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Dynamic fetal heart magnetic resonance imaging (MRI) presents unique challenges due to the fast heart rate of the fetus compared to adult subjects and uncontrolled fetal motion. This requires high temporal and spatial resolutions over a large field of view, in order to encompass surrounding maternal anatomy. In this work, we introduce Dynamic Cardiac Reconstruction Attention Network (DCRA-Net) - a novel deep learning model that employs attention mechanisms in spatial and temporal domains and temporal frequency representation of data to reconstruct the dynamics of the fetal heart from highly accelerated free-running (non-gated) MRI acquisitions. DCRA-Net was trained on retrospectively undersampled complex-valued cardiac MRIs from 42 fetal subjects and separately from 153 adult subjects, and evaluated on data from 14 fetal and 39 adult subjects respectively. Its performance was compared to L+S and k-GIN methods in both fetal and adult cases for an undersampling factor of 8x. The proposed network performed better than the comparators for both fetal and adult data, for both regular lattice and centrally weighted random undersampling. Aliased signals due to the undersampling were comprehensively resolved, and both the spatial details of the heart and its temporal dynamics were recovered with high fidelity. The highest performance was achieved when using lattice undersampling, data consistency and temporal frequency representation, yielding PSNR of 38 for fetal and 35 for adult cases. Our method is publicly available at https://github.com/denproc/DCRA-Net.

Two subsequent deep learning networks, one localizing the fetal chest and one identifying a set of landmarks on a coronal whole-uterus balanced steady-state free precession scan, were trained on 167 and 71 fetal datasets across field strengths, acquisition parameters, and gestational ages and implemented in a real-time setup. Next, a phase-contrast sequence was modified to use the identified landmarks for planning. The OWL pipeline was evaluated retrospectively in 10 datasets and prospectively in 7 fetal subjects (gestational ages between 36+3 and 39+3 weeks). The prospective cases were additionally manually planned to enable direct comparison both qualitatively, by scoring the planning quality, and quantitatively, by comparing the indexed flow measurements. OWL enabled real-time fully automatic planning of the 2D phase-contrast scans in all but one of the prospective participants. The fetal body localization achieved an overall Dice score of 0.94+-0.05 and the cardiac landmark detection accuracy was 5.77+-2.91 mm for the descending aorta, 4.32+-2.44 mm for the spine, and 4.94+-3.82 mm for the umbilical vein. For the prospective cases, overall planning quality was 2.73/4 for the automated scans, compared to 3.0/4 for manual planning, and the flow quantitative evaluation showed a mean difference of -1.8% (range -14.2% to 14.9%) by comparing the indexed flow measurements obtained from gated automatic and manual acquisitions. Real-time automated planning of 2D phase-contrast MRI was effectively accomplished for 2 major vessels of the fetal vasculature. While demonstrated here on 0.55T, the achieved method has wider implications, and training across multiple field strengths enables generalization. OWL thereby presents an important step towards extending access to this modality beyond specialised centres., Comment: 12 pages

This paper presents a novel approach for propagating uncertainties in dynamical systems building on high-order Taylor expansions of the flow and moment-generating functions (MGFs). Unlike prior methods that focus on Gaussian distributions, our approach leverages the relationship between MGFs and distribution moments to extend high-order uncertainty propagation techniques to non-Gaussian scenarios. This significantly broadens the applicability of these methods to a wider range of problems and uncertainty types. High-order moment computations are performed one-off and symbolically, reducing the computational burden of the technique to the calculation of Taylor series coefficients around a nominal trajectory, achieved by efficiently integrating the system's variational equations. Furthermore, the use of the proposed approach in combination with event transition tensors, allows for accurate propagation of uncertainties at specific events, such as the landing surface of a celestial body, the crossing of a predefined Poincar\'e section, or the trigger of an arbitrary event during the propagation. Via numerical simulations we demonstrate the effectiveness of our method in various astrodynamics applications, including the unperturbed and perturbed two-body problem, and the circular restricted three-body problem, showing that it accurately propagates non-Gaussian uncertainties both at future times and at event manifolds.

