: Characterizing a Long-Looped Nephron Quinton Neville Department of MSCS, St. Olaf College October 13, 2016 Northfield Undergraduate Math Symposium
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
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
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)
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)
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)
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
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
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 =0 0 0 0.1 0.2 0.3 0.4 0.5 τ
Results Discussion Long-Loop vs. Short-Loop Bifurcation γ 8 6 4 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 =0 0 0 0.1 0.2 0.3 0.4 0.5 τ 2 ρ n < 0 ρ 1 = 0 0 0 0.1 0.2 0.3 0.4 0.5 τ
Results Discussion Varying Lengths of Model THAL 0.125 0.5.25:1 0.25 0.5.5:1 0.5 0.5 1:1 1.0 0.5 2:1
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 =0 0 0 0.1 0.2 0.3 0.4 0.5 τ 2 ρ n <0 ρ 1 =0 0 0 0.1 0.2 0.3 0.4 0.5 τ
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
Results Discussion Thank you! Any Questions? Slides and references are available upon request: H. Ryu and A.T. Layton/ Math. Med. Biol. (2013)