# Three-dimensional reconstruction and characterization of bladder
deformations

Augustin C. Ogier<EMAIL_ADDRESS>Stanislas Rapacchi
<EMAIL_ADDRESS>Marc-Emmanuel Bellemare  marc-
<EMAIL_ADDRESS>Aix Marseille Univ, Universite de Toulon, CNRS,
LIS, Marseille, France  Aix Marseille Univ, CNRS, CRMBM, Marseille, France

###### Abstract

Background and Objective: Pelvic floor disorders are prevalent diseases and
patient care remains difficult as the dynamics of the pelvic floor remains
poorly known. So far, only 2D dynamic observations of straining exercises at
excretion are available in the clinics and the understanding of three-
dimensional pelvic organs mechanical defects is not yet achievable. In this
context, we proposed a complete methodology for the 3D representation of the
non-reversible bladder deformations during exercises, directly combined with
synthesized 3D representation of the location of the highest strain areas on
the organ surface. Methods: Novel image segmentation and registration
approaches have been combined with three geometrical configurations of up-to-
date rapid dynamic multi-slices MRI acquisition for the reconstruction of
real-time dynamic bladder volumes. Results: For the first time, we proposed
real-time 3D deformation fields of the bladder under strain from in-bore
forced breathing exercises. The potential of our method was assessed on eight
control subjects undergoing forced breathing exercises. We obtained average
volume deviation of the reconstructed dynamic volume of bladders around 2.5%
and high registration accuracy with mean distance values of 0.4 $\pm$ 0.3 mm
and Hausdorff distance values of 2.2 $\pm$ 1.1 mm. Conclusions: Immediately
transferable to the clinics with rapid acquisitions, the proposed framework
represents a real advance in the field of pelvic floor disorders as it
provides, for the first time, a proper 3D+t spatial tracking of bladder non-
reversible deformations. This work is intended to be extended to patients with
cavities filling and excretion to better characterize the degree of severity
of pelvic floor pathologies for diagnostic assistance or in preoperative
surgical planning.

###### keywords:

3D dynamic MRI, Bladder, Pelvic dynamics, Volume reconstruction

††journal: Computer Methods and Programs in Biomedicine

##  1 Introduction

Pelvic floor disorders affect one fourth of adult women and the prevalence
increases with age, up to 50% for women older than 80 years [1, 2]. Some
experts refer to them as hidden epidemics to describe the extent of these
diseases and their importance in the population [3]. These pathologies result
from various factors such as aging, pregnancy, childbirth, obesity, injuries
[4] and are mainly characterized by a weakening of the pelvic floor muscles
and ligaments leading to mechanical dysfunction of the pelvic organs support
structures. Symptoms range from urinary and bowel incontinence, sexual
dysfunction to abnormal descent of pelvic organs.

Besides the clinical examination, magnetic resonance imaging (MRI) has emerged
as one of the most attractive non-invasive methods for pelvic diagnosis [5].
Dynamic MRI examinations of a pelvic strain exercise are considered essential
[6] for the investigation of the pelvic area and guidelines emphasize their
importance to identify pathological organs deformations. Current clinical
practice involves 2D dynamic MRI acquiring a single sagittal plane during
straining exercises involving excretion to qualitatively assess the
displacement and deformations of the main pelvic organs (bladder, uterus-
vagina, rectum). During visual inspection, radiologists assess the
displacement of the pelvic organs relative to bone-anchored lines, such as the
pubococcygeal line [7]. Beyond manual measurements, automated measures of
strain-induced deformation features were proposed to build a quantitative
characterization of the pelvic organ dynamics to distinguish pathological
cases from healthy ones [8] and to graduate the severity of pelvic organ
prolapses [9].

The visualization and characterization of pelvic dynamics by MRI has so far
only been studied in the midsagittal plane. Although the majority of pelvic
movements are identifiable in this plane, an understanding of three-
dimensional deformities could provide full assessment of the pelvic organs
mechanical defects. Due to speed limitation in MRI acquisition, 3D imaging
have been exclusive to static pelvic MRI so far, to observe anatomy at rest
during instructed apnea. Ultrasound have been used to observe organs during
loading exercises [10], such as Valsalva maneuver for the diagnosis of
prolapse, but remains limited due to image interpretation difficulties and
restricted field of view. Although significant research has been performed,
the pelviperineal physiology and the anatomic basis of pelvic floor diseases
remain unclear [11, 12].

This misunderstanding is the main cause for repair surgery relapse [13]. To
better comprehend the complete mechanical functioning of the pelvic floor and
to provide better post-operative support, biomechanical models have been
proposed in recent years with the aim of providing patient-specific 3D
simulations of the pelvic region [14, 15]. Although pelvic system geometry
could be known from high-resolution MRI images, patient-specific boundary
conditions and material biomechanical properties are impossible to probe _in
vivo_ as it requires invasive procedures [16]. Biomechanical numerical
simulation could offer indirect characterization of pelvic organs mechanical
impairment and guide surgical intervention, anticipating patients recovery.
Based on shape descriptors and anatomical references approximation, comparison
criteria between simulation sequence and 2D midsagittal dynamic MRI of a same
patient was proposed to validate and enhance a physical model [17]. 2D dynamic
data are not sufficient to depict realistic 3D biomechanical models. Guiding
3D biomechanical models through a single 2D slice fails to fully simulate
pelvic organs dynamics as simulation errors remain in off-plane areas [18].
Recent attempts to provide a finite elements based simulation of organ
behaviours still lack realistic patient 3D data [19].

In light of the current state-of-the art in the study of pelvic disorders,
there is a definite need for methods to generate dynamic 3D volumes of pelvic
organs. Such approach would allow full visualization of the 3D dynamics of the
organs in order to help diagnosis and open the way to the geometric
characterization of real 3D deformations of the organs. A complete 3D
characterization of the pelvic floor would provide a better classification of
pathologies but also feed biomechanical simulators that could improve the
surgical management of pelvic organ prolapse.

Pelvis straining MRI falls within real-time applications along with bolus-
tracking perfusion MRI since organs do not recover the same position after
each strain exercises and intra-cavities fluids can be excreted [6]. Patients
cannot voluntarily strain their pelvis with the same intensity and regularity
during repetitive cycles of forced breathing. Current state-of-the-art
perfusion MRI methods rely either on multi-slices (3 to 6) dynamic
acquisitions in cardiac perfusion MRI, with an update rate of one heartbeat
($\approx$ 1 s) [20] or 3D dynamic acquisitions such as liver perfusion MRI,
with an update rate of few seconds per volume at best [21]. The targeted
spatial resolution combined with the extreme nature of the pelvic organ
unloading induced during these exercises requires an order of magnitude faster
imaging ($<$ 1 s) and prevents 3D real-time pelvis MRI.

