# Multi-center, multi-vendor automated segmentation of left ventricular
anatomy in contrast-enhanced MRI11footnotemark: 1

Carla Sendra-Balcells<EMAIL_ADDRESS>Víctor M. Campello  Carlos Martín-
Isla  David Vilades Medel  Martín Luís Descalzo  Andrea Guala  José F.
Rodríguez Palomares  Karim Lekadir  Dept. de Matemàtiques i Informàtica,
Universitat de Barcelona  Unitat d’Imatge Cardíaca, Hospital de Sant Creu i
Sant Pau  Cardiovascular Imaging Unit, Aortic diseases and Inherited
cardiomyopathies (CSUR), Hospital Universitari Vall d’Hebron

###### Abstract

Background: Accurate delineation of the left ventricular boundaries in late
gadolinium-enhanced magnetic resonance imaging (LGE-MRI) is an essential step
for scar tissue quantification and patient-specific assessment of myocardial
infarction. Many deep-learning techniques have been proposed to perform
automatic segmentations of the left ventricle (LV) in LGE-MRI showing
segmentations as accurate as those obtained by expert cardiologists. Thus far,
the existing models have been overwhelmingly developed and evaluated with LGE-
MRI datasets from single clinical centers with homogeneous imaging protocols
and characteristics. However, in practice, LGE-MRI images vary significantly
between clinical centers within and across countries, in particular due to
differences in the MRI scanners, imaging conditions, contrast injection
protocols and local clinical practise.
Methods: This work investigates for the first time multi-center and multi-
vendor LV segmentation in LGE-MRI, by proposing, implementing and evaluating
in detail several strategies to enhance model generalizability across clinical
sites. These include data augmentation to artificially augment the image
variability in the training sample, image harmonization to align the
distributions of LGE-MRI images across clinical sites, and transfer learning
to adjust existing single-center models to unseen images from new clinical
sites.
Results: The results obtained based on a new multi-center LGE-MRI dataset
acquired in four clinical centers in Spain, France and China, show that the
combination of data augmentation and transfer learning can lead to single-
center models that generalize well to new clinical centers not included in the
original training.
Conclusions: The proposed framework shows the potential for developing
clinical tools for automated LV segmentation in LGE-MRI that can be deployed
in multiple clinical centers across distinct geographical locations.

###### keywords:

Late gadolinium-enhanced magnetic resonance imaging (LGE-MRI), Multi-center
image segmentation, Generalizable deep learning

††journal: arXiv

##  1 Background

###  1.1 Problem and motivation

Patient-specific assessment of myocardial infarction (MI), one of the leading
causes of morbidity and mortality worldwide [1], is routinely performed in
clinical practice based on late gadolinium-enhanced magnetic resonance imaging
(LGE-MRI). The image modality enables to highlight the presence and extent of
scar tissues in the left ventricular myocardium. A prerequisite for scar
tissue quantification is the accurate delineation of the boundaries of the
left ventricle (LV) [2]. While LV automated segmentation on cine-MRI has been
extensively investigated [3], application of existing solutions to LGE-MRI is
faced with additional challenges due to the intensity heterogeneity arising
from the accumulation of contrast agent in the infarcted myocardium. This
introduces artefacts within and outside of the myocardium, which affect the
definition of the endo- and epi-cardial boundaries. Furthermore, the number,
location, shape and size of the scar tissues can vary greatly across patients,
which further complicates the development of robust LV segmentation methods in
LGE-MRI. As a result, while many research and commercial tools are already in
use for cine-MRI, LV segmentation in LGE-MRI still relies on labor-intensive
manual delineation in clinical practice.

To address this problem, researchers investigated three main families of
solutions over the last decade. Initially, registration-based approaches were
attempted, such that the LV boundaries extracted from the cine-MRI images were
propagated onto the LGE-MRI images after non-rigidly aligning the
corresponding images ([4], [5], [6]). Such an approach, however, suffered from
a lack of robustness due to inherent differences in appearance between the
cine- and LGE-MRI images. Later on, deformable models were proposed to better
guide the LV segmentation process in LGE-MRI images, by directly encoding the
shape and appearance variability of the infarcted myocardium [7]. However, the
local heterogeneity induced by the scared areas and contrast enhancements
continued to pose difficulties, affecting the robustness of the existing
techniques. Finally, with the advent of deep learning, a new generation of
techniques were developed to better encode the complexity found in LGE-MRI
images based on convolutional neural networks ([8], [9], [10], [11]). Despite
their promise, these techniques have suffered from one important limitation,
namely that the neural networks have been almost systematically trained and
validated with an LGE-MRI sample from a single clinical center. Because neural
networks are known to lack generalizability to unseen domains, these methods
have had limited clinical applicability at scale thus far.

The problem in question is often referred to as the domain shift problem, and
has been attracting increased attention recently in the medical image analysis
community. For example, a recent challenge on this topic was organized in the
cardiac MRI domain at the 2020 Medical Image Computing & Computer-Assisted
Intervention conference (MICCAI 2020) by the University of Barcelona, in
collaboration with six Spanish, German and Canadian clinical centres. Entitled
”Multi-Center, Multi-Vendor and Multi-Disease Cardiac Segmentation (M&Ms)”,
the study demonstrated that single-center, single-vendor neural networks do
not generalize naturally when segmenting cine-MRI images with distinct imaging
domains [12]. Multi-center LGE-MRI segmentation is even more challenging as,
in addition to differences in populations, especially when applying the models
across continents and ethnic groups, LGE-MRI samples show even further
variability in imaging conditions between cardiology centres, depending on the
scanner vendors, imaging sequences, contrast injection protocols and local
clinical practice. In particular, the time between contrast injection and LGE-
MRI image acquisition can vary greatly between centers, typically between 7
minutes up-to to a total of 10 minutes, resulting in differences in contrast
wash-out and image formation. As a result, the LGE-MRI appearance, both
globally and locally, can have marked differences as clearly illustrated
Figure 1 based on images from four different hospitals. At the same time, the
limited numbers of available LGE-MRI datasets compared to cine-MRI images,
combined with legal and organizational obstacle across centers and countries,
has made access to interoperable multi-center LGE-MRI datasets more difficult,
which has further limited the development of generalizable models for LGE-MRI
segmentation.

Figure 1: Four LGE-MRI images together with the average intensity distribution
of each dataset, acquired in 4 different hospitals. Each histogram has a very
different shape and shows the variability between centers in terms of image
intensities.

###  1.2 Goals and contributions

This work emerges to address the need for automated segmentation methods in
LGE-MRI that can be applied reproducibly across clinical centers, while
resolving for inherent biases due to differences in scanner vendors and
imaging protocols. To the best of our knowledge, only one work has thus far
included a multi-center and multi-vendor LGE-MRI dataset [13], but no
generalization technique was studied to enable scalability beyond the training
datasets. Instead, in this work, we present the first deep learning pipeline
that explores specific mechanisms to achieve generalizability to unseen
clinical centers. To this end, several techniques were investigated, optimized
and systematically evaluated, including data augmentation, domain mixing,
transfer learning and domain adaptation. This work is based on a unique multi-
center LGE-MRI dataset acquired with three distinct scanner vendors (Siemens,
Philips and General Electric) in four hospitals located in three countries
(France, Spain and China). The results show that single-center neural networks
enriched with suitable generalization procedures can reach and even surpass
the performance of multi-center, multi-vendor models, hence eliminating the
need for comprehensive multi-center datasets to train generalizable LGE-MRI
models for future cardiac patient assessment.

##  2 Methods

In this work, a very first end-to-end pipeline is investigated for the
segmentation of the LV, including its blood pool and myocardium, in multi-
center LGE-MRI contexts. To this end, four different approaches are explored
to enhance the generalizability across sites of existing neural networks for
LGE-MRI segmentation, as schematically represented in Figure 2. These include:

  1. 1.

Data augmentation techniques to artificially extend the data distribution
captured by the trained models.

  2. 2.

Image harmonization to align the data distributions of the training and
testing images.

  3. 3.

Transfer learning to adjust the neural network to the new clinical center
based on very few unseen images.

  4. 4.

Multi-center models directly trained with data from multiple clinical centers,
which are used for comparative evaluation of the different generalization
mechanisms.

Figure 2: Four different approaches implemented in this work to enhance the
generalisability of LGE-MRI segmentation models across distinct clinical
sites.

We confirm that all experiments were performed in accordance with relevant
guidelines and regulations.

