11institutetext:  Department of Integrative Oncology, BC Cancer Research
Institute, Vancouver, BC, Canada 22institutetext: BC Cancer Research
Institute, Vancouver, BC, Canada 33institutetext: BC Cancer, Vancouver, BC,
Canada 44institutetext: Functional Imaging, BC Cancer, Vancouver, BC, Canada
55institutetext: Department of Radiology, University of British Columbia,
Vancouver, BC, Canada 66institutetext: Department of Biomedical Engineering,
University of British Columbia, Vancouver, BC, Canada 77institutetext:
Department of Physics and Astronomy, University of British Columbia,
Vancouver, BC, Canada

# How To Segment in 3D Using 2D Models: Automated 3D Segmentation of Prostate
Cancer Metastatic Lesions on PET Volumes Using Multi-Angle Maximum Intensity
Projections and Diffusion Models

Amirhosein Toosi 1155 0000-0001-6432-9428 Sara Harsini 22 0000-0001-6196-6982
François Bénard 3355 0000-0001-7995-3581 Carlos Uribe 114455
0000-0003-3127-7478 Arman Rahmim 11556677 0000-0002-9980-2403

###### Abstract

Prostate specific membrane antigen (PSMA) positron emission
tomography/computed tomography (PET/CT) imaging provides a tremendously
exciting frontier in visualization of prostate cancer (PCa) metastatic
lesions. However, accurate segmentation of metastatic lesions is challenging
due to low signal-to-noise ratios and variable sizes, shapes, and locations of
the lesions. This study proposes a novel approach for automated segmentation
of metastatic lesions in PSMA PET/CT 3D volumetric images using 2D denoising
diffusion probabilistic models (DDPMs). Instead of 2D trans-axial slices or 3D
volumes, the proposed approach segments the lesions on generated multi-angle
maximum intensity projections (MA-MIPs) of the PSMA PET images, then obtains
the final 3D segmentation masks from 3D ordered subset expectation
maximization (OSEM) reconstruction of 2D MA-MIPs segmentations. Our proposed
method achieved superior performance compared to state-of-the-art 3D
segmentation approaches in terms of accuracy and robustness in detecting and
segmenting small metastatic PCa lesions. The proposed method has significant
potential as a tool for quantitative analysis of metastatic burden in PCa
patients.

###### Keywords:

Diffusion models Segmentation Prostate Specific Membrane Antigen Positron
Emission Tomography Maximum Intensity Projections

##  1 Introduction

Prostate cancer (PCa) is the second most prevalent cancer and the fifth
leading cause of cancer-related mortality among men [17]. Despite progress in
conclusive local treatments like radical prostatectomy (RP) and radiation
therapy (RT), an estimated 20-50% of patients will experience biochemical
recurrence (BCR), marked by increasing levels of prostate-specific antigen
(PSA). [5] The recurrence of prostate cancer may manifest as metastasis in the
regional lymph nodes and bone structures. As the disease progresses,
involvement of the liver and lungs, among other sites, may also occur [1].
Depending on which site is involved with the disease, different type of
treatment might be required.

PSA level raise is considered as the primary biomarker for following up on
prostate cancer treatment response and monitoring disease recurrence in
prostate cancer patients [2]. However, it cannot localize the recurrence of
the disease. Therefore, the precise identification of recurrence locations
becomes significantly important for therapeutic decision-making processes. As
a result, employing a diagnostic imaging modality that possesses both high
sensitivity and specificity is crucial for differentiating between local
relapse, oligometastatic disease, and extensive disease, hence to enable
individualized treatment plans for patients. Recent advancements in Positron
Emission Tomography (PET) imaging has led to improved detection and
quantification of many types of primary and metastatic lesions. Design of
recent PET radiopharmaceuticals that are able to target the prostate-specific
membrane antigen (PSMA), such as [18F]DCFPyL, with much higher sensitivity and
specificity compared to conventional imaging modalities has opened a new era
in the diagnosis, treatment decision-making, and patient management in
prostate cancer [16].

Deep learning algorithms, have shown great potential in computer-aided
diagnosis [12]. Yet, challenges arise from the nature of the imaging modality
that AI-based image recognition algorithms must cope with, including low
contrast, large intra- and inter-patient heterogeneity, and blurring and noise
in the images. The unique specifications of biochemical recurrent prostate
cancer metastatic lesions, including tumors with very small sizes, low-to-
moderate radiopharmaceutical uptake, especially compared to the high
biological uptake of the bladder and kidneys, make them hard targets to detect
[4]. Added to that, local tumor recurrence adjacent to the urinary bladder, or
in the abdomen area with high background noise and high uptake regions such as
ureters, further complicate the detection of PCa lesions even for physicians,
making the task of manual segmentation time and labor-intensive [7]. As such,
localizing the lesions in the image could help physicians save time and
increase the accuracy of the task.

