# Gadolinium dose reduction for brain MRI using conditional deep learning

Thomas Pinetz  Institute for Applied Mathematics, University of Bonn  Erich
Kobler  Department of Neuroradiology, University Hospital Bonn  Robert Haase
Department of Neuroradiology, University Hospital Bonn  Julian A. Luetkens
Diagnostic and Interventional Radiology, University Hospital Bonn  Mathias
Meetschen  Diagnostic and Interventional Radiology, University Hospital Essen
Johannes Haubold  Diagnostic and Interventional Radiology, University Hospital
Essen  Cornelius Deuschl  Diagnostic and Interventional Radiology, University
Hospital Essen  Alexander Radbruch  Department of Neuroradiology, University
Hospital Bonn  Katerina Deike  Department of Neuroradiology, University
Hospital Bonn  Alexander Effland  Institute for Applied Mathematics,
University of Bonn

###### Abstract

Recently, deep learning (DL)-based methods have been proposed for the
computational reduction of gadolinium-based contrast agents (GBCAs) to
mitigate adverse side effects while preserving diagnostic value. Currently,
the two main challenges for these approaches are the accurate prediction of
contrast enhancement and the synthesis of realistic images. In this work, we
address both challenges by utilizing the contrast signal encoded in the
subtraction images of pre-contrast and post-contrast image pairs. To avoid the
synthesis of any noise or artifacts and solely focus on contrast signal
extraction and enhancement from low-dose subtraction images, we train our DL
model using noise-free standard-dose subtraction images as targets. As a
result, our model predicts the contrast enhancement signal only; thereby
enabling synthesization of images beyond the standard dose. Furthermore, we
adapt the embedding idea of recent diffusion-based models to condition our
model on physical parameters affecting the contrast enhancement behavior. We
demonstrate the effectiveness of our approach on synthetic and real datasets
using various scanners, field strengths, and contrast agents.

This work has been submitted to the IEEE for possible publication. Copyright
may be transferred without notice, after which this version may no longer be
accessible.

##  1 Introduction

Figure 1: Visualization of our dose reduction and contrast enhancement DL
approach. The inference process is illustrated at the top using solid arrows,
while the additional processing steps for training are shown at the bottom. In
both cases, pairs of pre-contrast $\mathbf{x}_{\mathrm{PC}}$ and contrast-
enhanced images $\mathbf{x}_{\\{\mathrm{LD},\mathrm{SD}\\}}$ are preprocessed
to obtain subtraction images $\mathbf{z}_{\\{\mathrm{LD},\mathrm{SD}\\}}$. The
pairing of low-dose subtraction and low-dose image
$\\{\tilde{\mathbf{z}}_{\mathrm{LD}},\mathbf{x}_{\mathrm{LD}}\\}$ is then
processed by a conditional CNN generating a contrast signal
$\hat{\mathbf{y}}_{\mathrm{SD}}$ that approximates the target
$\mathbf{y}_{\mathrm{SD}}$. This target is obtained by removing the noise from
the standard-dose subtraction image $\mathbf{z}_{\mathrm{SD}}$. At inference
the predicted contrast signal $\hat{\mathbf{y}}_{\mathrm{SD}}$ is added to the
pre-contrast $\mathbf{x}_{\mathrm{PC}}$ or low-dose image
$\mathbf{x}_{\mathrm{LD}}$ to obtain a synthesized standard-dose image
$\hat{\mathbf{x}}_{\mathrm{SD}}$ or a further contrast enhanced image
$\hat{\mathbf{x}}_{\mathrm{SD}+}$.

Gadolinium-based contrast agents (GBCAs) are indispensable in first-line
diagnosis and follow-up imaging of various neuropathologies ranging from
multiple sclerosis to metastasis and tumors. However, GBCAs are costly, may
cause patient discomfort, can accumulate in the body independent of kidney
function [1], and have a potential risk of causing nephrogenic systemic
fibrosis in patients with impaired renal function [2]. Furthermore, the
gadolinium administered during magnetic resonance imaging (MRI) examinations
accumulates in the drinking water system [3]. Therefore, current guidelines
recommend administering the lowest dose of GBCA required for a reliable
diagnostic result [4].

In recent years, deep learning has had a prominent impact on diagnostics in
medicine and radiology in particular. Currently, major MRI vendors offer deep-
learning (DL) solutions to common imaging problems, such as denoising, super-
resolution, and motion correction [5]. However, DL image generation approaches
tend to create unrealistically smooth images. In the worst case, this could
result in pathologies becoming invisible. To find a tradeoff between noise
removal and small structure/texture preservation, distributional losses such
as perceptual or adversarial losses [6, 7, 8] or local Wasserstein distances
[9] have been used. These losses are known to cause hallucinations [10, 11],
as the generative models need to synthesize noise and artifacts to match the
statistics of real MRI images. In turn, these artifacts or noise pixels can be
interpreted as small pathologies [12], which could result in misdiagnosis.

Recent DL-based contrast agent dose reduction methods in MRI learned to
predict a synthetic contrast-enhanced (CE) standard-dose image
$\hat{\mathbf{x}}_{\mathrm{SD}}$ from at least a pre-contrast
$\mathbf{x}_{\mathrm{PC}}$ and low-dose CE $\mathbf{x}_{\mathrm{LD}}$ image
pair [13, 8, 14]. Their deep convolutional neural networks (CNNs) were trained
to directly approximate the corresponding standard-dose image
$\mathbf{x}_{\mathrm{SD}}$ and are therefore prone to the previously discussed
tradeoff. In contrast, we propose to disentangle contrast signal enhancement
from the synthesis of realistic standard-dose images. To this end, we first
extract the noise and contrast signal from a low-dose image by subtracting the
corresponding pre-contrast image, i.e., acquired before administering any
GBCA, as illustrated in Figure 1. The CNN processes the resulting low-dose
subtraction image to remove the noise and enhance the contrast signal. An
image equivalent to the standard-dose image is obtained by adding the
predicted contrast signal $\hat{\mathbf{y}}_{\mathrm{SD}}$ to the pre-contrast
image $\mathbf{x}_{\mathrm{PC}}$. We train the CNN by penalizing the deviation
of its output from a contrast enhancement image $\mathbf{y}_{\mathrm{SD}}$.
This target image is computed by extracting the CE regions from the standard-
dose subtraction image using a simple statistical model suppressing noise. The
focus on subtraction images enables the synthesis of CE images beyond the
standard dose by adding the predicted CE image
$\hat{\mathbf{y}}_{\mathrm{SD}}$ to the low-dose image
$\mathbf{x}_{\mathrm{LD}}$ instead of the pre-contrast image
$\mathbf{x}_{\mathrm{PC}}$.

Various physical factors such as the scanner’s field strength as well as the
dose and relaxivity of the administered contrast agent influence the general
enhancement behavior within CE images [15, 16, 17]. However, this has not been
explored by recent DL-based dose reduction methods due to the lack of multi-
center data for the evaluation of domain shifts. We propose to explicitly
condition the CNN onto these physical quantities for improved generalization.
In particular, we transferred the conditioning approach of denoising diffusion
models [18] to our setting. Using a non-linear embedding of these physical
factors, each layer of our CNN can be adapted, thus promoting visual analysis
of the learned embedding and increasing the model’s interpretability.