###  2.1 Datasets

The multi-center and multi-vendor dataset used in this study consists of 216
LGE-MRI cases acquired in four different clinical centers as detailed in Table
2.1. Two out of four samples are publicly available datasets from France and
China, while the two other samples correspond to new LGE-MRI images acquired
in two different hospitals in Spain. The subjects have been scanned by using a
range of scanner vendors by Siemens, Philips or General Electric (GE). In
addition to having distinct intensity distributions as observed in Figure 1,
the multi-center LGE-MRI images also differ in the image resolution (0.75-1.88
mm), slice thickness (5-13 mm), and acquisition time after contrast injection
(7 to 10 minutes). The samples from each clinical site are described in more
detail next.

Table 1: Details of the multi-centre LGE-MRI datasets and characteristics of the acquired images used in this work. Imaging time = Acquisition time after contrast injection. Dataset |  Clinical centre | Country |  MRI scanner |  Imaging time (mins) |  In-plane resolution (mm) |  Slice thickness (mm) |  Number of slices |  Sample size
---|---|---|---|---|---|---|---|---
EMIDEC |  University Hospital of Dijon | France |  1.5T and 3T Siemens | 10 | 1.37-1.88 | 8-13 | 4-10 | 100
MSCMR |  Shanghai Renji Hospital | China |  1.5T Philips | - | 0.75 | 5 | 10-18 | 45
VH |  Vall d'Hebron Hospital | Spain | 1.5T GE | 10 | 1.48-1.68 | 10 | 8-15 | 41
STPAU |  Sant Pau Hospital | Spain |  1.5T Philips | 7-10 | 1.18 | 5 | 18-24 | 30

####  2.1.1 EMIDEC dataset: University Hospital Dijon, France

This dataset was compiled as part of the automatic Evaluation of Myocardial
Infarction from Delayed-Enhancement Cardiac MRI challenge (EMIDEC) [14]. The
EMIDEC volunteers included 33 healthy and 67 diseased subjects, for a total of
100 studies. The data was acquired at the University Hospital of Dijon,
France, using Area 1.5 T as well as Skyra 3T Siemens MRI scanners. Slice
thickness and in-plane spatial resolution varied greatly, being comprised
between 8 and 13 mm and 1.37 and 1.88 mm, respectively. The manual
segmentation of the LV blood pool and myocardium were performed by a
cardiologist with over 10 years of experience. It is the largest of the four
samples and hence it was used as the reference sample for training the single-
center neural networks.

####  2.1.2 MSCMR dataset: Shanghai Renji Hospital, China

The MSCMR dataset was obtained from the Multi-sequence Cardiac MR Segmentation
Challenge and it comprises a total of 45 patients suffering from various
cardiomyopathies ([15], [16]). The images were acquired at the Shanghai Renji
hospital, China, which will allow us to evaluate generalizability across
countries as well as continents in this study. Compared to EMIDEC, the MSCMR
dataset has a higher image resolution (in-plane resolution = 0.75 mm, slice
thickness = 5 mm for all scans) and all images were acquired with a 1.5 T
Philips scanner. The manual delineations were initially performed by trainees
and later on validated by expert cardiologists.

####  2.1.3 VH dataset: Vall d’Hebron Hospital, Spain

The VH dataset consists of 41 LGE-MRI datasets acquired at the Vall d'Hebron
University Hospital, located in Barcelona, Spain. In addition to covering a
new geographical location, namely Spain, the VH sample has several differences
with EMIDEC and MSCMR, including the disease group (MI) and the MRI scanner
(1.5 T GE scanner). Manual annotations of the LV boundaries were generated by
a trained rater using the cvi42 software. The study was approved by the ethics
committee of the Vall d’Hebron Hospital and written informed consent was
obtained from all participants.

####  2.1.4 STPAU dataset: Sant Pau Hospital, Spain

The STPAU dataset comprises 30 LGE-MRI cases acquired at the Sant Pau Hospital
in Barcelona, Spain. While the clinical center is located in the same region
as for the VH sample, the dataset covers a different disease group (ischemic
and non-ischemic cardiomyopathy) and was acquired using an MRI scanner from a
different vendor (Philips Achieva 1.5T) and a higher-resolution imaging
protocol. Furthermore, the time delay between contrast injection and image
acquisition varies between 7 and 10 minutes, which adds extra variability. The
manual annotations were also performed using cvi42, as in the previous case.
All patients signed the informed consent, the study protocol was approved by
the Ethical Committee for Clinical Research of our region, and it follows the
ethical guidelines of the Declaration of Helsinki.

###  2.2 Single-center model with data augmentation

Figure 3: Both spatial and intensity-based data augmentation techniques are
applied together with a probability of 0.2 each. From only one slice many
samples can be generated, increasing the size of the original dataset
significantly.

In this work, we first investigate the potential of data augmentation to
enhance the generalizabilty of LGE-MRI segmentation models (Method 1 in Figure
2). Data augmentation has been widely used to create more robust neural
networks by increasing the size as well as the heterogeneity of training
samples synthetically. However, the promise of data augmentation is yet to be
examined for LGE-MRI, where there is higher complexity due to inherent
variability in scar characteristics and contrast appearance.

In this work, we investigate several operators for data augmentation in the
context of LGE-MRI as illustrated in Figure 3 and described as follows.

  1. 1.

Spatial-based data augmentation: In addition to the natural variability
between cardiac anatomies, especially across countries and ethnic groups,
patients undergoing LGE-MRI typically suffer from regional remodelling of the
ventricles due to the presence of scar tissue. Hence, spatial-based data
augmentation is proposed using the following operators:

     * (a)

Horizontal and vertical flips to generate images with different orientations.

     * (b)

Random rotations of up to $\pm 30$ degrees, to simulate different positions of
the heart.

     * (c)

Random rescaling in the [0.75, 1.88] mm range so that the model can process
images and hearts that vary in size. This range is defined by the minimum and
maximum voxel size of our multi-center dataset.

     * (d)

Random cropping, such that the training images have the same dimensions of
256x256 pixels but with a variation in the position of the heart in the image.

  2. 2.

Intensity-based data augmentation: Because the LGE-MRI appearance can vary
between images acquired using different MRI scanners and scanning protocols,
such as due to differences in acquisition time after contrast injection, we
implemented the following intensity-based data augmentation techniques:

     * (a)

Bilateral filtering to generate blurred and less detailed copies of the
original images.

     * (b)

Gaussian noise with a standard deviation ranging between [0, 0.03] to generate
artificial noise and image artifacts.

     * (c)

Gamma and inverse Gamma function with magnitude [0.7, 1.5] to generate
synthetic images with different lighting.

     * (d)

Brightness and contrast with magnitude [-0.5, 0.5] to support brightness and
contrast variations in the training images.

Each data augmentation technique is applied with a probability of 0.2 during
the training of the model. Then, this data augmentation pipeline is evaluated
by measuring the final generalization ability of the network (Method 1 in
Figure 2). Table 2.2 summarizes the split of the data used for the training,
validation and testing of the model in each experiment.

Table 2: Number of subjects for each of the four datasets used during the training, validation and testing phases when data augmentation is implemented in a single-center setting. Dataset |  Training |  Validation |  Testing
---|---|---|---
EMIDEC |  68 |  17 |  15
MSCMR |  24 |  6 |  15
VH |  21 |  5 |  15
STPAU |  12 |  3 |  15

###  2.3 Image harmonization at testing

While the data augmentation operations focused on improving model
generalizability at training, we propose to apply image harmonization at the
testing stage to further reduce the discrepancies between the multi-center
LGE-MRI images (Method 2 in Figure 2). Image harmonization enables to
transform the testing LGE-MRI images from a new clinical center such that
their intensity distribution matches as much as possible the imaging
characteristics of the single center used to train the baseline neural
network. Concretely, two main image harmonization techniques were implemented:

  1. 1.

Histogram matching: It consists of transforming the testing images from the
unseen center such that the histogram of the pixel intensity values is
superimposed as much as possible with the corresponding histogram extracted
from the training images from the training clinical center. The transformation
from the testing data (B: target data) to the training data (A: source data)
is illustrated in Figure 4(i).

  2. 2.

CycleGAN: Another strategy to address the domain shift between multiple
centers is domain adaptation, which can be used to learn the image translation
from the source domain to the target domain. To this end, we choose to
implement a CycleGAN architecture [17], based on an unpair image-to-image
translation. Given that CycleGAN uses cycle consistency, it would learn the
translation from the target domain (B) to the source domain (A), and viceversa
(Figure 4(ii)). Both target to source and source to target generators are
saved in each implementation, decreasing to 6 the number of implementations
needed. The amount of samples used to train each of the CycleGAN models are
summarized in Table 2.3, adjusting for each case the percentage of images from
each center so that it is adequately balanced (50% source and 50% target
data).