As mid-sagittal acquisitions, at about 1 frame per second, is the consensus
for exploration of the lesser pelvis region[6], we have kept these features in
our imaging schemes. In particular, we focused on maintaining the frame rate
of one image per second. However, to gain 3D information, it was necessary to
complete the spatial information, but with a short temporal footprint to
minimize errors due to deformations. For this purpose, we tested 3 spatial
multi-slices configurations among the most obvious ones, that can be easily
reproduced in clinical set-ups. We compared them to estimate the errors on
organs tracking induced by privileging spatial coverage or spatial resolution.
This shall give us an idea of the best configuration to use to see the bladder
but also possibly the main neighboring organs. Paired with a 3D static
acquisition, the proposed acquisition aims to provide volumetric dynamic
reconstruction. This study was a complete extension of a previous seminal work
[22].

Initially the bladder was the main organ concerned as it is straightforward
and the simplest organ to consider being homologous to the sphere. However we
have described our methodology in a generic way as we intend to apply it in
future studies for the visualization of the uterus and rectum. We leveraged a
semi-automated segmentation method, based on a fusion of image registration
approaches, for the tracking of organs in the dynamic planes, regardless of
their spatial orientation. Linear interpolations on the geodesic path of each
multi-planar segmentation have been proposed to refine the time scale
inherited from the acquisition methods. Finally, we built dynamic 3D
representations of organs through non-linear geometric registrations between
each dynamic partial volumes and the complete static volume of the high-
resolution static acquisition. As this study produced dynamic 3D bladder
volumes that can’t be directly compared to a ground truth impossible to
acquire, we emphasized metrics to assess the adequacy of the volume
reconstructions: accuracy of the registration and conservation of the bladder
volume.

To our knowledge, this study is the first to propose both dynamic 3D
visualization of the pelvic region during strain exercise and a dynamic 3D
tracking of the organs. Furthermore, the organ reconstruction process allowed
a synthesized characterization of the 3D deformations undergone by the pelvic
organs during loading exercises. Analysis of the deformation fields allowing
the reconstruction of organs, and more particularly the study of the resulting
Jacobians, revealed local volume changes over time and provided a high-level
3D representation of the location of the highest strain areas on the organs
surface. Specifically, analysis of Jacobians validated that our complete
reconstruction methodology produced similar results between different
geometric configurations of dynamic acquisitions.

##  2 Methods

###  2.1 MRI acquisitions

Both static and dynamic acquisition methods implemented in the context of this
study were performed using a 1.5T MRI scanner (MAGNETOM Avanto, Siemens AG,
Healthcare Sector, Erlangen, Germany) using a spine/phased array coil
combination and $T_{1}/T_{2}$ weighted balanced steady-state free precession
sequences ($T_{1}/T_{2}W$ bSSFP). In comparison to the $T_{2}W$ sequence
commonly used for pelvic area MRI, the $T_{1}/T_{2}W$ bSSFP sequence allows an
optimal acquisition speed combined with a strong contrast between body fluids
and the different pelvic organs tissues: uterus in dark gray, bladder in white
(hyper-signal), and rectum and viscera (dark gray tissue and light gray
content).

Figure 1: Axial (left) and sagittal (right) views of pelvic floor acquired
during a maximum expiration apnea with delineations of the major pelvic
organs: bladder (red), uterus/vagina (blue), and rectum (green).

####  2.1.1 Static acquisition configuration

A quasi-isotropic 3D static image was acquired, as depicted in Fig. 1. High
resolution in the sagittal plane and high contrast allowed the anatomy of the
pelvic region to be precisely delineated. The 3D T1/T2W bSSFP image (TR = 2.21
ms, echo time: 0.99 ms, flip angle: 32∘, field of view: 277 mm $\times$ 320 mm
$\times$ 88 mm, voxel size: 0.83 mm $\times$ 0.83 mm $\times$ 2.0 mm, GRAPPA =
2) was recorded during a maximum expiration apnea of 18 seconds. Static
acquisition was performed to allow precise three-dimensional manual
segmentation of the pelvic organs used as references for the reconstruction
scheme detailed in section 2.5.

####  2.1.2 Dynamic acquisition configurations

Dynamic sequences provided a lower spatial resolution and a reduced contrast
compared to static sequences, but allowed the monitoring of pelvic dynamics
during organ loading exercises. Analysis of the biomechanical deformations
induced during these exercises should allow to identify elasticity defects in
pelvic floor tissues. Three-dimensional dynamic MRI can be acquired either
using slab excitations or multi-planar slice excitations. Due to the extended
temporal footprint of 3D excitations ($>$1s), blurring can occur when
observing rapid deformations. Thus, fast acquisition ($\approx$ 100 ms) of
multi-planar 2D slices was preferred to follow the motion of the pelvic
organs. In this study, we sought to harness the potential of high-density coil
arrays to accelerate MRI using parallel imaging techniques.

$C_{lines}$

$C_{grid}$

$C_{star}$

Figure 2: Geometrical configuration of the planes for the three dynamic multi-
planar acquisitions sequences.

Non-linear reconstruction techniques, such as Compressed Sensing [23], were
not considered in this preliminary work to maintain reliability of the imaging
information and clinical transferability. The sliding-window temporal-GRAPPA
(TGRAPPA) technique [24] allowed for a significant acceleration. In TGRAPPA,
consecutive images share low-frequency information to provide basis for image
reconstruction. Assuming the pelvic deformations do not modify the low-
frequency image information within the range of a second, the full coverage of
the pelvic region was guaranteed using up to a maximum of 12 slices,
regardless of the geometry configurations, within a second. Another
interesting feature of this multi-planar setup is the T1-recovery of the
signal between images, which boosts signal-to-noise ratio and offers a
reinforced contrast between tissues and liquids.

####  2.1.3 Spatial order of dynamic acquisitions

Three dynamic sequences in multi-planar configurations were acquired, with
three different geometric configurations that varied plane number and
locations but maintained the time for acquiring one set of all planes within
1s. We denoted the different geometries as $C_{star}$, $C_{grid}$ and
$C_{lines}$ (cf. Fig. 2). The $Star$ configuration consisted of one coronal
plane, the mid-sagittal plane and 2 others oriented at about 45 degrees to the
sagittal. The $Grid$ configuration featured three sagittal and two coronal
planes. The $Lines$ configuration was made of several parallel sagittal planes
covering the body from the right lateral side to the left lateral side.
Whatever the configuration, the number of planes of one acquisition cycle was
noted $N_{p}$. For $C_{star}$ and $C_{grid}$, $N_{p}$ was equal to 4 and 5,
respectively. The number of planes for $C_{lines}$ was defined according to
the corpulence of each subject to cover the pelvic area as much as possible
and was typically set at 10 ($\pm$ 2).

