Introduction
Mathematical Analysis
Model Results
Mathematical Modeling in the Kidney:
Characterizing a Long-Looped Nephron
Quinton Neville
Department of MSCS, St. Olaf College
October 13, 2016
Northfield Undergraduate Math Symposium
Quinton Neville, St. Olaf College
Mathematical Modeling in the Kidney
Introduction
Mathematical Analysis
Model Results
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
Quinton Neville, St. Olaf College
Mathematical Modeling in the Kidney
Introduction
Mathematical Analysis
Model Results
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
Quinton Neville, St. Olaf College
Mathematical Modeling in the Kidney
Introduction
Mathematical Analysis
Model Results
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)
Quinton Neville, St. Olaf College
Mathematical Modeling in the Kidney
Introduction
Mathematical Analysis
Model Results
Mathematical Equations
That’s riveting, but where’s the math?
Mathematical Analysis
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)
Quinton Neville, St. Olaf College
Mathematical Modeling in the Kidney
Introduction
Mathematical Analysis
Model Results
Mathematical Equations
Chloride and Flow Rate Model Equations
C : Chloride Concentration, F : Fluid Flow Rate
R : Tubular Radius, P : Permiability
∂
∂
C(x, t) = −F (t) C(x, t)
∂t
∂x
Vmax (x)C(x, t))
− πR(x)
+ P (x)(C(x, t) − Ce (x)) (Conservation)
KM + C(x, t)
πR2 (x)
F (t) = 1 + K1 tanh(K2 (Css − C(1, t − τ )))(TGF Response)
Quinton Neville, St. Olaf College
Mathematical Modeling in the Kidney
Introduction
Mathematical Analysis
Model Results
Mathematical Equations
Derivation of the Characteristic Equation
Model equations for C(x, t) and F (t) ⇓
Nondimensionalization ⇓
Linearization; C(x, t) = S(x) + D(x, t) ⇓
λt
Seperation of Variables; D(x, t) = f (x)e
⇓
Quinton Neville, St. Olaf College
Mathematical Modeling in the Kidney
Introduction
Mathematical Analysis
Model Results
Mathematical Equations
Characteristic Equation
C : Chloride Concentration, R : Tubular Radius
S : Steady State Chloride Concentration, P : Permiability
Z 1
Z
−γe−λτ 1
0
1=
R(x) · exp −
f (R, Vmax , S, P, Ce ) dy dx.
R(1) 0
x
Key Parameters
Delay : Time (τ )
Gain : Sensitivity (γ)
Solution Behavior : λ = ρ + iω (λ ∈ C)
Real Part : Growth/Decay (ρ)
Imaginary Part : Frequency (ω)
Quinton Neville, St. Olaf College
Mathematical Modeling in the Kidney
Introduction
Mathematical Analysis
Model Results
Results
Discussion
Bifurcation Diagram or Rorschach Test?
Bifurcation Diagram
10
ρ6 = 0
8
6
ρ5 = 0
ρ4 = 0
ρ3 = 0
γ
ρ2 = 0
4
2
ρn < 0
ρ1 = 0
0
0
0.1
0.2
0.3
0.4
0.5
τ
Quinton Neville, St. Olaf College
Mathematical Modeling in the Kidney
Introduction
Mathematical Analysis
Model Results
Results
Discussion
Long-Loop vs. Short-Loop Bifurcation
Long−Looped Nephron
8
8
ρ5 = 0
6
γ
ρ2 = 0
4
4
ρn < 0
2
ρ1 = 0
0
0
0.1
ρ3 > 0
6
ρ3 = 0
γ
2
ρ >0
5
Q1
ρ4 = 0
0.2
0.3
0.4
0.5
τ
Quinton Neville, St. Olaf College
0
0
ρ3 > 0
ρ5 = 0
C : IRT
ρ6 = 0
ρ4 = 0
ρ4 > 0
ρ3 = 0
ρ2 > 0
P1
ρ5 > 0
ρ2 = 0
ρ1 > 0
ρ1 = 0
ρn < 0
0.1
0.2
τ
0.3
Mathematical Modeling in the Kidney
0.4
0.5
Introduction
Mathematical Analysis
Model Results
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
Quinton Neville, St. Olaf College
Mathematical Modeling in the Kidney
Introduction
Mathematical Analysis
Model Results
Results
Discussion
Bifurcation Variance in Model Cases
Base Case (.25:1)
Case 3 (2:1)
10
10
ρ6 = 0
ρ5 = 0
ρ6 = 0
8
6
8
ρ5 = 0
ρ4 = 0
ρ3 = 0
γ
ρ3 = 0
6
γ
ρ2 = 0
4
2
ρ4 = 0
ρ2 = 0
4
ρn < 0
ρn < 0
2
ρ1 = 0
ρ1 = 0
0
0
0.1
0.2
0.3
0.4
0.5
τ
Quinton Neville, St. Olaf College
0
0
0.1
0.2
0.3
τ
Mathematical Modeling in the Kidney
0.4
0.5
Introduction
Mathematical Analysis
Model Results
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
Quinton Neville, St. Olaf College
Mathematical Modeling in the Kidney
Introduction
Mathematical Analysis
Model Results
Results
Discussion
Thank you!
Any Questions?
Slides and references are available upon request:
H. Ryu and A.T. Layton/ Math. Med. Biol. (2013)
Quinton Neville, St. Olaf College
Mathematical Modeling in the Kidney