Numerical continuation techniques are powerful tools that have been extensively used to identify particular solutions of nonlinear dynamical systems and enable trajectory design in chaotic astrodynamics problems such as the Circular Restricted Three-Body Problem. However, the applicability of equilibrium points and periodic orbits may be questionable in real-world applications where the uncertainties of the initial conditions of the spacecraft and dynamical parameters of the problem (e.g., mass ratio parameter) are taken into consideration. Due to uncertain parameters and initial conditions, the spacecraft might not follow the reference periodic orbit owing to growing uncertainties that cause the satellite to deviate from its nominal path. Hence, it is crucial to keep track of the probability of finding the spacecraft in a given region. Building on previous work, we extend numerical continuation to moments of the distribution (i.e., stochastic continuation) by directly continuing moments of the probability density function of the spacecraft state. Only assuming normality of the initial conditions, and leveraging moment-generating functions, Isserlis' theorem, and the algebra of truncated polynomials, we propagate the distribution of the spacecraft state at consecutive surface of section crossings while retaining a symbolic map of the final moments of the distribution that depend on the initial mean and covariance matrix only. The goal of the work is to offer a differential algebra-based general framework to continue 3D periodic orbits in the presence of uncertain dynamical systems. The proposed approach is compared against traditional Monte Carlo simulations to validate the uncertainty propagation approach and demonstrate the advantages of the proposed in terms of uncertainty propagation computational burden and access to higher-dimensional problems., Comment: Paper presented and published in the proceedings of the 2024 ISSFD Conference. (Darmstadt, Germany)

Data-driven modelling techniques provide a method for deriving models of dynamical systems directly from complicated data streams. However, tracking and forecasting such data streams poses a significant challenge to most methods, as they assume the underlying process and model does not change over time. In this paper, we apply one such data-driven method, the Koopman autoencoder (KAE), to high-dimensional oscillatory data to generate a low-dimensional latent space and model, where the system's dynamics appear linear. This allows one to accurately track and forecast systems where the underlying model may change over time. States and the model in the reduced order latent space can then be efficiently updated as new data becomes available, using data assimilation techniques such as the ensemble Kalman filter (EnKF), in a technique we call the KAE EnKF. We demonstrate that this approach is able to effectively track and forecast time-varying, nonlinear dynamical systems in synthetic examples. We then apply the KAE EnKF to a video of a physical pendulum, and achieve a significant improvement over current state-of-the-art methods. By generating effective latent space reconstructions, we find that we are able to construct accurate short-term forecasts and efficient adaptations to externally forced changes to the pendulum's frequency., Comment: 44 pages, 23 figures, 1 table

Localized patterns are coherent structures embedded in a quiescent state and occur in both discrete and continuous media across a wide range of applications. While it is well-understood how domain covering patterns (for example stripes and hexagons) emerge from a pattern-forming/Turing instability, analyzing the emergence of their localized counterparts remains a significant challenge. There has been considerable progress in studying localized patterns over the past few decades, often by employing innovative mathematical tools and techniques. In particular, the study of localized pattern formation has benefited greatly from numerical techniques; the continuing advancement in computational power has helped to both identify new types patterns and further our understanding of their behavior. We review recent advances regarding the complex behavior of localized patterns and the mathematical tools that have been developed to understand them, covering various topics from spatial dynamics, exponential asymptotics, and numerical methods. We observe that the mathematical understanding of localized patterns decreases as the spatial dimension increases, thus providing significant open problems that will form the basis for future investigations., Comment: Review paper, 76 pages, 51 figures

