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Stochastic Modeling of Coupled Nephrons

Stochastic Modeling of Coupled Nephrons. Saziye Bayram * Bruce E. Pitman ** * SUNY-Buffalo State College ** SUNY-University at Buffalo. Overview. Anatomy and Physiology of the Kidney Structural Anatomy and Physiology of Nephrons Tubuloglomerular Feedback (TGF) Mechanism

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Stochastic Modeling of Coupled Nephrons

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  1. Stochastic Modeling of Coupled Nephrons Saziye Bayram* Bruce E. Pitman** *SUNY-Buffalo State College **SUNY-University at Buffalo

  2. Overview • Anatomy and Physiology of the Kidney • Structural Anatomy and Physiology of Nephrons • Tubuloglomerular Feedback (TGF) Mechanism • Experimental Findings • Earlier Mathematical Models of Nephron’s TGF Mechanism • Stochastic Models of Nephron’s TGF Mechanism • Goals and Physiological Relevance

  3. KIDNEYS • Filter waste materials out of the blood & eliminate them as urine from the body. • Homeostatic (regulates, balance the state) devices of the body. • Single human kidney consists of ~106 nephrons. • Filter 180-200 liters of blood daily. Figure:www.nlm.nih.gov/.../ ency/fullsize/8819.jpg

  4. Anatomy of Kidney Figure:www.ams.sunysb.edu/.../ SCICOMP/Kidney.index.html

  5. Cortex and Nephrons Figure:www.ams.sunysb.edu/.../ SCICOMP/Kidney.index.html

  6. Structural Anatomy of a Nephron Within 24 hrs, kidneys reclaim: ~1,300 g of NaCl (~97% of Cl) ~400 g NaHCO3 (100%) ~180 g glucose (100%) ~almost all of the180 liters of water that entered the tubules (excrete ~0.5 liter only) Each nephron processes a very small fraction of the total blood flow to the kidney, typically 200-300 nl/min for a rat kidney. Figure:anatomy.iupui.edu/.../ urinaryf04/urinaryf04.html

  7. TGF System of a Nephron • Regulates tubular fluid flow of nephron by monitoring [NaCl] at MD, with a delay. • [NaCl] at MD ↑ Diameter of AA ↓ Blood flow ↓ Pressure in capillaries ↓ Rate of filtration ↓ Transit time ↑ [NaCl] at MD ↓ Macula Densa EE AA Bowman’s Capsule Glomerulus PT Figure:ccollege.hccs.cc.tx.us/. ../kidneypict.htm

  8. Schematic Diagram of a Nephron

  9. Experimental Findings(By Just et al., Cupples et al., Leyssac, Holstein-Rathlou et al., and Casellas et al.) • TGF-mediated fluid flow in normotensive rat nephron either approximates a steady state or exhibits limit cycle oscillations (LCO) (20-50 mHz). • Irregular and chaotic flow oscillations observed in hypertensive rats. • Evidence of interaction between neighboring nephron: 60-70% of nephrons occur in pairs and triples. • Sustained oscillations in one nephron can propagate to the coupled nephron. Resultant oscillations are roughly synchronous.

  10. Interaction between paired nephrons Types of coupling: A- Vascular Coupling: Electrotonic in Nature B- Hemodynamic Coupling: Pressure related • Berne, R.M., and Levy, M.M. (1996), “Principles of Physiology”, Mosby-Year Book, Inc, MO

  11. The tubular pressure oscillations of a pair of neighboring nephrons

  12. Single Nephron ODE Model:

  13. The first mathematical models of the TGF system • were deterministic. • were complex but still a simplification of the real system. • did not capture the irregularities have seen in the experiments with hypertensive rats.

  14. Deterministic to Stochastic • In reality, there are a variety of factors that change over time, and can neither be controlled nor measured but nevertheless leave their mark on the experimental output. • In this regard, we will model certain parameters as random processes of some convenient form (e.g. by adding dynamic noise).

  15. In our case: Gain, Delay, and Coupling parameters are of interest because • these are the key parameters in understanding the stability of the pressure and flow regulation in renal dynamics • these were the main bifurcation parameters and have been considered constant in the former models. • computer simulations show that oscillations in the TGF system occur if the feedback gain is above a critical value.

  16. Goals and Physiological Relevance • To include noise in models of physiological systems, to provide more realistic representation of the process under study, and to contribute to a deeper understanding of the underlying mechanisms. • As a stochastic approach, we will hypothesis that gain, delay and/or coupling parameters vary randomly with time. (In fact, the gain magnitude is influenced by a variety of influences, such as arterial blood pressure, which changes over time.) • The constructed SDE will be simulated by the Monte Carlo methods and results will be compared with the experimental data. • Will estimate one or more of the parameters that determines the dynamics of the TGF mechanism. • Will be able to estimate the physiological parameters, and perhaps help to identify the underlying mechanisms of hypertension.

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