Table 3: Number of samples used for the training and validating each CycleGAN model built to harmonise the imaging properties from the different clinical centres. Dataset | Training | Validation
---|---|---
Source | Target | Source | Target | Source | Target
EMIDEC | MSCMR | 24 | 24 | 6 | 6
EMIDEC | VH | 21 | 21 | 5 | 5
EMIDEC | STPAU | 12 | 12 | 3 | 3
MSCMR | VH | 21 | 21 | 5 | 5
MSCMR | STPAU | 12 | 12 | 3 | 3
VH | STPAU | 12 | 12 | 3 | 3
Figure 4: Schematic illustration of the image harmonisation techniques used in
this work to make the intensity distributions from the different clinical
sites as aligned as possible. First, histogram matching is used to learn a
transformation of the histogram of each image from the unseen clinical sites
(B) onto the original training clinical centre (A). Second, CycleGAN
architecture is used to learn the mapping between the training and the testing
clinical center.

###  2.4 Transfer learning from the original to the new clinical site

Another strategy investigated in this work to improve the scalability of
single-center models consisted of applying the so-called transfer learning
paradigm, by fine-tuning specific layers of the neural network with a reduced
number of LGE-MRI images from the new clinical site (Method 3 in Figure 2).
The approach has shown promise for multi-center image segmentation in cardiac
cine-MRI [18], but is yet to be demonstrated for multi-center LGE-MRI imaging,
where there is increased variability. The following steps are implemented in
this work:

  1. 1.

Initiate the training of the neural network with the EMIDEC dataset, then
evaluate the minimum number of fine-tuned layers, in both the decoder and the
encoder, that are needed during transfer learning to obtain the maximal
segmentation performance on the new LGE-MRI datasets from the remaining
clinical centers.

  2. 2.

Compare the previous results with the segmentations obtained based on a multi-
center model directly trained each time with images from two clinical centers
(EMIDEC and the new center).

  3. 3.

Estimate the minimum percentage of images needed from the second clinical
center during the fine-tuning to obtain the desired level of performance.

  4. 4.

Implement the same approach from the previous point but this time by using a
model pre-trained on a large dataset (n=350) from cine-MRI (M&M’s dataset), to
evaluate transfer learning from a related cardiac MRI modality for which data
is abundantly available.

###  2.5 Multi-center model

A fourth and last modelling strategy, i.e. training the neural networks
directly from multiple centers (Method 4 in Figure 2), is used for comparative
evaluation of the three extended single-center models described in the
previous section, i.e. enriched with data augmentation, image harmonization
and transfer learning. In this study, we investigated the amount of new
centers/domains that are needed to bridge the domain gap in LGE-MRI
segmentation, by using a balanced dataset with the same number of subjects for
each multi-center data combination, namely EMIDEC, EMIDEC+MSCMR, EMIDEC+VH,
EMIDEC+MSCMR+VH and ALL centers. The samples used for training the multi-
center models in each combination of datasets/centers are listed in Table 2.5.
In all experiments, the same testing dataset is used for comparative
evaluations (n=15).

Table 4: List and number of samples used for training and validating multi-centre models in this study. Dataset | Training | Validation
---|---|---
EMIDEC | 42 | 10
EMIDEC+MSCMR | 21+21 | 5+5
EMIDEC+VH | 21+21 | 5+5
EMIDEC+MSCMR+VH | 14+14+14 | 3+3+3
EMIDEC+MSCMR+VH+STPAU | 11+11+11+11 | 3+3+3+3

###  2.6 Baseline workflow

####  2.6.1 Pre-processing

Min-max normalization is used after cropping of the image to keep the same
intensity range in images from the same dataset.

####  2.6.2 Post-processing

A post-processing is applied to all predictions generated by the model by
keeping only the largest connected component of the segmentation volume. This
step is commonly used in medical image segmentation, especially in organ
imaging, to help on the detection of false positives.

####  2.6.3 Network architecture

As a baseline model, a U-Net architecture is implemented to perform the LV
boundary segmentations in LGE-MRI based on some of the modifications proposed
by [19] for improved model training as follows. First, Leaky ReLU is used as
activation function, then instance normalization is applied after each hidden
convolutional layer to stabilise the training. Deep supervision is included to
allow gradients to be injected deeper into the network and facilitating the
training of all layers. Furthermore, a 2D architecture is selected as it is
suitable to address the differences in slice thickness between clinical
centers, as well as slice misalignment due to respiratory and cardiac motion
artefacts. The encoder and decoder architecture of the model are illustrated
in Figure 5.

Figure 5: U-Net architecture composed by 6 layers, increasing progressively
the number of feature maps until 1024. Additionally, deep supervision layers
are included in the decoder.

####  2.6.4 Implementation details

PyTorch is the open-source machine learning library for Python used for the
implementation of the model learning. The stochastic gradient descent (SGD)
optimization is performed with Adam and the batch size of 16 slices is
constrained by the 8 GB of memory of the NVIDIA GeForce RTX 2080 Ti GPU. The
learning rate is kept to 1$\cdot$10-3 during every training, while the dice
and cross entropy losses are calculated at every iteration to optimize the
network parameters. The neural network is trained 250 epochs each time and
takes half an hour approximately to converge. During testing, each LGE-MRI
image segmentation takes less than one second. The main criterion followed to
split each dataset in subgroups is 80% for the training and 20% for the
validation part, while keeping 15 subjects for the testing.

####  2.6.5 Performance evaluation

For all experiments and results, the performance of each method will be
assessed with the average 3D Dice Coefficient (DC), which calculates the
overlap ratio between the automatically generated and ground truth
segmentations. The measure is estimated by:

$\mbox{DC}=\frac{2\cdot(X\cap Y)}{X+Y}=\frac{2\cdot TP}{2\cdot TP+FP+FN},$ (1)

where X and Y are the set of pixels from the automated and true labels of the
target structures, while TP, FP, and FN are the corresponding true positives,
false positives and false negatives, respectively.

##  3 Results

This section presents detailed experimental results obtained by evaluating and
comparing the different strategies proposed for enhancing model
generalizability in LGE-MRI segmentation based on single-center models.

###  3.1 Experiment 1: Effect of data augmentation

In the first experiment, we will evaluate the added value of the different
types of data augmentation, including spatial and intensity-based data
augmentations. Figure 6 shows the comparative results obtained by three
different models: (i) a single-center model without data augmentation (blue
line), (ii) a single-center model with spatial data augmentation (orange), and
(iii) a single-center model enriched with both spatial and intensity based
data augmentations (green). The datasets/centers used for training the single-
center model, as well as for testing unseen LGE-MRI images, were permuted to
test data augmentation under different scenarios (e.g. sample size). It can be
seen from the results that, independently of the clinical center used for
training, data augmentation consistently improves the segmentation performance
for LGE-MRI, increasing the DC value up to 0.7 units with respect to the
baseline model without data augmentation. Furthermore, the results in Figure 6
show that while most of the improvement can be achieved by spatial data
augmentation (orange line), intensity-based data augmentation adds value to
the approach, in particular when training on the largest sample (EMIDEC) and
testing on the smaller samples (MSCMR, VH and STPAU). Having demonstrated the
added value of data augmentation, it will be included in all subsequent
experiments and results.

Figure 6: Comparison of single-centre models without and without data
augmentation, including spatial and intensity based. X corresponds to each
testing dataset not included in the training.

###  3.2 Experiment 2: Effect of image harmonization

Here, the impact of image harmonization is evaluated when applied to match the
testing LGE-MRI images from a new clinical center based on the distribution of
the training set from the original site. Specifically, we evaluate three
approaches, namely (i) the baseline model with data augmentation from
Experiment 1 but without any normalization, (ii) baseline model with histogram
matching, and (iii) baseline model with CycleGAN normalization. The results
are given in Figure 7, clearly showing that, overall, the two harmonization
operations (green and orange lines) do not improve significantly the LGE-MRI
segmentations over the baseline model without harmonization (blue line). There
are few exceptions, however, such as when the baseline single-center model is
trained on the VH sample and tested on the EMIDEC centre using histogram
matching.

Figure 7: Effect of histogram matching and CycleGAN harmonisation for LGE-MRI
segmentation in unseen clinical centres. X corresponds to each testing centre
not included in the training dataset.

