The kidney model as an inverse problem

Breinbauer, M.; Lory, Peter

doi:10.1016/0096-3003(91)90058-u

Cited by 7 publications

(2 citation statements)

References 54 publications

(108 reference statements)

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“…Thus, for example, Breinbauer (1988), and Breinbauer and Lory (1991) used an nonlinear optimization technique to compute, for a model of the renal medulla of the rabbit, a set of parameters that produced a maximum osmolality difference along the inner medulla; and Tewarson and coworkers (Kim and Tewarson, 1996;Tewarson, 1993b,a;Tewarson and Marcano, 1997) used nonlinear least square techniques to obtain model parameters that approximate tubular solute concentrations and water flow profiles along the rat renal medulla.…”

Section: Introductionmentioning

confidence: 99%

An Optimization Algorithm for a Distributed-Loop Model of an Avian Urine Concentrating Mechanism

Marcano

Layton

2006

Bull. Math. Biol.

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To better understand how the avian kidney's morphological and transepithelial transport properties affect the urine concentrating mechanism (UCM), an inverse problem was solved for a mathematical model of the quail UCM. In this model, a continuous, monotonically decreasing population distribution of tubes, as a function of medullary length, was used to represent the loops of Henle, which reach to varying levels along the avian medullary cones. A measure of concentrating mechanism efficiency - the ratio of the free-water absorption rate (FWA) to the total NaCl active transport rate (TAT) - was optimized by varying a set of parameters within bounds suggested by physiological experiments. Those parameters include transepithelial transport properties of renal tubules, length of the prebend enlargement of the descending limb (DL), DL and collecting duct (CD) inflows, plasma Na(+) concentration, length of the cortical thick ascending limbs, central core solute diffusivity, and population distribution of loops of Henle and of CDs along the medullary cone. By selecting parameter values that increase urine flow rate (while maintaining a sufficiently high urine-to-plasma osmolality ratio (U/P)) and that reduce TAT, the optimization algorithm identified a set of parameter values that increased efficiency by approximately 60% above base-case efficiency. Thus, higher efficiency can be achieved by increasing urine flow rather than increasing U/P. The algorithm also identified a set of parameters that reduced efficiency by approximately 70% via the production of a urine having near-plasma osmolality at near-base-case TAT. In separate studies, maximum efficiency was evaluated as selected parameters were varied over large ranges. Shorter cones were found to be more efficient than longer ones, and an optimal loop of Henle distribution was found that is consistent with experimental findings.

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Section: Introductionmentioning

confidence: 99%

An Optimization Algorithm for a Distributed-Loop Model of an Avian Urine Concentrating Mechanism

Marcano

Layton

2006

Bull. Math. Biol.

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“…As we have previously noted (Marcano-Velázquez and Layton, 2003; Marcano et al, 2006), one can distinguish between those model studies on the UCM that have sought to investigate sensitivity to parameters by varying one parameter at a time (e.g., Layton et al (2000), Layton and Layton (2005b), Layton et al (2004), and Wexler et al (1991)), and those that have incorporated algorithms that allow multiple parameters to vary simultaneously in an attempt to optimize a measure of model performance (e.g., Breinbauer (1988), Breinbauer and Lory (1991), Kim and Tewarson (1996), Tewarson (1993b), Tewarson (1993a), and Tewarson and Marcano (1997)). In the present study, as in our studies of the avian UCM (Marcano-Velázquez and Layton, 2003; Marcano et al, 2006), we applied the latter approach, because it allows one to identify the synergistic, and perhaps nonlinear effects of interacting parameters, which an organism may be able to adjust and coordinate to meet its functional objectives.…”

Section: Introductionmentioning

confidence: 99%

Maximum Urine Concentrating Capability in a Mathematical Model of the Inner Medulla of the Rat Kidney

Marcano

Layton

2009

Bull. Math. Biol.

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In a mathematical model of the urine concentrating mechanism of the inner medulla of the rat kidney, a nonlinear optimization technique was used to estimate parameter sets that maximize the urine-toplasma osmolality ratio (U/P) while maintaining the urine flow rate within a plausible physiologic range. The model, which used a central core formulation, represented loops of Henle turning at all levels of the inner medulla and a composite collecting duct (CD). The parameters varied were: water flow and urea concentration in tubular fluid entering the descending thin limbs and the composite CD at the outer-inner medullary boundary; scaling factors for the number of loops of Henle and CDs as a function of medullary depth; location and increase rate of the urea permeability profile along the CD; and a scaling factor for the maximum rate of NaCl transport from the CD. The optimization algorithm sought to maximize a quantity E that equaled U/P minus a penalty function for insufficient urine flow. Maxima of E were sought by changing parameter values in the direction in parameter space in which E increased. The algorithm attained a maximum E that increased urine osmolality and inner medullary concentrating capability by 37.5% and 80.2%, respectively, above base-case values; the corresponding urine flow rate and the concentrations of NaCl and urea were all within or near reported experimental ranges. Our results predict that urine osmolality is particularly sensitive to three parameters: the urea concentration in tubular fluid entering the CD at the outer-inner medullary boundary, the location and increase rate of the urea permeability profile along the CD, and the rate of decrease of the CD population (and thus of surface area) along the cortico-medullary axis.

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A mathematical modeling technique for renal counterflow systems

Lory¹

1993

Applied Mathematics Letters

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Realistic and comprehensive mathematical models of the renal concentrating mechanism lead to large Systems of nonlinear differential equations. Their complexity can be considerably reduced by combining like tubules into common structures. Then the different lengths of the combined tubules have to be modeled by the shunt flow technique. The present paper investigates the validity of this methodology.

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The kidney model as an inverse problem

Cited by 7 publications

References 54 publications

An Optimization Algorithm for a Distributed-Loop Model of an Avian Urine Concentrating Mechanism

An Optimization Algorithm for a Distributed-Loop Model of an Avian Urine Concentrating Mechanism

Maximum Urine Concentrating Capability in a Mathematical Model of the Inner Medulla of the Rat Kidney

A mathematical modeling technique for renal counterflow systems

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