Abstract. The interactions between incompressible fluid flows and immersed structures are nonlinear multi-physics phenomena that have applications to a wide range of scientific and engineering disciplines. In this article, we review representative numerical methods based on conforming and non-conforming meshes that are currently available for computing fluid-structure interaction problems, with an emphasis on some of the recent developments in the field. A goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study in fluid-structure interactions.

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Diabetes increases the reabsorption of Na(+) (TNa) and glucose via the sodium-glucose cotransporter SGLT2 in the early proximal tubule (S1-S2 segments) of the renal cortex. SGLT2 inhibitors enhance glucose excretion and lower hyperglycemia in diabetes. We aimed to investigate how diabetes and SGLT2 inhibition affect TNa and sodium transport-dependent oxygen consumption [Formula: see text] along the whole nephron. To do so, we developed a mathematical model of water and solute transport from the Bowman space to the papillary tip of a superficial nephron of the rat kidney. Model simulations indicate that, in the nondiabetic kidney, acute and chronic SGLT2 inhibition enhances active TNa in all nephron segments, thereby raising [Formula: see text] by 5-12% in the cortex and medulla. Diabetes increases overall TNa and [Formula: see text] by ∼50 and 100%, mainly because it enhances glomerular filtration rate (GFR) and transport load. In diabetes, acute and chronic SGLT2 inhibition lowers [Formula: see text] in the cortex by ∼30%, due to GFR reduction that lowers proximal tubule active TNa, but raises [Formula: see text] in the medulla by ∼7%. In the medulla specifically, chronic SGLT2 inhibition is predicted to increase [Formula: see text] by 26% in late proximal tubules (S3 segments), by 2% in medullary thick ascending limbs (mTAL), and by 9 and 21% in outer and inner medullary collecting ducts (OMCD and IMCD), respectively. Additional blockade of SGLT1 in S3 segments enhances glucose excretion, reduces [Formula: see text] by 33% in S3 segments, and raises [Formula: see text] by <1% in mTAL, OMCD, and IMCD. In summary, the model predicts that SGLT2 blockade in diabetes lowers cortical [Formula: see text] and raises medullary [Formula: see text], particularly in S3 segments.

Layton AT, Vallon V, Edwards A. Modeling oxygen consumption in the proximal tubule: effects of NHE and SGLT2 inhibition. Am J Physiol Renal Physiol 308: F1343-F1357, 2015. First published April 8, 2015 doi:10.1152/ajprenal.00007.2015.-The objective of this study was to investigate how physiological, pharmacological, and pathological conditions that alter sodium reabsorption (T Na) in the proximal tubule affect oxygen consumption (QO 2 ) and Na ϩ transport efficiency (TNa/QO 2 ). To do so, we expanded a mathematical model of solute transport in the proximal tubule of the rat kidney. The model represents compliant S1, S2, and S3 segments and accounts for their specific apical and basolateral transporters. Sodium is reabsorbed transcellularly, via apical Na ϩ /H ϩ exchangers (NHE) and Na ϩ -glucose (SGLT) cotransporters, and paracellularly. Our results suggest that TNa/QO 2 is 80% higher in S3 than in S1-S2 segments, due to the greater contribution of the passive paracellular pathway to TNa in the former segment. Inhibition of NHE or Na-K-ATPase reduced T Na and QO 2 , as well as Na ϩ transport efficiency. SGLT2 inhibition also reduced proximal tubular T Na but increased QO 2 ; these effects were relatively more pronounced in the S3 vs. the S1-S2 segments. Diabetes increased TNa and QO 2 and reduced TNa/QO 2 , owing mostly to hyperfiltration. Since SGLT2 inhibition lowers diabetic hyperfiltration, the net effect on TNa, QO 2 , and Na ϩ transport efficiency in the proximal tubule will largely depend on the individual extent to which glomerular filtration rate is lowered. sodium transport; glucose; metabolism; diabetes DESPITE INTENSE RESEARCH, the mechanisms underlying the development of chronic kidney diseases remain incompletely understood. Renal hypoxia is thought to be a unifying pathway to chronic kidney disease (15) and, in general, is due to a mismatch between changes in renal oxygen delivery and oxygen consumption (8). Oxygen consumption in the kidney serves in large part to actively reabsorb Na ϩ . Since the proximal tubule is where more than half the filtered load of Na ϩ is reabsorbed, the goal of this study was to investigate how physiological and pathological changes in sodium transport alter O 2 consumption in the proximal tubule.Sodium reabsorption along the proximal tubule is coupled to HCO 3 Ϫ and Cl Ϫ transport: early NaHCO 3 reabsorption raises the luminal concentration of Cl Ϫ and enhances the driving force for paracellular NaCl reabsorption in the later part of the tubule. Notably, changes in O 2 consumption (Q O 2 ) do not always correlate positively with changes in Na ϩ reabsorption. Deng et al. (4) observed that blocking carbonic anhydrase with benzolamide (a proximal tubule diuretic) lowered the energy efficiency of Na ϩ reabsorption in the kidney, as it simultaneously decreased net Na ϩ reabsorption and increased overall O 2 consumption. These effects were abolished by drugs that suppress Na ϩ /H ϩ exchange or basolateral Na ϩ -HCO 3 Ϫ cotransport. Deng et al. (4) surmised that benzolamide...

