Download Vascular impedance

醫用流體力學
Physiological Fluid Dynamics
Arterial Hemodynamics
邵耀華
台灣大學應用力學研究所
Hemodynamics is concerned with the forces
generated by the heart and the resulting motion
of blood through the cardiovascular system.
Blood flow in living animal is far from the
idealized motion of flow through smooth
cylindrical tubes. *Non-homogeneous materials.
*Viscous fluid. *Viscoelastic blood vessels.
Biophysics of the circulation.
Physicists
: Mathematicians
Physicians : Physiologists
Hemodynamics
 Physical properties of the heart and
blood vessels (Anatomy, Physiology)
 The relationship between the above
properties to the circulation of blood
 Applications of the above results to
physiological research or clinical science.
Background
 The pressure and flow
in blood vessels are
pulsatile and with
periodic waveforms.
 Arterial hemodynamics
describes the PressureFlow relationship in the
frequency domain.
Laminar vs. turbulent flow:
Steady flow (Poiseuille’s Law)
 Relation between volume flow (Q) and the
pressure drop (P) along a tube of length (L)
and inner diameter (D)
Q
Q  P D4 / L
 Poiseuille’s experiments (1846) showed that
Q =  P D4/128 L ; =1.3077 centi-poise
Note: Girard (1813), Navier (1827): Q  D3 ;
Hagen (1839): Q  D4.
Validity of Poiseuille’s Law in vivo
 Newtonian fluid
 Laminar flow
 No slip at the vascular wall
 Steady flow
 Cylindrical shape
 Rigid wall
Vascular resistance (R= P/Q)
R= P/Q (electric analogy) R = 128 L/  D4
R1
R2
R3
Re=R1+R2+R3
R1
1/Re=1/R1+1/R2+1/R3
R2
R3
Poiseuille’s Law underestimates the ratio of pressure
gradient to the flow in a blood vessel in vivo.
Hydraulic energies
 Pressure energy (Wp= P ·Vol)
 Kinetic energy (Wp= ½ Vol · v2)
 Gravitational energy (Wg= gh ·Vol)
 Total hydraulic energies
WT= (P + ½ v2+ gh ) ·Vol
Bernoulli’s Law (flow through orifices)
 Continuity
Q= A1 V1 =A2 V2
A2 ; V2
A1 ; V1
 Conservation of total hydraulic energy
P1 + ½ v12+ gh1 = P2 + ½ v22+ gh2
1
1  2
P2  P1   2  2 Q
2  A1 A2 
Implications
. if flow Q is constant: an  in radius (area)
will result in a  in flow velocity
 Resistance = 128L / D4
 Resistance to flow in a single vessel is:


increased with  viscosity and  length
decreased with  diameter to 4th power.
 For elliptical cross section
 ab
Q
P
2
2
4L a  b
3 3
Vascular Wall Properties
 Law of Laplace (wall tension, T=P D/2)
 Circumferential incremental Young’s
modulus of a thick walled isotropic elastic
cylindrical tube
2

Ri (1   )
Ro  P

R (1  2 ) 

2
2 
R  R
( Ro  Ri ) 
2
Einc
where  is the Poisson ratio, Ro the outer
radius, Ri the inner radius, R the radial
displacement and P is the pressure change,
Kelly (1994)
7.4
7.3
7.2
7.1
7
6.9
6.8
Diameter (mm)
Pressure (mmHg)
Laurent & Safar (1994)
CCA pressure-diameter
105
100
Pressure Diameter Curve
102
100
98
95
96
94
90
92
90
85
88
6.8
6.9
7
7.1
7.2
7.3
Arterial elasticity
and Pulse Wave velocity
 Moen-Korteweg equation
c
Eh
Eh

Di
2 Ri
 Modified Moen-Korteweg equation (thick wall)
Eh
c
2 Ro
2
h 
 (2  )
Ro 
3
(Bergel)
Eh
c
2 Rm
2
h 
 (2  )
Ro 
3
(Gow)
Pulse Wave Velocity (PWV)
in Vena Cava
canine’s vein
Pulsatile pressure and flow
(Electrical Analogy)
 Windkessel model