Table 1: Acquisition parameters of the three dynamic MRI sequences Configuration |  $C_{star}$ |  $C_{grid}$ |  $C_{lines}$
---|---|---|---
Slice thickness (mm) |  5 |  6 |  4
Number of cycles |  100 |  100 |  60
Time repetition (ms) |  2.9 |  2.6 |  1.9
Slice acquistion time (ms) |  183.72 |  124.95 |  91.61
Field of view (mm) |  302 $\times$ 350 |  300 $\times$ 350 |  299 $\times$ 350
In-plane resolution (mm) |  1.09 $\times$ 1.09 |  1.36 $\times$ 1.36 |  1.82 $\times$ 1.82

####  2.1.4 Temporal order of dynamic acquisitions

Magnetic resonance imaging involves a close relationship between spatial
resolution, temporal resolution and contrast of the acquired image. This leads
to a trade-off where modification of one of these parameters induces a change
in the others. Increasing the duration of a single image plane acquisition
(_i.e._ the sequence-specific slice acquisition time [TS]) resulted in an
acquired image with improved contrast and better spatial resolution. To follow
the motions of the pelvic organs, an acquisition cycle, which acquired all the
planes one after the other, had to be acquired in less than a second. Thus,
defining the timeframe of a cycle by $N_{p}\cdot\mathrm{TS}$, each plane of a
given configuration had to be acquired in less than $1/N_{p}$ seconds.
Therefore, assuming the same field of view, the lower the $N_{p}$ required by
the geometric configuration, the longer the $\mathrm{TS}$ was, and therefore
the more precise the spatial resolution, as detailed in the table 1, and
better the contrast, as illustrated in Fig. 3. To allow for clinical-grade
signal-to-noise ratio, the $\mathrm{TS}$ of $C_{Lines}$ was set at a minimum
of 110ms, even with a $N_{p}>10$.

The number of cycles, $N_{c}$, acquired for each dynamic configuration was
determined to set the total acquisition time (_i.e._
$N_{p}\cdot\mathrm{TS}\cdot N_{c}$ ) around 1:20 min in order to record
several deformation phases of pelvic organs during loading exercises.

Regardless of dynamic configuration, for each
$p\in\mathopen{[}0\,;N_{p}\mathclose{[}$, the spatial indexes of acquisition
planes, we denoted $\mathcal{P}_{p}$ the set of the acquired planes
$\\{\mathcal{P}^{k}_{p}\\}$ with $k\in\mathopen{[}0\,;N_{c}\mathclose{[}$ the
temporal index of the acquisition cycle.

$C_{lines}$

$C_{grid}$

$C_{star}$

Figure 3: Difference in contrast and resolution between similarly located
sagittal planes from the different dynamic acquisition configurations.

###  2.2 Mathematical formalism

We introduce here the mathematical formalism of diffeomorphic registration
widely used in the following sections. A diffeomorphism, $\phi$, allows to
transform a target image _I_ towards a source image _J_ , and to find the
correspondence between these two shapes, such as $\phi_{x}^{t}I\approx J$. The
shape I is warped into the new coordinate system by
$\phi_{x}^{t}I=I\circ\phi(x,t)$, where the geodesic path $\phi(x,t)$ is
described in [25]. The diffeomorphic function, $\phi(x,t)$, parameterized by
time, $t\in[0,1]$ and $x$, spatial coordinate, is defined by
$\delta\phi^{t}_{x}/\delta t=v_{t}(\phi^{t}_{x})$ and $\phi^{0}_{x}$ = Id,
with $v(x,t)$ a time-dependent velocity field.

###  2.3 Pelvic organ segmentation

####  2.3.1 State of the art

Automatic segmentation of main pelvic organs is a challenging task due to
overlapping MRI contrasts between the uterus, vagina, rectum and surrounding
soft tissues. Only the bladder is natively contrasted due to the liquid it
contains. To enhance the contrast of the other organs, extrageneous liquid
could be inserted in the pelvic cavities, but the clinical procedure loses its
non-invasive nature.

Several methods have been proposed for the segmentation of pelvic organs using
MRI but mainly on 3D static axial images [26, 27, 28] and only a few semi-
automatic methods have already been developed for segmentation of dynamic 2D
sagittal T2W MRI images [29, 30, 9]. Although promising, these methods still
require manual initialization and have not been able to obtain segmentations
with a average Dice similarity coefficient (DSC) greater than 0.90 for the
bladder [29, 9] or with a mean Hausdorff distance lower than 9 mm [30].
Recently, convolutional networks have shown good results in addressing the
complex segmentation of pelvic organs in MRI but mostly with static
volumes[31]. Overall, the segmentation of multi 2D dynamic MRI datasets is
still challenging due to the combined non-rigid deformations and natively
depleted contrasts. As a result, the segmentation process of pelvic organs has
mainly been performed manually or semi-automatically in studies that have
focused on pelvic dynamics [8, 18].

####  2.3.2 Our method

In this study, the segmentation stage is the first step of a complete process
allowing characterization of pelvic organ deformations (cf. Fig. 5). This step
is crucial, as incorrect segmentations would lead to false deformation
estimations. So, for the segmentation of pelvic organs, a semi-automated
segmentation method was proposed on each set of 2D dynamic planes
$\mathcal{P}_{p}$ independently, to balance between high segmentation accuracy
and user’s dedication. For each set of acquired planes $\mathcal{P}_{p}$,
segmentation masks $\mathcal{M}_{p}$ were manually depicted on a few slices
only, at regular temporal intervals. The contours were then automatically
propagated over all the remaining frames according to the method described in
[32]. The manually segmented slices were denoted ${\mathcal{M}_{p}}^{i\cdot
n}$, with $i\in\mathbb{N}$, and $n$ the time step of the mandatory manual
segmentation. According to our method, the manual segmentation of the slices
${\mathcal{M}_{p}}^{i\cdot n}$ and ${\mathcal{M}_{p}}^{(i+1)\cdot n}$ allowed
the automatic segmentation of the $n-1$ slices of the interval using a fusion
of different propagation schemes.