There are limited number of works that tried to tackle the problem of PCa
tumor/metastatic lesion detection and segmentation using the power of AI.
Prior works mainly focused on the local primary (intra-prostatic) tumor
segmentation [11] which is a relatively less challenging task, given the
locality of the disease occurrence. Only in [10] and [21] authors evaluated
the performance of CNN-based segmentation models for PCa lesions segmentation,
however only the dataset used in [21] is for PCa recurrence patients.

In this work, we introduce an innovative approach for automated detection and
segmentation of biochemically recurrent PCa metastatic lesions on PSMA-PET
images using a 2D Diffusion-based segmentation model. This approach includes
novel use of the ordered-subset expectation-maximization (OSEM) algorithm
applied to 2D segmentations of multi-angle maximum intensity projections (MA-
MIPs) to generate 3D segmentation of metastatic lesions in PET images, while
taking advantage of the computational efficiency and performance benefits of
training a 2D diffusion-based segmentation model on MA-MIPs. We show that our
method outperforms its state-of-the-art 3D rivals in terms of various
segmentation metrics on the target dataset.

##  2 Methods and Materials

###  2.1 Dataset

This is a post-hoc sub-group analysis of a prospective clinical trial.
Inclusion criteria were: (1) histologically proven prostate cancer with
biochemical recurrence after initial curative therapy with radical
prostatectomy, with a PSA > 0.4 ng/mL and an additional measurement showing
increase; (2) histologically proven prostate cancer with biochemical
recurrence after initial curative therapy with RT, with a PSA level > 2 ng/mL
above the nadir after therapy. [6]. Overall, $510$ whole-body [18F]DCFPyL
PSMA-PET/CT images were chosen. Each trans-axial PET image has a matrix size
of $192\times 192$ pixels, with each pixel covering $3.64mm^{2}$ in physical
space. All active lesions were manually delineated by an expert nuclear
medicine physician. On average each image had $1.92\pm 1.21$ PCa lesions with
an average active volume of $4.03\pm 7.02ml$ and long axis diameter of
$12.96\pm 10.11mm$ (on CT). The average maximum standard uptake value (SUVmax)
and SUVmean of all the lesions were $9.64\pm 10.04$ and $4.4\pm 3.55$,
respectively.

###  2.2 Data Preprocessing

PSMA-PET activity concentration values (Bq/ml) of all PSMA PET voxels were
converted to Standard Uptake Value (SUV). To decrease the contrast between
high uptake normal organs and the small lesions, SUV values were clipped to a
range of 0 to 25. CT images had an original voxel size of $(0.98\times
0.98\times 3.27)mm^{3}$, and the PET images had a voxel size of $(3.64\times
3.64\times 3.27)mm^{3}$. All PET/CT images were resampled to a voxel size of
$(2.0\times 2.0\times 2.0)mm^{3}$ using a third-order spline method for both
CT and PET images and further cropped to have matrix size of $250\times 250$.
72 axial rotations of the PSMA PET volumes were computed in every $5^{\circ}$
degrees of axial rotation, and the maximum intensity projections (MIPs) of all
72 volumes (the original volume and all 71 rotated ones) were computed, in
order to cover one complete $360^{\circ}$ axial rotation of the volume. Since
preserving the information of soft tissues in the CT equivalent of MIP
projections are not feasible, in order to provide more context information to
the DDPM network, 4 different MIP projections per each axial rotation were
computed: First, the normal MIP taking by computing the maximum value of each
ray in the rotated 3D volume (PET-MIP); second, projecting the maximum
intensity after further clipping the PSMA-PET voxel to the range of 0 - 10
(PET10-MIP); third, further clipping the voxels to the range of 0 - 5 and
capturing MIP (PET5-MIP); and finally, the depth location of the voxels with
maximum intensity along each ray (DEPTH-MIP). To the best of our knowledge,
this is the first time that such derived modality is being used for
segmentation in the related literature.

