Solitary Waves in Blood Vessels
•
heart sends a pressure wave in the arteries, inducing a local expansion of the vessel.
•
the deformation in the elastic vessel tube is felt as the pulse.
•
Q: why is it possible to feel the blood pulse at the wrist or ankle?
•
Q: how are blood pressure and speed of pulse related?
•
Q: how do the dicrotic pulses interact?
Pouya Bastani - APMA 935 - Spring 2008
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Modelling
Assumptions
•
blood is an incompressible fluid (constant fluid density ρ0 )
•
the artery is an infinitely long circular elastic cylinder
•
there are no viscous effects
•
the flow is 1D – ie. localized pressure increase causes radially symmetric expansion
2
Modelling
Fundamental Equations
•
conservation of mass
•
conservation of momentum
•
Newton’s second law
∂A
∂
+
(Av) = 0
∂t
∂x
∂v
∂v
1 ∂p
+v
=−
∂t
∂x
ρ0 ∂x
∂ 2 r(x, t)
ρR h l
ds = l p(x, t) ds − 2Tc sin
∂t2
„
dθ
2
«
ρR is the density of artery wall
Simplifications
•
use sin(dθ/2) ≈ dθ/2 with dθ = ds/a and A − πa2 ≈ 2πr(r − a) where a is the equilibrium
radius of the ring. Also introduce Young’s modulus
E=
•
stress
Tc
2πa
=
strain
hl 2π(r − a)
nondimensionalize
2
A → πa A
p → (Eh/2a)p
q
x → x ρR ah/2ρ0
v→v
t→
q
q
ρR a2 /Et
Eh/2aρ0
3
Modelling
Simplified Non-dimensional Equations
At + (Av)x = 0
vt + vvx = −px
p = (A − 1) + Att
Linearization
•
~ = (A, v, p) = (1, 0, 0)
linearize about equilibrium values S
Ãt + ṽx = 0
ṽt = −p̃x
p̃ = Ã + Ãtt
•
use S~0 ei(kx−ωt) as the Ansatz, and require the determinant to vanish, giving
2
ω=
k
1 + k2
!1
2
k3
∼k−
2
k1
•
k = 0 is the unset of instability
•
group velocity = 1 − k2 /2, so in the absence of nonlinearity the wave would disperse
•
3
change variables to ξ = k(x − t) and τ = k3 t implied by eik(x−t)+itk /2 and perturb about
the equilibrium solution (only even powers of k occur in the equations)
p = k2 p1 + k4 p2 + · · ·
A = 1 + k 2 A1 + k 4 A2 + · · ·
v = k 2 v1 + k 4 v2 + · · ·
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Modelling
Nonlinear Correction
•
O(k2 )
∂A
∂v
∂v
∂p
− ∂ξ1 + ∂ξ1 = 0
− ∂ξ1 + ∂ξ1 = 0
p1 = A1
this gives p1 = A1 = v1 + φ(τ ) where I take the arbitrary φ(τ ) ≡ 0
•
O(k4 )
∂A
∂A
∂v
∂ (A v ) +
2
− ∂ξ2 + ∂τ1 + ∂ξ
1 1
∂ξ = 0
∂v
∂v
∂v
∂p
− ∂ξ2 + ∂τ1 + v1 ∂ξ1 = − ∂ξ2
∂ 2A
1
p2 = A2 +
2
∂ξ
the equations can be combined to eliminate A2 , v2 , and p2 giving
∂p1
3 ∂p1
1 ∂ 3 p1
+ p1
+
=0
∂τ
2
∂ξ
2 ∂ξ 3
Korteweg-de Vries Equation
•
dropping the subscripts gives
pτ + 32 ppξ + 21 pξξξ = 0
5
Modelling
Soliton Solution
•
solitons travel unchanged in shape at speed c, so write p(ξ, τ ) = p(η) where η = ξ − cτ
3 0
1 000
0
−cp + pp + p = 0
2
2
•
for a localized solution p, pξ , pξξξ → 0 as η → ±∞, so integration constants = 0
•
integrate thrice to obtain
3 2
1 00
−cp + p + p = a1 = 0
4
2
1 3
1 0 2
1 2
− cp + p + (p ) = a2 = 0
2
4
4
2
p(τ, ξ) = 2c sech
hq
c
2 (ξ − cτ )
i
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Interpretation
Pressure Pulse
•
in the original variables
2s
chE
2
p(x, t) =
sech 4
a
k2 cρ0
x−
2ahρR
s
3
k2 cE
2
(1 + k c) t5
2
4a ρR
•
smoking → arteries become more rigid → E increases → blood pressure and speed increases
•
fat accumulation → thickness of artery wall h increases → pulse width & blood pressure increase
