# DDGS-CT: Direction-Disentangled Gaussian Splatting for Realistic Volume
Rendering

Zhongpai Gao∗ Benjamin Planche Meng Zheng
Xiao Chen Terrence Chen Ziyan Wu
United Imaging Intelligence, Boston, MA
<EMAIL_ADDRESS>
Equal contribution

###### Abstract

Digitally reconstructed radiographs (DRRs) are simulated 2D X-ray images
generated from 3D CT volumes, widely used in preoperative settings but limited
in intraoperative applications due to computational bottlenecks, especially
for accurate but heavy physics-based Monte Carlo methods. While analytical DRR
renderers offer greater efficiency, they overlook anisotropic X-ray image
formation phenomena, such as Compton scattering. We present a novel approach
that marries realistic physics-inspired X-ray simulation with efficient,
differentiable DRR generation using 3D Gaussian splatting (3DGS). Our
direction-disentangled 3DGS (DDGS) method separates the radiosity contribution
into isotropic and direction-dependent components, approximating complex
anisotropic interactions without intricate runtime simulations. Additionally,
we adapt the 3DGS initialization to account for tomography data properties,
enhancing accuracy and efficiency. Our method outperforms state-of-the-art
techniques in image accuracy. Furthermore, our DDGS shows promise for
intraoperative applications and inverse problems such as pose registration,
delivering superior registration accuracy and runtime performance compared to
analytical DRR methods.

##  1 Introduction

#### Motivation.

Digitally reconstructed radiographs (DRRs) are simulated (in silico) 2D X-ray
images rendered from 3D computational tomography (CT) volumes. While DRRs are
widely utilized in preoperative settings, such as optimizing dose delivery and
in radiation oncology [20, 38, 6], their potential intraoperative applications
remain underexplored due to computational bottlenecks and naive modeling
capability [1, 26]. For instance, real-time multimodal registration for image-
guided procedures is currently impractical because of the time-consuming
process of generating DRRs and integrating them into slice-to-volume
registration [34].

The advent of GPU-accelerated computing has significantly improved the
efficiency of radiography simulators [3, 36, 5], though they still fall short
of meeting the stringent requirements of real-time applications. Recent
developments in inverse graphics have led to the creation of differentiable
DRR renderers (_e.g_., DiffDRR [11], X-Gaussian [7], GaSpCT [24]), which show
promise for inverse problems such as 2D/3D CT image registration [12].
However, these methods do not fully capture the complexities of X-ray image
formation. They typically use computationally efficient ray-tracing techniques
that model the attenuated photon fluence at each detector pixel by
accumulating Hounsfield unit (HU) values along the 3D view-ray through the
voxelized CT volume. This approach, while fast, cannot account for complex
noise-inducing physical effects such as Compton scattering and beam hardening
[14], crucial for accurate X-ray imaging.

Physics-based Monte-Carlo simulations [2, 15, 3, 4] do not suffer from these
limitations. Relying on the decomposition of CT volumes into realistic
materials via HU thresholding, they accurately simulate single-photon
transport and probabilistically evaluate photon-matter interactions
(photoelectric effect, Compton scattering, Rayleigh scattering, _etc_.) to
determine the final attenuation and hit-region of each photon. This approach
provides a more accurate representation of X-ray image formation; but the
computational intensity of these methods—requiring >$10^{8}$ iterations to
render an image—makes them impractical for real-time applications and inverse
problems.

In this paper, we propose a novel, more elegant balance between (a) realistic
physics-inspired X-ray simulation and (b) efficient, differentiable DRR
generation, as illustrated in Figure 1.

#### Contributions.

Our approach leverages 3D Gaussian splatting (3DGS) [17] for efficient DRR
generation from CT volumes, similar to X-Gaussian [7] and GaSpCT [24].
However, we introduce a crucial adaptation: our direction-disentangled 3DGS
(DDGS) method accounts for the dependencies of X-ray image formation on light
directionality. Specifically, we decompose the radiosity contribution of 3D
regions into two independent components: a non-directional component modeling
the isotropic X-ray interactions with the corresponding material, and a
direction-dependent component approximating higher-dimensional anisotropic
interactions, such as scattering. Unlike the more generic view-dependent
spherical-harmonics decomposition adopted by common 3DGS solutions [10], ours
is tuned to the specific behaviors of high-energy photons. This formulation
allows our method to implicitly learn and reproduce the effects of scattering
on X-ray images without the need for complex photon-transport simulations at
runtime.

Additionally, we adapt the initialization phase of 3DGS to account for the
properties of tomography data. We adopt a geometry- and intensity-based dual
strategy to sample 3D points that correspond to: (a) contact regions between
different anatomical elements, where photon transport is likely to show
discontinuities; and (b) homogeneous regions, where the primary attenuation
contribution can be sparsely modeled but could benefit from view-dependent
noise modeling.

We demonstrate that our solution achieves higher image accuracy compared to
state-of-the-art methods across various medical benchmarks Additionally, we
showcase the efficient application of our method to inverse problems, such as
pose registration, highlighting its potential for real-time intraoperative use
cases. Our DDGS achieves better registration accuracy and runtime performance
than analytical DRR methods.

Figure 1: Proposed pipeline of Direction-Disentangled 3D Gaussian Splatting
(DDGS).

##  2 Related Work

The generation of DRRs and the simulation of X-ray images have been subjects
of extensive research due to their critical roles in medical imaging,
particularly in radiotherapy planning and diagnostic imaging [20, 38, 6, 40].
In this section, we provide some insight on X-ray imaging, traditional
simulation tools, and the recent integration of machine-learning techniques.

#### X-ray Image Simulation.

X-ray imaging relies on the interactions of high-energy photons—emitted by the
X-ray source of the scanner—with the matter (_e.g_., body part) placed in
between the X-ray source of the scanner and its radiation-sensitive detector
cells (dexels). Imaging contrast arises from differential absorption and
scattering of X-ray photons by tissues, primarily through the photoelectric
effect, Compton scattering, and, to a lesser extent, beam hardening for
polychromatic scanners. The photoelectric effect involves the absorption of an
X-ray photon by an atom and is more pronounced in higher atomic number
materials like bone, causing them to appear whiter on X-ray images111X-ray
film is historically white, turning black when exposed to high-energy photon.;
whereas Compton scattering occurs when an X-ray photon collides with an outer
electron, resulting in the photon being deflected and losing energy, which
contributes to image noise and reduced contrast [14, 28].

A variety of Monte Carlo (MC) simulation suites [2, 15, 3, 4] model these
X-ray interactions and light transport. They trace the paths of a large number
of individual photons as they undergo probabilistic interactions within the
material. By simulating numerous photon trajectories, Monte Carlo methods
account for the random nature of photoelectric absorption and Compton
scattering, as well as secondary effects such as fluorescence and electron
transport. These MC solutions produce highly realistic simulations of
radiographic images, though at a high computational cost which makes them
inadequate for most DRR use-cases. Furthermore, for these tools to render
realistic DRRs from CT volumes, one must first pre-process the CT data to
assign proper materials and their corresponding physical properties to each
voxel, which is essential for realistic simulation. This step is often
approximated by decomposing the volume into a small set of predefined
materials according to user-provided HU ranges (_e.g_., “air” material
assigned to voxels with HU value $<-800$, “lung” assigned to voxels in range
$]-800,-200]$, “fat” for $]-200,-100]$, _etc_.). This process can be time-
consuming and prone to errors, affecting the fidelity of MC-based DRR
generation [9].

#### Efficient and Differentiable DRR Generation.

Analytical DRR rendering methods [36, 35, 5] have been developed in parallel
to the aforementioned MC suites for real-time or near-real-time applications.
These methods have a much lower computational footprint, as they do not
simulate individual photon interactions but instead use mathematical models to
directly calculate the primary paths and approximated attenuations of X-rays.
The seminal work of Siddon [30] on computing line integrals through a
discretized CT volume is still at the core of most recent DRR solutions,
_e.g_., which proposed differentiable formulation of the rendering steps [11]
or integrated neural networks to add realistic noise to the output [33].
However, the inherent approximations in these methods can affect their
accuracy, particularly in modeling artifacts present in heterogeneous tissues
or complex anatomical structures, _e.g_., caused by scattering (non-primary
light contributions).

