Martin A. Grepl

Albumin Is Recycled from the Primary Urine by Tubular Transcytosis

Under physiologic conditions, significant amounts of plasma protein pass the renal filter and are reabsorbed by proximal tubular cells, but it is not clear whether the endocytosed protein, particularly albumin, is degraded in lysosomes or returned to the circulatory system intact. To resolve this question, a transgenic mouse with podocyte-specific expression of doxycycline-inducible tagged murine albumin was developed. To assess potential glomerular backfiltration, two types of albumin with different charges were expressed. On administration of doxycycline, podocytes expressed either of the two types of transgenic albumin, which were secreted into the primary filtrate and reabsorbed by proximal tubular cells, resulting in serum accumulation. Renal transplantation experiments confirmed that extrarenal transcription of transgenic albumin was unlikely to account for these results. Genetic deletion of the neonatal Fc receptor (FcRn), which rescues albumin and IgG from lysosomal degradation, abolished transcytosis of both types of transgenic albumin and IgG in proximal tubular cells. In summary, we provide evidence of a transcytosis within the kidney tubular system that protects albumin and IgG from lysosomal degradation, allowing these proteins to be recycled intact.

The glomerular filtration barrier function: new concepts

Purpose of reviewEach day, the human kidneys filter about 140 l of primary urine from plasma. Although this ultrafiltrate is virtually free of plasma protein, the glomerular filter never clogs under physiological conditions. Upto today it is still not entirely resolved as to how the kidney accomplishes this extraordinary task. Most of the proposed models for glomerular filtration have not considered electrical effects. Recent findingsIn micropuncture studies, we have directly measured an electrical field across the glomerular filtration barrier. This potential difference is most likely generated by forced passage of the ionic solution of the plasma across the charged glomerular filter (‘electrokinetic potential’). As all plasma proteins are negatively charged, the electrical field across the glomerular filtration barrier is predicted to drive plasma proteins from the filter toward the capillary lumen by electrophoresis. SummaryIn this review, we examine our novel model for glomerular filtration in more detail. We outline the physical mechanisms by which electrokinetic effects (streaming potentials) are generated. We investigate the potential impact of the electrical field on the passage of albumin across the glomerular filtration barrier. We review the mathematical heteroporous model including electrical effects and analyse a selection of experimental studies for indications that electrical effects influence glomerular permeability significantly.

Model Order Reduction of Parametrized Nonlinear Reaction-Diffusion Systems

Abstract We present a model order reduction technique for parametrized nonlinear reaction–diffusion systems. In our approach we combine the reduced basis method – a computational framework for rapid evaluation of functional outputs associated with the solution of parametrized partial differential equations – with the empirical interpolation method – a tool to construct “affine” coefficient-function approximations of nonlinear parameter dependent functions. We develop an efficient offline–online computational procedure for the evaluation of the reduced basis approximation: in the offline stage, we generate the reduced basis space; in the online stage, given a new parameter value, we calculate the reduced basis output. The operation count for the online stage depends only on the dimension of the reduced order model and the parametric complexity of the problem. The method is thus ideally suited for the many-query or real-time contexts. We present numerical results for a non-isothermal reaction–diffusion model to confirm and test our approach.

Approximation of Parametric Derivatives by the Empirical Interpolation Method

We introduce a general a priori convergence result for the approximation of parametric derivatives of parametrized functions. We consider the best approximations to parametric derivatives in a sequence of approximation spaces generated by a general approximation scheme, and we show that these approximations are convergent provided that the best approximation to the function itself is convergent. We also provide estimates for the convergence rates. We present numerical results with spaces generated by a particular approximation scheme—the Empirical Interpolation Method—to confirm the validity of the general theory.

Reduced basis approximation and a posteriori error bounds for 4D-Var data assimilation

We propose a certified reduced basis approach for the strong- and weak-constraint four-dimensional variational (4D-Var) data assimilation problem for a parametrized PDE model. While the standard strong-constraint 4D-Var approach uses the given observational data to estimate only the unknown initial condition of the model, the weak-constraint 4D-Var formulation additionally provides an estimate for the model error and thus can deal with imperfect models. Since the model error is a distributed function in both space and time, the 4D-Var formulation leads to a large-scale optimization problem for every given parameter instance of the PDE model. To solve the problem efficiently, various reduced order approaches have therefore been proposed in the recent past. Here, we employ the reduced basis method to generate reduced order approximations for the state, adjoint, initial condition, and model error. Our main contribution is the development of efficiently computable a posteriori upper bounds for the error of the reduced basis approximation with respect to the underlying high-dimensional 4D-Var problem. Numerical results are conducted to test the validity of our approach.

