Dominik Schillinger

A bstract: Optimality principles are widely used to understand branching networks, yet they are tested on individual branchings that cannot reliably distinguish competing scaling laws. We formulate the missing link between the laws and measurements by mapping each law to its predicted branch points through global optimization. Vascular branch points closely match weighted-length optima when each vessel is allowed its own fitted weight. Inferred scaling laws reveal type-specific geometric signatures hidden from local analysis. They beat all tested laws, including Murray’s law in 1,679 of 1,744 held-out hepatic and pulmonary trees. For coronary arteries on the heart surface, an exact discrete optimization model extends the framework beyond convex geometry and supports area-preserving scaling. The framework also identifies distinct laws for airways and plant shoots while rejecting weighted-length optimality for neuronal axons. Global optimization turns network geometry into a quantitative test of the principles that govern branching networks.

After brain injury, white matter changes are not confined to axons or tissue composition; glial cell bodies can reorganize spatially. Because glial cells respond partly to physical cues around them, this reorganization could alter the environment a cell experiences, even when whole tissue appears mechanically unchanged. Here, we used finite element models of corpus callosum white matter to test whether glial cell body arrangement alone can reshape this cellular environment. These models represented aligned axonal fibers, tissue matrix, and glial cell bodies. We compared tissue without glial cell bodies, one glial cell body with matched volume, and multiple glial cell bodies in clustered, dispersed, near boundary, or random patterns, while keeping total glial volume fixed. Loading was applied parallel and perpendicular to the axonal direction, and cell body stiffness was varied independently of arrangement. The tissue response changed only modestly, whereas stress and deformation energy around glial cell bodies were redistributed.

Visualization of cellular dynamics in microvascular networks is essential for deciphering physiology and disease. Existing imaging platforms commonly lack the spatiotemporal resolution and sensitivity for single-cell tracking in vivo, particularly beyond the penetration depth of optical microscopy. While recent fluorescence and optoacoustic methods allow detection of single microparticles in vivo, multiplexed imaging of distinct circulating cell populations remains unachieved. Here, we present an optimized labeling protocol using near-infrared (NIR) dyes ICG (indocyanine green) and DiR (1,1′-dioctadecyl-3,3,3′,3′-tetramethylindotricarbocyanine iodide) to generate spectrally distinct optoacoustic signatures for red blood cells (RBCs) and neutrophils. This enables noninvasive simultaneous tracking of both cell types in deep cerebrovascular networks of mice, supporting superresolution imaging with localization optoacoustic tomography and revealing differences in the velocities of different cell types inside opaque tissues. ICG-labeled RBCs are further detectable via diffuse optical localization imaging in the NIR-II window, permitting high-resolution visualization of cortical capillaries. In addition, incorporation of superparamagnetic nanoparticles allows noninvasive magnetic manipulation of the microcirculation alongside real-time single-cell monitoring, further providing a strategy for precise and reversible vascular occlusion in preclinical ischemic stroke models. Together, these capabilities provide a versatile platform for advancing vascular research at the single-cell level.

We propose a matrix-free inexact preconditioning strategy for elliptic partial differential equations discretized by the isogeometric Galerkin method on tensor-product spline spaces. We base our preconditioner on an approximation of the discrete linear operator by a sum of Kronecker product matrices. The action of its inverse on a vector of coefficients is approximated by an inner preconditioned conjugate gradient solve. The forward problem is solved by the inexact preconditioned conjugate gradient method. The complexity of the Kronecker matrix-vector products in the inner iteration is lower than the complexity of the matrix-vector products in the forward problem, leading to a reduced number of iterations and significant performance gains. We show the robustness, efficiency and effectiveness of our approach in test problems involving the Poisson equation and linear elasticity, and illustrate the performance gain with respect to preconditioning techniques based on fast diagonalization. The proposed method is implemented in our open-source framework for spline based discretization methods.

The simulation of tissue perfusion based on highly detailed synthetic vasculature that often consists of multiple supplying and draining trees with millions of vascular segments is computationally expensive. Converting highly detailed synthetic vasculature into a homogenized continuum flow representation offers a computationally efficient alternative. In this paper, we investigate such a modeling approach that retains the essential features of potentially deforming hierarchical vascular networks. It is based on multi-compartment homogenization, where each compartment represents homogenized perfusion via a Darcy-type flow model associated with vascular segments at a specific spatial resolution in one individual tree of the network. The compartments are coupled through a pressure-dependent mass exchange, applied in a smeared manner everywhere within the perfusion domain. Key parameters, namely the permeability tensors of each compartment and the intercompartmental perfusion coefficients, are estimated directly from the vascular segments of the synthetic vasculature using averaging techniques. Our approach leverages spectral decomposition and a reduced set of representative vessel segments to balance computational efficiency with physical fidelity. For scenarios involving deformation, such as in a pumping heart or a regenerating liver, we introduce a computationally efficient parameter update based on geometric mapping, avoiding full re-homogenization in nonlinear simulations. We demonstrate the effectiveness and accuracy of the approach for several benchmark examples, including a full-scale multi-compartment liver perfusion simulation that explicitly incorporates three non-intersecting vascular trees, reflecting the hepatic artery, portal vein, and hepatic vein.