Recently, researchers [7, 24] have tried to apply 3D Gaussian splatting (3DGS)
techniques [17] to DRR rendering, _i.e_., optimizing a cloud of 3D Gaussians
to approximate voxel data, thereby enabling faster rendering with minimal
accuracy loss. However, these early solutions overlooked the specific nuances
of X-ray imaging, directly applying 3DGS methods designed for natural imaging
[24] or oversimplifying the physical properties of X-ray attenuation [24],
leading to suboptimal modeling.

In this paper, we propose a novel formulation of 3D Gaussian splatting, tuned
to more efficiently approximate X-ray imaging (time- and quality-wise).

##  3 Methodology

###  3.1 Preliminaries

#### Gaussian Splatting.

3DGS [17] is a point-based rendering technique using 3D Gaussians to
explicitly represent scenes in a more compact manner than volumetric
representations. Each Gaussian
$g_{i}=\\{\bm{\mu}_{i},\bm{\Sigma}_{i},\bm{\alpha}_{i},\bm{f}_{i}\\}$ is
defined by its mean position $\bm{\mu}_{i}\in\mathbb{R}^{3}$, covariance
matrix $\Sigma_{i}\in\mathbb{R}^{3\times 3}$ (usually decomposed into separate
scale and rotation parameters), opacity $\bm{\alpha}_{i}\in\mathbb{R}$, and
radiance properties $\bm{f}_{i}\in\mathbb{R}^{k}$, where $k$ depends on the
light contribution model adopted. _E.g_., $k=(L+1)^{2}\times 3$ for
anisotropic, view-dependent RGB radiance decomposed into spherical harmonics
(SH) of degree $L$, a model commonly used for natural-imaging applications.
Images are rendered by casting view-rays through each pixel $p$ into the scene
and alpha-blending the Gaussian contributions to the final ray color $C$, as:

$C(p)=\sum_{j\in N}{c_{j}\sigma_{j}\prod_{l=1}^{j-1}{(1-\sigma_{l})}}\;\text{
with
}\sigma_{i}=\bm{\alpha}_{i}e^{-\frac{1}{2}(p-\widehat{\bm{\mu}}_{i})\widehat{\bm{\Sigma}}_{i}^{-1}(p-\widehat{\bm{\mu}}_{i})},$
(1)

where $N$ is the number of points sampled along the view-ray, $c_{j}$ is the
radiance (computed from $\bm{f}_{j}$, _e.g_., via SH) of the $j$th Gaussian on
the ray, and $\widehat{\bm{\mu}}_{j}$ and $\widehat{\bm{\Sigma}}_{j}$ are the
image-plane projections of $\bm{\mu}_{j}$ and $\bm{\Sigma}_{j}$.

Since the aforementioned rasterization process is differentiable w.r.t. to the
Gaussian parameters, the 3DGS representation of a scene can be learned via
gradient-descent, given a set of 2D scene observations $\bm{I}_{\text{GT}}$
and their corresponding camera parameters. A combination of $\ell_{1}$ and
SSIM [37] loss functions are commonly-adopted as criterion for this iterative
optimization process [17]:

$\mathcal{L}=(1-\lambda)\ell_{1}{(\bm{I}_{\text{GT}},\bm{I})}+\lambda\operatorname{SSIM}{(\bm{I}_{\text{GT}},\bm{I})},$
(2)

with $\bm{I}$ the rendered images (based on provided camera parameters), and
$\lambda$ a loss-weighting hyper-parameter. The initial state is typically
obtained by sampling relevant 3D points in the scene domain for the Gaussians,
_e.g_., from a structured-from-motion (SfM) point-cloud [32, 27].

#### Application to DRR.

The volumetric rendering performed by 3DGS methods is conceptually similar to
the one performed by analytical DRR methods [30, 11], _i.e_., aggregating the
attenuation values of light-rays cast through the voxelized CT volume. This
similarity has recently motivated researchers [7, 24] to apply 3DGS models to
DRR applications, driven by the compactness of 3DGS representations compared
to voxel data (addressing the $\mathcal{O}(n^{3})$ complexity inherent to
voxel rendering). By optimizing a 3DGS model to approximate the visual
properties of the CT volume, the resulting representation can render accurate
DRRs much faster than traditional methods, making it more suitable for real-
time intra-operative visualization. Consequently, the use of traditional,
slower rendering solutions can be limited to generating DRR targets
($\bm{I}_{\text{GT}}$) for the offline (_i.e_., pre-operative) training.

However, these early 3DGS-for-DRR solutions [30, 11] do not account for noise-
inducing photon interactions (_e.g_., scattering) when applying the analytical
methods [29, 5] to render their training data. Since scatter-free DRRs are not
affected by ray directions (_i.e_., the attenuation at each point is naively
considered isotropic), Cai et al. [7] simplified the radio-intensity function
of each Gaussian to $c_{i}=\operatorname{sigmoid}{(\bm{b}\cdot\bm{f}_{i})}$,
where $\bm{b}\in\mathbb{R}^{k}$ is a direction-independent optimizable basis
vector shared by all splats. Furthermore, their Gaussian point-cloud is
initialized via evenly-spaced Cai et al. [7] or random-uniform [24] sampling
of the scanned 3D space, _i.e_., ignoring relevant geometrical and material
properties of the target CT data. Inadequate initialization strategies
negatively impact the convergence of 3DGS models and the resulting image
accuracy [16]. In this work, we propose to reformulate both the initialization
strategy and the radiance function of the 3D Gaussians to address these
limitations.

###  3.2 Disentanglement of Isotropic and Anisotropic 3D Gaussians

We argue that, while X-ray scattering models are incompatible with the 3DGS
rendering formulation (secondary rays would have to be cast, increasing the
computational footprint exponentially), their anisotropic impact on X-ray
imaging can be approximated to some extent by direction-dependent radiance
functions. However, modeling this high-dimensional residual contribution is
only meaningful if it appears in the training DRRs, _i.e_., if the target
radiographs are rendered using physics-inspired simulation tools. This is not
always possible, _e.g_., if the mapping from HU values to materials for the
target CT scans is not provided. Ideally, in such cases, when only scatter-
free DRRs can be generated, the radiance function should gracefully degrade to
a lighter anisotropic formulation (_e.g_., similar to [7]), to avoid
unnecessary computations that could impact the model optimization, as well as
its runtime performance. Moreover, X-ray interactions with matter themselves
involve both isotropic (_e.g_., photoelectric absorption/fluorescence,
Rayleigh scattering) and anisotropic (_e.g_., Compton scattering) phenomena
[14, 22]. Therefore, for the sake of both modularity and accuracy, we propose
to decompose our DRR representation into isotropic Gaussians
$g_{i}^{\text{iso}}$ and anisotropic, direction-dependent ones
$g_{j}^{\text{dir}}$. We formulate their respective radiosity functions as:

$c_{i}^{\text{iso}}=\operatorname{sigmoid}{(\bm{b}^{\text{iso}}\cdot\bm{f}_{i}^{\text{iso}})}\;\;\;\;;\;\;\;\;c_{j}^{\text{dir}}=\operatorname{sigmoid}{\left(Y_{1..L}(\theta,\phi)\cdot\bm{B}^{\text{dir}}\bm{f}_{j}^{\text{dir}}\right)},$
(3)

where $\bm{f}_{i}^{\text{iso}},\bm{f}_{j}^{\text{dir}}\in\mathbb{R}^{k}$ are
feature vectors respectively encoding the contribution of isotropic and
anisotropic Gaussians, $\bm{b}^{\text{iso}}\in\mathbb{R}^{k}$ is a global
isotropic basis vector (similar to [7]), and
$\bm{B}^{\text{dir}}\in\mathbb{R}^{k_{L}\times k}$ is a second, anisotropic
basis matrix that interacts with the $L$-degree spherical-decomposition
$Y:\mathbb{R}^{2}\mapsto\mathbb{R}^{k_{L}}$ applied to the ray angles
$\theta,\phi$ without constant term (degree $=$ 0), resulting in
$k_{L}=L(L+2)$.

It is important to note that, while the usual spherical-harmonics
representation for Gaussian splatting also contains isotropic (degree $=$ 0)
and anisotropic (degree $>$ 0) components, the model assumes that both
contributions are co-located (_i.e_., they belong to the same 3D Gaussian
$g_{i}$ of position $\bm{\mu}_{i}$ and covariance $\bm{\Sigma}_{i}$). To
account for the complexity of X-ray light transport, we relax this co-location
constraint, _i.e_., modeling isotropic and anisotropic contributions via
distinct 3D Gaussians ($g_{i}^{\text{iso}}$ and $g_{j}^{\text{dir}}$). We
demonstrate in our evaluation that this disentanglement results in both higher
image quality and lighter representation.

