There have been major advances in the mathematical modelling of complex physiological and pathophysiological systems, with the ultimate aim to provide deeper clinical insight into various disease conditions. However, living tissue still lacks a unified system-level state description that remains meaningful under finite-resolution observation, intrinsic heterogeneity, and ongoing structural remodelling. Existing physical descriptors usually characterise specific mechanisms or responses, rather than providing an operational notion of tissue state that can be measured and compared consistently across conditions. Here, we introduce tissue state as an operational system-level state variable for biological matter. Tissue state is not defined as a purely mechanical quantity, nor is it intended to encode microscopic biological structure. Instead, it is formulated as an equivalence-class description, characterised by observable input-output behaviour within a calibrated physical description framework. In this work, mechanical response is one measurement channel through which tissue state can be accessed. Within a fixed geometric and loading framework, changes in observed motion responses are reflected not in unique constitutive parameter values, but in systematic changes in the posterior distribution over admissible parameter sets. This posterior distribution functions as a model-dependent representation of tissue state, rather than a literal reconstruction of tissue microstructure. By framing tissue behaviour in terms of state equivalence classes rather than single-valued material parameters, this perspective provides a physical basis for describing biological remodelling, heterogeneity, and history dependence under finite observation. It is relevant to the computational modelling of chronic organ disease, where tissue injury and remodelling may gradually reshape the admissible state of an organ before overt functional decline appears in standard clinical readouts. It also provides a state-based language for AI in therapeutic research and pharmaceutical safety assessment, including settings in which treatment effects or cross organ toxicity must be inferred from indirect, heterogeneous, and only partially identifiable measurements. More broadly, it suggests a way to bring living systems into physical theory, and to represent them more reliably in computational medicine, through appropriate system-level state variables and operational modes of measurement.

We introduce finite-element Gaussian processes (FEGPs), a novel physics-informed machine learning approach for solving inverse problems involving steady-state, linear partial differential equations (PDEs). Our framework combines a Gaussian process prior for the unknown solution function with a likelihood that incorporates the PDE in its weak form, using a finite-element approximation. This approach offers significantly better scalability than physics-informed Gaussian processes (PIGPs), which rely on the strong form of the PDE. Through numerical experiments on a range of synthetic benchmark problems, we show that FEGPs offer results which outperform PIGPs, and are competitive with physics-informed neural networks (PINNs) with improved uncertainty quantification.

There are many computational models set up to predict blood flow and pressure in vascular networks. Methods for a single forward solution of such models are well established but become problematic in clinical applications, where model calibration and patient-specific parameter estimation call for repeated forward simulations of the model requiring substantial computational costs. A potential workaround is emulation, which approximates the original mathematical model by a statistical or machine learning surrogate model. Our methodological framework is based on physics-informed neural networks, with a particular focus on patient-specific model calibration. Once fully trained, our machine learning model predicts flow and pressure waveforms in a fraction of the time required by the numerical solver, enabling fast parameter inference and inverse uncertainty quantification. The proposed framework is applied to clinical data from four patients diagnosed with a Double Outlet Right Ventricle (DORV), a congenital heart defect where both the aorta and main pulmonary artery connect to the right ventricle, potentially leading to insufficient oxygen delivery to the body and hence requiring careful blood flow monitoring. We assess the performance of our method in a comparative evaluation study that includes several alternative state-of-the-art machine learning methods, and we quantify the improvement achieved in terms of accuracy and efficiency gains.

PURPOSE:Accurate, patient-specific haemodynamic assessment is limited by the cost of computational solvers and the sparsity of clinical measurements. We demonstrate that physics-informed neural operator surrogates can emulate multi-vessel 1D haemodynamics with sufficient accuracy and speed to enable Bayesian parameter inference and non-invasive pressure estimation in a clinically relevant setting. METHODS:We construct neural operator surrogates for a 17-vessel 1D systemic arterial network with prescribed inflow and structured-tree outflow, using DeepONet, POD-DeepONet and Fourier Neural Operator (FNO) architectures that map inflow waveforms and biophysical parameters (vessel stiffnesses and microvascular properties) to flow and pressure fields. Conservation laws, PDE residuals and bifurcation conditions are incorporated via physics-informed loss terms. To generate realistic inflow boundary conditions from limited clinical data, we compare generative models and adopt a Wasserstein autoencoder. The best-performing surrogate is embedded in a Bayesian pipeline to perform MCMC-based parameter inference and non-invasive pressure prediction for two Fontan patients using sparse 4D flow MRI waveforms. RESULTS:The physics-informed FNO architecture achieved the lowest median relative errors across all vessels and markedly reduced maximum errors compared with purely data-driven training. In synthetic inverse tests, the PINO recovered vascular parameters more accurately than DeepONet and POD-DeepONet. For two Fontan patients, the calibrated model reproduced measured flow waveforms and yielded brachial pressure predictions consistent with cuff measurements, together with posterior uncertainty bands. CONCLUSIONS:Physics-informed neural operators can emulate multi-vessel haemodynamics with high accuracy at a fraction of the computational cost of traditional solvers. Coupled with Bayesian inference, the proposed framework enables practical, uncertainty-aware estimation of vascular parameters and non-invasive pressure waveforms from sparse clinical flow data.