Pressure vs. Velocity
•
1
assuming φ(τ ) = 0, we have p1 = v1 so in the dimensional form T
L = p0 p or to leading order
r
Ehρ0
v∼p
2a
•
for the femoral artery (a large one in the muscles of the thigh) of dogs
6
3
E ≈ 1.41 × 10 dyne/cm
h
≈ 0.12
a
vmax ≈ 500 cm/s
3
ρ0 ≈ 1.05 gr/cm
so we predict that pmax ≈ 111 mmHg in close agreement with experimental value 110 mmHg
•
note: 1 dyne = 1 g · cm/s2 and 1mmHg ≈ 1333 g/(cm · s2 )
7
Analysis
Weakly Damped KdV
•
what happens to the pulse if weak (0 < 1) dissipation due to blood viscosity is included?
Ut + 32 U Ux + 12 Uxxx + U = 0
Modulation Theory
•
expecting the amplitude to decay slowly with time, take c to be slowly varying
U (x, t) = 2c(τ )V (η)
2
where V (η) = sech(η) , η =
•
q
c(τ )
0
2 (x − Ω(τ )), τ = t and Ω(τ ) = c(τ ).
substitution gives
"
0
c
c0
0
+ V +
ηV = 0
c
2c
#
•
multiply by V and integrate η ∈ (−∞, ∞)
"
#Z
∞
3c0
2
V dη = 0
+
4c
−∞
•
since the integral = 2 we get c0 + 34 c = 0 and with φ(0) = 0
»
–
4 t
3c0
−4
t
−
Ω(τ ) =
c(τ ) = c0 e 3
1−e 3
4
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Numerics
Scheme
•
Fourth Order Runge-Kutta with FFT Integrating Factor
Damped KdV
•
error = | numerical solution − asymptotic solution | averaged in space at time T = 0.1
•
error convergence is O()
•
the soliton amplitude and speed drops as it evolves (expected from the analytical solution)
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Analysis
Higher Order Corrections
•
from error convergence, can expect asymptotic expansion c(τ ) = c0 (τ ) + c1 (τ ) + · · ·
•
substitution into the equation yields
–
»
–
»
1
c1
c1
−1
0
0
−1
0
0
0
+ c0 V +
(c0 + c1 + · · ·)(1 + + · · ·)
ηV = 0
(c0 + c1 + · · ·)(1 + + · · ·)
c0
2
c0
•
multiply by V and integrate as before
•
the first correction is
•
not really a correction, since we can include c1 in c0
•
same happens with c2 , and so on → there is no polynomial correction!!!
•
can explain the apparent paradox by extending error convergence computation
c00
4
c01
=
=−
c1
c0
3
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Numerics
Two-Soliton Interaction
•
arterial pulse is dicrotic: i.e. composed of a strong and weak pulse
•
solitons don’t change shape after interaction with one another
•
they slow down or speed up afterwards depending on their relative amplitude
•
u1 : single soliton with small amplitude
u3 = u1 + u2
•
larger amplitude solitons travel faster (speed ∝ amplitude) and have smaller width
•
steepening in arterial pulse wave coincides with the increase in pulse-wave velocity
u2 : single soliton with large amplitude
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References
[1] G.L. Lamb. Elements of Soliton Theory. Wiley-Interscience Publication, John Wiley & Sons, Inc., 1980.
[2] T. Dauxois, M. Peyrard. Physics of Solitons. Cambridge University Press, 2006.
[3] S. Yomosa. “Solitary Waves in Large Blood Vessels.” Journal of the Physical society of Japan. February
1987, vol. 56, pp. 506-520.
[4] H. Demiray. “A note on the solution of perturbed Korteweg-de Vries equation.” Applied Mathematics
and Computation. November 2002, vol. 132, pp. 643-647
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