Reduction of a model for sodium exchanges in kidney nephron

Marta Marulli; Vuk Miliši$\grave{\rm{c}}$; Nicolas Vauchelet; Marta Marulli; Vuk Miliši$\grave{\rm{c}}$; Nicolas Vauchelet

doi:10.3934/nhm.2021020

Networks and Heterogeneous Media

2021, Volume 16, Issue 4: 609-636. doi: 10.3934/nhm.2021020

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Reduction of a model for sodium exchanges in kidney nephron

1.
University of Bologna, Department of Mathematics, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
2.
Université Sorbonne Paris Nord, Laboratoire Analyse Géométrie et Applications (LAGA), CNRS UMR 7539, F-93430, Villetaneuse, France

Received: 01 June 2020 Revised: 01 April 2021 Published: 09 July 2021
Primary: 35L04, 35L60, 35L81; Secondary: 35Q92, 92C42

This work deals with a mathematical analysis of sodium's transport in a tubular architecture of a kidney nephron. The nephron is modelled by two counter-current tubules. Ionic exchange occurs at the interface between the tubules and the epithelium and between the epithelium and the surrounding environment (interstitium). From a mathematical point of view, this model consists of a 5 $\times$ 5 semi-linear hyperbolic system. In literature similar models neglect the epithelial layers. In this paper, we show rigorously that such models may be obtained by assuming that the permeabilities between lumen and epithelium are large. We show that when these permeabilities grow, solutions of the 5 $\times$ 5 system converge to those of a reduced 3 $\times$ 3 system without epithelial layers. The problem is defined on a bounded spacial domain with initial and boundary data. In order to show convergence, we use ${{{\rm{BV}}}}$ compactness, which leads to introduce initial layers and to handle carefully the presence of lateral boundaries. We then discretize both 5 $\times$ 5 and 3 $\times$ 3 systems, and show numerically the same asymptotic result, for a fixed meshsize.

Keywords:

Citation: Marta Marulli, Vuk Miliši $\grave{\rm{c}}$ , Nicolas Vauchelet. Reduction of a model for sodium exchanges in kidney nephron[J]. Networks and Heterogeneous Media, 2021, 16(4): 609-636. doi: 10.3934/nhm.2021020

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Abstract

Abbreviations: (α, β, γ, η, θ, λ, μ, a, b): Model parameters; R (t): Reliability function; MTBF (τ): the average Lifetime; PV: Photovoltaic; a-Si: Amorphous silicon; pc-Si: Polycrystalline silicon; mc-Si: Single crystalline silicon

1. Introduction

Many articles in literature have studied the degradation of photovoltaic modules when exposed to natural environments using accelerated tests to observe degradation in reality ^[1,2]. A study confirmed that after 20 years of continuous exposure a matrix of 70 polycrystalline silicon photovoltaic modules has undergone an average performance decay of 0.24% per year in a moderate subtropical climate environment ^[3,4]. Another study stated that after only one year of exposure in a tropical climate environment the electrical powers of two modules of type (a-Si) and (pc-Si) were degraded to 60% and 56% respectively of their initial values ^[5]. In addition to these results, another study has shown that some photovoltaic modules (mc-Si and pc-Si) had been degraded by 0.22% /year to 2.96% / year for the maximum power ^[6]. In the long run, the polycrystalline silicon modules have the best reliability with a degradation rate of 0.41% per year in a natural environment ^[7]. In a tropical environment (Ghana), the exposure of 14 polycrystalline silicon modules during a 19-year period recorded a degradation rate of 21% to 35% of nominal power ^[8]. The degradation is in the order of 1.2% per year for polycrystalline silicon modules and 0.8% per year for single crystalline silicon modules ^[9]. An important study that followed the degradation of 204 modules (123 mc-Si and 81 pc-Si) had revealed a degradation variance from 0 to 6% per year for exposure periods of 18 years to 24 years in a subtropical moderate environment ^[10]. In Saharan environment (southern Algeria for example) the degradation rate of polycrystalline silicon modules was very high ranging from 3.33% / year to 4.64% / year unlike mono-crystalline silicon modules which recorded a rate of 1.22% / year after 28 years of exposure ^[11,12]. Accelerated tests cannot evaluate totally the effect of natural environment on electrical and optical characteristics of a photovoltaic module ^[13]. But it is the only method to see the effect of single factor or limited number of climatic factors ^[13,14]. The return of experimental data within a period of operation in a natural environment allows to predict the lifetime and the degradation over the long term ^[15]. Our objective in this study is to search in the literature for an adequate model to simulate the reliability of photovoltaic modules (crystalline silicon) exposed in desert environments in order to probably estimate their degradation at any period of their operation. The method consists of using a genetic algorithm (artificial intelligence optimization method) to estimate the unknown parameters of the models and to check the competence of the simulation by comparing with feedbacks of experimental data.