The propagation from one manually segmented slice ${\mathcal{M}_{p}}^{i\cdot
n}$ to the next ${\mathcal{M}_{p}}^{(i+1)\cdot n}$ could be achieved, from an
Eulerian perspective, by successively estimating the correspondence maps
between the $n-1$ grey level $\mathcal{P}_{p}$ slices, pairwise. This process
could be done in both forward and backward strategies depending on the
starting slices. The forward strategy sought to estimate the diffeomorphism
$\phi^{k}_{E_{f}}$, given by:

$\phi^{k}_{E_{f}}=\phi^{i\cdot n+1}_{v_{f}}\circ\phi^{i\cdot
n+2}_{v_{f}}\circ\dotsc\circ\phi^{k}_{v_{f}}$ (1)

where $\phi^{j}_{v_{f}}$ is the deformation field mapping
$\mathcal{P}^{j-1}_{p}$ to $\mathcal{P}^{j}_{p}$. In a similar way, the
backward strategy was to estimate $\phi^{k}_{E_{b}}$, given by the combination
of the diffeomorphism $\phi^{j}_{v_{b}}$, mapping $\mathcal{P}^{j+1}_{p}$ to
$\mathcal{P}^{j}_{p}$. Therefore, for $k\in\mathopen{]}i\cdot n\,;(i+1)\cdot
n\mathclose{[}$ with $i\in\mathbb{N}$, the propagation to the missing
segmentations $M^{k}_{p}$ could have been performed by applying either
$\phi^{k}_{E_{f}}\mathcal{M}^{i\cdot n}_{p}$ or
$\phi^{k}_{E_{b}}\mathcal{M}^{(i+1)\cdot n}_{p}$. As registrations were
performed between successive MRI slices, these Eulerian approaches did follow
anatomical variations but rapidly diverged given that errors were accumulating
with the number of $\phi^{j}_{v}$ combinations. So, the accuracy decreased
with respect to the distance from the starting slice. To minimize this
divergence effect, the two approaches were merged and the missing
segmentations were generated by:

$\mathcal{M}^{k}_{p}=\left(\alpha_{k}\phi^{k}_{E_{f}}+\beta_{k}(\phi^{k}_{E_{b}}\circ\phi_{L_{f}})\right)M^{i\cdot
n}_{p}$ (2)

with $\phi_{L_{f}}$, the deformation field resulting from the geometric
registration between the two initial manually segmented masks. This
diffeomorphism $\phi_{L_{f}}$ allowed to perform a reference frame shift, and
merge the two Eulerian approaches. The determination of coefficients
$\alpha_{k}$ and $\beta_{k}$ were ruled by an $\arctan$ function to keep the
high accuracy of each Eulerian propagation process near its propagation
starting slice. The complete algorithm of the merging process and parametric
details of the diffeomorphic registrations are available in [32]. All the
segmentations were visually checked at the end of the process to ensure their
quality.

###  2.4 Temporal reconstruction of segmentations

After the segmentation process, $N_{p}$ datasets $\mathcal{M}_{p}^{k}$ were
obtained, each with a cardinality of $N_{c}$. The next step towards the 3D
reconstruction was to merge all $M_{p}$ datasets into a unique set,
$\mathcal{A}$, that we represented as a 2D matrix $[(t,p)]$. Taking $t_{0}$ as
the instant of the first image, we denoted, $t$ $=t_{0}+k\cdot\Delta t$ with
$\Delta t=\mathrm{TS}$. Such a matrix representation was chosen in order to
correctly arrange each segmentation mask $\mathcal{M}^{k}_{p}$ according to
the spatial and temporal parameters of the dynamic acquisitions. The spatial
index $p$ of each $\mathcal{M}^{k}_{p}$ indicated the temporal ordering of
acquisition between all planes acquired within each acquisition cycle and $k$
indexed the acquisition cycle. Therefore, as formulated in (3), each
$\mathcal{M}^{k}_{p}$ was associated to $\mathcal{A}_{p}^{k\cdot N_{p}+p}$.

From our 2D acquisition scheme, only one plane of a given configuration $C$
was available at a time $t$, as depicted in Fig. 4. As a consequence, for a
plane position $p$, the segmentation masks had to be interpolated to fulfill
the missing acquisition instants in a cycle. The missing segmentations in
$\mathcal{A}_{p}^{t}$ were reconstructed by linearly interpolating along the
geodesic path between the segmented mask of the previously acquired slice and
the next one, as formalized in (3). We advocated such a linear interpolating
scheme on the geodesic path since the motion of pelvic organs is assumed to be
continuous over the acquisition time of a cycle (less than 1 s).

$\displaystyle\begin{split}&\mathcal{A}_{p}^{t}=\begin{cases}\mathcal{M}_{p}^{(k//N_{p})}&\text{if}\
p=k{\@displayfalse\pmod{N_{p}}}\\\
\mathcal{M}_{p}^{k^{\prime}}\circ\phi(x,\frac{k-k^{\prime}}{N_{p}})&\text{otherwise}\end{cases}\\\
\hskip 5.0ptwith\hskip 10.00002pt&\\\
&\mathcal{M}_{p}^{k^{\prime}}\circ\phi(x,1)\approx\mathcal{M}_{p}^{k^{\prime\prime}}\\\
&k^{\prime}\hskip 1.99997pt=\big{(}(k-N_{p})//N_{p}\big{)}\cdot N_{p}+p\\\
&k^{\prime\prime}=k^{\prime}+N_{p}\end{split}$ (3)

Figure 4: Spatio-temporal configuration of the acquisition planes and temporal
reconstruction scheme, illustrated for $C_{star}$. Filled segmentations refer
to instants for which an image was acquired. Hatched segmentations correspond
to the temporally reconstructed ones. All segmentations of all acquisition
planes of a given instant $t$ are combined to provide the skeleton $S^{t}$ of
the organ volume at that time.

An illustration of the temporal reconstruction approach is given in Fig. 4
with the $C_{star}$ configuration. The scheme is identical whatever the
configuration with abscissa values ranged from $0$ to $N_{p}$ and with $t$ up
to $(N_{p}\cdot N_{c}-1)$. An horizontal line corresponds to the intermediate
masks generated by interpolation between two actually segmented masks.

###  2.5 Three-dimensional isotropic volume reconstructions

The temporal interpolation yields spatial and temporal samples of the
considered organ. In Fig. 4, the aggregation of masks on a vertical line
corresponds to the creation of what we called the skeleton of a dynamic volume
at time $t$. But all the masks belonged to 2D spatial domains of the different
acquisition planes. Therefore, the next step towards the 3D reconstruction of
the dynamic volumes was to compute the affine transformations required to
transfer the skeletons into a single 3D isotropic space. With all
segmentations defined in the same 3D domain, volume skeletons $S$ were
generated at each instant $t$ as the union of the segmentations from each
acquisition plane at that time:

$S^{t}=\bigcup_{p=0}^{N_{p}}{\mathcal{A}_{p}}^{t}\hskip 20.00003pt\forall
t\in\mathopen{[}N_{p}-1\,;N_{p}\cdot(N_{c}-1)+1\mathclose{]}$ (4)

Merging process was not performed for the first and last $N_{p}-1$ volumes
because segmentations were not available for all the acquisition planes at
these instants (cf. Fig. 4).

