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Postoperative pulmonary complications, particularly atelectasis, are common after cardiac surgery and may contribute to impaired gas exchange or acute lung injury (ALI). Neurally Adjusted Ventilatory Assist (NAVA) delivers ventilatory support proportional to the patient’s respiratory drive, offering theoretical advantages over Pressure Support Ventilation (PSV), including improved synchrony, enhanced diaphragmatic efficiency, and reduced risk of ventilator-induced lung injury. However, comparative data on gas exchange, dead space, and regional ventilation during weaning after cardiac surgery remain limited. This prospective crossover study evaluated 12 mechanically ventilated patients with mild ALI following cardiac surgery across three ventilation phases: two PSV phases (PSV1 and PSV2) separated by a phase of NAVA. Intrapulmonary shunt fraction was calculated from measurements obtained via a Swan-Ganz catheter. Physiological dead space fraction (VD/VT) was assessed using three methods: the Bohr–Enghoff equation, end-tidal CO₂-derived alveolar dead space fraction (AVDSf-ET), and a novel time-to-volume converted capnographic approach (VCAP-CALC). Regional ventilation was assessed using electrical impedance tomography (EIT), and neuroventilatory efficiency (NVE) was calculated from diaphragmatic electrical activity (EAdi). Data were analyzed using linear mixed-effects models to account for repeated measures and within-subject variability. VD/VT was significantly lower during NAVA compared with PSV1 and PSV2 when assessed by VCAP-CALC (58.5% vs. 63.8% and 61.3%, respectively; p < 0.001). The PaO₂/FiO₂ ratio and NVE were significantly higher during NAVA (p = 0.01 and p = 0.037, respectively). No significant difference in pulmonary shunt fraction was observed. EIT revealed a modest increase in dorsal end-expiratory lung volume during NAVA, without redistribution of tidal volume or Center of Ventilation. The VCAP-CALC method showed strong agreement with established dead space measures (R2 = 0.77–0.82) and demonstrated high repeatability (mean coefficient of variation 3.5%). NAVA is a safe and feasible ventilatory mode following cardiac surgery, associated with reduced dead space fraction, improved oxygenation and enhanced neuroventilatory efficiency. Given that shunt fraction remained unchanged, the observed improvement in ventilation–perfusion (V/Q) matching reflects a reduction in VD/VT. The potential implications for postoperative recovery and long-term outcomes merit evaluation in larger clinical studies. ClinicalTrials.gov: NCT03217305. Initial Release 21/06/2017.

We consider an inverse problem involving the reconstruction of the solution to a nonlinear partial differential equation (PDE) with unknown boundary conditions. Instead of direct boundary data, we are provided with a large dataset of boundary observations for typical solutions (collective data) and a bulk measurement of a specific realization. To leverage this collective data, we first compress the boundary data using proper orthogonal decomposition (POD) in a linear expansion. Next, we identify a possible nonlinear low-dimensional structure in the expansion coefficients using an autoencoder, which provides a parametrization of the dataset in a lower-dimensional latent space. We then train an operator network to map the expansion coefficients representing the boundary data to the finite element (FE) solution of the PDE. Finally, we connect the autoencoder's decoder to the operator network which enables us to solve the inverse problem by optimizing a data-fitting term over the latent space. We analyze the underlying stabilized finite element method (FEM) in the linear setting and establish an optimal error estimate in the H¹-norm. The nonlinear problem is then studied numerically, demonstrating the effectiveness of our approach.

We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that moves around inside/“on top” of it. Here the overlapping mesh is prescribed by a simple continuous motion, meaning that its location as a function of time is continuous and piecewise linear. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche’s method and also includes an integral term over the space-time boundary between the two meshes that mimics the standard discontinuous Galerkin time-jump term. The simple continuous mesh motion results in a space-time discretization for which standard analysis methodologies either fail or are unsuitable. We therefore employ what seems to be a relatively uncommon energy analysis framework for finite element methods for parabolic problems that is general and robust enough to be applicable to the current setting. The energy analysis consists of a stability estimate that is slightly stronger than the standard basic one and an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.

We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that evolves inside/“on top” of it. Here the overlapping mesh is prescribed by a simple discontinuous evolution, meaning that its location, size, and shape as functions of time are discontinuous and piecewise constant. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche’s method. The simple discontinuous mesh evolution results in a space-time discretization with a slabwise product structure between space and time which allows for existing analysis methodologies to be applied with only minor modifications. We follow the analysis methodology presented by Eriksson and Johnson (SIAM J Numer Anal 28(1):43–77, 1991; SIAM J Numer Anal 32(3):706–740, 1995). The greatest modification is the introduction of a Ritz-like “shift operator” that is used to obtain the discrete strong stability needed for the error analysis. The shift operator generalizes the original analysis to some methods for which the discrete subspace at one time does not lie in the space of the stiffness form at the subsequent time. The error analysis consists of an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.

The formation of buds on the cell membrane of budding yeast cells is thought to be driven by reactions and diffusion involving the protein Cdc42. These processes can be described by a coupled system of partial differential equations known as the Schnakenberg system. The Schnakenberg system is known to exhibit diffusion-driven pattern formation, thus providing a mechanism for bud formation. However, it is not known how the accumulation of bud scars on the cell membrane affect the ability of the Schnakenberg system to form patterns. We have approached this problem by modelling a bud scar on the cell membrane with a hole on the sphere. We have studied how the spectrum of the Laplace–Beltrami operator, which determines the resulting pattern, is affected by the size of the hole, and by numerically solving the Schnakenberg system on a sphere with a hole using the finite element method. Both theoretical predictions and numerical solutions show that pattern formation is robust to the introduction of a bud scar of considerable size, which lends credence to the hypothesis that bud formation is driven by diffusion-driven instability.