In this article we explore parameter inference in a novel hybrid discrete-continuum model describing the movement of a population of cells in response to a self-generated chemotactic gradient. The model employs a drift-diffusion stochastic process, rendering likelihood-based inference methods impractical. Consequently, we consider approximate Bayesian computation (ABC) methods, which have gained popularity for models with intractable or computationally expensive likelihoods. ABC involves simulating from the generative model, using parameters from generated observations that are “close enough” to the true data to approximate the posterior distribution. Given the plethora of existing ABC methods, selecting the most suitable one for a specific problem can be challenging. To address this, we employ a simple drift-diffusion stochastic differential equation (SDE) as a benchmark problem. This allows us to assess the accuracy of popular ABC algorithms under known configurations. We also evaluate the bias between ABC-posteriors and the exact posterior for the basic SDE model, where the posterior distribution is tractable. The top-performing ABC algorithms are subsequently applied to the proposed cell movement model to infer its key parameters. This study not only contributes to understanding cell movement but also sheds light on the comparative efficiency of different ABC algorithms in a well-defined context.

We introduce a framework for designing boundary constrained Gaussian process (BCGP) priors for exact enforcement of linear boundary conditions, and apply it to the machine learning of (initial) boundary value problems involving linear partial differential equations (PDEs). In contrast to existing work, we illustrate how to design boundary constrained mean and kernel functions for all classes of boundary conditions typically used in PDE modelling, namely Dirichlet, Neumann, Robin and mixed conditions. Importantly, this is done in a manner which allows for both forward and inverse problems to be naturally accommodated. We prove that the BCGP kernel has a universal representational capacity under Dirichlet conditions, and establish a formal equivalence between BCGPs and boundary-constrained neural networks (BCNNs) of infinite width. Finally, extensive numerical experiments are performed involving several linear PDEs, the results of which demonstrate the effectiveness and robustness of BCGP inference in the presence of sparse, noisy data.

The present paper investigates constrained global optimization techniques for computationally expensive black box functions that are globally defined but subject to some a priori unknown taboo regions. This challenge typically arises in healthcare applications, where the goal is to maximize the efficacy of a drug while staying within critical safety limits imposed by an external medical regulator. Motivated by the additional challenge where the constrained global optimum lies along the constraint boundary, we comparatively assess the performance of established optimization methods coupled with different acquisition functions in terms of accuracy and efficiency on the physiological application aforementioned and several benchmark problems representative of the complexity of the physiological application. We find the best method based on an average score computed across all applications. We also propose an ensemble method combining results from individual methods, which vastly outperforms the best average method. Furthermore, our study provides a thorough qualitative analysis of the optimization results, emphasizing the challenges a user may encounter when applying Bayesian optimization on constrained optimization problems.

Cardiac mechanics modelling promises to revolutionize personalized health care; however, inferring patient-specific biophysical parameters, which are critical for understanding myocardial functions and performance, poses substantial methodological challenges. Our work is primarily motivated to determine the passive stiffness of the myocardium from the measurement of the left ventricle (LV) volume at various time points, which is crucial for diagnosing cardiac physiological conditions. Although there have been significant advancements in cardiac mechanics modelling, the tasks of inference and uncertainty quantification of myocardial stiffness remain challenging, with high computational costs preventing real-time decision support. We adapt Gaussian processes to construct a statistical surrogate model for emulating LV cavity volume during diastolic filling to overcome this challenge. As the LV volumes, obtained at different time points in diastole, constitute a time series, we apply the Kronecker product trick to decompose the complex covariance matrix of the whole system into two separate covariance matrices, one for time and the other for biophysical parameters. To proceed towards personalized health care, we further integrate patient-specific LV geometries into the Gaussian process emulator using principal component analysis (PCA). Utilizing a deep learning neural network for extracting time-series left ventricle volumes from magnetic resonance images, Bayesian inference is applied to determine the posterior probability distribution of critical cardiac mechanics parameters. Tests on real-patient data illustrate the potential for real-time estimation of myocardial properties for clinical decision-making. These advancements constitute a crucial step towards clinical impact, offering valuable insights into posterior uncertainty quantification for complex cardiac mechanics models.