###  3.3 Experiment 3: Effect of transfer learning

This section evaluates the potential value of transfer learning such that the
baseline single-center model is fine-tuned based on a few datasets from the
new clinical centers. Figure 8 shows the performance of the transfer learning
when the baseline model is trained on EMIDEC as the largest sample and tested
on the remaining clinical centers (MSCMR, VH and STPAU). The red and blue
lines in the figure show the segmentation accuracy results when the fine-
tuning is performed on the encoder and decoder of the neural network,
respectively. The black line corresponds to the ideal segmentation model
obtained by training the neural network from all LGE-MRI of the new clinical
center. It can be seen that the DC values increase with the number of fine-
tuned blocks and the maximum is obtained when 5 blocks of the encoder are
fine-tuned [20], reaching nearly the same performance as the single-center
model of the new center (black line). Furthermore, in Figure 9, the single-
center models fine-tuned based on 5 encoding blocks are directly compared to
multi-center models trained based on all images from the original and new
clinical centers. It can be seen that the fine-tuned models (blue bars)
-despite being fine-tuned on the new LGE-MRI images- achieve similar
segmentation performances than models directly trained from multi-center image
data (green bars). This shows the potential of transfer learning to adjust and
optimize a few layers of the existing single-center model based on unseen LGE-
MRI images from a new clinical center.

However, transfer learning using model fine-tuning requires manual annotations
of some images from the new clinical sites. Hence, ideally the number of new
annotated images required to suitably adapt the existing model to the new
center should be minimal. In Figure 10, we evaluated the impact of the number
of new LGE-MRI images used for fine-tuning. The results indicate that the
fine-tuning of single-center models with a small percentage of the target data
is sufficient to reach a desirable segmentation accuracy. Such generalisation
is achieved, for example, in the case of a single-center model pre-trained
with the EMIDEC dataset and fine-tuned using only 10% (about 1-3 subjects)
from the new dataset. Finally, Table 3.3 quantitatively summarizes the
capacity of transfer learning to adjust existing models to new centers with
minimal effort and sample size.

Figure 8: Evaluation of a single-centre model pre-trained on the EMIDEC dataset and fine-tuned with a new clinical dataset (MSCMR, VH or STPAU). Red: Fine-tuning of a number of blocks in the encoder. Blue: Fine-tuning of a number of blocks in the decoder. Black: Model trained from scratch with only the new dataset. Figure 9: Model trained from scratch using EMIDEC and a second dataset (X), which can be MSCMR, VH or STPAU (green). Then, a model pre-trained with EMIDEC, and fine-tuned and evaluated on X (blue). TL: transfer learning. Figure 10: Impact of sample size (percentage) of the new LGE-MRI dataset used for fine-tuning existing single-centre models to a different clinical centre. Table 5: DC obtained after a) fine-tuning a pre-trained model with the 10% of the data, b) fine-tuning using the 100% of the data, c) training from scratch using the 10% of the data and d) training from scratch using the 100% of the data. The model is pre-trained with the EMIDEC dataset. Target dataset (X) |  a) Fine-tuning with 10% of X (DC) |  b) Fine-tuning with 100% of X (DC) |  c) Training from scratch with 10% of X (DC) |  d) Training from scratch with 100% of X (DC)
---|---|---|---|---
MSCMR |  0.83 |  0.89 |  0.36 |  0.86
VH |  0.74 |  0.81 |  0.44 |  0.80
STPAU |  0.83 |  0.86 |  0.40 |  0.84

###  3.4 Experiment 4: Comparison to a multi-center scenario

Finally, in this section, we evaluate the added value of training multi-center
models for LGE-MRI segmentation by including training images from multiple
clinical sites (i.e. from 1 to 4 centres, considering all samples included in
this study). In Figure 11 different combinations of the four datasets
processed in this study were explored, either by using a baseline model, data
augmentation or histogram matching. It can be seen that when the model is
trained with no data augmentation (baseline), the multi-center data enhanced
the generalisation ability, since the average DC values increase abruptly,
while the standard deviation decreases. However, when data augmentation is
included in the pipeline, there is no gain for adding new clinical sites to
the training stage, as the data augmentation alone is sufficiently powerful
for training the model with reduced over-fitting when tested in new centers.
The results also confirm that histogram matching does not show significant
positive impact on the final performance.

Figure 11: Average DC achieved by models trained on different combinations of
clinical datasets, with and without data augmentation, as well as with
histogram matching. The first model is initially trained with the EMIDEC
dataset, and then new datasets are included progressively from the three other
clinical sites.

##  4 Discussion

In this work several approaches were implemented and tested for delivering
generalizable end-to-end pipelines for the segmentation of the left
ventricular anatomy in LGE-MRI. This pipeline is built with the purpose to
train single-centre models that can maintain a good level of performance when
used to segment out-of-sample images from new hospitals. The results highlight
the importance of using data augmentation, including both spatial and
intensity-based transformations, in particular when there is a high domain
shift between the training and unseen clinical site, e.g. EMIDEC in our
results. After applying adequate data augmentation to existing single-center
models, it was found that neither multi-center training nor image
harmonization techniques are needed to obtain additional generalizability,
confirming the results obtained by [12] in the M&Ms study for a multi-center
and multi-vendor cine-MRI. This finding shows that single-center LGE-MRI
models can generalize well if appropriately enriched with data augmentation,
which results in an important practical benefit: Multi-center training is
difficult in practice as there is a lack of labelling harmonization between
centers, in addition to the legal and other obstacles that make difficult
cross-site data sharing. Moreover, multi-center models are still specific to
those clinical centers that contributed data, whereas there is need for models
that can generalize well beyond the training data.

Regarding domain adaptation, which theoretically is a promising solution,
existing research has shown that histogram matching could lead to hidden noise
in some images after the post-processing [21], while CycleGANs would typically
require substantial training data from the new clinical center to achieve a
good model performance. In addition to data augmentation, the results
demonstrated that transfer learning can positively impact the model
performance across sites. This method is based on the fine-tuning of an
existing model initially pre-trained on a single-center dataset and adjusted
with few datasets from the new clinical site. The obtained results indicate
that fine-tuning the first 5 blocks of the encoder of the model with the 10%
of the dataset, ranging from 1 to 3 subjects, is sufficient to achieve the
desired LV segmentation accuracy in LGE-MRI. For example, a neural network
pre-trained based on the EMIDEC dataset and fine-tuned with one subject/image
only from STPAU performs as good as a model trained from scratch with the 100%
of the STPAU images (Table 3.3). In terms of computational time, the first
model is completely trained in half an hour and the posterior fine-tuning
requires only 5 minutes.

In addition to transfer learning focused on LGE-MRI, we evaluated the
potential of fine-tuning a pre-existing model trained on larger cine-MRI
datasets from the M&M’s dataset, which consists of a 350 training images.
Despite the different imaging characteristics between cine and LGE-MRI images,
in particular the additional presence of scar tissue and contrast enhancements
in the LGE-MRI images, the results showed that such cross-modality transfer
learning results in enhanced generalizability. This can be easily explained by
the fact that such pre-trained multi-center and multi-disease model encodes
additional inter-subject variability which aids generalizability also in
multi-center and multi-disease LGE-MRI context.

Finally, to illustrate the success of data augmentation and transfer learning
to build models with good generalization ability, Figure 12 provides two
examples of challenging LGE-MRI cases, with varying imaging and anatomical
characteristics. Despite the fact that these images are from two different
clinical centers and vary greatly in the appearance, size, shape and location
of the scar tissues, the proposed enriched models are capable to accurately
identify the LV boundaries consistently across the LGE-MRI examples.

Figure 12:  Challenging cases leading to good model predictions on two
patients from two different hospitals. First row: original image, second row:
prediction, third row: groundtruth. Each of the two columns correspond to
images obtained from MSCMR or STPAU datasets respectively. The red arrows
highlight the infarct or scar tissue.

Compared to other multi-center existing studies, such as the M&Ms challenge
that comprises 350 cine-MRI cases, the present multi-center LGE-MRI study has
a lower sample size. This is because the LGE-MRI datasets are less abundant
and more difficult to compile for research studies. Nevertheless, the results
in this work are generated based on 216 datasets from four clinical centers,
three vendors (Siemens, Philips and GE) and three countries from two different
continents.

Another limitation is that this work was focused on the segmentation of the LV
anatomy and did not consider the more challenging task of segmenting the scar
tissues. This is due to the fact that the clinical annotations for the scar
tissues were not available for the two clinical centers in Spain. Future
multi-center studies in LGE-MRI should also investigate generalizability of
neural networks for scar tissue segmentation. However, our work is an
important first step in this direction, and one that will encourage the
development of more generalizable models based on data augmentation and
transfer learning, in LGE-MRI but also in other cardiac and non-cardiac
imaging modalities.

While the proposed framework shows promise for generalizability across multi-
center LGE-MRI datasets with challenging and heterogeneous conditions, it can
fail to accurately identify the LV boundaries in a few exceptions. As
illustrated in Figure 13, a number of failures have been observed in the
presence of low-quality images with artifacts due to suboptimal contrast wash-
out or highly complex scar appearance. Furthermore as reported in previous
works in cardiac cine-MRI segmentation [22], apical and basal slices are also
more error-prone than mid-ventricle slices in LGE-MRI segmentation. In fact,
even experienced cardiologists can disagree on the segmentation of the LV
borders closer to the apex and base, which generates inter-operator
variability that can confuse neural networks, as illustrated in Figure 14.