###  3.3 Initialization via Radiodensity-Aware Dual Sampling (RADS)

Our second main contribution targets the initialization of the point-cloud
that supports the Gaussian-mixture modeling of the 3D data. Since 3DGS was
defined for natural imaging (_e.g_., ignoring sub-surface light transport),
common initialization strategies limit their sampling to scene-surface points
(SfM- [17] or depth-guided [23] subsampling). Such sampling of the 3D space
is, however, inadequate for volumetric data, ignoring possibly salient
regions. While uniform sampling of the CT space could be considered [7, 24],
we argue that such a strategy is also suboptimal for anatomical data,
discarding domain-relevant properties. We demonstrate empirically (_c.f_.
Section 4) that uniform sampling results in slower convergence of the 3DGS
representation and degraded runtime performance.

Therefore, we propose a novel twofold sampling strategy that accounts for the
specific nature of CT data and for our dual isotropic/directional 3DGS model.
Based on the radiodensity values contained in the target CT volumes, our
solution samples two set of 3D points
$\mathbf{P}^{\text{mc}}\in\mathbb{R}^{n_{1}\times 3}$ and
$\mathbf{P}^{\text{dw}}\in\mathbb{R}^{n_{2}\times 3}$ with distinct,
complementary distributions ($n_{1},n_{2}$ scalar hyper-parameters).

Not unlike [23], the first set of points is sampled by applying the marching-
cubes algorithm [21] to extract points at the interface between materials with
different physical properties (_i.e_., distinct HU responses in the CT
volume). These interface regions typically need careful modeling when
simulating X-ray imaging, as they result in discontinuous photon transport.

To complement this first set of points, the second set is semi-randomly
sampled from the voxel centroids, according to a uniform distribution weighted
by the voxels’ radiodensities. _I.e_., voxels with higher radiodensity,
contributing to the X-ray attenuation, are more likely to be picked to
initialize the Gaussians. In other words, this radiodensity-weighted strategy
identifies additional points within the homogeneous regions of the CT volume
that also contributes to the imaging process.

Based on their distinct properties, we assign the entire
$\mathbf{P}^{\text{mc}}$ to initialize the isotropic Gaussians, whereas we
equally split $\mathbf{P}^{\text{dw}}$ to initialize both isotropic and
direction-dependent Gaussians. All in all, our dual radiodensity-aware
strategy improves the initialization of the 3DGS representation and is more
explainable than prior work (see Figure 2).

Figure 2: Illustration of the different sampling strategies for 3DGS
initialization.

##  4 Experiments

###  4.1 Experimental Protocol

#### Implementation.

For evaluation, we set the decomposition degree to $L$=1, the feature
dimension to $k$=8, and initial cloud sizes to $n_{1}$=15,000 and
$n_{2}$=10,000. We apply a loss weight $\lambda=0.2$. We choose the default
threshold level (_i.e_., the average of the minimum and maximum volume) for
the marching-cubes algorithm. DDGS training is performed on a single NVIDIA
A100 using an Adam optimizer [18] with a learning rate of $1.25\times 10^{-4}$
for $\bm{b}^{\text{iso}}$ and $\bm{B}^{\text{dir}}$ and $2.5\times 10^{-3}$
for $\bm{f}^{\text{iso}}$ and $\bm{f}^{\text{dir}}$. Default 3DGS [17]
learning rates are applied to the remaining parameters.

#### Datasets.

We consider 4 datasets. (1) NAF-CT [39] includes four CT images of abdomen,
chest, foot, and jaw. For each, we adopt TIGRE [5] to sample 50 evenly-
distributed projections for training and 50 randomly-distributed ones for
testing, in the range of [$None$, $None$] (scatter-free DRRs). (2) CTPelvic1K
[19] is a large dataset of pelvic CT images. We consider the first 10 scans of
the sub-dataset6 in our experiments. We adopt DeepDRR [33] to generate 60
training (evenly-distributed) and 60 (randomly-distributed) testing DRRs (in
the range of [$None$, $None$]), taking advantage of the scattering-modeling
capability of DeepDRR to render realistic X-ray data. (3) Ljubljana [25] is a
clinical dataset of 10 patients undergoing neurovascular surgery. Each patient
underwent one CT and two X-ray angiography scans. The pose of each X-ray is
also provided. Following DiffPose strategy [12], we randomly sample 900
projections for training and 100 for testing. (4) Finally, we provide some
additional qualitative results on DeepFluoro [13], a collection of pelvic
X-rays and CT images from six cadavers.

#### Metrics.

We evaluate our method in two settings: novel-view synthesis and
intraoperative 2D/3D image registration (_i.e_., pose estimation). For novel-
view synthesis, we use the standard PSNR and SSIM as metrics. For
intraoperative 2D/3D image registration, we consider the error both in terms
of rotation (angular distance) and translation (Euclidean distance). We
further measure the clinically-relevant registration accuracy of key
anatomical landmarks for Ljubljana scans [25] where they are provided.
Following [12], we compute the target registration error (TRE) w.r.t. the
positioning of these landmarks after registration.

###  4.2 Novel-View Synthesis

We first validate our contributions in terms of the quality of rendered DRRs
and compactness of our dual Gaussian-mixture representation.

Table 1: Comparison of Gaussian splatting-based DRR rendering techniques, on 2
datasets.

|  | 3DGS ($L=1$) | X-Gaussian ($k=32$) | DDGS (Ours, $L=1,k=8$)
---|---|---|---|---
|  | # points$\downarrow$ | PSNR$\uparrow$ | SSIM$\uparrow$ | # points$\downarrow$ | PSNR$\uparrow$ | SSIM$\uparrow$ | # points$\downarrow$ | PSNR$\uparrow$ | SSIM$\uparrow$
NAF-CT | abdomen | 11,149 | 47.43 | 0.994 | 10,802 | 47.17 | 0.993 | 13,928 | 48.09 | 0.994
chest | 13,669 | 44.42 | 0.988 | 16,568 | 43.33 | 0.987 | 13,533 | 44.50 | 0.989
foot | 8,616 | 44.51 | 0.984 | 10,909 | 44.34 | 0.985 | 10,786 | 44.70 | 0.985
jaw | 17,902 | 40.47 | 0.973 | 22,318 | 40.02 | 0.972 | 19,665 | 40.57 | 0.974
avg | 12,834 | 44.21 | 0.985 | 15,149 | 43.72 | 0.984 | 14,478 | 44.47 | 0.986
CTPelvic1K-Dataset6 | 001 | 53,988 | 35.40 | 0.971 | 50,059 | 36.81 | 0.979 | 41,947 | 37.88 | 0.984
002 | 49,099 | 37.03 | 0.982 | 48,113 | 37.79 | 0.986 | 43,933 | 38.43 | 0.988
003 | 60,755 | 35.73 | 0.973 | 59,822 | 36.46 | 0.977 | 48,536 | 38.28 | 0.983
004 | 42,349 | 38.87 | 0.982 | 37,243 | 39.84 | 0.985 | 39,176 | 40.35 | 0.986
005 | 42,482 | 39.37 | 0.984 | 46,014 | 39.30 | 0.984 | 39,570 | 40.36 | 0.987
006 | 53,832 | 37.14 | 0.983 | 57,197 | 37.42 | 0.983 | 47,398 | 38.26 | 0.986
007 | 45,360 | 37.09 | 0.980 | 48,454 | 37.62 | 0.983 | 41,032 | 38.87 | 0.987
008 | 51,211 | 38.03 | 0.980 | 45,750 | 38.70 | 0.982 | 43,577 | 39.81 | 0.986
009 | 44,060 | 38.19 | 0.981 | 41,279 | 38.92 | 0.985 | 41,389 | 39.87 | 0.987
010 | 38,691 | 37.80 | 0.984 | 40,030 | 37.55 | 0.985 | 39,691 | 38.73 | 0.987
avg | 48,183 | 37.47 | 0.980 | 47,396 | 38.04 | 0.983 | 42,625 | 39.08 | 0.986

#### Comparative Evaluation.

We compare to 3DGS [17] and X-Gaussian [7] on the NAF-CT [39] and CTPelvic1K
[19] datasets for the novel-view synthesis. For each method, we adopt the
hyper-parameters recommended by their respective authors (_e.g_., $L=1$ and
$3$ for normal 3DGS [17] and $k=8$ and $32$ for X-Gaussian [7]).