Dynamic Contrast Enhanced Magnetic Resonance Imaging (DCE-MRI) can be used as a non-invasive method for the assessment of myocardial perfusion. The acquired images can be utilized to analyze the spatial extent and severity of myocardial ischaemia (regions with impaired microvascular blood flow). In the present paper, we propose a novel generalisable spatio-temporal hierarchical Bayesian model (GST-HBM) to automate the detection of ischaemic lesions and improve the in silico prediction accuracy by systematically integrating spatio-temporal context information. We present a computational inference procedure with an adequate trade-off between accuracy and computational efficiency, whereby model parameters are sampled from the posterior distribution with Gibbs sampling, while lower-level hyperparameters are selected using model selection strategies based on the Watanabe Akaike information criterion (WAIC). We have assessed our method on both synthetic (in silico) data with known gold-standard and 12 sets of clinical first-pass myocardial perfusion DCE-MRI datasets. We have also carried out a comparative performance evaluation with four established alternative methods: Gaussian mixture model (GMM), opening and closing operations based on Gaussian mixture model (GMMC&Omax), Markov random field constrained Gaussian mixture model (GMM-MRF) and model-based hierarchical Bayesian model (M-HBM). Our results show that the proposed GST-HBM method achieves much higher in silico prediction accuracy than the established alternative methods. Furthermore, this method appears to provide a more robust delineation of ischaemic lesions in datasets affected by spatially variant noise.

One-dimensional (1D) cardiovascular models offer a non-invasive method to answer medical questions, including predictions of wave-reflection, shear stress, functional flow reserve, vascular resistance and compliance. This model type can predict patient-specific outcomes by solving 1D fluid dynamics equations in geometric networks extracted from medical images. However, the inherent uncertainty in in vivo imaging introduces variability in network size and vessel dimensions, affecting haemodynamic predictions. Understanding the influence of variation in image-derived properties is essential to assess the fidelity of model predictions. Numerous programs exist to render three-dimensional surfaces and construct vessel centrelines. Still, there is no exact way to generate vascular trees from the centrelines while accounting for uncertainty in data. This study introduces an innovative framework employing statistical change point analysis to generate labelled trees that encode vessel dimensions and their associated uncertainty from medical images. To test this framework, we explore the impact of uncertainty in 1D haemodynamic predictions in a systemic and pulmonary arterial network. Simulations explore haemodynamic variations resulting from changes in vessel dimensions and segmentation; the latter is achieved by analysing multiple segmentations of the same images. Results demonstrate the importance of accurately defining vessel radii and lengths when generating high-fidelity patient-specific haemodynamics models. KEY POINTS: This study introduces novel algorithms for generating labelled directed trees from medical images, focusing on accurate junction node placement and radius extraction using change points to provide haemodynamic predictions with uncertainty within expected measurement error. Geometric features, such as vessel dimension (length and radius) and network size, significantly impact pressure and flow predictions in both pulmonary and aortic arterial networks. Standardizing networks to a consistent number of vessels is crucial for meaningful comparisons and decreases haemodynamic uncertainty. Change points are valuable to understanding structural transitions in vascular data, providing an automated and efficient way to detect shifts in vessel characteristics and ensure reliable extraction of representative vessel radii.

The objective of this research is to estimate the specific biophysical parameters of a cardiac mechanics model using a time series of variables that can be acquired in the clinic.This method is driven by the need to infer the passive stiffness of the myocardium to diagnose cardiophysiological diseases, which requires the measurement of the volume of the left ventricle (LV) of the heart at different time points.Although there have been many advancements in cardiac models, their computation is complex and costly.To overcome this challenge, we propose a method that utilizes a Gaussian process to construct a statistical model for emulation.Since the LV volumes are acquired in a time series, we employ the Kronecker product to compute two covariance matrices separately for time and biophysical parameters.Once we construct an accurate emulator to represent the passive filling process of the cardiac mechanics model during diastole, we can estimate the biophysical parameters inversely.We also aim to evaluate the impact of increasing the number of time points on reducing the uncertainty of the inverse estimation in this study.

Fluorescence decay of amino acids in protein is a complex process for which multiple models have been proposed.Likelihood function evaluation for certain models can be computationally expensive, and as such surrogate models may be introduced to speed up inference.In this paper, Gaussian processes are implemented in likelihood estimation of a range of models defined by convolutions of an initial excitation input and a decay function using both synthetic and real world data.Parameter inference and model selection using the surrogate models are performed and compared against the exact results.Model selection when incorporating surrogate models into the inference process is shown to be consistent.