To assess the efficacy of our approach, we perform a qualitative and
quantitative comparison to recent DL-based dose reduction methods. More
precisely, we used the openly accessible brain tumor segmentation metastasis
(BraTS-METS) dataset [19] to synthesize low-dose CE images using [9]. In
addition, we train and test all methods on an in-house dataset containing real
low-dose CE images at different dose levels. For testing the generalization to
different settings, we further compiled a dataset from an external site
exhibiting diverse pathologies. The numerical results on all datasets show
that our approach outperforms recent approaches qualitatively and
quantitatively.

##  2 Related Work

To account for the as low as reasonable possible dosage recommendations,
various gadolinium dose reduction techniques have been proposed. While initial
research has focused on developing new contrast agents with increased
relaxivity [20], the advances of deep learning [21] enabled a complementary
research direction called virtual contrast [22, 23]. In these approaches, an
contrast-enhanced T1-weighted (T1w) image with substantially reduced GBCA dose
is acquired along with pre-contrast images such as T1w, T2w, or FLAIR. These
images are preprocessed and fed into a deep neural network to extract the
subtle CE signal and synthesize an artificial CE T1w image resembling a
corresponding standard-dose image.

In 2018, Gong et al. [13] introduced the first virtual dose reduction approach
that combined a 2D U-Net [24] with residual learning to reduce the GBCA dosage
by $90\%$. Their model was limited to processing each 2D slice individually.
Thus, no information across slices was processed. Later, Pasumarthi et al. [8]
introduced a 2.5D approach by feeding 7 neighboring slices into the U-Net-like
model to extract relevant information from adjacent slices. To further exploit
3D information, they performed a multi-planar reconstruction approach, in
which predictions for different orientations and rotations of the input images
were computed and the pixel-wise average defined the model output. In
addition, they used a combination of perceptual and adversarial losses to
counteract a smooth appearance of the synthesized images, which is advocated
by the $\ell_{1}$ and structured similarity index loss functions originally
used in [13].

In parallel, Bône et al. [25] developed a fully 3D U-Net-like CNN for a dose
reduction factor of $75\%$. In contrast to the previous two approaches, Bône
et al. processed the images using 3D convolution layers and residual blocks in
their architecture. Their model was trained to synthesize standard-dose images
that minimize the $\ell_{2}$-distance to the corresponding standard-dose
images from 4 input sequences (pre-contrast T1w, low-dose T1w, FLAIR, ADC).

To evaluate the medical benefit, several reader studies were performed:
despite the impressive image quality of the synthesized CE images, significant
performance issues in small and weakly-enhancing lesions [26, 14, 12] as well
as hallucinations of lesions [14, 12] limited clinical applicability. In
addition, the resulting synthesized images are severely smoothed and could be
distinguished from the real ones [14].

All previously discussed methods were trained on datasets that consist of
matched pre-contrast, low- and standard-dose scans. The compilation of such
datasets is time- and resource-intensive since low-dose CE images are not
acquired in clinical practice. To avoid the acquisition of real low-dose
images, recently methods synthesizing low-dose images from pre-contrast and
standard-dose image pairs were proposed. Mingo et al. [17] proposed an
approximative physical model to generate half-dose images by averaging pre-
contrast and standard-dose images. In contrast, Pinetz et al. [9] facilitated
generative adversarial networks to synthesize low-dose images at various dose
levels. To faithfully predict the noise and low-dose contrast signal, they
proposed a novel perceptual loss function based on the Wasserstein distance of
local patches. They evaluated the benefit of synthetic low-dose images for
training the model of Pasumarthi et al. [8]. It turned out that training on
just synthetic low-dose images is inferior compared to real-world low-dose
images. However, augmenting the training data by synthetic low-dose images
from different sites led to improved results. Similarly, Wang et al. [27]
proposed to synthesize low-dose CE images from pre-contrast and standard-dose
image pairs using an iterative transformer model. They considered a global
attention mechanism on subsampled feature maps in combination with rotational
shifts to focus the model on the overall contrast uptake behavior. They
evaluated the efficacy of the synthetic low-dose images for the downstream
tasks of standard-dose image synthesis and subsequent tumor segmentation.

A different line of research [28, 29, 30] focuses on synthesizing CE standard-
dose images from just pre-contrast images, i.e., without administering any
GBCA contrast agent. The study of Preetha et al. [29] demonstrated that
synthetic CE standard-dose images could potentially be used for tracking tumor
progression. However, the performance of these methods is not sufficient for
more general clinical settings [22]. Furthermore, Liu et al. [30] found that
synthesizing CE T1w images from pre-contrast images only is a hard problem and
significantly deteriorates automatic tumor segmentation.

##  3 Method

In this work, our main goal is to extract and enhance the differential
contrast signal encoded in subtraction images of a contrast-enhanced (CE) T1w
image and the corresponding _pre-contrast_ T1w image
$\mathbf{x}_{\mathrm{PC}}\in\mathbb{R}^{n}$ (i.e. acquired before injecting
any contrast agent), where $n$ denotes the number of pixels or voxels. More
precisely, we consider CE images that are acquired after administering the
_standard-dose_ $\mathbf{x}_{\mathrm{SD}}\in\mathbb{R}^{n}$ or a _lower-dose_
$\mathbf{x}_{\mathrm{LD}}\in\mathbb{R}^{n}$. We denote the administered dose
of a CE scan by $d$ and set the weight-dependent standard dose to
$d_{\mathrm{SD}}=1$. Thus, low-dose CE images $\mathbf{x}_{\mathrm{LD}}$ are
acquired with dose levels $d_{\mathrm{LD}}\in(0,1)$.

Our approach consists of three essential concepts, which are related in Figure
1. First, a contrast-preserving preprocessing yields subtraction images
$\mathbf{z}_{\mathrm{LD}}=\mathbf{x}_{\mathrm{LD}}-\mathbf{x}_{\mathrm{PC}}$
and
$\mathbf{z}_{\mathrm{SD}}=\mathbf{x}_{\mathrm{SD}}-\mathbf{x}_{\mathrm{PC}}$.
Second, a conditional convolution neural network (CNN) removes noise and
artifacts from the low-dose subtraction image $\mathbf{z}_{\mathrm{LD}}$ and
predicts the contrast signal $\hat{\mathbf{y}}_{\mathrm{SD}}$. This network is
conditioned on metadata to account for physical parameters that affect the
general contrast enhancement behavior. Besides the relative administered dose
$d_{\mathrm{LD}}$, we consider the field strength of the scanner
$B\in\\{1.5,3\\}$ and the field strength-dependent relaxivity of the used
contrast agent $r_{B}\in\mathbb{R}_{+}$. In addition, we include an estimate
of the noise level $\sigma_{\mathrm{LD}}$ of the low-dose subtraction image
$\mathbf{z}_{\mathrm{LD}}$ motivated by the recent success of diffusion models
[18]. Third, during training we process $\mathbf{z}_{\mathrm{SD}}$ to extract
the contrast signal image $\mathbf{y}_{\mathrm{SD}}$, which only contains the
noise-free contrast signal.