The article at hand reviews the state-of-the-art in finite element methods based on transfinite function spaces, which were originally introduced by Coons and subsequently refined by Gordon and co-workers. The main objective is to present the benefits of transfinite interpolation techniques, its connection and its use in popular approaches that include isogeometric elements, NURBS-enhanced finite elements, and advanced transition elements following the p N h - or x N y -frameworks. Note that a secondary objective is to revitalize interest in this element-type, which, despite its remarkable advantages, has not gained widespread adoption. To shed light on this aspect, our review addresses the historical reluctance to embrace transfinite interpolation, attributing it to the perceived mathematical complexity of shape functions and the limited disclosure of element formulation details in initial publications due to industrial confidentiality. By elucidating these features, we aspire to rekindle appreciation for one of the most powerful element formulations in the finite element method, leading to renewed interest in and exploration of its application. Finally, we summarize the unique key features of transfinite elements , which are: (i) Conventional finite elements of Lagrange, Serendipity, and isogeometric types are included as special cases in the formulation of transfinite elements; (ii) Polytope elements, enhancing mesh generation flexibility (e.g., image-based analysis), are generated in a straightforward manner; (iii) Transition elements capable of coupling different element families and/or meshes with varying element sizes can be easily derived; and (iv) Exact geometry representations using the blending function method, which is a direct application of the transfinite interpolation concept to the geometry mapping, are achieved.

Perfusion through a liver lobule is influenced by a range of physiological parameters such as lobule geometry, permeability, blood viscosity, and pressure gradients. We present a hybrid reduced-order modeling approach for real-time simulation of perfusion profiles in 2D lobular structures of the liver that addresses all these parametric aspects. Building on our previous work in Siddiqui et al. (Adv Model Simul Eng Sci 11:22, 2024), we extend the methodology to incorporate a broader range of physiological parameters and multiple interconnected lobules, and introduce hybrid reduced-order representations, thereby achieving even greater computational speedups. We illustrate our reduced-order representations through a stepwise construction, sequentially incorporating different aspects of the parameterized perfusion model. To demonstrate the viability of our approach, we first focus on a model of a single lobule. It is then extended to a model of several interconnected lobules via the formulation of suitable coupling terms. Our simulation results show that our hybrid reduced-order modeling approach is up to three orders of magnitude faster than corresponding high-fidelity finite element computations. At the same time, it maintains responsiveness to physical stimuli and produces results for variations in health conditions that are consistent with clinical experience.

In this work, we compare the nodal and isogeometric spatial discretization schemes for the nonlinear formulation of shear‐ and torsion‐free rods introduced in Gebhardt and Romero (see Reference no. 31). We investigate the resulting discrete solution space, the accuracy, and the computational cost of these spatial discretization schemes. To fulfill the required continuity of the rod formulation, the nodal scheme discretizes the rod in terms of its nodal positions and directors using cubic Hermite splines. Isogeometric discretizations naturally fulfill this with smooth spline basis functions and discretize the rod only in terms of the positions of the control points (see Nguyen et al. in Reference no. 41), which leads to a discrete solution in multiple copies of the Euclidean space . They enable the employment of basis functions of one degree lower, that is, quadratic splines, and possibly reduce the number of degrees of freedom (dofs). When using the nodal scheme, since the defined director field is in the unit sphere , preserving this for the nodal director variable field requires an additional constraint of unit nodal directors. This leads to a discrete solution in multiple copies of the manifold ; however, it results in zero nodal axial stress values. Allowing arbitrary length for the nodal directors, that is a nodal director field in instead of as within discrete rod elements, eliminates the constrained nodal axial stresses and leads to a discrete solution in multiple copies of . To enforce the unit nodal director constraint, we discuss two approaches using the Lagrange multiplier and penalty methods. We compare the resulting semi‐discrete formulations and the computational cost of these discretization variants. We numerically demonstrate our findings via examples of a planar roll‐up, a catenary, and a mooring line.