2. Research method

Two kinds of methods in the literature are used to predict the duration of good operation and the reliability of a photovoltaic module exposed in a natural environment, the first that uses the return of experiments, or the second that utilizes accelerated tests ^[16]. In this study, we use the feedback data that are practically measured in the desert of California and in the Algerian Sahara (Adrar region), extracted from references ^[16,17,18] to estimate the lifespan of photovoltaic modules (single crystalline silicon) in these environments. In order to calculate the parameters of models we will use a genetic algorithm. The iterative stochastic genetic algorithm uses an initial population to reach an optimal solution of any problem ^[19]. The initially chosen population has evolved from generation to generation where the most suitable individuals have a great chance of breeding. This mechanism of intelligence is realized by respecting the following steps ^[20,21]:

1. Creation of an initial population

2. Assessment of individuals in population

3. Selection of adapted individuals

4. Reproduction by crossing and mutation

5. Formation of a new generation

This process is circulated until an optimal solution is obtained. Practically, we represent these steps according to the flowchart below (Figure 1):

Figure 1. Flowchart of the genetic algorithm.

DownLoad: Full-Size Img PowerPoint

Originally, the reliability concerned the high technology systems (nuclear, aerospace...) to guarantee their operational safety. Today, all areas are interested in the study of reliability to make decisions on ratio Cost / gain and to control the failure sources ^[22,23]. Reliability of a system is a quantity characterizing the safety of operation or measuring the probability of operation of an appliance according to prescribed standards (definition presented in 1962 by the Academy of Sciences). Reliability (or survival function) is expressed by:

$R\left( t \right) = {r^{ - \smallint _0^th\left( x \right)dx}}$

(1)

h(t): Instant failure rate (probability of seeing a failure in a short interval after instant t.

The average time of operation (lifetime) which is the Mean Time Before Failure (MTBF) is given by:

$MTBF = \int_0^{ + \infty } {R\left( t \right)dt}$

(2)

According to their instantaneous rates of failure, the parametric reliability models are classified in the literature as follows ^[24,25,26]:

1. Models of constant rate: Exponential model.

2. Models of monotone rate: Weibul model, gamma model, Gompertz-Makeham model, exponential Weibul model, Mix of exponential models.

3. Models of rates of a bathtub shape: Modified Weibul model, exponential power model, quadratic model, and uniform model.

4. Models of rate in bell form: Generalized Weibul model, normal model, log-normal model, log logistic model, extreme values model.

The characteristics of chosen models are presented in the table 1 below:

Table 1. Parametric models and their characteristics.