Table 2: Novel-view synthesis evaluation on CTPelvic1K data.

| 500 | 2000 | 7000 | 15000 | 30000
---|---|---|---|---|---
|  | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM
DDGS (Ours) ($L=1,k=8$) | 001 | 25.77 | 0.908 | 30.46 | 0.944 | 35.18 | 0.975 | 36.91 | 0.981 | 38.04 | 0.984
002 | 27.64 | 0.921 | 31.60 | 0.954 | 35.33 | 0.977 | 36.90 | 0.984 | 38.30 | 0.987
DDGS (Ours) ($L=3,k=8$) | 001 | 26.20 | 0.909 | 31.50 | 0.950 | 35.65 | 0.978 | 37.68 | 0.983 | 38.15 | 0.985
002 | 27.87 | 0.923 | 32.23 | 0.956 | 36.02 | 0.979 | 37.87 | 0.986 | 38.97 | 0.989
3DGS ($L=1$) | 001 | 23.56 | 0.873 | 27.52 | 0.914 | 31.66 | 0.951 | 33.64 | 0.963 | 35.36 | 0.970
002 | 24.55 | 0.891 | 29.17 | 0.932 | 32.82 | 0.965 | 35.33 | 0.977 | 37.25 | 0.982
3DGS ($L=3$) | 001 | 23.55 | 0.872 | 27.54 | 0.914 | 33.34 | 0.962 | 36.01 | 0.975 | 37.12 | 0.980
002 | 24.55 | 0.891 | 28.96 | 0.931 | 34.50 | 0.973 | 37.72 | 0.985 | 38.75 | 0.988
X-Gaussian ($k=8$) | 001 | 17.31 | 0.794 | 24.79 | 0.877 | 29.00 | 0.923 | 30.72 | 0.933 | 33.39 | 0.947
002 | 13.45 | 0.742 | 27.19 | 0.904 | 32.72 | 0.962 | 34.52 | 0.973 | 36.04 | 0.978
X-Gaussian ($k=32$) | 001 | 26.34 | 0.854 | 31.28 | 0.950 | 33.56 | 0.968 | 35.07 | 0.974 | 36.83 | 0.979
002 | 28.32 | 0.868 | 33.44 | 0.966 | 35.85 | 0.980 | 36.55 | 0.984 | 37.89 | 0.986

As shown in Table 1, for NAF-CT [39] (without scatter-related noise), our DDGS
outperforms both 3DGS and X-Gaussian in terms of PSNR and SSIM. For CTPelvic1K
[19] (with scatter), our DDGS significantly outperforms 3DGS and X-Gaussian in
both PSNR and SSIM while using considerably fewer points. Therefore, our DDGS
method is more effective in simulating realistic X-ray images from CT scans.

Furthermore, we compare our DDGS (with degrees $L=1$ and $L=3$ and feature
dimension $k=8$) with traditional 3DGS (degrees $L=1$ and $L=3$) and
X-Gaussian (feature dimensions $k=8$ and $k=32$) on the 001 and 002 data of
CTPelvic1K [19] at iterations of 500, 2000, 7000, 15,000, and 30,000, as shown
in Table 2. Our model with $L=1$ and $k=8$ performs comparably to 3DGS with
$L=3$ and outperforms X-Gaussian with $k=32$ after 15,000 iterations. Our
model with $L=3$ achieves the best performance overall. Figure 3 presents a
qualitative comparison of the testing set, illustrating both the synthetic
views and the generated 3D Gaussian points.

Figure 3: Visualization of the Gaussian cloud optimization for different
methods. Table 3: Ablation study w.r.t. the proposed disentangled
isotropic/anisotropic representations.

| DDGS (Ours) | direct-entangled | direct-independent | direct-dependent | 3DGS-disentangled
---|---|---|---|---|---
| PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM
001 | 38.04 | 0.984 | 36.17 | 0.977 | 35.93 | 0.976 | 32.48 | 0.942 | 35.32 | 0.971
002 | 38.30 | 0.987 | 37.32 | 0.984 | 36.96 | 0.983 | 33.04 | 0.953 | 35.38 | 0.973

Table 4: Impact of initialization strategies on final image quality, measured
CTPelvic1K scans.

| 001 | 002 | 003 | 004 | 005 | 006 | 007 | 008 | 009 | 010 | avg
---|---|---|---|---|---|---|---|---|---|---|---
Radiodensity-based (Ours) | 37.88 | 38.43 | 38.28 | 40.35 | 40.36 | 38.26 | 38.87 | 39.81 | 39.87 | 38.73 | 39.08
Random (uniform dis.) | 37.17 | 38.36 | 38.15 | 40.09 | 39.97 | 38.36 | 38.69 | 39.37 | 39.85 | 38.84 | 38.89
Even sampling | 37.33 | 38.51 | 37.83 | 39.58 | 39.12 | 38.27 | 38.87 | 39.88 | 39.46 | 38.95 | 38.78

Table 5: Impact of $k$ dimensionality on image quality, measured on 2
CTPelvic1K scans (001, 002).

| $k=1$ | $k=4$ | $k=8$ | $k=16$ | $k=32$
---|---|---|---|---|---
| PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM
001 | 34.60 | 0.965 | 36.81 | 0.979 | 38.04 | 0.984 | 38.41 | 0.986 | 38.27 | 0.987
002 | 33.66 | 0.964 | 37.08 | 0.982 | 38.30 | 0.987 | 38.76 | 0.989 | 38.89 | 0.989

Figure 4: Visualization of the isotropic and anisotropic X-ray contributions
to the rendered DRRs. Figure 5: Illustration of the pose-registration
convergence using various differentiable DRR renderers. Table 6: Accuracy of
image registration on the CTPelvic1K [19] dataset.

| 3DGS | X-Gaussian ($k$=8) | X-Gaussian ($k$=32) | DDGS (Ours)
---|---|---|---|---
| 001 | 002 | 003 | 001 | 002 | 003 | 001 | 002 | 003 | 001 | 002 | 003
Rot. error (deg) | 0.134 | 0.062 | 0.229 | 0.326 | 0.163 | 0.312 | 0.097 | 0.070 | 0.266 | 0.067 | 0.067 | 0.152
Trans. error (mm) | 1.788 | 1.163 | 3.108 | 4.037 | 2.144 | 3.781 | 1.378 | 1.288 | 3.483 | 1.285 | 1.092 | 2.180

Table 7: Registration evaluation w.r.t. anatomical landmark accuracy and time efficiency, on Ljubljana. Method | DRR render time (ms) ↓ | total optimization time (s) ↓ | TRE (mm) ↓
---|---|---|---
mean | med | std | mean | med | std | mean | med | std
DiffDRR | 30.88 | 31.51 | 3.25 | 431.98 | 316.54 | 261.70 | 2.14 | 0.72 | 5.40
DDGS | 7.46 | 6.94 | 2.67 | 121.45 | 39.93 | 117.67 | 0.80 | 0.65 | 0.51

#### Ablation Study.

Table 3 demonstrates the effectiveness of our direction-disentangled
representation for 3D Gaussians. For several settings, we observe performance
degradation when learning the direction-dependent and direction-independent
components in the same 3D Gaussians (direct-entangled), learning the
direction-independent 3D Gaussians alone (direct-independent), or learning the
direction-dependent 3D Gaussians alone (direct-dependent). Additionally, we
decouple the direction-dependent and direction-independent components in 3DGS
[17] (3DGS-disentangled), which also showed lower performance compared to our
learnable features. Figure 4 shows the rendered views from disentangled 3D
Gaussians, highlighting that the direction-dependent 3D Gaussians focus more
on bones or anatomical structures, while the direction-independent 3D
Gaussians focus more on the background.

Table 4 shows the effectiveness of our proposed initialization scheme
(radiodensity-aware) compared to other initialization methods. Our scheme
outperforms both the uniformly random sampling used in 3DGS [17] and the even
sampling method used in X-Gaussian [7].

At last, we conduct experiments on the impact of feature dimension $k$, as
shown in Table 5. Our results indicate that increasing the feature dimension
generally improves overall performance. However, the improvement becomes
marginal once the feature dimension exceeds $k=16$.

