algorithm for dose reduction with

21 Purpose: Fluence–modulated proton computed tomography (FMpCT) using pencil 22 beam scanning aims at achieving task–speciﬁc image noise distributions by modulating 23 the imaging proton ﬂuence spot–by–spot based on an object–speciﬁc noise model. In 24 this work we present a method for ﬂuence ﬁeld optimization and investigate its perfor- 25 mance in dose reduction for various phantoms and image variance targets. 26 Methods: The proposed method uses Monte Carlo simulations of a proton CT (pCT) 27 prototype scanner to estimate expected variance levels at uniform ﬂuence. Using an 28 iterative approach, we calculate a stack of target variance projections that variance level. For a more heterogeneous head phantom, dose reduction increased to 41 16 . 0 % for the same task. Prescribing two diﬀerent ROIs resulted in dose reductions 42 between 25 . 7 % and 40 . 5 % outside of the ROI at equal peak variance levels inside the 43 ROI. Imaging doses inside the ROI were increased by 9 . 2 % to 19 . 2 % compared to the 44 uniform ﬂuence scan, but can be neglected assuming that the ROI agrees with the 45 therapeutic dose region. Agreement of resulting variance maps with the prescriptions 46 was satisfactory. 47 Conclusions: We developed a method for ﬂuence ﬁeld optimization based on a noise 48 model for a real scanner used in proton computed tomography. We demonstrated that 49 it can achieve prescribed image variance targets. A uniform ﬂuence ﬁeld was shown 50 not to be dose optimal and dose reductions achievable with the proposed method for 51 ﬂuence–modulated proton CT were considerable, opening an interesting perspective 52 for image guidance and adaptive therapy. 53

The simulation framework outputs data in the same format as the prototype scanner. It  Assume a voxel centered in (u, v, d) in the three-dimensional distance-driven projection, 131 where d is the binning depth and u and v are the coordinates normal to it. We can identify a 132 set of protons such that the most likely path of every proton crosses the voxel volume around 133 (u, v, d). The number of protons in that set is then referred to as the "counts" C(u, v, d). 134 These counts only consider protons used for image reconstruction and therefore are reduced 135 compared to the incident protons due to interactions with the object and cuts on the data. 136 In contrast to that, counts in the absence of interactions and cuts are referred to as F (u, v, d)  In the simulation study, three different phantoms with a physical counterpart were used. The 141 water phantom is a cylindrical container made from polystyrene (outer diameter 150.5 mm, wall thickness 6.35 mm) and filled with distilled water. The CTP404 phantom (Phantom (diameter 150 mm) and several tissue-equivalent inserts and two cylinders filled with air. For each pencil beam b, this resulted in a three-dimensional experimental counts map We fitted the Gaussian model to each pencil beam's C b , where N 0 is the total number of protons per pencil beam, and 161 (u 0 (d), v 0 (d)) is the pencil beam center at depth d. The pencil beam center is assumed 162 to diverge linearly with the binning depth, such that u 0 (d) = u 0 · (1 + δ u · d) and v 0 (d) 163 analogously, where (u 0 , v 0 ) is the pencil beam center at d = 0 and δ u and δ v are the linear 164 divergence factors. By construction, the isocenter-beam for u 0 = v 0 = 0 is parallel to the 165 d-axis. The σ u = σ u · 1 + δ 2 u u 2 0 and σ v analogously are the beam widths projected to a 166 plane normal to the d-axis while σ u and σ v are the actual beam widths. This resulted in 167 a fit with seven open parameters (N 0 , u 0 , v 0 , σ u , σ v , δ u , δ v ), which was performed for each 168 individual pencil beam by minimization of the squared deviation. The parameters σ u , σ v , δ u 169 and δ v were not specific to one pencil beam, and estimates for them were therefore found as All datasets were generated by shooting a regular grid of simulated proton pencil beams.

175
At d = 0, neighboring pencil beams were interspaced by ∆ PB,u = 12 mm along u and 176 ∆ PB,v = 8 mm along v. The pencil beam grid was offset in u by ∆ PB,u /4 = 3 mm so that 177 the summed fluence from two opposing angles was homogeneous. This helped to reduce the 178 total number of pencil beams and thereby reduce the complexity of the optimization. In 179 the simulation platform, protons were emitted from a point (  Using the F b as basis functions, it is possible to generate an arbitrary counts field C α for 201 rotation angle α by finding weights w α b , such that C α is expressed as a linear combination of 202 the reference counts F b from equation (2). Weights were found by minimizing the squared 203 deviation, and therefore (3)

205
Integration was performed over u and v, but only the isocenter binning depth d = 0 was  3. Calculate the pixel-wise counts target C α target (u, v, d). Then, optimize weights that 220 yield the counts target according to equation (3).