Model	Model of two parameters
Model	Reliability function	Average lifetime
Exponential model	$R\left(t \right) = {e^{ - \lambda t}}\; with\; \lambda > 0$	$MTBF = \frac{1}{\lambda }$
Weibul model	$R\left(t \right) = {e^{ - {{\left({\frac{t}{\eta }} \right)}^\beta }}}; \beta, \eta > 0$	$MTBF = \eta \mathit{\Gamma }\left({1{ + ^1}{/_\beta }} \right)$
Gamma model	$R\left(t \right) = 1 - \frac{1}{{\mathit{\Gamma }\left(\mu \right)}}\int_0^{\theta t} {{x^{\mu - 1}}{e^{ - x}}dx; \left({\mu, \theta } \right) > 0}$	$MTBF = \frac{\mu }{\theta }$
Exponential power model	$R\left(t \right) = {e^{1 - e \left({\lambda t} \right)\alpha }}\; with\alpha > 0;\lambda, \alpha > 0$	$MTBF = \int_0^{ + \infty } {{e^{1 - e \left({\lambda t} \right)\alpha }}dt}$
Normal model	$R\left(t \right) = 1 - \frac{1}{{\sqrt {2\pi } \sigma }}\int_0^{ + \infty } {{e^{ - \frac{{{{\left({x - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}dx}$	$MTBF = \mu$
Log-normal model	$R\left(t \right) = 1 - \frac{1}{{\sqrt {2\pi } \sigma }}\int_{ - \infty }^{Int} {{e^{ - \frac{{{{\left({x - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}dx}$	$MTBF = {e^{\mu + \frac{{{\sigma ^2}}}{2}}}$
Log logistic model	$R\left(t \right) = \frac{{{\alpha ^\beta }}}{{{\alpha ^\beta } + {t^\beta }}}\; or\; \alpha > 0;\beta > 1$	$MTBF = \int_0^{ + \infty } {\frac{{{\alpha ^\beta }}}{{{\alpha ^\beta } + {t^\beta }}}dt}$
Uniform model	$R\left(t \right) = \frac{{b - t}}{{b - a}}; for\; t \in \left[{a, b} \right]$	$MTBF = \int_a^b {\frac{{b - t}}{{b - a}}} dt$
Extreme values model	$R\left(t \right) = {e^{ - \alpha \left({{e^{\beta t}} - 1} \right)}}\; with\; \alpha > 0, \beta > 0$	$MTBF = \int_0^{ + \infty } {{e^{ - \alpha \left({{e^{\beta t}} - 1} \right)}}dt}$
Gompertz-Makeham model	$R\left(t \right) = {e^{ - at - \frac{b}{{Inc}}\left({{c^t} - 1} \right)}}$	$MTBF = \int_0^{ + \infty } {{e^{ - at - \frac{b}{{Inc}}\left({{c^t} - 1} \right)}}dt}$
exponential Weibul model	$R\left(t \right) = 1 - {\left\{ {1 - {e^{ - {{\left({\frac{t}{\eta }} \right)}^\beta }}}} \right\}^\mu }\; or\; \eta, \beta, \mu > 0$	$MTBF = \int_0^{ + \infty } {1 - {{\left\{ {1 - {e^{ - {{\left({\frac{t}{\eta }} \right)}^\beta }}}} \right\}}^\mu }dt}$
Mix of exponential models	$\begin{array}{l} R\left(t \right) = {a_1}{e^{ - \frac{t}{{{\theta _1}}}}} + \left({1 - {a_1}} \right){e^{ - \frac{t}{{{\theta _2}}}}}; \\ {\theta _1}, {\theta _2} > 0;\; 0 < {a_1} < 1 \end{array}$	$MTBF = {a_1}{\theta _1} + \left({1 - {a_1}} \right){\theta _2}$
Modified Weibul model	$R\left(t \right) = {e^{ - {{\left({\frac{t}{\eta }} \right)}^\beta }{e^{\mu t}}}}\; with\; \left({\eta; \beta; \mu > 0} \right)$	$MTBF = \int_0^{ + \infty } {{e^{ - {{\left({\frac{t}{\eta }} \right)}^\beta }{e^{\mu t}}}}} dt$
Quadratic model	$R\left(t \right) = {e^{ - \left({\alpha t + \frac{\beta }{2}{t^2} + \frac{\gamma }{3}{t^3}} \right)}}\alpha, \gamma > 0; - 2\sqrt {\gamma \alpha } \le \beta \le 0$	$MTBF = \int_0^{ + \infty } {{e^{ - \left({\alpha t + \frac{\beta }{2}{t^2} + \frac{\gamma }{3}{t^3}} \right)}}} dt$
Generalized Weibul model	$R\left(t \right) = {e^{1 - {{\left({1 + {{\left({\frac{t}{\eta }} \right)}^\beta }} \right)}^{\frac{1}{\gamma }}}}}\; with\; \left({\eta, \beta, \gamma } \right) > 0$	$MTBF = \int_0^{ + \infty } {{e^{1 - {{\left({1 + {{\left({\frac{t}{\eta }} \right)}^\beta }} \right)}^{\frac{1}{\gamma }}}}}} dt$

| Show Table

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3. Results and discussion

After filtering we present only the cases where the calculated average error is less than 2%. This choice is solely made to limit the size of the study. The other cases are not interesting as our goal is to visualize the most adequate model having the least error.

3.1. Simulation of reliability by modified Weibul model

Estimated parameters and simulated reliability graphs are shown below (Table 2, Figures 2 and 3):

Table 2. Parameters of modified Weibul model in the two areas.