Volume Complicance C=dV/dP,
Resistance R=P/Q
The rate of outflow equals to the volume
change, Q=dV/dt
1
dP dV

RC
PdV dt

dP P
Qin  C

dt R
Pressure of windkessel declines exponentially
P(t )  Po e
 ( t / RC )
Womersley number for fundamental
harmonic in some mammals (Aorta)
Weight (kg)
Radius (cm)
HR (bpm)
2
0.017
0.035
500
1.4
Rat
0.6
0.13
350
4.3
Cat
3
0.21
140
4.4
Rabbit
4
0.23
280
6.8
Dog
20
0.78
90
13.1
Man
75
1.5
70
22.2
Ox
500
2.0
52
25.6
Species
Mouse
2 =  Re Sh =  (UD/) (f D/U)
Pulsatile pressure and flow
(Electrical Analogy)  Noordergraaf (1963)
dP
dQ

 RQ  L
dx
dt
dQ
dP

 GP  C
dx
dt
dV
dI

 R ' I  L'
dx
dt
dI
dV

 G 'V  C '
dx
dt
Longitudinal Vascular Impedance :
ZL = -(dP/dx) /Q
Input Impedance
:
Zi = P/Q
Transverse Impedance
:
Zw = P/(dQ/dx)
Pulsatile pressure and flow
(Electrical Analogy)
Hemodynamics
Pressure (P)
Flow (Q)
scaling
Electrical
dyne/cm2
Voltage (V)
volt
cm3/sec
Current (I)
ampere
Resistance (R) dyne sec/cm5 Resistance (R’)
ohm
8/R4
gm/cm4
Inductance (L’)
henry
/R2
cm5/dyne
Capacitance (C’)
farad
d(R2) /dP
Inertance (L)
Compliance (C)
Leakage (G)
cm5/dyne sec Conductance (G’)
Longitudinal Vascular Impedance :
Input Impedance
:
ohm-1
ZL = R+jL
Zi = P/Q
Analogic Electric System
d 2P
dP
dQ

RC

P

L
 RQ
2
dt
dt
dt
dQ
Pi  Pi 1  Ri Qi  Li
dt
Vi
Pi 
C
dVi
Qi  Qi 1 
dt
Governing Equations in integral form
 Transport of Mass:

dV    ( v  nˆ )dA  0

A
dt V
 Transport of Momentum:

 vdV   v( v  nˆ )dA  Fnet

A
dt V
 Transport of Energy

dQ dW
edV   e(v  nˆ )dA 


A
dt V
dt
dt
Human Circulatory System
 Fundamental Variables
Pressure、 Flow
 Geometrical Variables
Size、 Thickness 、
Length、 Curvature
 Mechanical Properties
Stiffness 、 ViscoElasticilty
Vascular Impedance as an index for
arterial occlusion due to atherosclerosis
dP(t ) P(t )
Q(t )  C

