Mathematical Modeling in the Kidney. Characterizing a Long-Looped Nephron

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1 : Characterizing a Long-Looped Nephron Quinton Neville Department of MSCS, St. Olaf College October 13, 2016 Northfield Undergraduate Math Symposium

2 Kidney and Nephron Tubuloglomerular Feedback (TGF) System What is this Kidney you speak of? Production of urine Functional unit: Nephron Nephron Kidney as the Neuron Brain

3 Introduction Kidney and Nephron Tubuloglomerular Feedback (TGF) System Structure of the Long-Looped Nephron G : Glomerulus, THAL : Thin Ascending Limb, TAL : Thick Ascending Limb, MD : Macula Densa

4 Kidney and Nephron Tubuloglomerular Feedback (TGF) System Tubuloglomerular Feedback (TGF) System Purpose Regulate chloride concentration Key Features of Long-Looped Model Long-looped nephrons slightly higher concentrated urine Thin ascending limb (THAL)

5 Mathematical Equations That s riveting, but where s the math? Partial Differential Model of Long-Looped Nephron Simplify PDE ODE Bifurcation Analysis of the TGF System Key Parameters of Analysis Delay : TGF time delay (τ) Gain : TGF sensitivity to signal (γ)

6 Mathematical Equations Chloride and Flow Rate Model Equations C : Chloride Concentration, F : Fluid Flow Rate R : Tubular Radius, P : Permiability πr 2 (x) C(x, t) = F (t) C(x, t) t ( x ) Vmax (x)c(x, t)) πr(x) K M + C(x, t) + P (x)(c(x, t) C e(x)) (Conservation) F (t) = 1 + K 1 tanh(k 2 (C ss C(1, t τ)))(tgf Response)

7 Mathematical Equations Derivation of the Characteristic Equation Model equations for C(x, t) and F (t) Nondimensionalization Linearization; C(x, t) = S(x) + ɛd(x, t) Seperation of Variables; D(x, t) = f(x)e λt

8 Mathematical Equations Characteristic Equation C : Chloride Concentration, R : Tubular Radius S : Steady State Chloride Concentration, P : Permiability 1 ( 1 ) 1 = γe λτ R(x) exp f(r, V R(1) max, S, P, C e ) dy dx. Key Parameters 0 Delay : Time (τ) Gain : Sensitivity (γ) Solution Behavior : λ = ρ + iω (λ C) Real Part : Growth/Decay (ρ) Imaginary Part : Frequency (ω) x

9 Results Discussion Bifurcation Diagram or Rorschach Test? 10 Bifurcation Diagram 8 ρ 6 =0 ρ 5 =0 γ 6 4 ρ 4 =0 ρ 3 =0 ρ 2 =0 2 ρ n <0 ρ 1 = τ

10 Results Discussion Long-Loop vs. Short-Loop Bifurcation γ ρ 6 =0 ρ 5 =0 ρ 4 =0 Long Looped Nephron ρ 3 =0 ρ 2 =0 C : IRT ρ 5 = 0 ρ 8 5 > 0 Q1 ρ 4 = 0 6 ρ 3 > 0 ρ 3 = 0 γ 4 P1 ρ 2 > 0 ρ 2 = 0 ρ 3 > 0 ρ 5 > 0 ρ 4 > 0 ρ 1 > 0 2 ρ n <0 ρ 1 = τ 2 ρ n < 0 ρ 1 = τ

11 Results Discussion Varying Lengths of Model THAL : : : :1

12 Results Discussion Bifurcation Variance in Model Cases 10 8 ρ 6 =0 ρ 5 =0 Base Case (.25:1) 10 ρ 6 =0 ρ 5 =0 8 ρ 4 =0 Case 3 (2:1) γ 6 4 ρ 4 =0 ρ 3 =0 ρ 2 =0 γ 6 4 ρ 3 =0 ρ 2 =0 2 ρ n <0 ρ 1 = τ 2 ρ n <0 ρ 1 = τ

13 Results Discussion But what does it all mean?! Bifurcation Analysis Higher tendency towards oscillatory solutions in Long Loop vs. Short Loop Within Long Loop, longer THAL = more stable TGF system What s Next? Numerical analysis of full model equations Verify bifurcation analysis results

14 Results Discussion Thank you! Any Questions? Slides and references are available upon request: H. Ryu and A.T. Layton/ Math. Med. Biol. (2013)