	In Adrar Sahara	In the desert of California
Parameters model (η, β, μ)	(71, 1.35, 0.03)	(49, 1.05, 0.01)
Average lifetime MTBF (years)	28.75	30.47
Uncertainty	0.0039	0.0059

| Show Table

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Figure 2. Reliability of photovoltaic module in Adrar Sahara by modified Weibul model.

DownLoad: Full-Size Img PowerPoint

Figure 3. Reliability of photovoltaic modules in the desert of California by Weibul modified model.

DownLoad: Full-Size Img PowerPoint

After 20 years of operation in Adrar region we observe that the Weibul modified model predicts a 30% degradation of starting value of electrical power for this type of photovoltaic modules while the degradation is approximately of 38% in the desert of California.

3.2. Simulation of reliability by Generalized Weibul model

In the Table 3 and Figures (4, 5) we present the estimated parameters and the simulated reliability:

Table 3. Parameters of generalized Weibul model in the two areas.

	In Adrar Sahara	In the desert of California
Parameters model (η, β, γ)	(38, 2.5, 0.5)	(77, 1.2, 0.4)
Average lifetime MTBF (years)	23.45	26.09
Uncertainty	0.0184	0.0092

| Show Table

DownLoad: CSV

Figure 4. Reliability of photovoltaic modules in Adrar Sahara by generalized Weibul model.

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Figure 5. Reliability of photovoltaic modules in the desert of California by generalized Weibul model.

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In this case the generalized Weibul model predicts a degradation of 35% in Adrar region after 20 years of operation while the degradation is approximately of 44% in the desert of California.

3.3. Simulation of reliability by exponential Weibul model

The Table 4 and the Figures 6, 7 show the estimated parameters and the simulated reliability of PV module:

Table 4. Parameters of exponential Weibul model in the two areas.

	In Adrar Sahara	In the desert of California
Parameters model (η, β, μ)	(28, 4.05, 0.66)	(37, 2.65, 0.45)
Average lifetime MTBF (years)	22.57	23.40
Uncertainty	0.0254	0.0028

| Show Table

DownLoad: CSV

Figure 6. Reliability of photovoltaic module in Adrar Sahara by exponential Weibul model.

DownLoad: Full-Size Img PowerPoint

Figure 7. Reliability of photovoltaic module in the desert of California by exponential Weibul model.

DownLoad: Full-Size Img PowerPoint

By this model the degradation is about of 38% in Adrar region after 20 years of operation while the degradation is approximately of 46% in the desert of California.

3.4. Simulation of reliability by Extreme values model

Estimated parameters and simulated reliability graphs are shown below (Table 5, Figures 8 and 9):

Table 5. Parameters of extreme values model in the two areas.

	In Adrar Sahara	In the desert of California
Parameters model (α, β)	(0.015, 0.68)	(0.02, 0.9)
Average lifetime MTBF (years)	50.09	31.99
Uncertainty	0.0081	0.0227

| Show Table

DownLoad: CSV

Figure 8. Reliability of photovoltaic modules in Adrar Sahara by extreme values model.

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Figure 9. Reliability of photovoltaic modules in the desert of California by extreme values model.

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By model of extreme values, the degradation is about of 22% in Adrar region after 20 years of operation while it is approximately of 37% in the desert of California.

3.5. Simulation of Reliability by Uniform model

In this case the estimated parameters and the simulated reliability are shown below (table 6, figure 10 and 11):

Table 6. Parameters of uniform model in the two areas.

	In Adrar Sahara	In the desert of California
Parameters model (a, b)	(1.6, 77)	(0.4, 48)
Average lifetime MTBF (years)	37.70	23.80
Uncertainty	0.0074	0.0139

| Show Table

DownLoad: CSV

Figure 10. Reliability of photovoltaic module in Adrar Sahara by uniform model.

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Figure 11. Reliability of photovoltaic modules in the desert of California by uniform model.

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Finally, the uniform model predicts a degradation of 24% in Adrar region After 20 years of operation while it is approximately of 41% in the desert of California.

These results are summarized in the following Table 7.

Table 7. Summary of results.

Model of Reliability	Average life time (years)		Error Means (%)
	in Adrar	in California
modified Weibul model	28.75	30.47	0.4
Uniform model	37.70	23.80	1.1
Generalized Weibul model	23.45	26.09	1.3
exponential Weibul model	22.57	23.40	1.4
Extreme values model	50.09	31.99	1.5
Average lifetime calculated	32.51	27.15
Average lifetime calculated	29.83 ≈ 30years

| Show Table

DownLoad: CSV

The above results indicate that:

1. The experimental data used have guided us to predict the future of solar panels operating in desert environments. We therefore believe that more return data will give us confidence in models and methods.

