END_REF] . These strategies are used to detect target molecules [START_REF] Drummond | Electrochemical DNA Sensors[END_REF][START_REF] Ikebukuro | Novel Electrochemical Sensor System for Protein Using the Aptamers in Sandwich Manner[END_REF][START_REF] Cash | An Electrochemical Sensor for the Detection of Protein-Small Molecule Interactions Directly in Serum and Other Complex Matrices[END_REF] , hybridization events [START_REF] Millan | Sequence-Selective Biosensor for DNA Based on Electroactive Hybridization Indicators[END_REF][START_REF] Campuzano | Ternary Monolayers as DNA Recognition Interfaces for Direct and Sensitive Electrochemical Detection in Untreated Clinical Samples[END_REF] , single base mismatches 31 as well as for catalytic oxidation of guanine. [START_REF] Armistead | Electrochemical Detection of Gene Expression in Tumor Samples: Overexpression of Rak Nuclear Tyrosine Kinase[END_REF] Following the DNA immobilization, we measured the current by square wave voltammetry (0-0.6 V, step size: 4 mV, amplitude: 50 mV, frequency: 50 Hz) in the absence and presence of aTF. The current is greatest for the DNA modified sensor and decreases upon addition and formation of the DNA-SRTF1 complex (Figure 3b). Upon addition of PRG, the current increases and returns to its DNA baseline. Use of the scrambled/non-specific DNA results in no change in the current in the presence of SRTF1 or PRG (Figure 3b). Similarly, use of DNA with a non-specific aTF (HucR, a uric acid transcription factor), which does not recognize the DNA binding site, affords no change in the current in the presence of this aTF or PRG. Further, in the absence of SRTF1, no significant change in current is observed on addition of PRG to DNA functionalized electrodes (Figure S9). Increasing the number of SRTF1 binding sites on the DNA from one to two and three increases the magnitude of the current change upon addition of SRTF1. The biosensors with DNA 2 and DNA 3 exhibit a 20% and 35% decrease in current upon aTF complexation, respectively. This result is consistent with greater protein surface coverage impeding ferro/ferricyanide mobility to the electrode surface for electron transfer, and a lower background signal. However, only in the DNA 2 system, does the current return to a baseline level in the presence of PRG. We suspect that the larger DNA 3 -SRTF1 complex on the surface inhibits PRG binding, and thus this reaction does not readily go to completion. Next, we evaluated the analytical response of the system towards PRG by quantifying the current changes of the biosensor system in the presence of varying concentrations of PRG (0-10 µM) in Tris buffered saline, pH 7.4 and purchased artificial urine spiked with 2 µM BSA to better mimic the physiological conditions. We plotted the peak currents before and after addition of PRG. The difference in peak currents positively correlates with the PRG concentration. The limit of detection for the DNA 1 -SRTF1 biosensor in buffer is 47 nM. Whereas in artificial urine, the LOD increases to 56 nM (Figure 4a; Table S4). There is no significant effect to the addition of BSA and PRG in absence of SRTF1 to DNA functionalized electrode (Figure S9 andS10). A bottleneck for achieving lower limit of detection of sensor is the relatively small change in current signal from the baseline observed at lower PRG concentration. We addressed this challenge using the DNA 2 -aTF biosensor, which possesses two aTF binding sites per DNA -i.e., increased number of binding sites for the same electrode surface area. At a low concentration of PRG, more SRTF1 unbinds from the DNA 2 than DNA 1 , affording a greater current change from the baseline and lowering the LOD. The LOD for the DNA 2 -SRTF1 decreases to 17 nM in buffer and to 29 nM in artificial urine (Figure 4b; Table S4). From a woman's health perspective, the current PRG detection range is suitable for monitoring all trimesters of fetus development which span a urine PRG concentration from 30 nM to 1 µM (Table S5). These results are comparable to our previously developed FRET based sensor that had an LOD of 15 nM in artificial urine (Table S1). [START_REF] Grazon | A Progesterone Biosensor Derived from Microbial Screening[END_REF] Efforts are ongoing to further advance the design and components as well as the type of electrochemical measurement to lower the LOD so as to identify optimal times for conception based on a PRG measurement (Table S5).