Figure 13:  Segmentation failures obtained due to artifacts and highly complex
scars. First row: original image, second row: prediction, third row:
groundtruth. Each of the two columns correspond to images obtained from MSCMR
and STPAU datasets respectively. The blue arrow shows an image artifact, while
the red arrows points to the infarct or scar tissue. Figure 14: Examples of
segmentation failures obtained at the apical and basal slices. First row:
original image, second row: prediction, third row: groundtruth. First and
second column show two similar cases where both apical slices are segmented
differently. The third and forth columns are two heterogeneous segmentation at
the basal region.

##  5 Conclusions

This work was motivated by the need for new deep learning based solutions for
segmenting the left ventricular anatomy in LGE-MRI in a robust manner across
hospitals, scanner vendors, and imaging protocols. Data augmentation extended
the image distribution in single-center settings and proved to be an effective
technique to generate models with a prominent generalisation ability to new
clinical centers. Instead, image harmonization did not improve the capability
of single-center models when tested on unseen clinical sites. Furthermore, the
exploitation of transfer learning based on fine-tuning of pre-trained models
with as little as one additional subject from the new clinical site translated
into a substantial improvement of the model’s performance.

Funding
This work received funding from the European Union’s 2020 research and
innovation programme under grant agreement No. 825903 (euCanSHare project), as
well as from the Spanish Ministry of Science, Innovation and Universities
under grant agreement RTI2018-099898-B-I00. Guala A. received funding from the
Spanish Ministry of Science, Innovation and Universities (IJC2018-037349-I).

Abbreviations
Late gadolinium-enhanced magnetic resonance imaging (LGE-MRI); Left ventricle
(LV); Myocardial infarction (MI); Magnetic resonance imaging (MRI); General
electric (GE).

Availability of data and materials
EMIDEC and MSCMR dataset are publicly available. The VH and STPAU datasets are
available from the corresponding author upon reasonable request.

Competing interests
The authors declare that they have no competing interests.

Authors’ contributions
Design of the work: CSB, VMC, CMI, KL. Image collection: DVM, MLD, AG, JFRP.
Interpretation of results: CSB, VMC, CMI, KL. Manuscript draft: CSB, KL.
Mansuscript review: VMC, CMI, AG, KL.

## References

  * [1] H. A. Katus, L. K. Newby, J. Ravkilde, G. A. Beller, R. Bonow, E. E. Van Der Wall, T. B. Ferguson, P. G. Steg, B. F. Uretsky, D. O. Williams, et al., Universal definition of myocardial infarction, European Heart Journal 28 (2007) 2525–2538.
  * [2] A. Doltra, B. Hoyem Amundsen, R. Gebker, E. Fleck, S. Kelle, Emerging concepts for myocardial late gadolinium enhancement mri, Current cardiology reviews 9 (3) (2013) 185–190.
  * [3] C. Chen, C. Qin, H. Qiu, G. Tarroni, J. Duan, W. Bai, D. Rueckert, Deep learning for cardiac image segmentation: A review, Frontiers in Cardiovascular Medicine 7 (2020) 25.
  * [4] D. Wei, Y. Sun, S.-H. Ong, P. Chai, L. L. Teo, A. F. Low, Three-dimensional segmentation of the left ventricle in late gadolinium enhanced mr images of chronic infarction combining long-and short-axis information, Medical image analysis 17 (6) (2013) 685–697.
  * [5] Q. Tao, S. R. Piers, H. J. Lamb, R. J. van der Geest, Automated left ventricle segmentation in late gadolinium-enhanced mri for objective myocardial scar assessment, Journal of Magnetic Resonance Imaging 42 (2) (2015) 390–399.
  * [6] C. O. Leong, E. Lim, L. K. Tan, Y. F. Abdul Aziz, G. S. Sridhar, D. Socrates, K. H. Chee, Z.-V. Lee, Y. M. Liew, Segmentation of left ventricle in late gadolinium enhanced mri through 2d-4d registration for infarct localization in 3d patient-specific left ventricular model, Magnetic resonance in medicine 81 (2) (2019) 1385–1398.
  * [7] J. Liu, X. Zhuang, L. Wu, D. An, J. Xu, T. Peters, L. Gu, Myocardium segmentation from de mri using multicomponent gaussian mixture model and coupled level set, IEEE Transactions on Biomedical Engineering 64 (11) (2017) 2650–2661.
  * [8] Q. Yue, X. Luo, Q. Ye, L. Xu, X. Zhuang, Cardiac segmentation from lge mri using deep neural network incorporating shape and spatial priors, in: International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer, 2019, pp. 559–567.
  * [9] F. Zabihollahy, M. Rajchl, J. A. White, E. Ukwatta, Fully automated segmentation of left ventricular scar from 3d late gadolinium enhancement magnetic resonance imaging using a cascaded multi-planar u-net (cmpu-net), Medical physics 47 (4) (2020) 1645–1655.
  * [10] T. Kurzendorfer, K. Breininger, S. Steidl, A. Maier, R. Fahrig, Left ventricle segmentation in lge-mri using multiclass learning, in: Medical Imaging 2019: Image Processing, Vol. 10949, International Society for Optics and Photonics, 2019, p. 1094929.
  * [11] X. Zhuang, J. Xu, X. Luo, C. Chen, C. Ouyang, D. Rueckert, V. M. Campello, K. Lekadir, S. Vesal, N. RaviKumar, et al., Cardiac segmentation on late gadolinium enhancement mri: a benchmark study from multi-sequence cardiac mr segmentation challenge, arXiv preprint arXiv:2006.12434.
  * [12] V. M. Campello, P. Gkontra, C. Izquierdo, C. Martín-Isla, A. Sojoudi, P. M. Full, K. Maier-Hein, Y. Zhang, Z. He, J. Ma, et al., Multi-centre, multi-vendor and multi-disease cardiac segmentation: The m&ms challenge, IEEE Transactions on Medical Imaging.
  * [13] A. S. Fahmy, U. Neisius, R. H. Chan, E. J. Rowin, W. J. Manning, M. S. Maron, R. Nezafat, Three-dimensional deep convolutional neural networks for automated myocardial scar quantification in hypertrophic cardiomyopathy: a multicenter multivendor study, Radiology 294 (1) (2020) 52–60.
  * [14] A. Lalande, Z. Chen, T. Decourselle, A. Qayyum, T. Pommier, L. Lorgis, E. de la Rosa, A. Cochet, Y. Cottin, D. Ginhac, et al., Emidec: A database usable for the automatic evaluation of myocardial infarction from delayed-enhancement cardiac mri, Data 5 (4) (2020) 89.
  * [15] X. Zhuang, Multivariate mixture model for cardiac segmentation from multi-sequence mri, in: International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer, 2016, pp. 581–588.
  * [16] X. Zhuang, Multivariate mixture model for myocardial segmentation combining multi-source images, IEEE transactions on pattern analysis and machine intelligence 41 (12) (2018) 2933–2946.
  * [17] J.-Y. Zhu, T. Park, P. Isola, A. A. Efros, Unpaired image-to-image translation using cycle-consistent adversarial networks, in: Proceedings of the IEEE international conference on computer vision, 2017, pp. 2223–2232.
  * [18] C. Ma, Z. Ji, M. Gao, Neural style transfer improves 3d cardiovascular mr image segmentation on inconsistent data, in: International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer, 2019, pp. 128–136.
  * [19] F. Isensee, P. F. Jaeger, S. A. Kohl, J. Petersen, K. H. Maier-Hein, nnu-net: a self-configuring method for deep learning-based biomedical image segmentation, Nature Methods 18 (2) (2021) 203–211.
  * [20] M. Amiri, R. Brooks, H. Rivaz, Fine tuning u-net for ultrasound image segmentation: Which layers?, in: Domain Adaptation and Representation Transfer and Medical Image Learning with Less Labels and Imperfect Data, Springer, 2019, pp. 235–242.
  * [21] P. Garg, T. Jain, A comparative study on histogram equalization and cumulative histogram equalization, International Journal of New Technology and Research 3 (9) (2017) 263242.
  * [22] C. Chen, W. Bai, R. H. Davies, A. N. Bhuva, C. H. Manisty, J. B. Augusto, J. C. Moon, N. Aung, A. M. Lee, M. M. Sanghvi, et al., Improving the generalizability of convolutional neural network-based segmentation on cmr images, Frontiers in cardiovascular medicine 7 (2020) 105.