Once the CNN is trained it predicts a noise-free contrast enhancement signal
image $\hat{\mathbf{y}}_{\mathrm{SD}}\in\mathbb{R}^{n}$. A particular
advantage of this approach is that this enhancement signal image can be added
to the pre-contrast _or_ low-dose image to generate anatomical images
$\hat{\mathbf{x}}_{\mathrm{SD}}\in\mathbb{R}^{n}$ or
$\hat{\mathbf{x}}_{\mathrm{SD}+}$, as shown in the top right of Figure 1.
Thus, we can use our model simultaneously for GBCA dose reduction and contrast
signal enhancement beyond standard-dose images.

###  3.1 Contrast-preserving preprocessing

To compute the subtraction images $\mathbf{z}_{\\{\mathrm{LD},\mathrm{SD}\\}}$
from a pre-contrast $\mathbf{x}_{\mathrm{PC}}$ and CE
$\mathbf{x}_{\\{\mathrm{LD},\mathrm{SD}\\}}$ image pair, we perform the
preprocessing pipeline illustrated in Figure 2. First, the brain region
$\mathbf{b}\in\\{0,1\\}^{n}$ is extracted by an affine registration of an
atlas brain towards the pre-contrast image $\mathbf{x}_{\mathrm{PC}}$ to
minimize the effects of motion artifacts from facial regions in the subsequent
steps. Second, the itk-elastix library is used to co-register the low-dose
$\mathbf{x}_{\mathrm{LD}}$ or standard-dose $\mathbf{x}_{\mathrm{SD}}$ images
to the corresponding pre-contrast images. Third, each scan is coarsely
normalized by mapping its $95\%$ intensity percentile of the brain region to
$1$. This coarse alignment of the intensity values is refined by a radiometric
registration step computing a scalar to minimize a robust distance as
performed in [9].

Figure 2: Illustration of the preprocessing steps for extracting the initial
low-dose subtraction image $\tilde{\mathbf{z}}_{\mathrm{LD}}$ from the pre-
contrast $\mathbf{x}_{\mathrm{PC}}$ and low-dose $\mathbf{x}_{\mathrm{LD}}$
image. The top images in the steps show the transformed low-dose image while
the bottom images visualize the effects on the subtraction image.

Let us denote these initial subtraction images by
$\tilde{\mathbf{z}}_{\\{\mathrm{LD},\mathrm{SD}\\}}$. We further processed
them to normalize the noise level within the images, which is often spatially
varying in parallel imaging [31]. Since the contrast signal is positive in the
subtraction images, we use the negative values in the brain region to roughly
estimate the noise levels $\sigma_{\mathrm{LD}},\sigma_{\mathrm{SD}}$ of the
low-dose and standard-dose subtraction image, respectively. Then, we mask out
potential contrast-enhancing pixels and artifacts by removing pixels that are
outside of $\pm 2\sigma_{\\{\mathrm{LD},\mathrm{SD}\\}}$. On this masked
subtraction image, we estimate the local mean
$\mathbf{m}_{\\{\mathrm{LD},\mathrm{SD}\\}}\in\mathbb{R}^{n}$ and standard
deviation $\mathbf{s}_{\\{\mathrm{LD},\mathrm{SD}\\}}\in\mathbb{R}^{n}$ using
large Gaussian filter kernels with a standard deviation of $16$ pixels. These
local noise characteristics are then used to correct field-inhomogeneities and
normalize the noise level within the subtraction images, such that

$\displaystyle\mathbf{z}_{\mathrm{LD}}\coloneqq\frac{\tilde{\mathbf{z}}_{\mathrm{LD}}-\mathbf{m}_{\mathrm{LD}}}{\mathbf{s}_{\mathrm{LD}}}\sigma_{\mathrm{LD}}\approx\sigma_{\mathrm{LD}}\frac{\tilde{\mathbf{z}}_{\mathrm{SD}}-\mathbf{m}_{\mathrm{SD}}}{\mathbf{s}_{\mathrm{SD}}}\eqqcolon\mathbf{z}_{\mathrm{SD}}.$
(1)

Note that we multiply both sides with the estimated noise level of the low-
dose subtraction image $\sigma_{\mathrm{LD}}$ to homogenize the noise
characteristics in both subtraction images. Therefore, the two subtraction
images differ essentially only in the regions where GBCA is accumulated or
artifacts occurred. This effect is illustrated by comparing the different
empirical densities in Figure 3.

Figure 3: Visualization of the empirical densities of the different
subtraction images of a prototypical case. The plot on the left depicts the
densities of the _initial_ subtraction images
$\tilde{\mathbf{z}}_{\mathrm{LD}}$ and $\tilde{\mathbf{z}}_{\mathrm{SD}}$
along with the fitted Gaussian model plotted in pink. The associated densities
of the _normalized_ subtraction images
$\mathbf{z}_{\mathrm{LD}},\mathbf{z}_{\mathrm{SD}}$ are shown on the right.
The normalization ensures that both densities overlap more nicely on the right
for small intensities. For larger intensities, the contrast-enhancing pixels
of the standard-dose subtraction images dominate. The pink line on the right
illustrates the decision of the discriminative CE signal extraction function
(2).

###  3.2 Contrast signal extraction

The results of the previous preprocessing steps are subtraction images
$\mathbf{z}_{\mathrm{LD}}$ and $\mathbf{z}_{\mathrm{SD}}$ after administering
a low or standard dose, respectively. These images have approximately the same
noise characteristics due to normalization.

Next, we extract the _contrast signal_ image $\mathbf{y}_{\mathrm{SD}}$ from
the standard-dose subtraction image $\tilde{\mathbf{z}}_{\mathrm{SD}}$. This
image serves as the ground truth for training the CNN. To prevent the CNN from
learning to synthesize noise, we aim for noise-free target images. To this
end, we use a simple discriminative model to compute a CE mask

$\displaystyle\mathbf{p}_{\mathrm{SD}}=f(\mathbf{z}_{\mathrm{SD}})\coloneqq\mathrm{sigmoid}(w\mathbf{z}_{\mathrm{SD}}+b)\in[0,1]^{n},$
(2)

where the parameters $w,b\in\mathbb{R}$ of the pixel-wise function $f$ are
chosen such that $f(\sigma_{\mathrm{LD}})=0.01$ and
$f(4\sigma_{\mathrm{LD}})=0.99$. These parameters represent a trade-off
between noise suppression and preservation of faint CE signals. The pink line
on the right of Figure 3 visualizes the resulting discriminative model. This
model quickly saturates to 1 and thereby preserves the CE signal for a wide
range. Finally, the target images are computed by

$\mathbf{y}_{\mathrm{SD}}=\mathbf{p}_{\mathrm{SD}}\odot\mathbf{z}_{\mathrm{SD}},$

where $\odot$ denotes the Hadamard product.