###  4.3 Application to Downstream Task (2D/3D CT Image Registration)

We validate our solution in terms of applicability to downstream tasks,
considering here the registration of intraoperative 2D X-ray images to
preoperative 3D CT volumes; a task crucial to a variety of clinical
applications.

We choose the first five testing images on the 001, 002, and 003 CT data of
CTPelvic1K [19] for image registration. Considering this task as an
optimization-based problem that can be solved via inverse graphics (_i.e_.,
rendering synthetic images based on predicted poses and comparing to the
target image, then backpropagating the difference to improve the pose
prediction), we adopt the framework from iComMA [31], replacing their image
rendered by DDGS. We use Adam optimizer with a learning rate of 0.05. We use
the isocenter pose as the initialization pose. As shown in Table 6 and Figure
5, our DDGS achieves the lowest rotation and translation errors.

Finally, we also consider the experimental protocol proposed in DiffPose [12],
where the authors rely on a pretrained pose-regression CNN to get a first
rough estimation of the scanner pose, before refining said pose via a
gradient-descent based optimization, leveraging their differentiable DRR
renderer [11]. We adopt their pose-initialization network and testbed for the
Ljubljana dataset, replacing their DiffDRR analytical renderer [11] by our
DDGS, and compare to them in terms of runtime performance and landmark-
registration accuracy. The results, reported in Table 7, highlight the
convergence boost brought by our lighter solution.

##  5 Conclusion

We proposed DDGS, a novel 3DGS-based representation for efficient and
realistic DRR generation from CT volumes. We extended traditional 3DGS by
considering light directionality, crucial for accurate X-ray image modeling.
By decomposing radiosity into non-directional and direction-dependent
components, DDGS captures isotropic interactions and anisotropic effects.
Unlike generic view-dependent spherical-harmonics decomposition, DDGS
implicitly learns scattering effects, substantially improving the speed and
accuracy of downstream clinical applications such as intraoperative 2D/3D
registration.

Limitations. It should be noted that, similar to previous methods, our model
expects an offline DRR renderer to produce realistic ground truth to train
DDGS, which can be a limitation for some scenarios (_e.g_., when such renderer
or certain parameters required for realistic rendering are not available). Our
direction-dependent function relating to anisotropic X-ray effects may also
fail to approximate complex multi-bounce light scenarios. Recent developments
in $N$-dimensional Gaussian splatting [8] show promising potential for
modeling such cases.

Societal Impact. The above limitation may restrict the adoption of DDGS to
specific use cases, and our method should undergo a clinical evaluation of the
generated images in terms of anatomical accuracy. Nonetheless, we believe that
our proposed method can positively impact both the computer vision and medical
communities by providing a more efficient and versatile DRR tool.

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# Pixelated silicon detector for radiation beam profile measurements

J. Tikkanen<EMAIL_ADDRESS>S. Kirschenmann  N. Kramarenko  E.
Brücken  P. Koponen  P. Luukka  T. Siiskonen  R. Turpeinen  J. Ott  Radiation
and Nuclear Safety Authority (STUK), Jokiniemenkuja 1, FI-01370 Vantaa,
Finland  Helsinki Institute of Physics, Gustaf Hällströmin katu 2, FI-00014
University of Helsinki, Finland  University of Helsinki, Department of
Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland  Lappeenranta-
Lahti University of Technology LUT, Yliopistonkatu 34, FI-53850 Lappeenranta,
Finland  Santa Cruz Institute for Particle Physics, University of California
Santa Cruz, 1156 High Street, CA-95064, USA

###### Abstract

A pixelated silicon detector, developed originally for particle physics
experiments, was used for a beam profile measurement of a Co-60 irradiator in
a water phantom. The beam profile was compared to a profile measured with a
pinpoint ionization chamber. The differences in the pixel detector and
pinpoint chamber relative profiles were within approximately 0.02, and after
calculating correction factors with Monte Carlo simulations for the pixel
detector, the differences were decreased to almost less than 0.005. The
detector’s capability to measure pulse-height was used to record an electron
pulse-height spectrum in water in the Co-60 beam, and the results agreed well
with simulations.

###### keywords:

Radiotherapy dosimetry, beam profile measurement, position sensitive detector,
pixel detector, pulse-height spectrum

††journal: Radiation Measurements

##  1 Introduction

An accurate estimation of dose delivered to the patient in radiation therapy
requires measurement of dose-distribution, or beam profile, in a water
phantom. Whereas determination of the dose at the center of the beam for field
sizes of for example 10 cm$\times$10 cm can be done with traditional farmer
type ionization chambers with nominal volume of 0.6 cm3, small and conformal
beams used in modern radiotherapy require a small detector to minimize the
averaging of dose in the detection volume [1]. Therefore, the quality
assurance requirements for the modern radiotherapy beams are challenging to
fulfill with traditional measurements based on ionization chambers. Small
pinpoint chambers or diamond detectors may be used [1], but the measurements
can be cumbersome. Position sensitive detectors with a good position
resolution and well-known energy and dose rate dependence would be a valuable
addition to rapidly scan the beam to obtain dose profiles.

Figure 1: Schematic image of the data acquisition chain. Power cables are not
depicted.

An alternative to moving a detector with a single radiation detecting volume
is to use detectors divided into segments, either strips or pixels. Pixelated
detectors can provide an excellent two-dimensional spatial resolution, while a
sub-millimeter pixel size helps to reduce the individual pixel count rates
that can be very high in radiotherapy beams. Acquisition of the energy
spectrum would possibly give additional information about possible changes in
the beam quality.

Pixelated semiconductor detectors have been used as tools in particle physics,
for example at the CMS experiment at CERN for tracking charged particles. In
tracking sensors collision products ideally do not experience significant
energy loss, hence the material of choice should be light. In radiation
therapy, when measuring dose-to-water, the detector should not affect the
particle fluence significantly compared to only water being present, and hence
similar considerations apply [2]. The atomic composition and density of
silicon are not as close to water as those of ionization chambers, but due to
a high amount of ion-hole pairs created per energy deposited, the detector can
be kept thin to minimize the effect on the particle fluence. Also, silicon as
a semiconductor material allows fast signal processing, and acquisition of a
pulse-height spectrum.

In this study, we used a silicon pixel detector, to measure the beam profile
of a Co-60 irradiator. This detector-type was originally fabricated for the
R&D phase of the first upgrade of the inner tracking detector of the CMS
experiment [3]. Co-60 irradiators were common in radiation therapy before the
development of linear accelerators, and nowadays, they are used mostly for
detector calibration purposes. The energies of photons emitted by a Co-60
source (1.173 MeV and 1.332 MeV) are close to the mean energy of a 6 MV linear
accelerator beam [4]. The beam profile measured with the pixel detector was
compared to a profile measured with a pinpoint ionization chamber. Also, we
measured a 2D-profile to demonstrate the capability of the pixelated detector
for high resolution planar profile measurements. The pulse-height measurement
feature of the detector was used to record a pulse-height spectrum of the
detector in water in the Co-60 beam, and the results were compared to a
simulated spectrum.

##  2 Measurement setup and methods

In the beam profile measurements, we used a GBX200 (Best Theratronics Ltd,
Ottawa, Canada) Co-60 irradiator at the premises of the Finnish Radiation and
Nuclear Safety Authority (STUK) dosimetry laboratory. The irradiator contained
a high-activity Co-60 source, and the radiation beam was collimated with
inter-meshing lead and tungsten bars. The radiation beam shape was
rectangular, beam axis horizontal, and the nominal field size 10 cm$~{}\times$
10 cm at 100 cm source-to-detector distance.

The dose profiles were measured in a 30 cm$~{}\times$ 30 cm$~{}\times$ 30 cm
water phantom with a 1 cm thick front wall. The phantom wall material was
poly-methyl-methacrylate (acrylic glass, PMMA), and because of the horizontal
beam direction, the beam passed through the front wall before reaching water.
Distance from the source to the phantom wall (source to surface distance, SSD)
was 95 cm. The water temperature during the measurements was $19\pm
1~{}^{\circ}$C.