Biologically, transcription factors control gene expression and are a mainstay of many synthetic biology experiments performed in whole cell (WCB) and cell-free platforms. [START_REF] Gui | The Application of Whole Cell-Based Biosensors for Use in Environmental Analysis and in Medical Diagnostics[END_REF][START_REF] Karig | Cell-Free Synthetic Biology for Environmental Sensing and Remediation[END_REF][START_REF] Garenne | Cell-Free Gene Expression[END_REF] Transformation of these strategies to biosensing of small analytes is of significant interest. [START_REF] Jung | Cell-Free Biosensors for Rapid Detection of Water Contaminants[END_REF][START_REF] Fernandez-López | Transcription Factor-Based Biosensors Enlightened by the Analyte[END_REF][START_REF] Voyvodic | Plug-and-Play Metabolic Transducers Expand the Chemical Detection Space of Cell-Free Biosensors[END_REF] Successes in WCB include detection of environmental pollutants [START_REF] Fernandez-López | Transcription Factor-Based Biosensors Enlightened by the Analyte[END_REF][START_REF] Bereza-Malcolm | Environmental Sensing of Heavy Metals Through Whole Cell Microbial Biosensors: A Synthetic Biology Approach[END_REF][START_REF] Kim | Synthetic Biology for Microbial Heavy Metal Biosensors[END_REF][START_REF] Cheng | Transcription Factor-Based Biosensors in High-Throughput Screening: Advances and Applications[END_REF] and diagnostic biomarkers [START_REF] Gui | The Application of Whole Cell-Based Biosensors for Use in Environmental Analysis and in Medical Diagnostics[END_REF][START_REF] Courbet | Detection of Pathological Biomarkers in Human Clinical Samples via Amplifying Genetic Switches and Logic Gates[END_REF] via regulation of a reporter gene followed by subsequent expression of, for example, green fluorescent protein (GFP). [START_REF] Fernandez-López | Transcription Factor-Based Biosensors Enlightened by the Analyte[END_REF][START_REF] Wan | Engineering Metabolite-Responsive Transcriptional Factors to Sense Small Molecules in Eukaryotes: Current State and Perspectives[END_REF][START_REF] Lee | In Vitro Use of Cellular Synthetic Machinery for Biosensing Applications[END_REF] WCBs offer signal amplification through gene expression, high selectivity, and high reproducibility. A recent study describes a PRG WCB based on transcription of GFP due to a slight conformational change of a nonnatural TF upon PRG binding. [START_REF] Liu | De Novo Design of a Transcription Factor for a Progesterone Biosensor[END_REF] However, challenges include maintenance of cells, costs, biocontainment and regulatory concerns, as well as difficulty in detection of analytes that are toxic to cells. Whereas, cell-free systems hold promise of safety, high sensitivity, and fast response time with examples of detecting virus, nucleic acids, antibiotics and biomarkers. [START_REF] Voyvodic | Plug-and-Play Metabolic Transducers Expand the Chemical Detection Space of Cell-Free Biosensors[END_REF][START_REF] Zhang | Advances in Cell-Free Biosensors: Principle, Mechanism, and Applications[END_REF][START_REF] Pardee | Low-Cost Detection of Zika Virus Using Programmable Biomolecular Components[END_REF][START_REF] Mitchler | Transcription Factor-Based Biosensors: A Molecular-Guided Approach for Natural Product Engineering[END_REF] The majority of the cell-free systems are based on protein synthesis using crude or purified components with optical output signals based on fluorescence, colorimetric, and bioluminescence reporters. [START_REF] Gui | The Application of Whole Cell-Based Biosensors for Use in Environmental Analysis and in Medical Diagnostics[END_REF][START_REF] Lopreside | Comprehensive Profiling of Diverse Genetic Reporters with Application to Whole-Cell and Cell-Free Biosensors[END_REF] On the electrochemical front, Barton et al. [START_REF] Gorodetsky | Electrical Detection of TATA Binding Protein at DNA-Modified Microelectrodes[END_REF] , describe a transcriptional activator TATA binding protein (TBP) that binds and kinks a redox molecule labeled DNA, which significantly attenuates charge transport through DNA. Recently, Mousavi et al., report an