An integrated muscle mechanicfluid dynamic model of lamprey swimming

1
An integrated muscle mechanic-fluid dynamic model
of lamprey swimming

• Chia-Yu Hsu
• CCS, Mathematics, Tulane University
• Nov.8, 2009

SEAMS Workshop (Cha-Cha Days)
2
Collaborators
Avis H. Cohen and Eric Tytell University of
MarylandPhilip Holmes Princeton UniversityLex
Smits Princeton University Tyler McMillen Cal
State FullertonThelma Williams University of
LondonBoyce Griffith - NYU Lisa Fauci Tulane
University
3
Coupled System
4
Goal

• To couple these systems of neuronal networks,
  mechanical muscular and the hydro-dynamics to
  generate the lamprey swimming pattern.

Movie courtesy from Lex Smits, Megan Leftwich,
Melissa Green
5
Central Pattern Generator
6
Methods for CPG
Avis Cohen and Eric Tytell, UMD
7
Ultrastructure of vertebrate skeletal muscle
Action potential bursts - Calcium release from
SR Calcium binds to the muscle filaments ,
causing conformational changes in thick filaments
which form cross bridges force is generated .
Calcium then resequestered by SR, and muscle
relaxes.
8
Ultrastructure of vertebrate skeletal muscle
Action potential bursts - Calcium release from
SR Calcium binds to the muscle filaments ,
causing conformational changes in thick filaments
which form cross bridges force is generated .
Calcium then resequestered by SR, and muscle
relaxes.
9
Ultrastructure of vertebrate skeletal muscle
Action potential bursts - Calcium release from
SR Calcium binds to the muscle filaments ,
causing conformational changes in thick filaments
which form cross bridges force is generated .
Calcium then resequestered by SR, and muscle
relaxes.
10
Ultrastructure of vertebrate skeletal muscle
Action potential bursts - Calcium release from
SR Calcium binds to the muscle filaments ,
causing conformational changes in thick filaments
which form cross bridges force is generated .
Calcium then resequestered by SR, and muscle
relaxes.
11
Simplified Muscle Model
Passive elastic forces by Hookes Law
12
Muscular activation on isshown in red.

• 280 muscle segments on each side of body
• The muscles in the head do not contract.
• Force generated depends on the activation wave,
  length of each muscle segment, speed of
  lengthening or shortening of muscle

13
From Motoneuron to Muscle via Calcium Dynamics
The Hill muscle model produces forces according
to
This model is a modified version of that fitted
to single myotome data it incorporates nonlinear
length and velocity dependence.
Williams, Bowtell Curtin, J. Exp. Biol. 201,
869-875, 1998
14
Mathematical Model
Flow is governed by the incompressible Navier
Stokes equations (Fluid coupled with elastic
structure)
Fk is a delta function layer of force exerted
by the Kth segment of the body on the fluid.
15
Immersed Boundary Framework
Transmit f(X(t), t) to grid
f (X(t), t)
X (t?t) X (t) U(X(t),t) ?t
Solve Navier -Stokes on grid
Grid-based
U(X (t), t)
Interpolate grid velocity
C.Peskin, Acta Numerica, 2002, 479-517
16
Numerical Solution/IBAMR

• Immersed boundary adapted mesh refinement method

Three levels of mesh refinement grids
B.Griffith, et all, J.Comput. Phys. 22310-49,
2007
17
Free swimming
Movie courtesy from Cohen and Tytell
18
Simulation of steady swimming
Fq1Hz, L4pi cm Re6100, St0.53, Mech/Act0.82
19
Swimming with different activation frequencies
Fq2 Re1200 St1.23
Fq0.5 Re3800 St0.71
20
Swimming in different viscosities
Water
25 pct
Movies courtesy from Eric Tytell
21
Simulations-Swimming in different viscosities
22
Simulation-Turning

• Asymmetric muscle contraction ratio

23
3D lamprey model and simulation

• Circular links
• Lateral links
• Sinusoidal wave propagation

24
Conclusion and Future work

• Phase lag shown on the tapered lamprey
• swimming model
• Integrative 3D model with muscular calcium
  dynamics
• Characterize wake structure as function of model
  parameters

25
Acknowledgements

• Dr. Boyce Griffith
• (IBAMR support)
• Dr. Hideki Fujioka
• (Parallel computation support )
• Center for Computational Science, Mathematics,
  Tulane University
• Louisiana Optical Network Initiative (LONI)
• Grant support by NSF DMS-0652775 and
• NIH(CRCNS)RO1NS05427102.

26
Thank You!
27
Coupled System-See You in APS