The coordinate system in the rest of this paper is the following: horizontal
(x) axis is the horizontal axis perpendicular to the beam axis, and the depth
(z) axis is the beam axis. The y-axis is vertical. To move detectors inside
the water phantom, a scanning device capable of moving the detector in three
dimensions was built from a 3D-printer (Creality Ender 5 plus, Shenzhen,
China), named DOSCAN [5]. The scanner used stepper motors, and self-made
control electronics.

###  2.1 Silicon pixel detector

Figure 2: Measurement setup used for the beam profile scans. The pixel
detector is mounted inside the acrylic box. The detector box is attached to
the 3D scanning system surrounding the phantom via a PMMA rod. On the right,
in addition to the detector chip, printed circuit board (PCB, yellow),
horseshoe connector (green), and the LEMO cable for detector bias voltage can
be seen. On the left, partly visible are the GBX200 Co-60 irradiator and the
readout electronics on top of the scanner.

To fulfill the requirements for the performance of the detector, a 280
$\mathrm{\SIUnitSymbolMicro m}$ thick n+ in n type silicon-based pixel sensor
design with a rectangular pitch of $100\times 150$ $\mathrm{\SIUnitSymbolMicro
m}$2 and $52\times 80$ pixels, with a size of 1$\times$1 cm2, and active area
of 0.78$\times$0.8 cm2 was chosen. The detector is described in more detail in
[3]. Si pixel detectors of this design are used in CMS tracker at CERN, which
consists in total of 66560 pixels organized into 16 sections of $52\times
80=4160$ pixels each. The sensor used for the beam profile measurements forms
a Single Pixel Chip Module (SCM) when bonded with a single readout chip (ROC),
using indium solder bumps.

The ROC (psi46dig2.1-r [6]) contains an individual readout for each pixel,
called a pixel unit cell (PUC). The pulse-height of charge created by a
charged particle is measured in each PUC. The pulse processing chain includes
pre-amplification, pulse shaping circuits and a charge discriminator, and the
measured pulse-height is converted into a digital signal (8-bit). The readout
from the PUCs is done in double columns (2$\times$80 pixels). If a pixel has
measured a signal (has been hit), it sends the pulse-height information to the
double-column readout, from where it is sent further into a readout buffer.
The data is written finally into event frames, where one event contains the
pixel addresses (column and row) and the pulse-heights from all pixels hit in
a 25 ns time window.

The full electronic readout chain in the measurements (shown in Fig. 1)
consisted of an adapter card that connects the pixel detector via standard
SCSI cable to the so-called Detector-Test-Board (DTB) based on an Altera FPGA
[7]. The DTB was connected via a USB2 cable (15 m long) to the data
acquisition PC. The bias was applied from the back side of the detector with a
Keithley 6478 Picoammeter/Voltage source via a high voltage filter
(Particulars F-01A) close to the detector. The data acquisition PC as well as
the voltage source were located in a separate room to control the setup
remotely during measurements when the experimental area is closed due to
radiation protection.

To characterize and train the pixel detector before the measurement, certain
tests are undertaken, which are described extensively in [8]. These tests
ensure the functioning state of the ROC, calibrate the pixels and deliver
additional information, such as the trimming value, which is set above the
noise level to prevent false detection. In the measurements, the trimming
value was set to 60 VCAL, which corresponds to low energy threshold of
approximately 10 keV. Unfortunately, the ROC does not allow self-triggering.
Instead, trigger signals are sent at a constant rate of 100 kHz, and only the
latest event is read out with each trigger.

###  2.2 Measurements with the pixel detector

For the beam profile measurements we used the setup shown in Fig. 2. The
silicon pixel detector was inserted in a custom designed and watertight CNC-
machined box of acrylic glass (PMMA). The box was shaped to minimize the
amount of air around the detector chip while maintaining enough space to have
a low probability of breaking the detector during setup, and there was 5 mm of
air in front, and 1 mm behind the detector. The box was attached via a PMMA
rod to the DOSCAN scanner. Two thin plastic bags were placed around the
casing, the rod, and cables to ensure water-tightness of the system. The
detector shorter edge (7.9 mm) was along the horizontal axis.

The detector was still during, and moved between single measurements. The
detector was at 5 cm distance (depth) from the phantom front wall, and the
step size of the detector 4 mm, approximately half of the silicon chip width,
leading to overlap of the detector between measurement points. The depth of
the detector in the phantom was measured with a Micro-epsilon (Ortenburg,
Germany) IDL1700 laser displacement sensor, with a knowledge of the distance
of the detector chip from the front wall of the casing. The stepper motors
were turned off during the measurements, as described in [5], due to noise
induced in the detector.

A 2D scan of the beam profile was made of the upper left corner of the beam at
5 cm depth. The scan area was approximately -8 cm to center on the x-axis and
from center to 8 cm on the y-axis, the step size being 8 mm on both axes. The
horizontal axis was scanned first before taking a step in the vertical
direction, leading to minimal movement of the less-ideally functioning
vertical axis [5].

A laptop handled data acquisition through the pXar software [8], and the
scanner was operated with a Raspberry Pi 4B (Raspberry Pi Foundation,
Cambridge, UK). The laptop gave a signal to the Raspberry, through a socket
connection when the data acquisition finished, and the Raspberry to the laptop
after the detector had been moved to the correct position. The code on the
laptop was written in C++, utilising pXar and Root libraries [9], and the code
on Raspberry with python.

###  2.3 Measurements with an ionization chamber

The GBX200 beam profile was also measured with an ionization chamber and the
DOSCAN scanner. The chamber employed was a PTW 31015 pinpoint (Freiburg,
Germany), with cavity radius of 1.45 mm and length of 5 mm. The chamber
current was measured with a Keithley 6517B electrometer (Tektronix, Beaverton,
USA). The chamber reference point was placed at 5 cm depth from the phantom
wall and the current was measured at 1 mm intervals on the x-axis. The field
size and phantom orientation were the same as in the pixel detector
measurements.

In a 2D scan of the corner of the beam with the pinpoint chamber, the step
sizes were 2 mm on the horizontal, and 4 mm on the vertical axis. The detector
was in the beginning of the scan placed in the center of the beam, indicated
by the cross-hair of the GBX200. The scanning order was, again, horizontal
axis first.

###  2.4 Simulations

The silicon pixel detector inside the watertight casing was modelled with
EGSnrc radiation transport simulation software [10], using the C++ class
library and egs_chamber user code [11]. A model of the GBX200 irradiator,
constructed with BEAMnrc user code [12], was used as a shared library source
in the simulations. The modeling of the GBX200 irradiator will be described in
a further study.

The model of the detector included the silicon chip, gap for the bump-bonds,
ROC, PCB, horseshoe-connector, and the casing. The connector and cables were
not modeled, and those areas were filled with PMMA. These areas had more than
5 mm of plastic in front of the detector, and although the simplification
affects the photon flux slightly, the electrons from these regions do not
reach the silicon chip. The dose was simulated in a 0.9 mm $\times$ 6 mm
$\times$ 0.28 mm region, the size of $6\times 60$ pixels, at the center of the
chip. The detector was shifted in 2 mm intervals on the x-axis from the center
of the beam to 78 mm.

The variance reduction techniques (VRT) for the egs_chamber user code are
described in [11]. An intermediate phase space file was scored to a volume
that enveloped at least 1.5 cm of the plastic in the casing around the hole
cut for the detector. The cross-section enhancement VRT was used in a 2.5 cm
sized cube around the detector chip with an enhancement factor of 64. Russian
roulette VRT was also used, with air around the detector as cavity geometry,
rejection factor of 128, water as rejection range medium, and Esave parameter
of 521 keV. Electron and photon cut-offs (ECUT and PCUT) were 521 keV and 10
keV.

The simulation was also run including only the detector chip, ROC and PCB in
the water phantom to estimate the performance of the detector with an ideal
(water equivalent) casing. Also, a dose-to-water profile was simulated in 0.2
cm $\times$ 0.6 cm $\times$ 0.1 cm (xyz) sized water voxels along the x-axis.

A pulse-height spectrum was simulated using the egs_phd user code. The active
volume was the size of the detector chip excluding the outer rim of pixels.
The bin-size was 10 keV, the electron cut-off parameter 0.531 MeV, and photon
cut-off energy 10 keV. Monte Carlo transport parameters common to all the
simulations are given in Table 1.