dt
R
Harmonic Analysis of Pulsatile
Flow Waves
Harmonic Amplitude of Flow Wave
Harmonic Analysis of Pulsatile
Pressure Waves
Vascular Impedance
• Vascular impedance characterize the resultant
of interactions of cardiac output with various
organs and tissues
• Input Impedance:
Zi = P / Q
• Longitudinal Impedance: Zl = (-dP/dx)/Q
(correlation of pressure gradient to the flow)
• Transverse Impedance:
Zt = P / (-dQ/dx)
(correlation of pressure to flow gradient)
Quantification of Input Impedance
(Z=P/Q)
Aortic Input Impedance
• Resonance
Frequency
• Impedance
Matching
• Flow
Distribution
Effects of exercise on
arterial pressure-flow waveforms
EKG
0.58 sec
0.83 sec
Carotid a. pressure
Carotid a. flow
Radial a. pressure
Radial a. flow
Dorsalis Pedis a. pressure
Dorsalis Pedis a. flow
Effects of Exercise on Doppler Spectra
Vascular impedance
 Vascular impedance gives the changes
in harmonic amplitudes thus provides
more information than typical clinical
indexes such as PI and RI !
 Can it be accessed non-invasively ?
 Color Duplex Sonography !
Non-invasive Impedance Measurements
Ultrasound M-Mode
Non-invasive Impedance Measurements
Ultrasound Doppler
M-Mode image processing
D=P
(Elastic)
=D2/2Eh
Doppler Image Processing
Pressure-Flow waveforms
頸動脈阻抗頻譜分析結果
Resonance
Measurement of Mechanical Properties for
Blood Vessel and Soft-Tissues
Correlation of Waves
measured at Two Sites
x
PWV 
t
PWV 
Eh
D
PWV of Arterial Tree
wave speed (m / sec)
10
carotid + thoracic + abdominal + femoral + dosalis pedis
8
radial artery
experimental data
[Milnor 1989]
6
4
radial
artery
carotid
artery
Arch
Thoracic abdominal Iliac
artery
artery
artery
femoral
artery
dosalis
pedis
artery
Significance of
Aortic Impedance
clinical risk factors for
developing foot ulceration
Avolio, A et al. (1994) Circulation
Comparative differences in changes
in oscillatory and steady components
of arterial hemodynamics in the early
stages of cardiac failure in dogs
The importance of the pulsatile
arterial function on the heart but also
show that these occur before changes
in peripheral resistance
 Major change in the paced dogs is
an increase in aortic impedance
(i.e, characteristic impedance)
average characteristic impedance is
shown to increase from 121 to 186
dyne- s- cm-5 (an increase of 54%)
 Pacing also produced a
significant fall in
mean arterial pressure
a decrease in mean pressure from
a baseline of 90 to 75 mm Hg after
pacing (a reduction of 17%).
 implications in understanding the
adaptive changes both in the heart
and in the complex arterial load
significance of passive effects on the
aortic impedance in relation to the
changes in arterial compliance
 a reduction in mean arterial pressure
should lead to a decrease in
characteristic impedance due to
passive effects of distending
pressure
 vascular compliance may have
actually decreased to a much greater
degree than that determined by
characteristic impedance
Active effects of vascular tone or structural
changes, with ensuing speculations
involving the distribution of angio-tensin II
receptors throughout the arterial tree.
The increase in aortic characteristic
impedance can be almost totally explained
by the passive effect of reduction in aortic
diameter due to the decrease in mean
pressure
A. Avolio (1994)
Water hammer formula
where
Zc (characteristic impedance),
 (blood density),
c (wave velocity),
A (lumen area), and R (radius)
Moens-Korteweg relation
where
 (blood density),
E (Young's modulus)
h (arterial wall thickness)
R (radius)
It is unlikely that during the 48 hours of
pacing, would resulted in substantial
structural changes in the material of the
aortic wall to cause changes in E or h as
well as blood density
where K is a constant.
Conclusion: the significant changes
that occur in the aortic impedance
spectrum in the canine model of
ventricular failure relate more to distal
effects, such as wave reflection,
caused by changes in caliber of prearteriolar vessels, possibly reflex
mediated, rather than substantial
changes in proximal conditions such
as characteristic impedance.
Development and Validation
of a Noninvasive Method to
Determine Arterial Pressure
and Vascular Compliance
Brinton, T.(1997) American Journal of Cardiology
The ability not only to record
automated systolic and diastolic
pressure, but also to derive
measurements of the rate of pressure
change during the cardiac cycle,
would have great potential clinical
value.
Physical model
Derivation of noninvasive pressure
measurements is based on a T tube
aorta and straight tube brachial artery,
and assumes that the systolic phase of
the supra-systolic cuff signal and the
diastolic phase of the sub-diastolic cuff
signal most closely approximate systolic
and diastolic aortic pressures,
respectively.