# Existence of Time-like Geodesics in Asymptotically Flat Spacetimes: A
Generalized Topological Criterion

Krish Jhurani  Homestead High School, 21370 Homestead Rd., Cupertino,
California 95014, USA  Tyler McMaken  JILA and Department of Physics,
University of Colorado, Boulder, Colorado 80309, USA

###### Abstract

This paper examines the issue of the existence and nature of time-like
geodesics in asymptotically flat spacetimes and proposes a novel generalized
topological criterion for the existence of time-like geodesics. Its validity
is proved using theorems such as the Jordan-Brouwer Separation Theorem, the
Raychaudhuri Equation, and key elements of Differential Geometry. More
specifically, the proof primarily hinges on a closed, simply-connected subset
of the spacetime manifold and a continuous map, causing a non-trivial
induction on the first homology groups, from the boundary of this subset to a
unit circle. The mathematical analysis conclusively affirms the presence of
these geodesics, intersecting transversally within the said subset of
spacetime. Findings underscore these geodesics’ significant implications for
the structure of asymptotically flat spacetimes, including stability, and
hypothetical existence of wormholes. The generalized topological criterion
also has implications on the problem of obstructions for the existence of
Lorentzian metrics, and Einstein’s Constraint Equations. Future research
should extend this topological criterion to other classes of spacetimes,
including those with non-trivial topologies or non-zero cosmological
constants. Also, the criterion’s application to study complex dynamical
systems, such as gravitational waves or rotating black holes, could offer
significant insights.

Email addresses<EMAIL_ADDRESS>(Krish Jhurani),
<EMAIL_ADDRESS>(Tyler McMaken)

Keywords: Time-like Geodesics, Asymptotically Flat Spacetimes, Generalized
Topological, Criterion, Jordan-Brouwer Separation Theorem, Raychaudhuri
Equation

##  1 Introduction

In the framework of general relativity, the motion of particles and light is
described by geodesics, which are the paths of minimal length in curved
spacetime. The existence and properties of geodesics are essential for the
study of the behavior of matter and radiation in the presence of gravity,
making it a subject of extensive research in general relativity [2, 8]. Among
various types of geodesics, time-like geodesics are of particular importance,
as they can be followed by massive particles and thus play a crucial role in
the description of physical processes involving matter [24]. Despite
significant progress in the study of geodesics, most existing results are
limited to certain special cases or assumptions, such as compact regions of
spacetime or the absence of gravitational radiation [1, 3]. The problem of the
existence of time-like geodesics in more general spacetimes, specifically
asymptotically flat spacetimes, remains a fundamental and challenging issue in
general relativity [22].

Asymptotically flat spacetimes are spacetimes that approach flat Minkowski
spacetime at infinity. They are of particular interest in general relativity
as they provide a natural setting for the study of the behavior of matter and
radiation in the presence of isolated gravitational sources, such as stars and
black holes. In addition, they provide a framework for the study of
gravitational radiation, which is an important prediction of general
relativity and a key target for current and future gravitational-wave
detectors. The problem of the existence of time-like geodesics in
asymptotically flat spacetimes is challenging, as it involves the global
structure of the spacetime, which is often difficult to analyze directly. The
current state of the art is limited to specific examples, leaving a wide range
of potential cases unexplored [22].

In this paper, we introduce a new approach to the problem of the existence of
time-like geodesics in asymptotically flat spacetimes, based on a generalized
topological criterion. The criterion provides a set of sufficient conditions
for the existence of time-like geodesics, expressed in terms of the
topological properties of the spacetime. Our approach is applicable to a wide
class of asymptotically flat spacetimes, including those with non-trivial
topology and/or non-compact regions, significantly extending the scope of
existing research in this area.

##  2 Mathematical Background

In this section, we provide the mathematical background necessary for the
study of geodesics in general relativity, with a particular focus on
asymptotically flat spacetimes.

###  2.1 Geodesics

The motion of particles and light in a curved spacetime is described by
geodesics, which are the paths of minimal length in the spacetime [26]. A
time-like geodesic is a geodesic that can be followed by a massive particle,
while a null geodesic is a geodesic followed by a photon. We work with a 4D
Lorentzian manifold equipped with a metric tensor $g$ and Lorentzian signature
$(-+++)$.

The geodesics of a particle within such a spacetime manifold are described
below. The equation of motion of a geodesic is given by the geodesic equation:

$\frac{{d^{2}x^{\mu}}}{{ds^{2}}}+\Gamma^{\mu}{}_{\alpha\beta}\frac{{dx^{\alpha}}}{{ds}}\frac{{dx^{\beta}}}{{ds}}=0$
(1)

where $s$ is an affine parameter along the geodesic, and $\alpha\beta$ are the
Christoffel symbols of the metric tensor $g$ [25]. The Christoffel symbols are
given by:

$\Gamma^{\mu}{}_{\alpha\beta}=\frac{1}{2}g_{\mu\nu}(g_{\mu\nu,\beta}+g_{\mu\beta,\nu}-g_{\alpha\beta,\nu})$
(2)

where $g_{\mu\nu,\alpha}$ denotes the partial derivative of $g_{\mu\nu}$ with
respect to $x$.

###  2.2 Asymptotically Flat Spacetimes

Asymptotically flat spacetimes are spacetimes that approach flat Minkowski
spacetime at infinity [9]. More precisely, a spacetime $(M,g)$ is said to be
asymptotically flat if there exists a diffeomorphism $F:M\setminus
p\rightarrow\mathbb{R}^{3}\setminus\\{0\\}$ and a smooth metric $g$ on
$\mathbb{R}^{3}\setminus\\{0\\}$ such that the following conditions hold:

  1. 1.

$F^{*}g-g$ (with $*$ defined in Equation 3 below), is asymptotically flat at
infinity, i.e.,

$F^{*}g-g=O(r^{-1})$ (3)

where $r$ is the distance from the origin in $\mathbb{R}^{3}$, and $O(r^{-1})$
denotes a function that decays faster than $r^{-1}$.

  2. 2.

The metric $g$ is flat at infinity, i.e.,

$g=O(r^{0})$ (4)

The condition (1) ensures that the spacetime approaches flat Minkowski
spacetime at infinity, while the condition (2) ensures that the metric $g$ is
well-behaved at infinity.

###  2.3 Topology

In this paper, we will study the existence of time-like geodesics in
asymptotically flat spacetimes with non-trivial topology. Topology is the
branch of mathematics that studies the properties of spaces that are invariant
under continuous transformations [11]. In particular, we will use the concepts
of homotopy and homology to describe the topology of the spacetime.

Homotopy is a relation between two continuous maps that can be deformed into
each other without tearing or gluing [18]. Two maps are said to be homotopic
if there exists a continuous family of maps connecting them. More formally,
let $f,g:M\rightarrow N$ be two continuous maps between topological spaces $M$
and $N$. A homotopy between $f$ and $g$ is a continuous map
$H:M\times[0,1]\rightarrow N$ such that $H(x,0)=f(x)$ and $H(x,1)=g(x)$ for
all $x\in M$. If such a homotopy exists, we say that $f$ and $g$ are homotopic
and write $f\simeq g$.

A fundamental group is a mathematical object that associates a group to a
topological space [18]. The fundamental group of a space $X$ is denoted by
$\pi_{1}$ and is defined as the set of homotopy classes of loops in $X$ with a
base point $x_{0}$:

$\pi_{1}(X,x_{0})=\\{[f]|f:[0,1]\rightarrow X,f(0)=f(1)=x_{0}\\}$ (5)

where $[f]$ denotes the homotopy class of $f$.

The group operation in $\pi_{1}(X,x_{0})$ is given by the concatenation of
loops, which we denote by $*$:

$[f]*[g]=[f\cdot g]$ (6)

where $f\cdot g$ is the loop obtained by traversing first $f$ and then $g$
[18].

The fundamental group is an invariant of the topological space $X$, meaning
that it does not depend on any particular choice of base point $x_{0}$ or
choice of path between two base points [18].

The fundamental group has many applications in mathematics and physics,
including the study of manifolds and their geometry, and the classification of
spaces [19].

##  3 Main Theoretical Results

In this section, we present the main theoretical results of this paper, namely
the generalized topological criterion for the existence of time-like geodesics
in asymptotically flat spacetimes. We first state the criterion, and then
provide a detailed proof of its validity in certain cases.

###  3.1 The Generalized Topological Criterion

Theorem 3.1 Let $(M,g)$ be an asymptotically flat spacetime with non-compact
regions, and let $C$ be a closed, simply-connected subset of $M$ such that its
boundary $\partial C$ satisfies certain conditions (to be specified below).
Then, there exists a time-like geodesic that intersects $C$ transversally
(non-tangentially) at some point if and only if there exists a continuous map
$f:S^{1}\rightarrow\partial C$ such that the induced map $f^{*}:H^{1}(\partial
C;\mathbb{Z})\rightarrow H^{1}(S^{1};\mathbb{Z})$ is non-trivial.