Table 1:  Monte Carlo transport parameters for the EGSnrc simulations. Global SMAX | 1e10
---|---
ESTEPE | 0.25
XImax | 0.5
Skin depth for BCA | 3
Boundary crossing algorithm | EXACT
Electron-step algorithm | PRESTA-II
Spin effects | On
Brems angular sampling | KM
Brems cross sections | NIST
Photon cross sections | mcdf-xcom
Electron Impact Ionization | On
Triplet production | On
Radiative Compton corrections | On
Bound Compton scattering | On
Pair angular sampling | KM
Pair cross sections | NRC
Photoelectron angular sampling | On
Rayleigh scattering | On
Atomic relaxations | On
Photonuclear attenuation | On
Figure 3: Visualisation of the Monte Carlo model of the detector and the
waterproof casing.

##  3 Results and discussion

###  3.1 Comparison of measured dose-profiles

The relative beam profiles measured with the ionization chamber and the pixel
detector are shown in Fig. 4. The relative profiles are denoted as
$P_{\textrm{IC}}(x)=I_{\textrm{IC}}(x)/I_{\textrm{IC}}(0)$ and
$P_{\textrm{pixel}}(x)=I_{\textrm{pixel}}(x)/I_{\textrm{pixel}}(0)$, where
$I_{\textrm{IC}}(x)$ is the current of the chamber, and
$I_{\textrm{pixel}}(x)$ the number of recorded events per pixel (hit-profile)
at position $x$. The values for the pixel detector are averaged over six
columns (area of 0.9 mm $\times$ 8 mm). For both profiles, the normalization
value ($I_{\textrm{IC}}(0)$ and $I_{\textrm{pixel}}(0)$) was an average from
the profile around the center of the beam. The uncertainty of the silicon
detector profile was calculated assuming Poisson statistics, meaning that the
standard uncertainty is the square-root of the number of hits. However, the
standard deviation of the hits in the pixels was larger, as can be seen in
Fig. 4. Even though the ionization chamber current profile should give a good
approximation of the dose-profile, charged particle fluence effects and dose
averaging in the chamber cavity volume affect the response of the detector at
the beam edges, and how much the chamber current profile differs from the
dose-profile is unknown to us.

Figure 4: The relative current-profile of the pinpoint ionization chamber
($P_{\textrm{IC}}$), and hit-profile of the pixel detector
($P_{\textrm{pixel}}$) in the GBX200 beam, at 5 cm depth in a water phantom,
and difference between the two profiles. The uncertainty ($k=1$) was
calculated assuming the chamber measurement uncertainty to be insignificant,
and the pixel detector uncertainty was calculated from Poisson statistics.

The field sizes (full width at half maximum, FWHM) were calculated by finding
the half-value at both edges of the beam from a second order polynomial to the
data around the half-value. The FWHM values for the profiles in Fig. 4 were
10.00 cm for the chamber and 10.01 cm for the pixel detector. The x-axis
values were shifted with the half-values so that the zero position was at
half-way between the half-value positions. This allowed the comparison of
different profile measurements without the effect of detector positioning.

The pixel detector measurements were in a non-ideal geometry due to the air
inside the water-tight casing. Although the casing material is relatively
water equivalent (PMMA), there was roughly 5 mm of air in front, and 1 mm
behind the detector. The air around the detector affects the electron and
photon flux compared to only water being present, and the effective depth of
the detector was not 5 cm.

From the simulations, we calculated two different correction factors for the
measured data: the ratio between dose-to-water and dose to detector relative
profiles

$k_{1}(x)=\frac{P_{D_{\textrm{w}}}(x)}{P_{D_{\textrm{Si}}}(x)}=\frac{D_{\textrm{w}}(x)/D_{\textrm{w}}(0)}{D_{\textrm{Si}}(x)/D_{\textrm{Si}}(0)},$
(1)

and between dose-to-detector without and with the casing

$\displaystyle k_{2}(x)=$ $\displaystyle\frac{P_{D_{\textrm{Si, no
casing}}}(x)}{P_{D_{\textrm{Si}}}(x)}$ $\displaystyle=$
$\displaystyle\frac{D_{\textrm{Si, no casing}}(x)/D_{\textrm{Si, no
casing}}(0)}{D_{\textrm{Si}}(x)/D_{\textrm{Si}}(0)},$ (2)

where $D$ refers to simulated dose. The factor $k_{1}$ is akin to a
calibration factor for the pixel detector. The second one is for speculation
about how the detector would perform in a more ideal geometry, without air, or
other non water-equivalent materials around the detector.

The difference of the relative silicon detector profile $P_{\textrm{pixel}}$
multiplied with the factors $k_{1}$ and $k_{2}$ to $P_{\textrm{IC}}$ is
depicted in Fig. 5. For the $k_{1}$ corrected profile, the difference stays
almost below 0.01 and for the most part is between $\pm$0.005. This means that
the deviation between the non-corrected pixel detector and chamber profile is
likely due to detector response, and not due to errors in the measurements. In
addition, it means that the detector is capable of measuring profiles with
sufficient accuracy if a calibration is given as a function of position, at
least in a Co-60 beam.

The $k_{2}$ corrected $P_{\textrm{pixel}}$ profile is also significantly
closer to $P_{\textrm{IC}}$. This means most likely that the air around the
detector affected the profile measurement, and with a more optimal casing, the
dose-profile could be measured with a higher accuracy without applying
position dependent correction factors.

Figure 5: The difference of the pixel detector beam profile, non-corrected and
multiplied with the factors $k_{1}$ and $k_{2}$ obtained from simulations, to
the pinpoint ionization chamber beam profile. Error bars are omitted for
clarity, but as in Fig. 4, the standard deviation of the data was
significantly larger than the statistical uncertainty.

Using the charge deposited in a pixel at each detector location (charge-
profile) instead of number of detection events (hit-profile) should give a
better estimation of the dose to the detector chip since this value is
approximately proportional to the energy imparted in the pixel. However, both
methods yielded similar results, except the charge-profile had a higher
variance. Therefore the hit-profile was used.

The results from the 2D scan of the corner of the beam are shown in Fig. 6.
The coordinates for both the pixel detector, and the chamber scans were
adjusted by assuming the half value of the profile to be 5 cm from the
midpoint of the beam on the x- and y-axis. The pixel detector results are
averaged over 6 $\times$ 10 pixels (0.9 mm $\times$ 1 mm). As with the
1D-profile, the differences are in the order of a few percent of the profile
maximum, although the variance here is higher due to smaller number of counts,
and less averaging of pixels. The profile has a structure along the x-axis at
the points measured with the edge of the silicon chip that might be a result
of charge balance effects at the chip borders. Otherwise, the comparison to
the chamber scan shows similar behaviour as the 1D-scan; the pixel detector
values at the edge are lower before, and higher beyond the half maximum of the
profile. No correction factors, such as $k_{1}$ and $k_{2}$, were calculated
for the 2D-profile.

(a)

(b)

Figure 6: a) 2D hit profile of a corner of the GBX Co-60 irradiator beam
measured with the silicon pixel detector, and b) difference to a ionization
chamber beam profile. The chamber profile between the measurement points was
interpolated to get the reference value for the pixel detector results. Note,
that due to smaller size of the chamber scan, the values could be compared
only on part of the profile.

###  3.2 Repeatability

The x-axis scan was repeated multiple times to test the repeatability of the
detector. Six profiles were measured during the same day with same detector
setup. The detector suffered from errors during some scans, more towards the
end of the day. The results from the position of the detector where errors
occurred were discarded, leaving gaps in some of the profiles. The data was
again averaged over six columns in the x-direction, and the maximum deviations
from the mean of all the scans are shown in Fig. 7. For most measurement
points, the difference to mean stays below 0.005. The highest deviations
occurred at the level part of the beam, where the deviations were caused (in
practice) solely by the detector, not the scanner. From [5], a 100
$\mathrm{\SIUnitSymbolMicro m}$ shift in the detector position would have
caused a 0.004 difference in the profile at highest gradients.

Figure 7: Minimum and maximum difference of the repeated scan results to the
mean of all the scans. The profile shape is visualized in Fig. 4a). The scan
was repeated six times. The gray line shows the standard deviation calculated
from the number of hits in each pixel.