Figure 4 illustrates the contrast signal extraction steps for a prototypical
case. As can be seen by the contrast signal image $\mathbf{y}_{\mathrm{SD}}$
on the right of the first row, the noise is almost completely suppressed,
while the CE signal is preserved. The second row compares the standard-dose
image $\mathbf{x}_{\mathrm{SD}}$, utilized as ground truth in previous
approaches, and our implicit standard-dose image generated by adding the CE
signal to the pre-contrast image, i.e.
$\mathbf{x}_{\mathrm{PC}}+\mathbf{y}_{\mathrm{SD}}$. These images essentially
differ only in the non-enhancing regions due to noise and inhomogeneities of
the bias field. By using the contrast signal $\mathbf{y}_{\mathrm{SD}}$ as
ground truth, we avoid learning to predict the degradations shown in the
bottom-right image.

Figure 4: The top row depicts the contrast signal extraction from the
standard-dose subtraction image $\mathbf{z}_{\mathrm{SD}}$. The pixel-wise
multiplication with the CE mask $\mathbf{p}_{\mathrm{SD}}$ yields the target
contrast signal image $\mathbf{y}_{\mathrm{SD}}$. The bottom row visualizes
the differences between the standard-dose image $\mathbf{x}_{\mathrm{SD}}$ and
the standard-dose image implicitly generated by adding the contrast signal to
the pre-contrast image $\mathbf{x}_{\mathrm{PC}}+\mathbf{y}_{\mathrm{SD}}$.
Note that both standard-dose images only differ by the noise in non-enhancing
regions.

###  3.3 CNN architecture

The target of the CNN is to suppress noise and artifacts within the low-dose
subtraction image $\mathbf{z}_{\mathrm{LD}}$ and enhance the contrast signal
to predict a synthetic contrast signal image $\hat{\mathbf{y}}_{\mathrm{SD}}$
resembling the target contrast signal $\mathbf{y}_{\mathrm{SD}}$. Besides
$\mathbf{z}_{\mathrm{LD}}$, we feed the low-dose image
$\mathbf{x}_{\mathrm{LD}}$ and the local standard deviation
$\mathbf{s}_{\mathrm{LD}}$ into the model for better distinction of contrast
signal from noise and artifacts.

As sketched in Figure 1, we use a U-Net-type architecture with residual
blocks, which has proven effective in various image processing tasks [32]. We
adopt only 3D local operators (convolution, pooling, interpolation, etc.) in
its design since contrast enhancement is a local phenomenon and the
acquisition orientation is arbitrary in clinical practice. In addition, we
perform an isotropic regridding if required. For downsampling, we use strided
convolutions and tri-linear interpolation for upsampling. At each scale, we
apply two conditional residual blocks (CRBs) before down- and after
upsampling.

To fuse the upsampled features with the skip connections, we propose a gated
upsampling block. Let $\mathbf{f}_{u}\in\mathbb{R}^{m}$ be the trilinearly
upsampled features, $\mathbf{f}_{s}\in\mathbb{R}^{m}$ the features from the
skip connection, and $\mathbf{f}_{u/s}=(\mathbf{f}_{u}\
\mathbf{f}_{s})^{\top}\in\mathbb{R}^{2m}$. Then, the fusion block combines the
features by the gating mechanism

$\mathbf{f}=\mathrm{sigmoid}(\widetilde{\mathbf{W}}_{1}\mathbf{f}_{u/s})\odot\mathbf{W}_{1}\mathbf{f}_{u}+\mathrm{sigmoid}(\widetilde{\mathbf{W}}_{2}\mathbf{f}_{u/s})\odot\mathbf{W}_{2}\mathbf{f}_{s},$

where $\widetilde{\mathbf{W}}_{\\{1,2\\}}\in\mathbb{R}^{2m\times m}$ and
$\mathbf{W}_{\\{1,2\\}}\in\mathbb{R}^{m\times m}$ are learned $1\times 1$
convolutions. Thereby, the block can adaptively select or combine both
features.

As discussed before, the overall contrast enhancement strength depends on
various physical factors such as the field strength $B$, the contrast agent
relaxivity $r_{B}$, or the relative administered dose $d_{\mathrm{LD}}$. To
condition the CNN on these metadata, we stack these scalars along with the
low-dose noise level estimate $\sigma_{\mathrm{LD}}$ into a condition vector
$\mathbf{c}\in\mathbb{R}^{4}$. This vector is processed by a two-layer fully
connected network and the resulting embedding $\mathbf{e}\in\mathbb{R}^{128}$
is injected into each CRB using projection matrices. For the CRB design, we
follow the approach of Ho et al. [18], which has proven to be effective for
diffusion-based generative models. In detail, the conditional information is
added as a bias term along the channel dimension in each CRB. Thereby, the
model selects the information that is relevant during training. The exact
model configuration can be found in the supplementary material or the
associated github repository (released upon acceptance).

###  3.4 Training and inference

Given a dataset
$\mathtt{DS}=(\mathbf{x}_{\mathrm{PC}}^{i},\mathbf{x}_{\mathrm{LD}}^{i},\mathbf{x}_{\mathrm{SD}}^{i},d_{\mathrm{LD}}^{i},B^{i},r_{B}^{i})_{i=1}^{N}$,
we learn the parameters $\theta$ of the CNN for contrast signal extraction and
enhancement by minimizing the loss

$\mathcal{L}(\theta)\coloneqq\sum_{i=1}^{N}\ell(\hat{\mathbf{y}}_{\mathrm{SD}}^{i}(\theta),\mathbf{y}_{\mathrm{SD}}^{i}),$

where the sample-wise loss reads as

$\ell(\hat{\mathbf{y}}_{\mathrm{SD}},\mathbf{y}_{\mathrm{SD}})=\sum_{j=1}^{n}\left([\mathbf{b}]_{j}+\lambda_{1}+\lambda_{2}[\mathbf{p}_{\mathrm{SD}}]_{j}\right)d\left([\hat{\mathbf{y}}_{\mathrm{SD}}]_{j},[\mathbf{y}_{\mathrm{SD}}]_{j}\right).$

Here $\mathbf{b}\in\\{0,1\\}^{n}$ is the registered brain mask of the atlas,
$\mathbf{p}_{\mathrm{SD}}\in[0,1]^{n}$ is the CE mask, and
$d:\mathbb{R}\times\mathbb{R}\to\mathbb{R}_{+}$ is the Huber loss function
with $\epsilon=0.1$. The hyperparameters $\lambda_{1},\lambda_{2}>0$ control
the weight on regions outside the brain mask and contrast-enhancing areas,
respectively. Note that we did not explicitly model any artifacts that
typically occur in MRI due to, for instance, patient motion. These artifacts
are typically scarce in the dataset and not correlated in the different
images. Thus, the loss function will not promote any CE in these regions.

To better match the standard-dose images $\mathbf{x}_{\mathrm{SD}}$, which are
not noise normalized, we revert the noise normalization (1) at inference. Let
$\hat{\mathbf{y}}_{\mathrm{SD}}$ denote the CNN’s noise-normalized output.
Then, the corresponding reverted output is defined as

$\tilde{\mathbf{y}}_{\mathrm{SD}}=\gamma\mathbf{s}_{\mathrm{LD}}\odot\hat{\mathbf{y}}_{\mathrm{SD}},$

where $\gamma>0$ accounts for different scaling factors in the brain on the
training set and is computed by

$\gamma=\frac{1}{N}\sum_{i=1}^{N}\frac{\langle\mathbf{b}^{i},\mathbf{s}_{\mathrm{SD}}^{i}\rangle}{\langle\mathbf{b}^{i},\mathbf{s}_{\mathrm{LD}}^{i}\rangle}.$

Then, the synthesized standard-dose and enhanced images are given by
$\tilde{\mathbf{x}}_{\mathrm{SD}}=\mathbf{x}_{\mathrm{PC}}+\tilde{\mathbf{y}}_{\mathrm{SD}},\
\tilde{\mathbf{x}}_{\mathrm{SD}+}=\mathbf{x}_{\mathrm{LD}}+\tilde{\mathbf{y}}_{\mathrm{SD}}$,
respectively.