electrochemical sensor to detect antibiotic genes by using a cell-free expression of restriction enzymes mediated by transcription factoranalyte binding. The restriction enzyme cleaves the reporter genes that binds to capture genes on the electrode producing a detectable electrochemical signal. [START_REF] Sadat Mousavi | Electrochemical Interface for Gene-Circuit-Based Sensors[END_REF] To date, the work on TFs based electrochemical sensors includes the capture of binding events between DNA and aTFs to detect transcription factors (example, NF-κβ) [START_REF] Gorodetsky | Electrical Detection of TATA Binding Protein at DNA-Modified Microelectrodes[END_REF][START_REF] Bonham | Quantification of Transcription Factor Binding in Cell Extracts Using an Electrochemical, Structure-Switching Biosensor[END_REF][START_REF] Guo | A Label-Free Electrochemical Biosensor for Detection of HIV Related Gene Based on Interaction between DNA and Protein[END_REF][START_REF] Li | In Vitro Analysis of DNA-Protein Interactions in Gene Transcription Using DNAzyme-Based Electrochemical Assay[END_REF][START_REF] Li | Novel Electrochemical Nuclear Factorkappa B Biosensor via Analyte-Restrained Peptide Nucleic Acid Displacement Reaction Constructed on Different Gold Nanoparticle-Deposited Carbon Electrodes[END_REF][START_REF] Murata | An Electrochemical Device for the Assay of the Interaction between a Dioxin Receptor and Its Various Ligands[END_REF][START_REF] Gorodetsky | DNA Binding Shifts the Redox Potential of the Transcription Factor SoxR[END_REF][START_REF] Yu | A One-Step Structure-Switching Electrochemical Sensor for Transcription Factor Detection Enhanced with Synergistic Catalysis of PtNi@MIL-101 and Exo III-Assisted Cycling Amplification[END_REF][START_REF] Tichý | Protein P53 Binding to Cisplatin-Modified DNA Targets Evaluated by Modification-Specific Electrochemical Immunoprecipitation Assay[END_REF] or interaction between analyte and transcription factor that result in conformational changes. Finally, Sode et al., report a glutamine binding protein tagged with a redox probe that undergoes conformational changes in the presence of L-glutamine altering the electrochemical current output. [START_REF] Takamatsu | Continuous Electrochemical Monitoring of L-Glutamine Using Redox-Probe-Modified L-Glutamine-Binding Protein Based on Intermittent Pulse Amperometry[END_REF] In summary, we report an electrochemical transcription factorbased biosensor for the detection of small molecule analytes. This aTF-DNA sensor joins the family of electrochemical-based biosensors, which are being increasingly investigated for health monitoring. [START_REF] Bollella | Beyond Graphene: Electrochemical Sensors and Biosensors for Biomarkers Detection[END_REF][START_REF] Cesewski | Electrochemical Biosensors for Pathogen Detection[END_REF][START_REF] Labib | Electrochemical Methods for the Analysis of Clinically Relevant Biomolecules[END_REF][START_REF] Wongkaew | Functional Nanomaterials and Nanostructures Enhancing Electrochemical Biosensors and Lab-on-a-Chip Performances: Recent Progress, Applications, and Future Perspective[END_REF][START_REF] Furst | Impedance-Based Detection of Bacteria[END_REF][START_REF] Furst | Label-Free Electrochemical Detection of Human Methyltransferase from Tumors[END_REF][START_REF] Singh | Hitting the Diagnostic Sweet Spot: Point-of-Care SARS-CoV-2 Salivary Antigen Testing with an off-the-Shelf Glucometer[END_REF] The advantages of such biosensors, generally, include rapid test-to-results, high selectivity/sensitivity, and low-cost of detection. Our sensor demonstrates clinically relevant sensitivity required for adaptation into a prototype POC biosensor. The key findings of this work are: 1) translation of transcription factor binding affinities (to target DNA and analyte) into an electrical readout; 2) increasing the number of aTF binding sites on DNA improves sensitivity; and, 3) detection of physiological levels of PRG in buffer and artificial urine samples. Our results hold promise for the development of a new suite of electrochemical biosensors based on natural or engineered bacterial aTFs.