###  2.3 Segmentation Network Architecture

The model used in this work for automated segmentation of the metastatic
lesions on MA-MIPs is a denoising diffusion probabilistic model (DDPM)
initially proposed in [14] and modified in [20] for brain tumor segmentation
on 2D trans-axial MRI slices. The general idea behind diffusion models
comprised of two chains of incrementally noising and denoising, known as
forward $(q)$ and reverse $(p)$ processes, respectively. The forward process
$p$ starts with adding a small amount of Gaussian noise to the input image $x$
over $T$ time steps, resulting in a series of noisy images
$x_{0},x_{1},\ldots,x_{T}$. Then, during the reverse process $p$, the model
which is a U-Net architecture based network, learns to predict the slightly
less noisy image $x_{t-1}$ from $x_{t}$ for each step $t\in\\{1,\ldots,T\\}$ .
Throughout the training of the diffusion model, the ground truth image
$x_{t-1}$ in each time step $t$ is known, and hence the model can be trained
using $L_{2}$ loss.

During test time, the sampling process $p$ starts from random Gaussian noise
$x_{T}\sim\mathcal{N}(0,\mathbf{I})$, and iteratively denoises it using the
trained U-Net model to generate a fake image $x_{0}$.

Writing the forward process $q$ as
$q(x_{t}|x_{0}):=\mathcal{N}(x_{t};\sqrt{\bar{\alpha}_{t}}x_{0},(1-\bar{\alpha}_{t})\mathbf{I})$
where $\alpha_{t}:=1-\beta_{t}$ and $\beta_{t}$ is the variance of the forward
process $q$ at the time step $t$, and
$\bar{\alpha}_{t}:=\prod_{s=1}^{t}\alpha_{s}$, then $x_{t}$ can be directly
expressed based on $x_{0}$ as:

$x_{t}=\sqrt{\bar{\alpha}_{t}}x_{0}+\sqrt{1-\bar{\alpha}_{t}}\epsilon,\ \ \
\text{where}\ \ \epsilon\sim\mathcal{N}(0,\mathbf{I})$ (1)

as shown in [8]. For the reverse (denoising) process, given the parameters of
the trained U-Net model ($\theta$), the learned reverse process $p_{\theta}$
can be written as
$p_{\theta}(x_{t-1}|x_{t}):=\mathcal{N}(x_{t-1};\mu_{\theta}(x_{t},t),\Sigma_{\theta}(x_{t},t))$.
Here $x_{t-1}$ can be predicted using the following formula as given in [8]:

$x_{t-1}=1\frac{1}{\sqrt{\alpha_{t}}}\left(x_{t}-\frac{1-\alpha
t}{\sqrt{1-\bar{\alpha}_{t}}}\epsilon_{\theta}(x_{t},t)\right)+\sigma_{t}\mathbf{z},\
\ \ \text{where}\ \ \mathbf{z}\sim\mathcal{N}(0,\mathbf{I})$ (2)

$\sigma_{t}$ is the variance scheme of the reverse process [14] and
$\mathbf{z}$ is the random component of the sampling process. The U-Net
denoted as $\epsilon_{\theta}$ at each time step $t$ takes $x_{t}$ (as defined
in equation1) as input, and learns the noise scheme
$\epsilon_{\theta}(x_{t},t)$. At the time step $t$ during sampling, the
predicted $\epsilon_{\theta}$ is subtracted from $x_{t}$ according to Equation
2 to construct $x_{t-1}$ which is a slightly less noisy version of the input
$x_{t}$.

In its classic form, a diffusion model trained on a dataset, during the
inference time, takes a random Gaussian noise $x_{T}$ as input, and generates
a synthetic image that fits in the distribution of the training dataset. In
case of semantic segmentation, starting from a random noise, the trained
diffusion model will generate a mask according to the distribution of the
segmentation masks it was trained on, but not necessarily the segmentation
mask of the test sample it has been given. As such, in [20] the authors
modified the forward and reverse processes of a classic diffusion model by
providing the brain MR axial slices as prior information and binding them to
the anatomical information. As denoted in [20], the trans-axial image $b$
corresponding to the ground truth segmentation mask $x_{b}$ are concatenated
together, $X:=b\oplus x_{b}$. The incremental noise during the forward process
$q$ is only added to the ground truth segmentation mask $x_{b}$, defined as
$x_{b,t}$. As a result, equation 1 is modified as follows:

$x_{b,t}=\sqrt{\bar{\alpha}_{t}}x_{b}+\sqrt{1-\bar{\alpha}_{t}}\epsilon,\ \ \
\text{where}\ \ \epsilon\sim\mathcal{N}(0,\mathbf{I})$ (3)

Accordingly, equation 2 of the reverse process $p$ is rewritten as follows:

$x_{b,t-1}=1\frac{1}{\sqrt{\alpha_{t}}}\left(x_{b,t}-\frac{1-\alpha
t}{\sqrt{1-\bar{\alpha}_{t}}}\epsilon_{\theta}(X_{t},t)\right)+\sigma_{t}\mathbf{z},\
\ \ \text{where}\ \ \mathbf{z}\sim\mathcal{N}(0,\mathbf{I})$ (4)

Here the input image prior $b$ is of size $(c,h,w)$, where each channel has
the size of $h\times w$. The corresponding ground truth segmentation mask
$x_{b}$ is of size $(1,h,w)$. Consequently, X has dimension $(c+1,h,w)$.
Figure 1 visually summarizes the forward and reverse processes during training
of the DDPM model for the task of segmentation.