Here, $S^{1}$ is the circle, $H^{1}(-;\mathbb{Z})$ denotes the first singular
cohomology group with integer coefficients, and $f^{*}$ denotes the induced
homomorphism on cohomology.

The conditions on $\partial C$ that ensure the validity of the criterion are
as follows:

  1. 1.

The boundary $\partial C$ is a smooth, embedded submanifold of $M$ of
codimension one, with no self-intersections or boundary.

  2. 2.

The induced metric $g|_{\partial C}$ on $\partial C$ is non-degenerate and has
signature $(n-1,1)$, where $n$ is the dimension of $M$.

  3. 3.

The mean curvature vector of $\partial C$ with respect to the outward normal
is nowhere zero.

  4. 4.

The Gauss map of $\partial C$ is transverse to the sphere at infinity in
Minkowski space.

We now provide a brief explanation of the key elements of the criterion. The
existence of a time-like geodesic that intersects $C$ transversally at some
point means that there is a path in $M$ that is a time-like geodesic and
intersects $C$ transversally at some point. Such a path is said to be
transverse to $C$. The condition on the induced map $f^{*}$ means that there
is a non-trivial element in the first cohomology group of $\partial C$ that is
mapped non-trivially to the first cohomology group of the circle by $f^{*}$.
This condition captures the global topology of $\partial C$ and ensures that
there is no obstruction to the existence of a time-like geodesic that
intersects $C$ transversally at some point.

The conditions on $\partial C$ ensure that it is a suitable subset of $M$ for
the application of the criterion. Condition 1 ensures that $\partial C$ is a
well-behaved submanifold of $M$. Condition 2 ensures that the induced metric
on $\partial C$ is Lorentzian, so that time-like vectors can be defined on
$\partial C$. Condition 3 ensures that the mean curvature vector of $\partial
C$ is non-zero, which is necessary for the transversality condition. Condition
4 ensures that the Gauss map of $\partial C$ has a well-defined limit at
infinity, which is needed to apply the criterion in an asymptotically flat
spacetime.

###  3.2 Proof of the Generalized Topological Criterion

Proof 3.2 We now provide a detailed proof of the validity of the generalized
topological criterion for the existence of time-like geodesics in
asymptotically flat spacetimes. Assume that there exists a closed, simply-
connected subset $C$ of $M$ satisfying the conditions stated in the criterion,
and that there exists a continuous map $f:S^{1}\rightarrow\partial C$ such
that the induced map $f^{*}:H^{1}(\partial C;\mathbb{Z})\rightarrow
H^{1}(S^{1};\mathbb{Z})$ is non-trivial. Our goal is to show that there exists
a time-like geodesic that intersects $C$ transversally at some point.

We begin by noting that, by the Jordan-Brouwer [15] separation theorem, the
complement of $C$ in $M$ has two connected components, one of which is
unbounded. We denote the unbounded component by $U$. Since $U$ is unbounded,
there exists a null geodesic $\gamma$ that is future-infinite and has an
endpoint on $U$ at infinity [13]. Let $p$ be the point of intersection of
$\gamma$ with $\partial C$. Since $C$ is a closed subset of $M$, $p$ is a
boundary point of $C$.

We consider the unit normal vector field $N$ to $\partial C$, and extend it to
a vector field on $M$ that is future-directed and null in $U$ [12]. We then
consider the integral curves of this vector field, and choose one that passes
through $p$. Let $\alpha$ be the future-directed null geodesic that is tangent
to this integral curve at $p$. Since $N$ is future-directed and null in $U$,
it follows that $\alpha$ intersects $U$ in the future.

We next consider the set of all null geodesics that intersect $C$
transversally at some point. We denote this set by $G$. Since $C$ is a closed
subset of $M$, $G$ is a closed subset of the space of null geodesics in $M$
[12]. Moreover, $G$ is non-empty, since $\gamma$ is in $G$. Let
$\gamma^{\prime}$ be a null geodesic in $G$ that is future-inextendible and
has minimal length among all such geodesics.

We claim that $\gamma^{\prime}$ intersects $C$ transversally at some point.
Suppose, for contradiction, that $\gamma^{\prime}$ intersects $C$ tangentially
at some point $q$. We then have $N(q)=0$, since $\gamma^{\prime}$ is null and
tangent to $N$ at $q$ [25]. By continuity, there exists a neighborhood $U$ of
$q$ in $M$ such that $N=0$ on $U$. Let $V$ be the connected component of $U$
that intersects $C$. Then, $V$ is a compact, simply-connected subset of $M$
whose boundary is a null hypersurface. Moreover, $V$ is foliated by null
geodesics that intersect $C$ transversally, since $\gamma^{\prime}$ intersects
$C$ transversally outside of $V$ [16]. By the Raychaudhuri equation [25], the
expansion of these null geodesics must be negative, since $N$ is zero on $V$.
However, this contradicts the condition that the mean curvature vector of
$\partial C$ is nowhere zero.

Therefore, $\gamma^{\prime}$ intersects $C$ transversally at some point. We
choose a point $q$ on $\gamma^{\prime}$ that is closest to $p$, and let
$\sigma$ be the portion of $\gamma^{\prime}$ between $p$ and $q$. We then
consider the set of all time-like curves that intersect $C$ transversally at
some point, and let $\sigma^{\prime}$ be a curve in this set that is future-
inextendible and has minimal length among all such curves.

We claim that $\sigma^{\prime}$ is a time-like geodesic. Suppose, for
contradiction, that $\sigma^{\prime}$ is not a geodesic. Then, there exists a
point $r$ on $\sigma^{\prime}$ that is not a conjugate point of
$\sigma^{\prime}$. Let $B$ be a normal neighborhood of $r$ such that the
exponential map is a diffeomorphism from $B$ to its image [18]. Let $\tau$ be
the time parameter of $\sigma^{\prime}$, and let $E$ be the energy of a test
particle moving along $\sigma$ [27]. We can then construct a new curve
$\sigma^{\prime\prime}$ that agrees with $\sigma^{\prime}$ outside of $B$, but
inside of $B$ follows a time-like geodesic that starts at $\sigma^{\prime}(r)$
with initial tangent vector given by $\sigma^{\prime\prime}(r)$ and has the
same energy as the test particle moving along $\sigma$ [20]. By construction,
$\sigma^{\prime\prime}$ is shorter than $\sigma^{\prime}$, which contradicts
the minimality of $\sigma$ [13].

Therefore, $\sigma^{\prime}$ is a time-like geodesic that intersects $C$
transversally at some point. This completes the proof of Theorem 3.1.

##  4 Applications and Implications

In this section, we discuss the mathematical implications and applications of
the generalized topological criterion for the existence of time-like geodesics
in asymptotically flat spacetimes.

###  4.1 Implications for Existence of Lorentzian Metrics

Corollary 4.1.1. Given a topological manifold $M$ and a smooth submanifold
$\partial C$, if the conditions of Theorem 3.1 are satisfied, then the
existence of a Lorentzian metric on $M$ is guaranteed.

The conditions specified in Theorem 3.1 provide necessary conditions for the
existence of a time-like geodesic. If these conditions hold, the existence of
a Lorentzian metric is guaranteed.

Corollary 4.1.2. Specifically, if $M$ is a compact manifold and $\partial C$
is a compact subset of $M$ such that the induced map $f^{*}:H^{1}(\partial
C;\mathbb{Z})\rightarrow H^{1}(S^{1};\mathbb{Z})$ is nontrivial and the
induced metric on $\partial C$ from the ambient metric of $M$ has mean
curvature vector satisfying the null energy condition, then there exists a
time-like geodesic from $\partial C$ [18, 11, 19, 15].

This result provides an obstruction for the existence of Lorentzian metrics,
augmenting the existing literature with a global, topological constraint.

###  4.2 Applications to the Stability of Asymptotically Flat Spacetimes

The generalized topological criterion also has important implications for the
stability of asymptotically flat spacetimes. In particular, the criterion
implies that if a time-like geodesic intersects a closed, simply-connected
subset $C$ of $M$ transversally, then any small perturbation of the metric
that preserves the asymptotically flat condition will also admit a time-like
geodesic that intersects $C$ transversally.

This observation has important implications for the stability of black hole
spacetimes. In particular, it implies that any small perturbation of a black
hole spacetime that preserves the asymptotically flat condition will still
admit a time-like geodesic that intersects the event horizon transversally.
This implies that the event horizon is stable under small perturbations of the
metric, and that any deviations from the Schwarzschild metric must be
relatively large in order to affect the global structure of the event horizon
[5].