ASSOCIATED CONTENT Supporting Information. is available free of charge on the ACS Publications website at DOI:

Materials and methods, DNA sequences, expression and assay conditions, electrode characterization (AFM, impedance spectroscopy), and binding kinetics.

Figure 1 .

 1 Figure 1. (a) Overview of the electrochemical biosensor: DNA-TF complex on gold electrode blocks mediator access giving low current output. (b) Presence of hormone is detected by a higher current output using square wave voltammetry. (c) DNA with 3 (DNA 3 ), 2 (DNA 2 ), and 1 (DNA 1 ) binding pockets.

Figure 2 .

 2 Figure 2. Bio-layer interferometry (a) of DNA 1 , DNA 2 , DNA 3 (1, 2, 3 binding sites respectively). (b) to calculate KD, different [SRTF1] were added to DNA 2 to measure the association constant and thereafter dipped in buffer solution for determination of the dissociation constant. (c) the DNA 2 -SRTF1 system responds to PRG in a dose-dependent manner. Please see SI page 3 for additional details.

Figure 3 .

 3 Figure 3. A typical square wave voltammogram (a). Current drops on SRTF1 addition and increases on PRG addition. Peak current of square wave voltammogram plotted (b) for DNA 1 , DNA 2 , DNA 3 , and scrambled DNA with addition of aTF and PRG. The error bars represent the standard deviation for N = 3 biological replicates, and statistical significance established according to a t-test. (* p < 0.05, *** p < 0.001)

Figure 4 .

 4 Figure 4. Calibration curve of the electrochemical sensor system. DNA 1 (a) and DNA 2 (b). Difference in current output before and after addition of PRG is plotted against concentrations of PRG in artificial urine and buffer. The error bars represent the standard deviation for N = 3 biological replicates.

Funding Sources

DARPA (W911NF-16-C-0044), Marie-Curie fellowship from the European Union under the program H2020 (Grant 749973), Motorola Foundation sponsored Society of Women Engineer's scholarship, and NIH (R01EB029795)

ACKNOWLEDGMENT

This work was supported in part by the DARPA (W911NF-16-C-0044; CMK, MWG, JEG), the Marie-Curie fellowship from the European Union under the program H2020 (Grant 749973; CG), the Motorola Foundation sponsored Society of Women Engineer's scholarship (KS), and the NIH (R01EB029795; CMK, JEG).

AUTHOR INFORMATION Corresponding Author

* James E. Galagan; jgalag@bu.edu Mark W. Grinstaff; mgrin@bu.edu Present Addresses R Baer's new address is Stemloop, Inc., Evanston IL 60204.

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

CONFLICT OF INTEREST

KS, RB, CG, CMK, JEG, and MWG are inventors on a patent describing this technology filed by Boston University, which is available for license.

ABBREVIATIONS

aTF, allosteric transcription factor; POC, point-of-care; PRG, progesterone; BLI, bio-layer interferometry; WCB, whole cell biosensor; PDG, pregnanediol glucuronide; LOD, limit of detection; SWV, square wave voltammetry.