As $S^{t}$ are partial volumes, the final step towards the 3D reconstruction
of the dynamic volumes was to complete these skeletons. To achieve the most
natural and realistic reconstruction possible, a subject-specific approach was
used. The 3D volume resulting from the segmentation of the finely resolved
static acquisition was used as a template to be registered to each partial
volume (cf. Fig. 5). The registration process involved two distinct steps to
allow independent analysis of two significant biomechanical mechanisms in the
study of pelvic organ disorders: organ displacement and organ deformation.
First, organ displacement was estimated at each time by aligning the mass
centers of the static 3D volume and each dynamic partial volume. Then, a 3D
nonlinear registration was performed between the pre‐aligned volume contours.
Nonlinear diffeomorphic model was motivated by the need to register organs
that might show dynamics with large deformations. To solve the many-to-few
problem of matching complete to partial point clouds, a point/landmark-based
similarity metric based on probabilistic estimate was used. Based on a bi-
directional expected matching term between structures to be registered, the
PSE metric introduced by Pluta _et al._ [33] minimizes the distance between
each point of a given point-set and the expected corresponding point from the
other point-set. Let {V} be the set of points describing the contour of the
complete static volume $V_{s}$ and {R} the set of points describing the
contour of a dynamic volume skeleton. Cardinalities of {V} and {R} were
defined as $m$ and $r$, respectively. Note that $r<<m$, as {R} described the
contour of a partially sampled volume. Our registration problem was therefore
to minimize the objective function :

$PSE(\\{V\\},\\{R\\})=-\frac{1}{r}\sum_{i=1}^{r}\left\lVert
R_{i}-\frac{1}{m}\sum_{j=1}^{m}G(R_{i};V_{j},\sigma_{v})V_{j}\right\rVert^{2}$
(5)

with $G(R_{i};V_{j},\sigma_{v})$ a normalized Gaussian with mean $V_{j}$ and
standard deviation $\sigma_{v}$. The standard deviation $\sigma_{v}$ was
empirically set to $1mm$. Implementation of the registration was performed
using the B‐Spline SyN process of the ANTs library [34]. Registration
processes produced 3D displacement vector fields
$u^{t}=(u_{x}^{t},u_{y}^{t},u_{z}^{t})$, for each instant $t$, enabling to map
the complete static volume $V_{s}$ to each spatially undersampled dynamic
volume $S^{t}$ through the transformations $h^{t}$ = $v+u^{t}(v)$ with $v$ a
voxel position set by $v=(v_{x},v_{y},v_{z})$. Hence, for
$t\in\mathopen{[}N_{p}-1\,;N_{p}\cdot(N_{c}-1)+1\mathclose{]}$, every complete
dynamic volumes were generated using $V_{s}\left(v+u^{t}(v)\right)$ for each
$v$.

Figure 5: Schematic overview of the reconstruction and characterization
pipeline. Reconstruction of the dynamic volumes was achieved by registrating
the 3D volume from the static acquisition to the partial volumes from the
dynamic acquisitions, obtained after the temporal reconstruction process. The
analysis of the resulting deformation fields over time provided the
localization of the deformed tissue areas. * static entities, † dynamic
entities.

###  2.6 Displacement and deformation characterization

The radiological interpretation of resultant 4D images can be tedious, thus a
quantitative synthesis was sought to facilitate and standardize diagnosis.
Taking advantage of the reconstruction process, two quantitative metrics may
be directly extracted. The organ displacement recovered from initial affine
transformation could be a simple way to characterize organ movement. This
metric remained crucial because clinical evaluation of the pelvic dynamics,
and hence, the assessment of the pelvic floor disorders is routinely based on
the observation of such anatomical landmarks [35]. Similar metrics have
already been studied with classic 2D dynamic MRI [8, 9]. But to exploit the
full potential of the three-dimensional aspect of the reconstruction, 3D organ
deformations were analysed using the deformation fields resulting from the
non-linear registration.

Jacobian maps were computed to characterize 3D deformations. Let us denote the
Jacobian matrix of $h$ as $Dh$ (with the $(i,j)$-$th$ element $\delta
h_{i}/\delta v_{j}$). The Jacobian map, defined as the determinant of the
Jacobian matrix $|Dh(v)|$, encoded the percentages of the dynamic local volume
variations at each instant with respect to the static volume image (_i.e._ a
value of 0.9 denotes 10% tissue loss, whereas 1.1 a 10% tissue increase). The
quantity of deformations at each voxel of the organ for each instant $t$ was
therefore assessed through Jacobian map $J^{t}(v)=\mathrm{det}\left(\delta
h^{t}(v)/\delta v\right)$. The projection of each Jacobian map onto the
surface of the static volume provided a visualization of the deformations
undergone by the organ at the different phases of the loading exercises in a
common domain. Information provided by all Jacobian maps has also been
synthesized into a single map to facilitate the visualization of the
deformation areas with a single representation. For this purpose, the
$\sigma_{J}$ map was calculated for which each voxel $v$ represented the
standard deviation (SD) of the values of each $J^{t}$ map estimated across all
$t$ at $v$. Calculation was as follow:

$\sigma_{J}(v)=\sqrt{\frac{1}{n}\sum_{t=0}^{n}\left(J^{t}(v)-\overline{J(v)}\right)^{2}}$
(6)

with $n=N_{c}\cdot N_{p}-2(N_{p}-1)$, the number of dynamic volumes
reconstructed. The $\sigma_{J}$ map projected on a mesh of the reference
static volume provided a high-level representation of the location of the most
deformed tissue area during a dynamic acquisition sequence.

##  3 Results

###  3.1 Datasets

Pelvis areas of eight healthy participants (five women) were imaged. Subjects
were 27.6 $\pm$ 3.8 years old and weighted 69.3 $\pm$ 11.0 kg. The static
acquisition was recorded first during a maximum expiration apnea of 18
seconds. Afterwards, the three dynamic sequences in multi-planar
configurations were acquired successively during forced breathing exercises of
1:20 minute each with a short rest between sequences. During these exercises,
after 10 seconds of free breathing, the subject alternately inspired and
expired at maximum capacity. Subjects were also instructed to increase the
pelvic pressure to the maximum inspiration and conversely to contract the
pelvic floor during the expiration. These actions increased the intra-
abdominal pressure, causing deformities of the pelvic organs. The study was
approved by the local human research committee and was conducted in conformity
with the Declaration of Helsinki.