Isolated patches of spatially oscillating pattern have been found to emerge near a pattern-forming instability in a wide variety of experiments and mathematical models. However, there is currently no mathematical theory to explain this emergence or characterise the structure of these patches. We provide a method for formally deriving radial amplitude equations to planar patterns via non-autonomous multiple-scale analysis and convolutional sums of products of Bessel functions. Our novel approach introduces nonautonomous differential operators, which allow for the systematic manipulation of Bessel functions, as well as previously unseen identities involving infinite sums of Bessel functions. Solutions of the amplitude equations describe fully localised patterns with non-trivial angular dependence, where localisation occurs in a purely radial direction. Amplitude equations are derived for multiple examples of patterns with dihedral symmetry, including fully localised hexagons and quasipatterns with twelve-fold rotational symmetry. In particular, we show how to apply the asymptotic method to the Swift--Hohenberg equation and general reaction-diffusion systems., Comment: 19 pages, 4 figures

Congenital Heart Disease (CHD) is a group of cardiac malformations present already during fetal life, representing the prevailing category of birth defects globally. Our aim in this study is to aid 3D fetal vessel topology visualisation in aortic arch anomalies, a group which encompasses a range of conditions with significant anatomical heterogeneity. We present a multi-task framework for automated multi-class fetal vessel segmentation from 3D black blood T2w MRI and anomaly classification. Our training data consists of binary manual segmentation masks of the cardiac vessels' region in individual subjects and fully-labelled anomaly-specific population atlases. Our framework combines deep learning label propagation using VoxelMorph with 3D Attention U-Net segmentation and DenseNet121 anomaly classification. We target 11 cardiac vessels and three distinct aortic arch anomalies, including double aortic arch, right aortic arch, and suspected coarctation of the aorta. We incorporate an anomaly classifier into our segmentation pipeline, delivering a multi-task framework with the primary motivation of correcting topological inaccuracies of the segmentation. The hypothesis is that the multi-task approach will encourage the segmenter network to learn anomaly-specific features. As a secondary motivation, an automated diagnosis tool may have the potential to enhance diagnostic confidence in a decision support setting. Our results showcase that our proposed training strategy significantly outperforms label propagation and a network trained exclusively on propagated labels. Our classifier outperforms a classifier trained exclusively on T2w volume images, with an average balanced accuracy of 0.99 (0.01) after joint training. Adding a classifier improves the anatomical and topological accuracy of all correctly classified double aortic arch subjects., Comment: Accepted for publication at the Journal of Machine Learning for Biomedical Imaging (MELBA) https://melba-journal.org/2023:015

Dynamic free-breathing fetal cardiac MRI is one of the most challenging modalities, which requires high temporal and spatial resolution to depict rapid changes in a small fetal heart. The ability of deep learning methods to recover undersampled data could help to optimise the kt-SENSE acquisition strategy and improve non-gated kt-SENSE reconstruction quality. In this work, we explore supervised deep learning networks for reconstruction of kt-SENSE style acquired data using an extensive in vivo dataset. Having access to fully-sampled low-resolution multi-coil fetal cardiac MRI, we study the performance of the networks to recover fully-sampled data from undersampled data. We consider model architectures together with training strategies taking into account their application in the real clinical setup used to collect the dataset to enable networks to recover prospectively undersampled data. We explore a set of modifications to form a baseline performance evaluation for dynamic fetal cardiac MRI on real data. We systematically evaluate the models on coil-combined data to reveal the effect of the suggested changes to the architecture in the context of fetal heart properties. We show that the best-performers recover a detailed depiction of the maternal anatomy on a large scale, but the dynamic properties of the fetal heart are under-represented. Training directly on multi-coil data improves the performance of the models, allows their prospective application to undersampled data and makes them outperform CTFNet introduced for adult cardiac cine MRI. However, these models deliver similar qualitative performances recovering the maternal body very well but underestimating the dynamic properties of fetal heart. This dynamic feature of fast change of fetal heart that is highly localised suggests both more targeted training and evaluation methods might be needed for fetal heart application.