Figure 1:  Visual explanation of how the forward and reverse processes of DDPM
model works, along with the input ground truth segmentation mask during the
forward process and the prior information of the anatomical/functional context
during the reverse process

The stochastic property of the DDPM enables us to generate more than one
segmentation mask prediction per each input image, thus, ensembling the
multiple predictions of the same input image can potentially improve the
segmentation performance. Therefore, during sampling we ensemble 10
segmentation mask variations and take the mean image as the predicted
segmentation for the input MA-MIP.

###  2.4 3D Reconstruction of 2D Masks Using OSEM Algorithm

The Ordered Subsets Expectation Maximization (OSEM) algorithm is an iterative
reconstruction technique widely used in medical imaging modalities such as
Positron Emission Tomography (PET) and Single-Photon Emission Computed
Tomography (SPECT) [9]. It reconstructs 3D volumetric images from a series of
2D tomographic projection data acquired at different angles around the
subject. OSEM is an accelerated variant of the Expectation Maximization (EM)
algorithm [13], which aims to estimate the unknown 3D radio-tracer
distribution within the subject by maximizing the likelihood of observing the
measured 2D projection data. The OSEM algorithm introduces an ordered subsets
approach to accelerate the convergence rate of the EM algorithm.

A novelty of our work is that OSEM algorithm, unlike conventionally applied to
acquired data to generate images, is applied to segmentations of MA-MIPs as
generated from the images. By utilizing the OSEM algorithm, the proposed
method in this study efficiently reconstructs the 3D segmentation volume from
the predicted 2D segmentation masks obtained from the MA-MIPs. The back-
projection step of the OSEM algorithm is used to map the 2D segmentation masks
onto the 3D volume, iteratively refining the estimate until convergence.

##  3 Experiment Details

We evaluated our proposed MIP-based segmentation method on the PSMA-PET image
dataset described in section 2.1. Per each patient, 72 axial rotated volumes
of the PSMA-PET image were generated. In order to avoid padding the images as
much as possible and providing more meaningful information to the network, all
the whole-body PSMA-PET volumes were divided into two section, upper body and
lower body, resulting in 144 volumes per each patient. For each volume, 4
different MIP images were generated, as described in section 2.2. These four
different MIPs were stacked to prepare the input data for training the DDPM
model, resulting in the size of $(4,250,250)$. These steps resulted in 66240
images from 460 PSMA-PET volumes of the same number of patients for
training/validation of the DDPM model and 7200 images for testing from 50
patients.

Backbone of the DDPM model used in this work is a U-Net architecture network
described in [20], and [14] with input size of five-channel $256\times 256$.
It uses six feature map with resolutions from $128\times 128$ to $4\times 4$,
two residual blocks, and one self-attention head with $16\times 16$
resolution. Similar to [20] we chose a 10000-steps linear noise schedule for
training/sampling. The model is trained for 72 hours (150,000 iterations) on
an NVIDIA V100 GPU 16 GB, with a learning rate of $10^{-4}$ using Adam
optimizer, and a batch size of 1.

In order to benefit from the ensemble of multiple predicted masks using the
stochastic property of the trained DDPM model, for each image in the test set,
10 variations of segmentation masks were predicted. The computed mean image of
the 10 masks were taken as the final segmentation per each projection, after
thresholding each of the 10 predicted mask at $0.5$.

Next, the 72 segmentation masks per each volume in the test dataset were
reconstructed to 3D using the OSEM algorithm, utilizing the PyTomography
toolbox [15]. For each 3D reconstructed mask, OSEM ran for 40 iterations and
20 subsets, which took around 3 minutes per each volume. Finally per each
patient in the test set, the two volumes of upper and lower body were stitched
together to make the whole-body segmentation mask.