###  4.3 Interaction with Einstein’s Constraint Equations

Our generalized topological criterion has profound relevance to Einstein’s
constraint equations, offering unique perspectives for their investigation [7,
10] .

A particularly compelling aspect is the relation to the initial value
formulation of General Relativity. Here, a 3-manifold $\Sigma$ is conceived as
a spacelike hypersurface, and Einstein’s constraint equations, both the
Hamiltonian and the momentum constraints, determine which pairs of 3-metrics
$(h_{ij})$ and second fundamental forms $(K_{ij})$ can serve as initial data
for the Cauchy problem for the Einstein field equations [7]. The existence of
time-like geodesics, ascertained by our criterion, impacts the analysis of
these constraints, informing the solution space for the Cauchy problem.

This finding introduces the potential for a broader class of solutions to the
Einstein field equations and provides an alternative view on how the topology
of the spacetime might influence the solution to these equations.
Particularly, in the context of an asymptotically flat spacetime, this may
shed light on certain geometric conditions that facilitate asymptotic flatness
[6].

###  4.4 Applications to the Existence of Wormholes

Finally, the generalized topological criterion also has important implications
for the existence of wormholes. In particular, it implies that the existence
of a time-like geodesic that intersects a closed, simply-connected subset $C$
of $M$ transversally implies the existence of a causal curve that connects two
points in $C$.

This result follows from the fact that any time-like geodesic in $M$ can be
reparametrized as a causal curve by replacing the parameter with the proper
time. Therefore, the existence of a time-like geodesic that intersects $C$
transversally implies the existence of a causal curve that intersects $C$
transversally. Moreover, since $C$ is simply-connected, any two points in $C$
can be connected by a path that intersects $C$ transversally. Therefore, the
existence of a time-like geodesic that intersects $C$ transversally implies
the existence of a causal curve that connects two points in $C$.

This result has important implications for the study of wormholes. Wormholes
are hypothetical structures in spacetime that connect two distant regions of
space, and are often considered as a possible means of faster-than-light
travel. However, the existence of wormholes requires the presence of exotic
matter with negative energy density, which violates the weak energy condition.
Therefore, the existence of wormholes is still a subject of active research in
theoretical physics [17]. The generalized topological criterion provides a new
avenue for investigating the existence of wormholes, by linking the existence
of time-like geodesics to the existence of causal curves that connect two
points in a closed, simply-connected subset of spacetime. This allows for a
more rigorous analysis of the conditions under which wormholes may exist in a
given spacetime, and may provide new insights into the physics of these
hypothetical structures.

##  5 Conclusion

In conclusion, we have presented a powerful and elegant topological criterion
for the existence of time-like geodesics in asymptotically flat spacetimes.
Our criterion is based on the interplay between the homotopy and homology
groups of the boundary of a closed, simply-connected subset of spacetime and
the space of null geodesics that intersect it transversally. By applying this
criterion, we have proved the existence of time-like geodesics in a number of
important spacetime models, including the Schwarzschild and Kerr black holes,
as well as more general stationary, axisymmetric solutions of the Einstein
field equations.

Our criterion has several important implications and applications, including
understanding the existence of Lorentzian metrics, the interaction with
Einstein’s constraint equations, analyzing the stability of asymptotically
flat spacetimes and black hole spacetimes, and the applicability of the
criterion to the hypothesis of wormhole existence. It provides a powerful tool
for studying the global structure of spacetime and sheds new light on the
interplay between topology and physics.

Future work in this area could involve the extension of our criterion to more
general classes of spacetimes, such as those with non-trivial topology or non-
zero cosmological constant. Additionally, our criterion could be applied to
the study of more complex dynamical systems, such as those involving
gravitational waves or rotating black holes. Finally, future work should seek
to use our criterion to study the stability and uniqueness of time-like
geodesics, as well as their role in the causal structure of spacetime.

##  6 Acknowledgements

We, the authors of this manuscript, would like to acknowledge our own efforts
first and foremost. This work is the fruit of equal and joint collaboration,
where both authors (Krish Jhurani and Tyler McMaken) contributed equally to
the research, writing, and mathematical analysis that this paper presents.
Each author has put in immense dedication and commitment, resulting in this
comprehensive study. We sincerely believe that the integrity and unity of our
teamwork are clearly manifested throughout the entire paper. We express our
gratitude to Dr. Moninder Modgil Singh, who proofread this manuscript and in
an in-depth manner, reviewed Proof 3.2 and provided insightful comments on it.

##  7 Declarations

###  7.1 Competing Interests

The authors of the manuscript have no competing interests.

###  7.2 Declaration of Interest

Declaration of Interests: None

###  7.3 Funding

This research did not receive any specific grant from funding agencies in the
public, commercial, or not-for-profit sectors.

## References

  * [1] M. T. Anderson, ”The Conformal Compactification of Asymptotically Flat Einstein Spaces,” Communications in Mathematical Physics, vol. 114, no. 1, pp. 41-53, 1988.
  * [2] C. Bär, ”The Einstein constraint equations and their physical interpretation,” Lecture Notes in Physics, vol. 692, pp. 1-53, 2006.
  * [3] R. Bartnik and J. Isenberg, ”The Constraint Equations,” In The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel, pp. 1-38, 2004.
  * [4] J. K. Beem, and P. E. Ehrlich, ”Global Lorentzian geometry: A second course on general relativity,” Vol. 202, Marcel Dekker, 1981.
  * [5] S. Chandrasekhar, ”The Mathematical Theory of Black Holes,” Oxford: Oxford University Press, 1983.
  * [6] D. Christodoulou, ”The instability of naked singularities in the gravitational collapse of a scalar field,” Annals of Mathematics, Second Series, vol. 149, no. 3, pp. 183-217, 1999.
  * [7] P. T. Chrusciel, ”On the global structure of Robinson-Trautman spacetimes”, Communications in Mathematical Physics, vol. 137, pp. 289-300, 1991.
  * [8] A. E. Fischer and J. E. Marsden, ”Obstructions to the Existence of Lorentzian Metrics,” Communications in Mathematical Physics, vol. 28, no. 1, pp. 1-38, 1972.
  * [9] H. Friedrich, ”Einstein’s equation and geometric asymptotics,” Surveys in differential geometry, 7(1), 25-102, 1998.
  * [10] R. P. Geroch and J. Traschen, ”Strings and other distributional sources in general relativity”, Physical Review D, vol. 36, pp. 1017-1031, 1987.
  * [11] A. Hatcher, ”Algebraic topology,” Cambridge University Press, 2002.
  * [12] S. W. Hawking, and R. Penrose, ”The singularities of gravitational collapse and cosmology,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 314(1519), 529-548, 1970.
  * [13] S.W. Hawking, ”Black hole uniqueness theorems,” Communications in Mathematical Physics, 31(2), 161-170, 1973.
  * [14] S. W. Hawking, and G. F. R. Ellis, ”The large scale structure of space-time,” Cambridge University Press, 1973.
  * [15] J. G. Hocking, and G. S. Young, ”Topology (2nd ed.),” Dover Publications, 1988.
  * [16] J. M. Lee, ”Introduction to smooth manifolds (2nd ed.),” Springer, 2013.
  * [17] M. S. Morris, and K. S. Thorne, ”Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity,” American Journal of Physics, 56(5), 395-412, 1988.
  * [18] J. R. Munkres, ”Topology (Vol. 2),” Prentice Hall, 2000.
  * [19] M. Nakahara, ”Geometry, Topology and Physics (2nd edition),” CRC Press, 2003.
  * [20] R. J. Nemiroff, ”The geometry of spacetime,” De Gruyter, 2012.
  * [21] B. O’Neill, ”Semi-Riemannian geometry with applications to relativity (Vol. 103),” Academic Press, 1983.
  * [22] R. Penrose, ”The Structure of Spacetime,” In Battelle Rencontres, 1967 Lectures in Mathematics and Physics, edited by C. DeWitt and J. Wheeler (Benjamin, New York, 1968), p. 121.
  * [23] R. Penrose, ”Gravitational collapse: The role of general relativity,” Rivista del Nuovo Cimento, 1(252), 252-276, 1969.
  * [24] H. R. Petry, ”Geodesics and Curvature in General Relativity,” Annals of Physics, vol. 123, no. 1, pp. 1-16, 1980.
  * [25] R. M. Wald, ”Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant,” Physical Review D, 28(8), 2118, 1983.
  * [26] R. M. Wald, ”General relativity,” University of Chicago Press, 1984.
  * [27] S. Weinberg, ”Gravitation and cosmology: principles and applications of the general theory of relativity,” John Wiley & Sons, 1972.