##  4 Numerical results

In this section, we evaluate our proposed approach quantitatively and
qualitatively for synthetic and real data from multiple clinical sites. In
particular, we focus on lesions and metastases, as previous approaches have
struggled to enhance these small and often subtle pathologies properly.

For the evaluation, we considered the recent approaches of Pasumarthi et al.
[8] and Ammari et al. [14]. We denote their methods as Pa-2.5D and Am-3D,
respectively. We used the available source code of Ammari et al., while we
reimplemented the method of Pasumarthi et al. To train both models, we used
the hyperparameters reported in their respective papers. Our models were
trained for $2\cdot 10^{5}$ iterations with a batch size of 16 and patches of
size $96^{3}$ on two A40 GPUs, which took approximately 4 days each. The
hyperparameters for the loss function are set to
$(\lambda_{1},\lambda_{2})=(0.01,1)$ to focus on contrast-enhancing regions in
the brain. The resulting loss was optimized using Adam [33] with a learning
rate of $10^{-4}$ and momentum parameters $(0.9,\,0.999)$. During training the
learning rate was annealed to $10^{-6}$ using a cosine schedule. To limit the
influence of artifacts induced by the boundary handling, we used an 8-pixel
margin for evaluating the loss.

To exclude errors due to intensity shifts in the quantitative evaluation, we
registered the outputs of Am-3D and Pa-2.5D radiometrically to the pre-
contrast images by minimizing the difference in non-CE regions in the brain as
in [9]. We consider the peak-signal-to-noise ratio (PSNR) evaluated on entire
images PSNRI, inside the brain PSNRB, or individual lesions PSNRL to assess
the general image quality. These regions are defined by the associated brain
and lesion segmentation masks. Furthermore, the CE of each lesion is assessed
by computing the mean and maximal relative enhancement defined as

$\overline{c}(\mathbf{x})=\frac{\sum_{i\in
L}[\mathbf{x}-\mathbf{x}_{\mathrm{PC}}]_{i}}{\sum_{i\in
L}[\mathbf{x}_{\mathrm{SD}}-\mathbf{x}_{\mathrm{PC}}]_{i}},\,\widehat{c}(\mathbf{x})=\frac{\max_{i\in
L}[\mathbf{x}-\mathbf{x}_{\mathrm{PC}}]_{i}}{\max_{i\in
L}[\mathbf{x}_{\mathrm{SD}}-\mathbf{x}_{\mathrm{PC}}]_{i}}.$

Here $L$, is the subset of pixels included in the segmentation mask of the
particular lesion.

###  4.1 Datasets

####  4.1.1 Synthetic low-dose METS

As a first dataset, we generated a synthetic low-dose brain metastasis (SLD-
METS) dataset to ease data access and reproducibility. To this end, we used
the openly available code of Pinetz et al. [9] to synthesize low-dose images
$\mathbf{x}_{\mathrm{LD}}$ from pre-contrast and standard-dose image pairs
$(\mathbf{x}_{\mathrm{PC}},\mathbf{x}_{\mathrm{SD}})$ of the BraTS-METS
dataset [19]. We selected this dataset since it includes various small
lesions, which are challenging for dose reduction techniques [12]. To match
the dose to the dominant one of the subsequent real low-dose dataset, we
generated low-dose scans at a dose of $33\%$ of the standard dose for all
$238$ samples in the BraTS-METS training set. We randomly divided the
resulting set further such that $190$ samples were used for training and the
remaining $48$ for testing. Note that the data of the BraTS-METS dataset does
not include any meta information such as the field strength of the scanner or
the administered contrast agent type.

####  4.1.2 Real low-dose

At the University Hospital Bonn, we prospectively collected $761$ scans with
multiple contrast agent doses using Ingenia $1.5/3$T and Achieva $1.5/3$T
scanners (Philips Healthcare, Best, Netherlands). For each sample, pre-
contrast $\mathbf{x}_{\mathrm{PC}}$, low-dose $\mathbf{x}_{\mathrm{LD}}$, and
standard-dose images $\mathbf{x}_{\mathrm{SD}}$ at the same resolution were
acquired in addition to the routine clinical protocol. The GBCA Gadobutrol
($r_{1.5}=4.6$ and $r_{3}=4.5$) or Gadoterate ($r_{1.5}=3.9$ and $r_{3}=3.4$)
were administered [15]. For training, validation and testing, we split the
dataset randomly into $80\%/10\%/10\%$ subsets. We denote this dataset as the
real low-dose (RLD) dataset.

$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\hat{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\hat{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\hat{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\hat{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\hat{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\hat{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\hat{\mathbf{x}}_{\mathrm{SD}}$ (ours)

Figure 5: Qualitative comparison for test samples of the SLD-METS dataset. The
zooms highlight the metastasis location. By design, our approach adds the CE
signal to the input image, thereby preserving its image quality. The input
images at the top have the same resolution, while they strongly differ at the
bottom. Thus, the image quality of our output $\hat{\mathbf{x}}_{\mathrm{SD}}$
is poor in the bottom row and decent in the top row.

####  4.1.3 Real low-dose METS

To investigate the generalization to different settings, we additionally
collected a real low-dose metastasis (RLD-METS) dataset prospectively at an
external site. In detail, we used $20$ samples with $48$ metastases in total.
Here the GBCA Gadoterate was administered and the patients were scanned using
$3$T Skyra or $1.5$T Aera (Siemens Healthcare, Erlangen Germany) MRI devices.
In addition, a radiology resident (three years of experience in neuroimaging)
segmented all metastasis reported in the diagnostic findings. Further details
of both real-world datasets can be found in the supplementary material.

###  4.2 Evaluation using synthetic low-dose images

For this experiment, we consider the SLD-METS dataset, where each sample
consists of a pre-contrast, low-dose, and standard-dose T1w image along with
pre-contrast T2w and FLAIR images. As this dataset neither includes ADC images
nor scan-specific metadata, we had to adapt the methods slightly. We replaced
the additional ADC input of Am-3D with the available T2w image. For our
method, we shrank the condition to just include the estimated noise level
$\sigma_{\mathrm{LD}}$ due to lacking information about the scanner and
contrast agent. Note that all low-dose images are synthesized for the dose
$d_{\mathrm{LD}}=0.33$. Thus, this dataset adheres to the assumptions of the
competing DL-based dose reduction methods. Furthermore, we discarded the noise
normalization step (1) of our model due to strong resolution mismatches of
pre-contrast and CE images in numerous samples of the BraTS-METS dataset.