Since no extrageneous liquid was injected into pelvic cavities in this study,
only the segmentation of the bladder was straightforward and the analysis
focused exclusively on this organ. For each subject, bladder volumes were
generated at a rate of about $1/\mathrm{TS}$ volumes per second. According to
the configurations, 394, 492, and approximately 680 volumes were reconstructed
for configurations $C_{Star}$, $C_{Grid}$ and $C_{Lines}$, respectively, at a
rate of about 5, 8, and 9 volumes per second. As the inspiration/expiration
phases of subjects were not paced, the number of respiratory cycles could
fluctuate between successive dynamic acquisitions. Depending on the sequences
and subjects, pelvic organs undergone from 4 to 12 inspiration/expiration
phases. The average post-processing time for the complete methodology was
around 18 min per reconstructed dynamic volume, regardless of the geometrical
configuration.

Dynamic configuration $C_{lines}$ has only been studied on 3 subjects due to a
lack of resources required to the manual segmentation of the dynamic images
needed to initialize the semi-automatic process.

Participants were instructed to go to the toilet 2 hours before the exam, to
drink moderately up to 30 minutes before and then to fast so that the bladder
would not fill during the exam. This way, organ volume did not vary between
the different MRI sequences. Bladder volume of each subject was computed from
the manual segmentation of the static acquisition. Subjects presented
heterogeneous values ranged from 48 cm3 to 403 cm3.

###  3.2 Spatial discrepancy between segmentations from secant acquisition
planes

Dynamic 2D segmentations were independently realized on differently oriented
planes. As only one plane could be imaged at a time, the segmentations from
these planes naturally had a time offset between them (cf. Fig. 4). The
evaluation of the discrepancy between segmentations from secant acquisition
planes aimed to validate the relevance of the temporal reconstruction process
presented in section 2.4 for the reduction of this temporal bias.

Let $L_{ij}$ be the line defined as the intersection of two secant acquisition
planes: $L_{ij}=\mathcal{P}_{i}\cap\mathcal{P}_{j}$. The intersection of
$L_{ij}$ with the segmentation mask $\mathcal{A}_{i}^{t}$ related to the plane
$\mathcal{P}_{i}$ provided the set of points
$I_{i,j}^{t}=\\{L_{ij}\cap\mathcal{A}_{i}^{t}\\}$. From this set, the inferior
and the superior intersection points were denoted as $(I_{i,j}^{t})^{sup}$ and
$(I_{i,j}^{t})^{inf}$, respectively. A schematic illustration is depicted in
Fig. 6. In this figure, for visual reasons, the two secant planes were taken
at very different times, each corresponding to one of the
inspiration/expiration phases. In practice, during this study, distances
between segmentation intersections remained narrower thanks to the high speed
of plane acquisitions.

Spatial discrepancy $\zeta$ between two segmentations from two secant planes
was defined as the sum of the two Euclidean distances between the superior and
inferior intersection points:

$\displaystyle\begin{split}\zeta(\mathcal{A}_{i},\mathcal{A}_{j})^{t}=\left\lVert(I_{i,j}^{t})^{sup}-(I_{j,i}^{t})^{sup}\right\rVert\\\
+\left\lVert(I_{i,j}^{t})^{inf}-(I_{j,i}^{t})^{inf}\right\rVert\end{split}$
(7)

Spatial discrepancy was calculated between each pair of secant planes for each
configuration except for $C_{lines}$ (which was only composed of parallel
planes). Spatial discrepancy scores for all subjects and all instants were
1.71 $\pm$ 1.24 mm for $C_{star}$ and 2.51 $\pm$ 1.57 mm for $C_{grid}$, which
corresponds to only a few pixels.

Figure 6: Schematic illustration of the spatial inconsistency between two
segmentations from two secant acquisition planes.

###  3.3 Consistency of three-dimensional reconstructions

The B‐spline SyN registration implemented for the 3D reconstruction process
was optimized as follow: the gradient step was 0.1 for each of the four multi-
resolution levels with shrink factors of {$6,4,2,1$} and no smoothing sigmas.
The number of iterations per level were {$250,200,150,100$} with a stepping
out criterion based on a convergence threshold of $10^{-6}$ within 15
iterations. Knot spacing was set to $44$ mm for the B-spline smoothing on the
update displacement field at the base level which is reduced by a factor of
two for each successive multiresolution level. This yielded a final knot
spacing of $5.5$ mm. Such low regularization scheme was retained because
pelvic organ deformations are mostly global and no high frequency deformations
had to be permitted in the registration process. Furthermore, this
optimization preserved the overall topology of the template static organ. No
restrictions were applied to the smoothing of the total displacement field as
this did not improve the results.

Table 2: Evaluation of three-dimensional reconstructions Configuration |  Subject | RAVD (%) | HD (mm) | MD (mm)
---|---|---|---|---
$C_{star}$ |  #1 | 1.64 $\pm$ 1.47 | 1.37 $\pm$ 0.24 | 0.21 $\pm$ 0.06
|  #2 | 1.88 $\pm$ 1.80 | 1.41 $\pm$ 0.28 | 0.18 $\pm$ 0.03
|  #3 | 1.95 $\pm$ 1.41 | 1.36 $\pm$ 0.27 | 0.17 $\pm$ 0.04
|  #4 | 2.82 $\pm$ 1.61 | 1.28 $\pm$ 0.22 | 0.18 $\pm$ 0.03
|  #5 | 3.40 $\pm$ 0.94 | 1.33 $\pm$ 0.23 | 0.17 $\pm$ 0.02
|  #6 | 1.66 $\pm$ 0.67 | 1.31 $\pm$ 0.23 | 0.16 $\pm$ 0.02
|  #7 | 5.94 $\pm$ 3.31 | 1.32 $\pm$ 0.23 | 0.17 $\pm$ 0.02
|  #8 | 1.58 $\pm$ 1.23 | 1.42 $\pm$ 0.29 | 0.19 $\pm$ 0.08
|  Overall | 2.61 $\pm$ 2.22 | 1.35 $\pm$ 0.25 | 0.18 $\pm$ 0.04
$C_{grid}$ |  #1 | 2.70 $\pm$ 2.49 | 2.44 $\pm$ 0.68 | 0.45 $\pm$ 0.17
|  #2 | 2.36 $\pm$ 2.12 | 2.07 $\pm$ 0.33 | 0.33 $\pm$ 0.09
|  #3 | 2.63 $\pm$ 1.68 | 2.86 $\pm$ 0.63 | 0.65 $\pm$ 0.15
|  #4 | 2.74 $\pm$ 2.13 | 1.76 $\pm$ 0.29 | 0.28 $\pm$ 0.07
|  #5 | 1.26 $\pm$ 0.67 | 1.65 $\pm$ 0.28 | 0.23 $\pm$ 0.05
|  #6 | 1.67 $\pm$ 1.03 | 1.82 $\pm$ 0.23 | 0.25 $\pm$ 0.05
|  #7 | 2.00 $\pm$ 1.46 | 1.76 $\pm$ 0.35 | 0.25 $\pm$ 0.05
|  #8 | 1.51 $\pm$ 1.21 | 1.75 $\pm$ 0.37 | 0.22 $\pm$ 0.09
|  Overall | 2.11 $\pm$ 1.79 | 2.01 $\pm$ 0.58 | 0.33 $\pm$ 0.17
$C_{lines}$ |  #5 | 1.40 $\pm$ 0.91 | 3.60 $\pm$ 0.70 | 0.85 $\pm$ 0.15
|  #6 | 1.03 $\pm$ 0.70 | 4.49 $\pm$ 0.89 | 1.07 $\pm$ 0.20
|  #7 | 5.28 $\pm$ 2.85 | 3.31 $\pm$ 0.60 | 0.75 $\pm$ 0.14
|  Overall | 2.57 $\pm$ 2.62 | 3.80 $\pm$ 0.90 | 0.89 $\pm$ 0.21
RVD = relative volume deviation, HD = Hausdorff distance, MD = mean distance.
Values are mean $\pm$ standard deviation of all reconstructed volumes.

