Download Analysis of Coronary Circulation: A Bioengineering Approach

Analysis of Coronary
Circulation: A
Bioengineering Approach
Ghassan S. Kassab
Dept. of Bioengineering, UC San Diego
Presented by M.S. Ju
1. Introduction
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Coronary blood circulation: supply O2
and nutrients to heart; remove waste
products.
Hemodynamics characteristics
– Phasic arterial inflow & venous outflow
– Spatial & temporal flow heterogeneity
– Vascular compliance & zero-flow pressure
– autoregulation
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Modeling of steady-state blood flow in
diastolic, maximally vasodilated state of
coronary vasculature
Effects of systole, vasoactivity & timevarying boundary conditions
Bioengineering approach
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Vascular geometry & branching pattern
Mechanical properties
Rheology of blood
Apply Physical laws to get governing equations &
boundary conditions
Goals
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Review morphometric data of coronary
vasculature & its hemofynamic
applications
Mechanical properties of coronary
vessels
New pressure-flow relationship by
considering interaction of blood flow &
vessel elasticity
2. Innovation in
Morphometry of Coronary
Vasculature
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Diameter-defined Strahler ordering
system
Distinction between series & parallel
vessel segments
Connectivity matrix - asymmetric
branching pattern
Longitudinal position matrix – position
of daughter vessel
Applications
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Anatomy of coronary vasculature in
pig and complete set of data
Capillaries & veins in health as well as
right ventricular hypertrophy
3. Anatomy of Coronary
Vasculature
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RCA- right coronary artery
LCCA- left common coronary artery
LAD- left anterior descending artery
LCx – left circumflex artery
Cast of porcine left anterior
descending artery
1cm
Cardiac Arterioles
Venule in porcine left
ventricle
Venous flow return to
heart
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R.1 great cardiac vein – posterior vein
of left ventricle – posterior
interventricular vein – oblique of
Marshal – anterior cardiac vein –
coronary sinus – right atrium
R.2 smallest cardiac veins of
Thebesius – endocardial surface &
drains – heart chamber (right ventricle)
Branching patterns of
cardiac vessels
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Porcine coronary arteries &
endocardial veins have tree-like
branching patterns
Coronary capillary vessels has nontree-like topology
4. Mathematical description
of coronary arterial and
venous trees
Order of vessels
„ Zero – capillary
„ +1 – smallest arterioles supplying blood to
capillaries
„ -1 – smallest venules draining capillaries
„ +2 – confluent of two arterioles of order 1 (if its
diameter > diameter of order 1)
„ -2 - confluent of two venules of order 1 (if its
diameter > diameter of order 1)
„ All arterioles: 1,2,3,4,5,…, n
„ All venules: -1, -2, -3, …., -n,
Anatomy of Coronary Blood Vessels
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Connectivity matrix
Longitudinal position matrix
Measured variables
– Order no.
– Diameter
– Length
– Connectivity matrix
– Longitudinal position matrix
– Fraction of vessels connected in series
5. Mathematical description
of capillary Network
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Capillary vessels – order # 0
Capillaries fed directly by arterioles C0a
Capillaries drained into venules - C0v
Capillaries connected to C0a & C0v – C00
Connection patterns: Y, T, H, HP
(hairpin); anastomosesed via
transverse cross-connection Ccc
6. Hemodynamic
Applications of
Morphometric Data
6.1 Analysis of total crosssectional area and blood
volume
Arteriole
total cross-sectional area
An =
π
4
Dn2 N n
Venule total cross-sectional area; ellipse major axis
a, minor axis b
π
bn
An = a ( )
an
4
2
n
Total blood volume
Vn = An Ln
6.2 Pressure & flow
resistance
6.2.1 Arterial tree
connectivity & longitudinal matrices and
vascular morphometric data can be
used to do hemodynamic analyses
∵ Reynold no. & Wormsley no.are small
∴ Poiseuille’s flow can be applied
(steady-state, laminar, Newtonian, rigid
vessels)
Volumetric flow between node I & node j
Qij =
π
128
∆Pij Gij
whereGij =
Dij4 µij
Lij
Viscosity 4.0 cp for order no. 11-5 and decreases linearly
with order no. to 2.0cp in Pre-capillary arteriole
mj
∑Q
i =1
ij
= 0 ( 4)
sign convention '+' into a node, '-' out of a node
mj
∑ [P − P ]G
i =1
i
j
ij
= 0 (5a )
Boundary conditions:
At Sinus of Valsalva P = 100 mmHg
At first bifurcation of capillary network P = 26mmHg
In matrix form
GP=G’P’ (6)
G~ nxn conductance matrix
P~ 1xm column vector of unknown nodal pressure
G’P’~ boundary pressure times conductance of attached
vessels
Remarks:
Re-examination of assumptions can be checked by
Reynold’s no. & Womersly’s no.