Table 1 contains a quantitative evaluation of the considered dose reduction
approaches. The PSNRI column reports the average score of the $48$ test
samples and the PSNRL represents the mean of all $351$ lesions in the test
set. While our approach is inferior in terms of PSNR score on the entire
image, we significantly outperform all competitors on lesions. A $t$-test of
our approach and the closest competitor yielded the $p$-values presented in
the last row. The worse PSNR scores on the entire images of our model
originate from the low resolution of many pre-contrast images in the SLD-METS
dataset compared to the available post-contrast images. By design, our model
only focuses on the accurate prediction of the CE signal. Thus, it cannot
change the overall image quality of the input images. Taking this into
account, the PSNRI score of our implicit ground truth
$\mathbf{x}_{\mathrm{PC}}+\mathbf{y}_{\mathrm{SD}}$ is $42.48$dB, which is the
upper bound for our approach and below the $44.02$dB of Ps-2.5D. This can also
be seen in the qualitative comparison depicted in the bottom row of Figure 5.
The image quality of $\mathbf{x}_{\mathrm{PC}}$ is worse compared to
$\mathbf{x}_{\mathrm{SD}}$. In particular, the boundaries between the white
and gray matter or the ventricles are hardly visible in
$\mathbf{x}_{\mathrm{PC}}$ as it was acquired with a lower in-plane
resolution. In contrast to Am-3D and Pa-2.5D, our approach cannot overcome
this lack of resolution. Nevertheless, the CE signal is also well captured by
our approach.

Table 1: Quantitative comparison on the SLD-METS dataset.

Model | PSNRI | PSNRL | $\overline{c}$ | $\widehat{c}$
---|---|---|---|---
$\mathbf{x}_{\mathrm{LD}}$ | $37.63\pm 2.82$ | $20.52\pm 2.90$ | $0.43\pm 0.04$ | $0.45\pm 0.04$
Am-3D [14] | $42.74\pm 2.62$ | $27.41\pm 4.74$ | $0.84\pm 0.14$ | $0.90\pm 0.08$
Pa-2.5D [8] | $\mathbf{44.02}\pm 1.94$ | $29.96\pm 3.74$ | $0.87\pm 0.10$ | $0.88\pm 0.08$
$\hat{\mathbf{x}}_{\mathrm{SD}}$ (ours) | $40.95\pm 4.88$ | $\mathbf{32.89}\pm 4.34$ | $\mathbf{0.98}\pm 0.05$ | $\mathbf{0.97}\pm 0.04$
$p<0.001$ | $p<0.001$ | $p<0.001$ | $p<0.001$

The average mean and maximal relative enhancement across the $351$ lesions is
listed in the $\overline{c}$ and $\widehat{c}$ columns of Table 1. Our
approach significantly improves the predicted CE strength compared to Am-3D
and Pa-2.5D. Note that $\overline{c}$ is not only sensitive to the CE strength
(as $\widehat{c}$), but also to a lesion’s internal morphology and border
delineation. A qualitative comparison of a typical lesion is visualized in the
zooms at the top row of Figure 5. While Am-3D struggles to predict the
internal morphology of the lesions, the CE strength of Pa-2.5D is too low. In
contrast, our approach predicts the right contrast strength, morphology, and
boundary of the lesion.

###  4.3 Evaluation on real low-dose images

$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$ (ours)Philips Achieva
3TGadobutrol $d_{\mathrm{LD}}=0.10$Philips Ingenia 1.5TGadoterate
$d_{\mathrm{LD}}=0.33$$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$ (ours)Philips Achieva
3TGadobutrol $d_{\mathrm{LD}}=0.10$Philips Ingenia 1.5TGadoterate
$d_{\mathrm{LD}}=0.33$$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$ (ours)Philips Achieva
3TGadobutrol $d_{\mathrm{LD}}=0.10$Philips Ingenia 1.5TGadoterate
$d_{\mathrm{LD}}=0.33$$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$ (ours)Philips Achieva
3TGadobutrol $d_{\mathrm{LD}}=0.10$Philips Ingenia 1.5TGadoterate
$d_{\mathrm{LD}}=0.33$$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$ (ours)Philips Achieva
3TGadobutrol $d_{\mathrm{LD}}=0.10$Philips Ingenia 1.5TGadoterate
$d_{\mathrm{LD}}=0.33$$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$ (ours)Philips Achieva
3TGadobutrol $d_{\mathrm{LD}}=0.10$Philips Ingenia 1.5TGadoterate
$d_{\mathrm{LD}}=0.33$$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$ (ours)Philips Achieva
3TGadobutrol $d_{\mathrm{LD}}=0.10$Philips Ingenia 1.5TGadoterate
$d_{\mathrm{LD}}=0.33$

Figure 6: Qualitative evaluation on the RLS dataset. The top row depicts a
$33\%$ low-dose sample, where the yellow zoom highlights the lesion. The
bottom row shows a case with a $10\%$ real low-dose image and the lesion is
marked by the yellow circles.

Next, Am-3D, Pa-2.5D, and our approach trained on the RLD dataset are
evaluated on the $77$ RLD test images. Here, ADC images are additionally fed
into Am-3D as required [14].

The quantitative results for this dataset are depicted in Table 2. In addition
to the structured similarity index (SSIM), we computed the mean absolute error
in CE brain regions MAECE as suggested in [9] to compare the contrast
enhancement due to a lack of lesion segmentation masks for this dataset. For
Am-3D, only a PSNR score in the brain region w.r.t. their corresponding ground
truth is reported as their preprocessing requires skull stripping and a non-
invertible contrast normalization step. The results in Table 2 show that our
approach significantly outperforms previous approaches in terms of PSNR score
in the brain and the MAE in the contrast-enhancing brain regions. The
corresponding $p$-values of a $t$-test are listed in the last row of Table 2.
As a result, our approach predicts a contrast enhancement signal closer to the
ground truth. Moreover, the PSNR values for the whole image and the SSIM are
also improved, although, unlike Pa-2.5D, the SSIM is not part of our loss
function. Note that this is in contrast to the PSNRI scores for the SLD-METS
dataset; however, in the RLD dataset, all T1w images are acquired with the
same resolution and have the same image quality.

Table 2: Quantitative evaluation on the RLD dataset.

Model | PSNRI | PSNRB | SSIM | MAECE
---|---|---|---|---
$\mathbf{x}_{\mathrm{LD}}$ | $29.32\pm 2.27$ | $29.96\pm 3.54$ | $.899\pm.038$ | $0.28\pm 0.014$
Am-3D [14] | $-$ | $30.77^{\ast}\pm 2.40$ | $-$ | $-$
Pa-2.5D [8] | $32.45\pm 3.33$ | $33.66\pm 4.36$ | $.916\pm.049$ | $0.17\pm 0.07$
$\tilde{\mathbf{x}}_{\mathrm{SD}}$ (ours) | $\mathbf{32.63}\pm 2.64$ | $\mathbf{35.05}\pm 4.19$ | $\mathbf{.919}\pm.038$ | $\mathbf{0.15\pm 0.05}$
$p=0.33$ | $p<0.001$ | $p=0.43$ | $p=0.011$
∗Am-3D is evaluated w.r.t. its associated ground truth.