Validation of the 3D organ reconstructions was a difficult process due to the
lack of ground truth as a point of comparison. Therefore, the evaluation
focused on the consistency of the reconstructed volumes through two metrics:
accuracy of the registration and conservation of the bladder volume. As the
bladder is an incompressible organ (filled with a liquid), its volume has to
remain unchanged during the registration process when the static volume is
deformed to match the shape of the partial dynamic volumes. Relative absolute
volume deviation (RAVD) was computed at each instant between the reference
volume and the reconstructed volume. The time averages of the values are
gathered in Table 2. Average volume deviations were quite similar for all
configurations with a mean value of $2.39\%\pm 2.16\%$. Outliers in Subject
7’s data might be attributed to the fact that the subject did not comply with
pre-acquisition fluid consumption guidelines and therefore had the lowest
bladder volume among all subjects.

Evaluation of the mapping of the deformed static volume onto the organ volume
skeletons provided the assessment of the registration accuracy. For each point
of the contour of the dynamic volume skeleton, the distance to the nearest
point of the contour of the reconstructed volume was calculated. From these
distances were extracted the maximum value and the average value, described as
the non-symmetric Hausdorff distance (HD) and the mean distance (MD),
respectively (Table 2). The average HD and MD over time reflected great
registration accuracy for all configurations with mean values of 2.19 $\pm$
2.09 and 0.41 $\pm$ 0.31 mm, respectively. It should be noted that comparison
of the results between configurations must take into consideration that HD and
MD were computed in the discrete 3D domains of the reconstructions and were
therefore directly related to their isotropic spatial resolution. As reported
in Table 1, voxel size was 1.09 mm for $C_{star}$, 1.36 mm for $C_{grid}$ and
1.82 mm for $C_{lines}$. These results once expressed in voxels are quite
similar between configurations and accuracy is about 1 or 2 voxels overall.

Figure 7: Anterior (left) and posterior (right) views for each subject and
configuration of the $\sigma_{J}$ map projected on the corresponding reference
static volumes. Reddest areas indicate the location of the highest strain
areas on the organ surfaces during dynamic acquisition sequences.

###  3.4 High-level representation of pelvic organ deformation zones

Reconstructed volumes over skeletons and projections of the associated
Jacobian maps are available for two subjects as supporting materials.
Projections of the $\sigma_{J}$ maps on meshes of the static reference volumes
are shown in Fig. 7 for each subject and each configuration. Visualizations
were performed using the Anatomist software [36]. At each point, the value for
the intersection between the mesh of the static reference volume and the
$\sigma_{J}$ map, was estimated as the mean of the values within a sphere of 4
mm radius, almost 3 voxels, around the intersection point. This projection
scheme was common to all configurations to provide a common reference for
comparison regardless the resolutions of the reconstructed volumes. The
reddest areas indicate the locations that underwent the largest deformations
during the loading exercises. Note that volume scales are not the same for
each subject in Fig. 7, _e.g._ bladder volume was 48 cm3 for subject 7 whereas
it was 403 cm3 for subject 8.

Forced breathing exercises with maximum contraction are hardly reproducible
for untrained subjects. Deformation magnitudes were different for each dynamic
sequence and consequently differed for each geometric configuration $C$. For
instance, in Fig. 7, subjects 4 and 5 did not exert the same intensities
during each loading exercise explaining the main scale differences. On the
whole, no visual correlation can be done between geometry configuration and
amplitude of the deformations. Dynamic sequences with the highest deformation
characteristics are $C_{star}$ for subjects 1, 2, 7, and 8, $C_{grid}$ for
subjects 3, 4, and 5, and $C_{lines}$ for subject 6. Intra-subject comparison
of deformation amplitudes might not be relevant and was not the focus of our
study. However, regardless of the load magnitude applied to the bladder, the
intra-subject locations of the most deformed tissue areas are quite similar
between dynamic sequences. Overall, tissue areas that have undergone
deformities are located on the anterior and superior parts of bladders due to
the abdominal viscera pushing against the bladder during loading exercises.
Few deformation zones are visible on the inferior part due to a strong pelvic
floor in our young healthy volunteers. However, inferior deformations in
patients could indicate pelvic floor weaknesses.