###  3.3 Count-rate performance

The performance of the detector in a radiation therapy (RT) photon beam could
not be tested within this study, but a crude approximation on the count rate
can be made with relative dose rate in the Co-60 and a linac beam. The RT
photon beams are pulsed, and a high dose rate is delivered during a single
pulse compared to the average dose rate. Usually, the average dose rate is
adjusted with the number of pulses in a time interval, and the dose rate
during a single pulse cannot be adjusted. The duration of a pulse is in the
range of few $\mathrm{\SIUnitSymbolMicro s}$, and the pause between pulses in
the order of milliseconds. From [13], the dose rate during a single pulse for
a Varian 21EX linac with 6 MV voltage at 10 cm depth in water is 60 Gy/s. The
dose rate in the Co-60 beam was approximately 6 mGy/s. With an approximation
that the dose rate is proportional to flux (which would be the case with an
identical electron spectrum), the rate of electrons hitting the detector would
be 10000 times higher in a RT photon beam during a pulse.

From simulations, the number of electrons hitting the detector was
approximately $7\cdot 10^{-8}$ per source photon. Using the activity of the
cobalt source of approximately 100 TBq, and taking into account that, in
practice, two photons are emitted per decay, on average 0.35 electrons are
hitting the detector in the Co-60 beam in the 25 ns event frame time window.
In a radiation therapy beam the same number would be, by using the relative
dose rate compared to Co-60 beam, 3500. Considering the total number of pixels
(4160), and the readout process, the detector would most likely not be able to
measure the beam profile in a radiation-therapy beam due to buffer overflows.
However, this is not to say that a similar detector with a smaller pixel size,
or different readout electronics would not function sufficiently well in a
linac beam.

###  3.4 Spectrum

The detector registers the charges deposited in individual pixels, from which
spectra can be constructed. The charges are originally measured with regard to
a calibration voltage VCAL and were calibrated to energy values by measuring
the spectra of Am-241 and Ba-133 radionuclide sources, and determining the
peak centroids of Am-241 gamma (26.3 keV and 59.5 keV), and Ba-133 X-ray (30.6
keV – 36.0 keV) and gamma (81.0 keV) peaks. Fig. 8 shows the radionuclide
spectra taken with the Si pixel detector and the energy calibration curve.

(a)

(b)

Figure 8: (a): Spectra of the Am-241 and Ba-133 radionuclide measurements with
the silicon pixel detector. (b): Energy calibration curve obtained from the
Am-241 (26.3 keV, 59.5 keV) and Ba-133 (30.6 keV – 36.0 and 81.0 keV) peaks.

The pulse-height data from a measurement point at the center of the beam was
studied more closely. The pixel hits and respective charges are registered and
stored in event frames.

(a)

(b)

Figure 9: Pixel charge and cluster charge. (a): Spectrum of the pixel charge
and cluster charge. (b): Cluster charge and the contribution of single hits
(cluster size=1) and clusters of size 2.

Fig. 9a shows the cumulative spectrum over all detector pixels (pixel charge)
and event frames. Looking at individual event frames, in most of the cases
($36\,\%$ in this data set), single pixels are hit. However, if neighbouring
pixels are hit, the combined charge from this cluster of pixels relates
usually to energy deposited by a single electron (or positron): In some cases,
the charge created by an electron within the volume of a single pixel can
diffuse and be registered by the readout of a neighboring pixel (charge-
sharing). Also, an electron track can encompass multiple pixels.

Figure 10: Pulse-height spectra from simulations and from measurements after
applying the clustering algorithm. The spectrum was simulated also for a
thicker detector to test a hypothesis for the origin of the peak at
approximately 95 keV for the original spectrum.

To account for the charge-sharing issues, and to measure the full energy
deposited by a single electron, a clustering algorithm was developed to sum up
the charge of pixels belonging to the same cluster: For every hit pixel, it
was checked, whether the nearest neighbours were also hit. This procedure was
continued, until no further hit in the vicinity was found (both ends of the
track). Charges registered in pixels belonging to the same cluster were then
summed, and considered as a single pulse when constructing the pulse height
spectrum. The spectrum after applying this clustering algorithm is akin to a
spectrum measured using the whole detector as an active volume, but without
the effect of summing of charge from multiple electrons hitting the detector
within one event.

Fig. 9a shows the spectrum before and after applying the described clustering
algorithm. The largest contribution to the cluster charge spectrum are single
pixel hits (cluster size 1), followed by clusters with two components. These
distributions are depicted in Fig. 9b. The hits in an individual event frame
can be visualised as event displays, discussed in section 3.5. For example,
the event frame depicted in Fig. 11b contains three separate clusters, one of
which contains only a single pixel.

Because the signal from each pixel is read only if a charge threshold is
exceeded in that pixel, regardless of whether it was exceeded in the
neighboring pixels, the energy measured for a single cluster is underestimated
in cases where a charge below the threshold is collected in one or more pixels
along the particle track. A similar effect may occur due to charge sharing.
Extrapolation of the energy calibration curve significantly beyond 81 keV may
also cause errors in the energy measured by the detector. Pile-up, and
counting two electron tracks crossing the same pixel as a single cluster is
also possible, but these events happen with low probability. With regard to
the detector design of $80\times 52=4160$ pixels, the event with the most hits
(51) at this position would relate to an occupancy of only 1.2%.

The distribution of the calculated cluster charge aligns quite well with
simulation results. This can be seen in Fig. 10, where the measured spectrum
after applying the clustering algorithm is compared to a simulated pulse-
height spectrum for the silicon detector.

The measured spectrum in Fig. 9 has a bump around 95 keV, which falls together
with a peak in the simulated spectrum at around the same value. The electron
stopping power for silicon is relatively constant in the energy range from 500
keV to 1500 keV, and electrons in this range can travel through the detector
chip without significant changes in direction. An 1.25 MeV electron would lose
approximately 98 keV along a 280 $\mathrm{\SIUnitSymbolMicro m}$ track. The
dot markers in Fig. 10 show the simulated spectrum for a $1.3\times
280\,$\mathrm{\SIUnitSymbolMicro m}$=364\,$\mathrm{\SIUnitSymbolMicro m}$$
thick detector. The bump for this spectrum is at approximately 125 keV, and
$1.3\times 95$ keV is equal to 123.5 keV. Therefore, the explanation for the
bump is most likely electrons passing through the detector in a close to a
straight line perpendicular to the detector surface.

###  3.5 Event displays

Figure 11: Event displays of three events with a high pixel hit rate, showing
several clusters. The color shows the charge deposited in the pixel in keV.

To visualise individual clusters in event frames, single event frames with a
comparatively higher than usual amount of hits were studied more closely and
depicted as event displays. We would like to point out here that clusters with
more than 10 constituents only contribute to about 1% of all clusters. Fig. 11
shows example event displays with clusters of this category. The amount of
energy deposited at each end of the electron track hints at the direction of
the particle: As the electron loses energy along the track, the stopping power
increases and more energy will be deposited per unit distance. Also, the
electron tracks become more curved at lower energies and electrons can travel
longer distances inside a single pixel. Thus, the ending with more energy
deposited per pixel is supposedly the ending of the track. However, we only
measure the energy imparted in the detector and not the total energy of the
electron initially created in a certain distance from the detector chip.
Hence, the energy reconstructed, in most cases, is only a fraction of the
original electron energy.

##  4 Conclusions

The silicon pixel detector was successfully used for a measurement of the beam
profile in a Co-60 irradiator beam. On an x-axis scan, the results agreed
within 0.02 to ionization chamber measurements, and the maximum difference was
reduced to approximately 0.01 when correcting for the influence of the
detector waterproof casing by Monte Carlo simulations. The average difference
was reduced even further by calculating a calibration factor along the scan
axis with MC.

We utilized the capability of the pixel detector to record pulse-heights in
each pixel, and measured a pulse-height spectrum in the water phantom. Also,
we applied an algorithm to analyze charge deposited in a group of neighboring
pixels as a single charge pulse. The spectrum constructed like this
corresponds approximately to a spectrum scored to the whole active volume of
the detector, without the effect of pile-up from multiple electrons hitting
the detector in the same event. The results agreed well with a Monte Carlo
simulated pulse-height spectrum.

Although the results from the Co-60 beam measurement show promise also for use
in a radiation therapy beam, by comparing the dose rates, the detector would
most likely not be able to measure a beam profile in a linac photon beam. The
situation might be remedied with a smaller pixel size and different
electronics.

## Acknowledgements

This study was performed in the framework of the Academy of Finland project,
number 314473, Multispectral photon-counting for medical imaging and beam
characterization, for which we would like to acknowledge the funding.

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