In problems with interfaces, the unknown or its derivatives may have jump discontinuities. Finite difference methods, including the method of A. Mayo and the immersed interface method of R. LeVeque and Z. Li, maintain accuracy by adding corrections, found from the jumps, to the difference operator at grid points near the interface and by modifying the operator if necessary. It has long been observed that the solution can be computed with uniform O.h 2 / accuracy even if the truncation error is O.h/ at the interface, while O.h 2 / in the interior.We prove this fact for a class of static interface problems of elliptic type using discrete analogues of estimates for elliptic equations. Moreover, we show that the gradient is uniformly accurate to O.h 2 log .1= h//. Various implications are discussed, including the accuracy of these methods for steady fluid flow governed by the Stokes equations. Two-fluid problems can be handled by first solving an integral equation for an unknown jump. Numerical examples are presented which confirm the analytical conclusions, although the observed error in the gradient is O.h 2 /.

The mammalian kidney is particularly vulnerable to hypoperfusion, because the O(2) supply to the renal medulla barely exceeds its O(2) requirements. In this study, we examined the impact of the complex structural organization of the rat outer medulla (OM) on O(2) distribution. We extended the region-based mathematical model of the rat OM developed by Layton and Layton (Am J Physiol Renal Physiol 289: F1346-F1366, 2005) to incorporate the transport of RBCs, Hb, and O(2). We considered basal cellular O(2) consumption and O(2) consumption for active transport of NaCl across medullary thick ascending limb epithelia. Our model predicts that the structural organization of the OM results in significant Po(2) gradients in the axial and radial directions. The segregation of descending vasa recta, the main supply of O(2), at the center and immediate periphery of the vascular bundles gives rise to large radial differences in Po(2) between regions, limits O(2) reabsorption from long descending vasa recta, and helps preserve O(2) delivery to the inner medulla. Under baseline conditions, significantly more O(2) is transferred radially between regions by capillary flow, i.e., advection, than by diffusion. In agreement with experimental observations, our results suggest that 79% of the O(2) supplied to the medulla is consumed in the OM and that medullary thick ascending limbs operate on the brink of hypoxia.

This paper concerns a class of deferred correction methods recently developed for initial value ordinary differential equations; such methods are based on a Picard integral form of the correction equation. These methods divide a given timestep [tn, tn+1] into substeps, and use function values computed at these substeps to approximate the Picard integral by means of a numerical quadrature. The main purpose of this paper is to present a detailed analysis of the implications of the location of quadrature nodes on the accuracy and stability of the overall method. Comparisons between Gauss-Legendre, Gauss-Lobatto, Gauss-Radau, and uniformly spaced points are presented. Also, for a given set of quadrature nodes, quadrature rules may be formulated that include or exclude function values computed at the left-hand endpoint tn. Quadrature rules that do not depend on the left-hand endpoint (which are referred to as right-hand quadrature rules) are shown to lead to L(α)-stable implicit methods with α ≈ π/2. The semiimplicit analog of this property is also discussed. Numerical results suggest that the use of uniform quadrature nodes, as opposed to nodes based on Gaussian quadratures, does not significantly affect the stability or accuracy of these methods for orders less than ten. In contrast, a study of the reduction of order for stiff equations shows that when uniform quadrature nodes are used in conjunction with a right-hand quadrature rule, the form and extent of order-reduction changes considerably. Specifically, a reduction of order to O( 2 ) is observed for uniform nodes as opposed to O( ∆t) for non-uniform nodes, where ∆t denotes the time step and a stiffness parameter such that → 0 corresponds to the problem becoming increasingly stiff. (2000): 65B05. AMS subject classification

Single-nephron proximal tubule pressure in spontaneously hypertensive rats (SHR) can exhibit highly irregular oscillations similar to deterministic chaos. We used a mathematical model of tubuloglomerular feedback (TGF) to investigate potential sources of the irregular oscillations and the corresponding complex power spectra in SHR. A bifurcation analysis of the TGF model equations, for nonzero thick ascending limb (TAL) NaCl permeability, was performed by finding roots of the characteristic equation, and numerical simulations of model solutions were conducted to assist in the interpretation of the analysis. These techniques revealed four parameter regions, consistent with TGF gain and delays in SHR, where multiple stable model solutions are possible: 1) a region having one stable, time-independent steady-state solution; 2) a region having one stable oscillatory solution only, of frequency f1; 3) a region having one stable oscillatory solution only, of frequency f2, which is approximately equal to 2f1; and 4) a region having two possible stable oscillatory solutions, of frequencies f1 and f2. In addition, we conducted simulations in which TAL volume was assumed to vary as a function of time and simulations in which two or three nephrons were assumed to have coupled TGF systems. Four potential sources of spectral complexity in SHR were identified: 1) bifurcations that permit switching between different stable oscillatory modes, leading to multiple spectral peaks and their respective harmonic peaks; 2) sustained lability in delay parameters, leading to broadening of peaks and of their harmonics; 3) episodic, but abrupt, lability in delay parameters, leading to multiple peaks and their harmonics; and 4) coupling of small numbers of nephrons, leading to multiple peaks and their harmonics. We conclude that the TGF system in SHR may exhibit multistability and that the complex power spectra of the irregular TGF fluctuations in this strain may be explained by switching between multiple dynamic modes, temporal variation in TGF parameters, and nephron coupling.