In order to evaluate the performance of our proposed method, we trained eight
3D-based segmentation methods among the state-of-the-art biomedical image
segmentation networks in the literature. Same dataset (PET and CT volumes,
before MA-MIP generation) were used for training and evaluating these models.
The input size of all methods were two-channel cropped volumes of size
$128\times 128\times 128$. Batch sizes, number of crops, and other hyper-
parameters were modified in a such way to let the 3D networks fit on four
NVIDIA V100 16GB gpus for training. Sliding window inferencing with size of
$192\times 192\times 192$ were used in test time for all models except the
ones with explicit requirement of having the same size of training sample
crops, $128\times 128\times 128$. All models were trained for 500 epochs and
best model based on the validation set loss were picked for inferencing.
Implementations were done using MONAI toolbox, python 3.10, and PyTorch 2.1,
on Ubuntu 18.04 LTS. Results are reported briefly in table 1 of the next
section.

##  4 Results and Discussion

In this section we report the result of the proposed MA-MIP based segmentation
method on our test set. As baseline comparison, we also show results for 8
state-of-the-art (SOTA) 3D segmentation methods in the literature.

Table 1: Performance evaluation of our method compared to the SOTA methods Model |  Dice$\uparrow$ |  HD95$\downarrow$ |  Jaccard$\uparrow$ |  %Vol. Error$\downarrow$
---|---|---|---|---
U-Net |  ${0.461}$ |  ${45.32}$ |  ${0.345}$ |  ${45.6\%}$
U-Net++ |  ${0.429}$ |  ${126.44}$ |  ${0.308}$ |  ${39.4\%}^{\star}$
Flexible U-Net (b1) |  ${0.451}$ |  ${26.06}^{\star}$ |  ${0.343}$ |  ${48.5\%}$
Attention U-Net |  ${0.467}$ |  ${135.77}$ |  ${0.337}$ |  ${48.4\%}$
SegResNet |  ${0.470}^{\star}$ |  ${64.74}$ |  ${0.359}^{\star}$ |  ${42.3\%}$
UNETR |  ${0.407}$ |  ${80.01}$ |  ${0.299}$ |  ${45.1\%}$
Swin UNETR |  ${0.438}$ |  ${56.60}$ |  ${0.328}$ |  ${43.6\%}$
V-Net |  ${0.426}$ |  ${57.80}$ |  ${0.325}$ |  ${51.7\%}$
MIP-DDPM (Ours) |  $\mathbf{0.532}$ |  $\mathbf{19.60}$ |  $\mathbf{0.433}$ |  $\mathbf{32.9\%}$
Figure 2:  Visual comparison of our method against SOTA on a sample case.

Figure 2 shows predicted segmentation masks using the proposed and 8 SOTA
techniques for a random test sample, also confirming visual improvements.
Table 1 summarizes the overall results of our proposed method in terms of Dice
score, 95 percentile Hausdorff Distance, Jaccard index, and Volume Error
Percentage. It is seen that our proposed MIP-DDPM model outperforms all SOTA
methods. PSMA PET images may enable earlier detection of metastases, which can
improve management decisions, especially in case of biochemical recurrent
disease. Hence, the primary clinical objective revolves around detecting both
local recurrence and distant metastases. In many cases, as demonstrated in
this study, the total tumor volume is small, further challenging this task.
Evaluating 17 SOTA object detection methods on the same dataset, the best
performing model achieved 0.75 recall [19, 18]. In fact, on the dataset used
in this study, a best performing prior automated segmentation method had
achieved a mean dice score of only 0.38 using a SOTA self-supervised pre-
training method [21]. A very recent work on this dataset using SOTA 3D
segmentation methods achieved a mean dice score of $\sim$ 0.47 using custom
volume-preserving loss functions [3]. These depict the complexity of the task,
and that our proposed framework in this work outperforms SOTA techniques
previously reported.

##  5 Conclusion

Our work introduces a novel method for segmentation of metastatic lesions in
3D volumetric PET images, utilizing 2D diffusion models. Instead of 2D trans-
axial slices or 3D volumes, our approach segments the lesions on multi-angle
maximum intensity projections of PET images. Final 3D segmentation masks are
then obtained through OSEM-based reconstruction of the segmented 2D MA-MIPs.
Our method demonstrates superior performance compared to SOTA segmentation
techniques. The proposed framework holds significant promise as a tool for
segmenting small metastatic lesions in medical images.

####  5.0.1 Acknowledgements

This work was supported by the Canadian Institutes of Health Research (CIHR)
Project Grant PJT-162216, as well as computational resources and services
provided by Microsoft AI for Health.

####  5.0.2

The authors have no competing interests to declare that are relevant to the
content of this article.

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