Certified Reduced Basis Methods for Parametrized Elliptic Optimal Control Problems with Distributed Controls.

In this paper, we consider the efficient and reliable solution of distributed optimal control problems governed by parametrized elliptic partial differential equations. The reduced basis method is used as a low-dimensional surrogate model to solve the optimal control problem. To this end, we introduce reduced basis spaces not only for the state and adjoint variable but also for the distributed control variable. We also propose two different error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. The reduced basis optimal control problem and associated a posteriori error bounds can be efficiently evaluated in an offline–online computational procedure, thus making our approach relevant in the many-query or real-time context. We compare our bounds with a previously proposed bound based on the Banach–Nečas–Babuška theory and present numerical results for two model problems: a Graetz flow problem and a heat transfer problem. Finally, we also apply and test the performance of our newly proposed bound on a hyperthermia treatment planning problem.

Offline Error Bounds for the Reduced Basis Method

The reduced basis method is a model order reduction technique that is specifically designed for parameter-dependent systems. Due to an offline-online computational decomposition, the method is particularly suitable for the many-query or real-time contexts. Furthermore, it provides rigorous and efficiently evaluable a posteriori error bounds, which are used offline in the greedy algorithm to construct the reduced basis spaces and may be used online to certify the accuracy of the reduced basis approximation. Unfortunately, in real-time applications a posteriori error bounds are of limited use. First, if the reduced basis approximation is not accurate enough, it is generally impossible to go back to the offline stage and refine the reduced model; and second, the greedy algorithm guarantees a desired accuracy only over the finite parameter training set and not over all points in the admissible parameter domain. Here, we propose an extension or “add-on” to the standard greedy algorithm that allows us to evaluate bounds over the entire domain, given information for only a finite number of points. Our approach employs sensitivity information at a finite number of points to bound the error and may thus be used to guarantee a certain error tolerance over the entire parameter domain during the offline stage. We focus on an elliptic problem and provide numerical results for a thermal block model problem to validate our approach.

Certified Reduced Basis Methods for Parametrized Distributed Elliptic Optimal Control Problems with Control Constraints

In this paper, we employ the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by scalar coercive elliptic partial differential equations. We consider the standard linear-quadratic problem setting with distributed control and unilateral control constraints. For this problem class, we propose two different reduced basis approximations and associated error estimation procedures. In our first approach, we directly consider the resulting optimality system, introduce suitable reduced basis approximations for the state, adjoint, control, and Lagrange multipliers, and use a projection approach to bound the error in the reduced optimal control. For our second approach, we first reformulate the optimal control problem using a slack variable, then develop a reduced basis approximation for the slack problem by suitably restricting the solution space, and finally derive error bounds for the slack based optimal control. We discuss benefits and drawbacks of b...

Efficient Reduced Basis Solution of Quadratically Nonlinear Diffusion Equations

Abstract We present reduced basis approximations and associated a posteriori error estimation procedures for a steady quadratically nonlinear diffusion equation. We develop an efficient computational procedure for the evaluation of the approximation and bound. The method is thus ideally suited for many-query or real-time applications. Numerical results are presented to confirm the rigor, sharpness and fast convergence of our approach.

Inverse correlation between vascular endothelial growth factor back-filtration and capillary filtration pressures

Background Vascular endothelial growth factor A (VEGF) is an essential growth factor during glomerular development and postnatal homeostasis. VEGF is secreted in high amounts by podocytes into the primary urine, back-filtered across the glomerular capillary wall to act on endothelial cells. So far it has been assumed that VEGF back-filtration is driven at a constant rate exclusively by diffusion. Methods In the present work, glomerular VEGF back-filtration was investigated in vivo using a novel extended model based on endothelial fenestrations as surrogate marker for local VEGF concentrations. Single nephron glomerular filtration rate (SNGFR) and/or local filtration flux were manipulated by partial renal mass ablation, tubular ablation, and in transgenic mouse models of systemic or podocytic VEGF overexpression or reduction. Results Our study shows positive correlations between VEGF back-filtration and SNGFR as well as effective filtration rate under physiological conditions along individual glomerular capillaries in rodents and humans. Conclusion Our results suggest that an additional force drives VEGF back-filtration, potentially regulated by SNGFR.