##  4 Discussion

While extensive research has been conducted on the movement of organs such as
the brain [37] and heart [38], little work has been done on the analysis of
the pelvic organ dynamics. Most of the studies have focused either on the
realization of biomechanical simulators generated from 3D static acquisition
[15], or on the characterization of organ deformations based on a single 2D
dynamic slice [8, 9]. Recently, Courtecuisse _et al._ [18] has proposed a
registration approach to deform pelvic organs acquired in 3D static
acquisition using single-plane 2D dynamic acquisition and _a priori_ knowledge
of boundary conditions and mechanical laws of the pelvic organs. This
biomechanical knowledge had to be injected into the registration model because
the ground truth information from a single 2D section is not enough to
accurately guide organs that deform nonlinearly in all 3 dimensions. This
approach remains limited because it depends on pelvic tissue characterization,
which remains unclear with high inter-subject variability, and the realization
of patient-specific models would require invasive procedures to probe the
biomechanical properties _in vivo_. In our study, we provided 3D spatial
coverage of the organs through the acquisition of several dynamic slices.
Dynamic volumes did not require to be simulated in a biomechanical sense and
were reconstructed independently of any mechanical model. And the subject-
specific initial geometric model required for the 3D reconstruction process
was readily available from a fast 3D static acquisition. Two possible static
states were available for each subject from maximum inspiration or maximum
expiration breath-hold. The latter one was retained because it was the state
closest to the natural condition of organs and, in pathological routines, will
be less painful than inspiration. In further studies with patient recruitment,
the deformation of the static model on dynamic volumes may require refinement
to account for pelvic prolapse. This study emphasized metrics to assess the
adequacy of bladder reconstructions. Using these metrics, we demonstrated the
appropriateness of the registration method for the reconstruction by
preserving an average RAVD value around $2.5\%$ and respecting the ground-
truth contours provided by the dynamic multi-slices with average HD values of
2.2 $\pm$ 1.1 mm and average MD values of 0.4 $\pm$ 0.3 mm.

One limitation is the semi-automated segmentation that requires a minimum user
interaction. For pelvic organs, existing automatic segmentation methods are
generally dedicated to a specific type of datasets, either static 3D images or
2D axial or sagittal images. Most of approaches are based on deformable models
[29] which are rather difficult to parameterize even for one type of images.
So, this step remained the only one that has not been fully automated in our
pipeline. Still, one of the originalities of our study was to propose dynamic
acquisitions under different planes with open 3D geometric configurations. The
advantages of the semi-automatic method that we propose are manifold: First,
the time spent to perform manual segmentation has been reduced by about $90\%$
as only one in ten slices was manually segmented; Second, the proposed method
is suitable on all planes, regardless of their orientations, resolutions, and
contrasts. Thus, the proposed method adjusts to all these acquisition
parameters by propagating organ contours that were manually initialized on a
few instants, from one slice to another similar one. In further studies, such
semi-automatic method of segmentation will facilitate the creation of labeled
datasets of 2D dynamic MRI and hence will allow to investigate the relevance
of deep-learning methods for pelvic organ segmentation. To date, these
approaches have been scarcely used and still provide insufficient results,
with average DSC around $85\%$ [39].

Within one cycle, planes are not acquired simultaneously, thus skeleton
definition is ill-posed due to the spatial discrepancy $\zeta$ between
crossing planes segmentations. This element was alleviated by proposed
temporal interpolation that joined virtually simultaneous crossing planes.

Dynamic volumes from each configuration could not be directly compared because
subjects could not reproduce same pelvic organ movements between each dynamic
acquisition. Frequency of pushing phases and magnitude of the load applied to
the organs were highly variable. Volumes, and associated $J$ maps, could not
be compared one to one from one configuration to another. However, the higher-
level $\sigma_{J}$ maps allowed to compare the spatial distribution of
deformed tissue areas. For all subjects, the spatial distribution of deformed
tissue regions was similar between configurations. Although $\sigma_{J}$ map
has no direct physical meaning, its projection on the reference volume
provided a preliminary mechanical characterization of the organs.

Overall, our processing pipeline proved to be compatible with all three
configurations in terms of bladder reconstruction and deformation
characterization, with equivalent end results. One of the three dynamic
acquisition planes configurations can be chosen to ensure either complete
spatial coverage ($C_{lines}$) or high spatial resolution ($C_{star}$), and
combined with our methodology to provide dynamic 3D representation of the
bladder during loading exercises. The proposed pipeline is directly applicable
in clinics and can be added to a routine examination as the acquisition time
required is minimal and sequences are standard. In practice, only a 20 seconds
static apnea sequence and a multi-planar acquisition of one straining exercise
are necessary.

We have designed a complete methodology for the 3D+t representation of pelvic
organs, during loading exercises, directly combined with a high-level 3D
representation of the most strained areas on the organ surfaces. To our
knowledge, this study is the first to propose a dynamic 3D real-time
observation of the pelvic region as well as a 3D representation of the organs
deformations. Our proposal opens novel perspectives towards real-time dynamic
3D pelvic imaging and biomechanical pelvic organs characterization.

In this study, three different geometrical configurations of dynamic multi-
planar acquisitions have been proposed to provide the first dynamic 3D
representation of pelvic organs. Two of them were particularly original,
$C_{star}$ and $C_{grid}$, because they were composed of planes oriented
differently from the sagittal plane as usually imagined in clinics. In this
study, we verified that all three configurations could provide deformation
characterization. In consequence, the choice of a multi-planar configuration
depends on the preference for spatial coverage or spatial resolution. In
clinical practice, it might be easier to configure $C_{lines}$, but
configuring $C_{star}$ should be straight-forward and enable improved
visualization of smaller structures.

To allow a direct transfer to clinics, only the acceleration methods available
on the clinical machines were considered. However, the recently developed
simultaneous multi‐slice acquisition would allow the simultaneous acquisition
of several parallel planes at the same time [40]. Besides, future studies will
be conducted with the ethical authorization to inject extrageneous liquid into
the rectal and vaginal cavities in order to segment all pelvic organs and
study the complete pelvic dynamics. The ability to now obtain dynamic 3D
volumes allows the development of more advanced 3D shape descriptors [41] to
better characterize the degree of severity of a pathology for diagnostic
assistance. These studies are a 3D extension of descriptors such as we already
proposed for 2D dynamic pelvic acquisitions [8]. All this leads to the
eventuality of a 3D ground truth that will allow the validation of future
biomechanical models at least.

##  5 Acknowledgements

This study was ethically approved by the following institutional review board:
“comité de protection des personnes Sud-Ouest et Outre-Mer 1”, from September
21st 2020, ID RCB : 2020-A00673-36. This research did not receive any specific
grant from funding agencies in the public, commercial, or not-for-profit
sectors.

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## Supporting materials caption

Animations describing the volume deformations of 2 observed bladders. Global
displacement is not reproduced, but colormap provided a visualization of the
local surface changes. First column: reconstruction of the 3D dynamic volumes
(blue) on the skeletons from the multi-slice acquisition segmentations (red).
Second column: Projection on the static volume of the Jacobian maps associated
with the registrations generating the reconstructed complete dynamic volumes.
Third column: Single projection of the $\sigma_{J}$ map on the static volume.
Volumes from each geometric configuration were placed in a common time frame
but the volume rate per second is different and each configuration was imaged
during independent acquisition. For each representation, the anterior (left)
and posterior (right) views are provided.