Brachial artery (BA) is assumed to be in close
proximity to the central T tube aorta which
minimizes the influence of reflective waves and
blood flow.
BA is a flexible tube filled with blood that allows
expansion and compression in the radial
direction, perpendicular to arterial wall, and
allows displacement of blood along the length of
the artery.
Physical model of the brachial artery
Assuming steady state conditions at MAP: where
Volume (V) = Ao Lc,
cross section (A) = Ao,
effective cuff length (Lc) = cuff width/[radical] 2
Graphic display of oscillometric signal
derived using a cuff sphygmomanometer
silicone membrane pressure transducer
8-bit A/D converter
• under dynamic conditions due to
the pulse cycle;
• P(t) = MAP + [Pin(t)-Pout(t)] = MAP + P(t),
where P(t): instantaneous pressure.
• Change in artery volume, dV/dt, may
be separated into a perpendicular
and parallel component:
• During the systolic cycle, dV||/dt is mainly
dependent on the blood flow which is at a
minimum.
• Therefore, only dV/dt is contributing to
the determination of arterial elasticity or
compliance:
• Where dV/dt is a positive change in arterial
volume due to a positive change in pressure with
respect to MAP, expanding the walls of the artery in
the radial direction (systolic cycle).
• Assuming elasticity, Ke and compliance
(C) = 1/Ke, which are dependent on the
structural properties of the artery, the
change in pressure, and corresponding
response of the artery, can be related.
Normalized pressure pulse from
the oscillometric signal
• A geometric weight factor was used to
normalize the pressure distribution for the
perpendicular component of the cylindrical
artery surface area
• S = 2  Ro Lc; (2 cuff bladder size )
• ST = 2  Ro2 + 2  Ro Lc.
To solve for compliance, C, we assumed a
sinusoidal pulse wave to approximate the
frequency /wave length relation for the systolic
cycle.
The maximum (peak positive) and minimum (peak
negative) dP/dt and dA/dt occur at the "zero" point,
• Assuming negligible influence from external
pressure during the subdiastolic pressure phase,
the cross-section variation dA(t), can be defined
as:
• Assuming R(t) = Ro sin(2 pi f t), where f is the
frequency of the pulse. Therefore:
• The pulse frequency of the human
cardiovascular system, f, is the frequency
of the systolic wave, 1/[2 tpp] in which tpp
is the time interval from (dP/dt)max to
(dP/dt)min. The peak-to-peak amplitude
[dP/dtMax - dP/dtMin] was defined as
(dP/dt)pp, and obtained from the systolic
wave.
Brachial artery diameter, Do, was generated by a
mathematical model in which the average size of
the vessel in an adult at MAP was scaled for body
surface area using the height and weight of each
patient. Do = 2 Ro.:
Comparisons of pressures obtained
noninvasively and by intraarterial cannulation
Measurements of the
rate of pressure change
Reproducibility of blood pressure and
vascular compliance measurements
Discussions
• This study demonstrate that this method
provides measurements of vascular
compliance from analysis of the oscillometric
signal.
• The data demonstrate a good correlation with
compliance measurements obtained from the
catheterization laboratory.
Discussions
• Because both hypertension and
atherosclerosis may be associated with
structural and functional vascular modifications,
• The compliance calculations should be of
value in the early detection and management
of these and related diseases
Discussions
• Since anti-hypertensive medications such as
alpha blockers and vasodilators affect vascular
tone, and angiotensin-converting enzyme
inhibitors may alter arterial structure, vascular
compliance should be useful in evaluating the
effectiveness of treatment.
Discussions
• These compliance measurements may also be
useful in evaluating the role of serum lipid
levels on structural changes in the arterial wall
and the effectiveness of lipid-lowering
treatment.
Limitations to this study
• First, measurements of vascular compliance
from the brachial artery were compared with
systemic measurements. However, that the
arterial pulse differs slightly between brachial
artery and aorta, a difference that likely
influenced the correlation between
noninvasive and invasive data.
Limitations to this study
• the volume of the brachial artery segment at
MAP was not measured, but rather estimated.
Direct evaluation of the artery segment volume
may have minimally altered our findings.
Concluding Remarks
• the measurement of absolute pressure and
rate of pressure change show good correlation
with catheter data and that vascular
compliance can be reliably assessed by this
new method.
• The technology should provide a valuable
noninvasive tool for the assessment of both
cardiac function and vascular properties.