2. The calculated mean error that present the average relative distance of the graph from the points of comparison is generally small (especially for the modified Weibul model). These reflect the skill of the optimization method used (the genetic algorithm).

3. Extrapolation of curves in longer durations allows informing on the reliability (outside of periods of real measurements).

4. The modified Weibul model is the most adequate of the models tested to simulate the reliability of photovoltaic modules (single crystalline silicon) and to estimate their lifetimes (MTBF) in the desert environments. It predicted a duration of nearly 30 years in the desert of California and of around 29 years for the Adrar area.

5. It should be noted that the degradation of electrical power of photovoltaic modules in Californian desert is significant compared to that of Adrar region in the first step (in the initial period of 30 years).

6. These obtained results are more or less comparable to those stated in references ^[11,12] (a degradation close to 1.53% /year in this study).

4. Conclusion

It has been confirmed in this article that the modified Weibul law is the most adequate model compared to other tested models to simulate the reliability function of photovoltaic modules and estimate their lifetime while operating in desert environments (California and Adrar). Using simulation findings, an average lifespan of about 30 years has been predicted for photovoltaic modules exposed in desert regions where the maximum power of the photovoltaic module is degraded to almost 46% of its initial value. The annual rate of degradation is in the order of 1.5% / year. This obtained result is more or less comparable to those presented in the literature. The prediction results must be taken into consideration for any study of construction of solar stations in the Saharan environments.

Acknowledgment

I thank my fellow researchers in Renewable Energy Research Unit in Saharan areas (URERMS) for all given help.

Conflict of interest

The authors declare there are no conflicts of interest in this paper.