A qualitative comparison of RLD test samples is shown in Figure 6. The
required skull stripping and non-linear intensity normalization of Am-3D are
readily apparent. Thus, a quantitative comparison to the target
$\mathbf{x}_{\mathrm{SD}}$ is not reasonable. Nevertheless, the CE signal
strength in pathological regions (highlighted by the yellow circles) is well
visible despite too smooth output images. The internal morphology and the
border delineation also exhibit blurring. Pa-2.5D yields better image quality
but the contrast strength in pathological regions is not well captured. In
particular, the contrast signal overshoots for the $33\%$ low-dose scan in the
first row and undershoots for the $10\%$ scan at the bottom, although the
training dataset contains low-dose images at both dose levels. Our approach
yields the highest image quality and predicts the contrast enhancement more
accurately due to the focus on the enhancement signal and the conditional
embeddings. As a result, our approach generates CE images preserving the image
quality of the inputs by avoiding the synthesis of anatomical or noise
patterns.

###  4.4 Generalization on real low-dose metastases

To analyze the generalization capabilities of the different approaches, we use
the RLD-METS dataset. In addition, we compare the performance of the models
trained on the SLD-METS and RLD training sets. To apply the models trained on
the SLD-METS dataset, we resampled all images to $1\text{mm}^{3}$ and
preprocessed them using the CaPTK111https://cbica.github.io/CaPTk/ toolbox
according to BraTS requirements. Note that this preprocessing failed on $4$
test samples with $5$ lesions in total, which we excluded for the evaluation
of the models trained on SLD-METS. Further, we exclude Am-3D [14] trained on
RLD due to limited comparability caused by the differences in preprocessing.
Consequently, the images were not skull-stripped and processed on the acquired
resolution for the Pa-2.5D and our model trained on the RLD dataset.

Table 3: Quantitative metrics on the RLD-METS dataset. The models in the top
row were trained on the SLD-METS dataset and the bottom rows were trained on
the RLS dataset.

| Model | PSNRB | PSNRL | $\overline{c}$ | $\widehat{c}$
---|---|---|---|---|---
SLD-METS | $\mathbf{x}_{\mathrm{LD}}$ | $27.37\pm 1.07$ | $19.70\pm 3.01$ | $0.36\pm 0.32$ | $0.46\pm 0.10$
Am-3D [14] | $26.50\pm 1.81$ | $23.41\pm 2.53$ | $0.82\pm 0.21$ | $\mathbf{0.85}\pm 0.19$
Pa-2.5D [8] | $28.27\pm 2.10$ | $25.75\pm 2.44$ | $0.77\pm 0.22$ | $0.82\pm 0.18$
$\tilde{\mathbf{x}}_{\mathrm{SD}}$ (ours) | $\mathbf{29.21}\pm 2.26$ | $\mathbf{26.28}\pm 2.78$ | $\mathbf{0.84}\pm 0.16$ | $0.84\pm 0.17$
$p<0.001$ | $p=0.186$ | $p=0.64$ | $p=0.49$
RLD | $\mathbf{x}_{\mathrm{LD}}$ | $31.13\pm 1.76$ | $23.52\pm 4.32$ | $0.39\pm 0.09$ | $0.54\pm 0.12$
Pa-2.5D [8] | $32.82\pm 1.98$ | $27.39\pm 3.47$ | $0.58\pm 0.38$ | $0.70\pm 0.27$
$\tilde{\mathbf{x}}_{\mathrm{SD}}$ (ours) | $\mathbf{33.13}\pm 2.45$ | $\mathbf{28.38}\pm 3.98$ | $0.78\pm 0.34$ | $0.83\pm 0.36$
$p=0.187$ | $p=0.047$ | $p<0.001$ | $p=0.004$
$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours) | $-$ | $-$ | $\mathbf{1.17}\pm 0.42$ | $\mathbf{1.33}\pm 0.42$

SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)SLD-
METS$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Am-3D
[14]Pa-2.5D [8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)RLD$\mathbf{x}_{\mathrm{PC}}$$\mathbf{x}_{\mathrm{LD}}$$\mathbf{x}_{\mathrm{SD}}$Pa-2.5D
[8]$\tilde{\mathbf{x}}_{\mathrm{SD}}$
(ours)$\tilde{\mathbf{x}}_{\mathrm{SD}+}$ (ours)

Figure 7: Qualitative comparison on the RLD-METS dataset. The top row depicts
the outputs for the models trained on the SLD-METS dataset. The bottom two
rows show the outputs of the models trained on the RLD dataset. The samples in
the first and second rows are identical. All images were acquired by a Siemens
Aera 1.5T scanner using $d_{\mathrm{LD}}=0.33$. The lesions are highlighted by
yellow zooms.

Table 3 lists the resulting PSNR and relative contrast enhancement scores. The
top rows show the metrics for the models trained on the SLD-METS dataset, and
the bottom rows those of the models trained using RLD. Our models yield the
best PSNR scores in the brain regions and for lesions. As before, we computed
$t$-test of our models with its closest competitors, The resulting $p$-values
are shown below the corresponding $\tilde{\mathbf{x}}_{\mathrm{SD}}$ (our)
rows. For our SLD-METS model, the PSNR improvement is significant in the brain
region, while our RLD model significantly outperforms the other models for
PSNRL. The outputs of Am-3D yield the best mean and maximal contrast
enhancement $\overline{c}/\widehat{c}$ for the models trained on the SLD-METS
dataset. However, the improvement is not significant compared to our approach.
For the models trained on the RLD dataset, our approach is superior by a clear
margin. In general, there is a consistent improvement in image quality for all
models when trained on real low-dose (RLD) images. Interestingly, the relative
contrast enhancement scores decrease for all methods, which might be because
the SLD-METS dataset only contains pathological images.

A qualitative comparison of the different models is visualized in Figure 7.
The first and second rows highlight the same lesion of a case. The first row
shows the outputs of the models trained on the SLD-METS dataset, using the
CaPTK preprocessing of BraTS. Am-3D[14] enhances the lesion the strongest but
also emphasizes streaking artifacts, resulting in degraded image quality. In
contrast, the middle row depicts the results of the models trained on the RLD
dataset. Comparing the zooms of the first and second rows, we see that the RLD
models generate a better image quality due to operating on higher resolutions.
In addition, the output of Pa-2.5D [8] in the second row shows a hardly
enhanced lesion, which might be due to multiple dose levels in the RLD
dataset. The last row presents the case in which our model performed the worst
in terms of PSNRL compared to Pa-2.5D. Nevertheless, the lesion is also
clearly visible in our output. Finally, we highlight that by adding the
predicted contrast signal $\tilde{\mathbf{y}}_{\mathrm{LD}}$ to the low-dose
image we obtain images $\tilde{\mathbf{x}}_{\mathrm{SD}}$ with a stronger CE.
The associated mean and maximal relative enhancements listed at the bottom of
Table 3 confirm that $\tilde{\mathbf{x}}_{\mathrm{SD}+}$ exceeds the CE of
standard-dose images.