221
The algorithm extends ideas from literature for x-ray CT 21,22 to requirements of pCT such 222 as the three-dimensional projections due to distance-driven binning 39 and a more complex can be calculated analytically as reconstruction was performed using filtered backprojection.

II.. MATERIALS AND METHODS
Finding a stack of variance projections V α target (u, v, d) whose variance reconstruction 33 yields 242 a given image variance target V target (x, y, z) is a problem with a large set of solutions. We 243 therefore aimed to find the inverse operation of variance reconstruction, 42 i.e. a "variance 244 forward projection." An initial guess V α 0 (u, v, d) could be obtained by performing ray-245 tracing 43 through the image variance target V target (x, y, z) followed by a ramp-filtration.

246
The additional filtration was motivated by the fact that variance reconstruction is very 247 close to a simple unfiltered backprojection. 42 Since ray-tracing is the inverse operation to 248 filtered backprojection, an additional ramp-filtration was required. While such forward-and 249 backprojection yield V target again, this often yields unphysical negative variance projection 250 values and amplifies noise. Therefore, a median filter was applied to the ramp-filtered 251 projections followed by thresholding to positive values.

252
To minimize the error introduced by thresholding, we employed an iterative approach by 253 applying variance reconstruction to the i-th set of variance projections the number of contributing protons C. Therefore, the pixel-wise counts required to achieve 262 the variance target could be calculated as (V α unit /V α target ) · C α unit . However, for low counts, we 263 need to consider that C follows a Poisson distribution (contrary to a normal distribution at 264 sufficiently high counts) and therefore an additional correction function needs to be introduced, where P C (n ) = C n exp(−C)/n ! is the Poisson probability of de- assumed k(C) = C for all C > 300. Furthermore, k(C) was thresholded to return at least 272 C min = 8 protons to avoid detector elements with missing information. 273 We used an optimization according to equation (3) to find pencil beam weights 274 w α b (C α target ) which achieve the pixel-wise projection counts target of Due to the fact that C α unit and C α target are both affected by interactions with the object, the 277 optimization also needed to be performed for unit fluence allowing for an elimination of the 278 effect of attenuation and scattering. This resulted in fluence modulation factors imaging fluence if its central axis intersects the ROI, and a low imaging fluence otherwise.

288
The fluence modulation factors were In a simulation study we prescribed three different image variance targets, which can be ap- in image variance cannot be achieved due to the ramp filtration involved in reconstruction.

308
Throughout this work we use the nomenclature "constant", "A" and "B" to refer to the 309 three prescriptions. For all phantoms we first simulated a high dose unit fluence dataset with m α i = 1.

311
The mean incident proton fluence was chosen to be 133 mm −2 such that it yielded a typical   counts roughly agreed with those using the optimization, but were uniform, as required.

390
Instead, using the optimization, a heterogeneous counts pattern was observed.

391
For the CTP404 phantom ( figure 7 (a,b)) and the head phantom ( figure 7 (c,d)), variance 392 increased around heterogeneities both in unit fluence and fluence-modulated scans. For the 393 head phantom in particular the palate exhibited locally elevated variance levels. The fluence 394 modulation with prescription A was less conformal, compared to those of the water phantom.

395
In particular for the CTP404 phantom variance contrast was impaired. Counts sinograms for 396 prescription A in figure 6 (c) and figure 7 (b,d) are similar, but phantom-specific differences 397 are noticeable.   case.
415 Figure 9 shows the head phantom with unit fluence (a,b) and for the constant variance      the optimal choice of the contrast in the image variance prescription should be studied in 492 the future, but is out of scope for this work.

493
Using a simple intersection-based approach also showed dose savings compared to unit 494 fluence acquisitions. However, dose savings were considerably less compared to the opti-495 mized FMpCT scans and conformity of variance with the prescription was degraded. By 496 construction, a prescription of constant variance is not possible with this approach.

497
Future work should also address the impact of iterative image reconstruction, which is 498 frequently used for pCT imaging. [44][45][46][47][48] In contrast to the direct filtered backprojection algo-499 rithm used in this study, iterative reconstruction employs a regularization method (typically 500 total variation), which reduces noise and whose optimal weight depends on the object and