References

[1]	Human nephron number: Implications for health and disease. Pediatr. Nephrol. (2011) 26: 1529-1533.
[2]	J. S. Clemmer, W. A. Pruett, T. G. Coleman, J. E. Hall and R. L. Hester, Mechanisms of blood pressure salt sensitivity: New insights from mathematical modeling, Am. J. Physiol. Regul. Integr. Comp. Physiol., 312 (2016), R451–R466. doi: 10.1152/ajpregu.00353.2016
[3]	C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4th edition, volume 325, Berlin: Springer, 2016. doi: 10.1007/978-3-662-49451-6
[4]	A. Edwards, M. Auberson, S. Ramakrishnan and O. Bonny, A model of uric acid transport in the rat proximal tubule, Am. J. Physiol. Renal Physiol., (2019), F934–F947.
[5]	Impact of nitric-oxide-mediated vasodilation and oxidative stress on renal medullary oxygenation: A modeling study. Am. J. Physiol. Renal Physiol. (2016) 310: F237-F247.
[6]	V. Giovangigli, Z.-B. Yang and W.-A. Yong, Relaxation limit and initial-layers for a class of hyperbolic-parabolic systems, SIAM J. Math. Anal., 50 (2018), 4655–4697. doi: 10.1137/18M1170091
[7]	E. Giusti, Minimal Surfaces and Functions of Bounded Variation, volume 80., Birkhäuser/Springer, Basel, 1984. doi: 10.1007/978-1-4684-9486-0
[8]	E. Godlewski and P.-A. Raviart, Hyperbolic Systems of Conservation Laws, volume 3/4., Paris: Ellipses, 1991.
[9]	A quantitative systems physiology model of renal function and blood pressure regulation: Application in salt-sensitive hypertension. CPT Pharmacometrics Syst. Pharmacol. (2017) 6: 393-400.
[10]	The multiplication principle as the basis for concentrating urine in the kidney. Journal of the American Society of Nephrology (2001) 12: 1566-1586.
[11]	M. Heida, R. I. A. Patterson and D. R. M. Renger, Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space, J. Evol. Equ., 19 (2019), 111–152. doi: 10.1007/s00028-018-0471-1
[12]	F. James, Convergence results for some conservation laws with a reflux boundary condition and a relaxation term arising in chemical engineering, SIAM J. Math. Anal., 29 (1998), 1200–1223. doi: 10.1137/S003614109630793X
[13]	S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Commun. Pure Appl. Math., 48 (1995), 235–276. doi: 10.1002/cpa.3160480303
[14]	A computational model for simulating solute transport and oxygen consumption along the nephrons. Am. J. Physiol. Renal Physiol. (2016) 311: F1378-F1390.
[15]	Mathematical modeling of kidney transport. Wiley Interdiscip Rev. Syst. Biol. Med. (2013) 5: 557-573.
[16]	A. T. Layton and A. Edwards, Mathematical Modeling in Renal Physiology, Springer, 2014. doi: 10.1007/978-3-642-27367-4
[17]	Distribution of henle's loops may enhance urine concentrating capability. Biophys. J. (1986) 49: 1033-1040.
[18]	P. G. LeFloch, Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves, Basel: Birkhäuser, 2002. doi: 10.1007/978-3-0348-8150-0
[19]	M. Marulli, A. Edwards, V. Milišić and N. Vauchelet, On the role of the epithelium in a model of sodium exchange in renal tubules, Math. Biosci., 321 (2020), 108308, 12 pp. doi: 10.1016/j.mbs.2020.108308
[20]	V. Milišić and D. Oelz, On the asymptotic regime of a model for friction mediated by transient elastic linkages, J. Math. Pures Appl. (9), 96 (2011), 484–501. doi: 10.1016/j.matpur.2011.03.005
[21]	Hormonal regulation of salt and water excretion: A mathematical model of whole kidney function and pressure natriuresis. Am. J. Physiol. Renal Physiol. (2014) 306: F224-F248.
[22]	R. Natalini and A. Terracina, Convergence of a relaxation approximation to a boundary value problem for conservation laws, Comm. Partial Differential Equations, 26 (2001), 1235–1252. doi: 10.1081/PDE-100106133
[23]	Transport efficiency and workload distribution in a mathematical model of the thick ascending limb. Am. J. Physiol. Renal Physiol. (2012) 304: F653-F664.
[24]	B. Perthame, Transport Equations Biology, Basel: Birkhäuser, 2007.
[25]	B. Perthame, N. Seguin and M. Tournus, A simple derivation of BV bounds for inhomogeneous relaxation systems, Commun. Math. Sci., 13 (2015), 577–586. doi: 10.4310/CMS.2015.v13.n2.a17
[26]	J. L. Stephenson, Urinary Concentration and Dilution: Models, American Cancer Society, (2011), 1349–1408.
[27]	M. Tournus, Modèles D'échanges Ioniques Dans le Rein: Théorie, Analyse Asymptotique et Applications Numériques, PhD thesis, Université Pierre et Marie Curie, France, 2013.
[28]	M. Tournus, A. Edwards, N. Seguin and B. Perthame, Analysis of a simplified model of the urine concentration mechanism, Netw. Heterog. Media, 7 (2012), 989–1018. doi: 10.3934/nhm.2012.7.989
[29]	A model of calcium transport along the rat nephron. Am. J. Physiol. Renal Physiol. (2013) 305: F979-F994.
[30]	A mathematical model of the rat nephron: Glucose transport. Am. J. Physiol. Renal Physiol. (2015) 308: F1098-F1118.
[31]	A mathematical model of the rat kidney: K $^{+}$ -induced natriuresis. Am. J. Physiol. Renal Physiol. (2017) 312: F925-F950.
[32]	W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3

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通讯作者: 陈斌, bchen63@163.com

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沈阳化工大学材料科学与工程学院沈阳 110142

Networks and Heterogeneous Media

1.3 1.9

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Networks and Heterogeneous Media

Reduction of a model for sodium exchanges in kidney nephron

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Abstract

1. Introduction

2. Research method

3. Results and discussion

3.1. Simulation of reliability by modified Weibul model

3.2. Simulation of reliability by Generalized Weibul model

3.3. Simulation of reliability by exponential Weibul model

3.4. Simulation of reliability by Extreme values model

3.5. Simulation of Reliability by Uniform model

4. Conclusion

Acknowledgment

Conflict of interest

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Networks and Heterogeneous Media

Reduction of a model for sodium exchanges in kidney nephron

Related Papers:

Abstract

1. Introduction

2. Research method

3. Results and discussion

3.1. Simulation of reliability by modified Weibul model

3.2. Simulation of reliability by Generalized Weibul model

3.3. Simulation of reliability by exponential Weibul model

3.4. Simulation of reliability by Extreme values model

3.5. Simulation of Reliability by Uniform model

4. Conclusion

Acknowledgment

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

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Figures and Tables

Other Articles By Authors

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