###  4.5 Embedding analysis

Next, we perform a qualitative analysis of the embedding. To this end, we
extracted the embedding vectors $(\mathbf{e}^{i})_{i=1}^{N}$ for all training
samples of the RLD-dataset. Using the principal component analysis (PCA) and
the t-distributed stochastic neighbor embedding (tSNE) [34], we computed
mappings from the $128$-dimensional embedding space into 2D, shown in Figure
8. The PCA plot on the left demonstrates an almost linear dependency of the
embedding on the field strength $B\in\\{1.5,3\\}$ and the administered dose
$d_{\mathrm{LD}}\in(0,1)$. In contrast, the tSNE plot on the right shows that
the embedding vectors form clusters depending on the contrast agent’s
relaxivity $r_{B}\in\mathbb{R}_{+}$. Furthermore, the standard deviation
$\sigma_{\mathrm{LD}}$ smoothly changes in each cluster. Thus, each
acquisition parameter affects the embedding, which in turn controls our
models’ output.

Figure 8: Visualization of the embedding vectors on the RLD training set. The
PCA plot on the left shows the dependency on the field strength $B$ and dose
$d_{\mathrm{LD}}$, while the tSNE plot on the right resolves the effect of the
contrast agent relaxivity $r_{B}$ and the noise level $\sigma_{\mathrm{LD}}$
on the embedding.

##  5 Conclusion

In this work, we presented a novel approach for GBCA dose reduction using
subtraction images. We demonstrated that our conditional CNN denoises and
enhances initially degraded subtraction images of low-dose and pre-contrast
images to extract the contrast signal. Moreover, we demonstrated that adding
the resulting contrast signal to the pre-contrast or low-dose image yields
realistic CE images, which can be enhanced beyond the standard dose. The
extensive numerical evaluations on synthetic and real low-dose images showed
that our approach significantly outperforms the state-of-the-art, particularly
for lesions in the brain. In addition, embedding meta-information of the
acquired images such as the injected contrast agent type and dose, or the
scanner’s field strength is beneficial for predicting the enhancement strength
more accurately. In the future, we aim to evaluate our approach clinically to
reduce GBCA administration and simultaneously increase the visibility of
pathologies.

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## Supplementary material

### CNN architecture details

Here, we provide additional details of the used CNN architecture, illustrated
in Figure 9. The model operates on 4 scales and each image/feature channel is
downsampled using down convolution. More precisely, we use pre-defined
$5\times 5\times 5$ smoothing kernels according to Zhang [35] to promote
shift-invariance. Further, we upsample lower-resolution features using
trilinear interpolation operators. We use $3\times 3\times 3$ convolution
filters in every convolution layer and extract on the finest scale $32$
feature channels. The number of features is doubled after every downsampling
operation. For each non-linearity, we use the SiLU activation function within
the network. At the model output, we apply a ReLU activation function to
ensure the positivity of the predicted contrast enhancement.

The main building blocks of the architecture are conditional residual blocks
(CRBs). Each of these blocks is conditioned onto the embedding by projecting
the embedding vector the corresponding number of feature channels before
adding the result to the residual path. This design is adapted from Ho et al.
[18] to account for 3D features. In addition, we removed the normalization
layers.

We introduced the gated upsampling block to merge upsampled features with the
skip connections. Let $\mathbf{f}_{u}\in\mathbb{R}^{m}$ be the trilinearly
upsampled features, $\mathbf{f}_{s}\in\mathbb{R}^{m}$ the features from the
skip connection, and $\mathbf{f}_{u/s}=(\mathbf{f}_{u}\
\mathbf{f}_{s})^{\top}\in\mathbb{R}^{2m}$. Then, the fusion layer combines the
features by the gating mechanism

$\mathbf{f}=\mathrm{sigmoid}(\widetilde{\mathbf{W}}_{1}\mathbf{f}_{u/s})\odot\mathbf{W}_{1}\mathbf{f}_{u}+\mathrm{sigmoid}(\widetilde{\mathbf{W}}_{2}\mathbf{f}_{u/s})\odot\mathbf{W}_{2}\mathbf{f}_{s},$

where $\widetilde{\mathbf{W}}_{\\{1,2\\}}\in\mathbb{R}^{2m\times m}$ and
$\mathbf{W}_{\\{1,2\\}}\in\mathbb{R}^{m\times m}$ are learned $1\times 1$
convolutions. Thereby, the layer can adaptively select or combine both
features.

Figure 9: Illustration of the conditional U-net-like CNN. The input
images$(\mathbf{z}_{\mathrm{LD}},\mathbf{x}_{\mathrm{LD}})$ are downscaled and
then fed into the conditional residual blocks (CRBs) of the corresponding
scale. The embedding $\mathbf{e}$ is projected to the corresponding number of
channels in each CRB and added to the residual path. To fuse features of
different scales, the gated upsampling blocks implement a pixel-wise gating
approach.

### Datasets

#### RLD dataset

For $761$ subjects, three T1-weighted (T1w) images with multiple GBCA dose
levels ($d_{\mathrm{PC}}=0$, $d_{\mathrm{LD}}\in[0.1,0.33]$,
$d_{\mathrm{SD}}=1$) were acquired at $2$ clinical locations within our
institution on $5$ different MRI systems and $2$ Gadolinium-based contrast
agent (GBCA) types. MRI systems included 3T Achieva and Ingenia as well as
1.5T Achieva and Ingenia (Philips Healthcare, Best, Netherlands). Each triplet
of T1w images was acquired at the same scanner within a time frame of 30min
using the same acquisition sequence. Table 4 lists the distribution of the
samples to the different settings.

Table 4: Overview of the real low-dose (RLD) dataset.

System | Field strength | GBCA | Sequence | $T_{E}$ | $T_{R}$ | $\alpha$ | Count
---|---|---|---|---|---|---|---
Achieva | 1.5 | Gadobutrol | T1-TFE | 3.26-3.45 | 7.23-7.55 | 8 | 178
3 | T1-TFE | 2.96-3.33 | 6.61-7.22 | 8 | 435
Ingenia | 1.5 | Gadoterate | T1-FFE | 3.59-3.60 | 15.15-15.51 | 30 | 119
3 | T1-TFE | 3.69-3.89 | 7.52-7.93 | 8 | 29
Total |  |  |  |  |  |  | 761

#### RLD-METS dataset

In contrast to the RLD dataset above, this dataset was acquired at an external
site. In detail, $20$ subjects were scanned to obtain three T1w scans with
different dose levels ($d_{\mathrm{PC}}=0$, $d_{\mathrm{LD}}=0.33$,
$d_{\mathrm{SD}}=1$) using the GBCA Gadoterate. Magnets included 3T Skyra and
1.5T Aera (Siemens Healthcare, Erlangen Germany). All images were acquired
using a 3D TurboFlash pulse sequence; the corresponding parameters along with
the distribution of the samples are depicted in Table 5 In total, there are
$47$ lesions in the RLD-METS dataset and each sample has at least one
metastasis and at most 5. Thus, these pathologies are homogeneously
distributed across the samples.

Table 5: Overview of the real RLD-METS test set.

System | Field strength | $T_{E}$ | $T_{R}$ | $\alpha$ | Count
---|---|---|---|---|---
Aera | 1.5 | 3.15 | 1920.0-1990.0 | 10 | 11
Skyra | 3 | 2.20 | 1640.0-2000.0 | 9 | 9
Total |  |  |  |  | 20