Re =
UD
ν
U : mean flow velocity;
D : blood vessel lumen diameter;
ν : kinematic viscosity of blood
D ω
Wm =
2 ν
ω : circular frequency of pulsatile flow
heart rate 110cycle/m in
FACT: Re < 120, Wm < 1 for n < 9
Correction of loss due to bifurcation: Bernoulli’s equation
6.2.2 Capillary Network
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Topology of coronary capillary blood vessel –
not tree-like structure
Cross connection make uniform distribution of
pressure and blood flow
Capillary network is simulated based on:
geometry, branching pattern, distensibility and
blood rheology
At capillary dimension, nature of blood cells is
important and it is non-Newtonian
Viscosity is not constant - apparent viscosity
1
U −2 2
µ app = [k1 + k 2 ( ) ]
D
U ~ mean velocity of blood
D ~ diameter of capillary vessels
k1 , k 2 depends on vessel diameter, hematocrit & shear rate
experimentally determined in rigid glass tube & in vivo
6.2.3 Venous tree
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Morphology and connectivity of
venous tree are used to simulate flow
by consider
– Vascular connectivity
– Longitudinal position of bifucation
– Vascular geometry
– Distensibility
– Boundary conditions
All together, one can simulate pressure-flow
relationship of coronary circulation
Coronary veins & venules are elliptical tubes
Modification of Poiseuille’s flow is necessary
2
v
y
w
b
u
a
x
z
2
 x  y
w = 2U [1 −   −   ] (9)
a b
dP
= µ ∇2w
dz
dP
a2 + b2
= −4µU ( 2 2 ) (10)
dz
a b
dP
−
a2 + b2
dz
( 2 2 )
∴U =
a b
4µ
dP
π a 3b 3 ( − )
π a 3b 3
dz
Q = π a bU =
=−
∆P
4µ (a 2 + b 2 )
4µ L (a 2 + b 2 )
− ∆P
=
R
π a 3b 3
∴R =
note : ∆P < 0
2
2
4µ L (a + b )
7. Distensibility of
coronary vessels
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Elasticity of blood vessels is important
determinant of pressure-flow
relationship
Pressure affects vessel diameter and
blood vessel diameter control pressure
distribution (interact through B.C.)
Important features of distensibility
– Linear in physiological range
– Compliance is small, i.e. epicardial
arteries are relative rigid in diastolic state
8. Steady Laminar Flow in
an Elastic Tube
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If distensibility of blood vessels is
known, mechanics of vessel is coupled
to mechanics of blood flow to yield
pressure-flow relationship
Assumptions:
– Tube is long & slender
– Flow is laminar & steady
– Disturbance due to entry & exit are
negligible
– Deformed tube is smooth & slender
dP 128µ
Q
−
=
4
dz π D
P ~ pressure, z ~ axial coordinate, Q ~ volume flow rate
D & µ ~ Diameter & viscosity
Note : D = D(z)
In physiological range
D - D* = α ( P − P * ) (14)
where D is diameter at pressure P
dP dP dD 1 dD
=
=
(15)
dz dD dz α dz
From (13)
1 dD 128µ
Q
=
4
α dz π D
D( L)
L
∫ D dD = ∫
4
D (0)
128µ α Q
0
D ( L ) − D ( 0) =
5
5
π
dz
128µ α Q L
π
(18)
Approximation when D(L)-D(0) is small
Let D( L) = D(0) + ε
D 5 ( L ) − D 5 ( 0) = ( D ( 0) + ε ) 5 − D 5 ( 0)
≈ D 5 (0) + 5 D 4 (0)ε + 10 D 3 (0)ε 2 + L − D 5 (0)
= 5 D 4 (0)ε + 10 D 3 (0)ε 2
∴ 5 D (0)ε + 10 D (0)ε =
4
3
2
640 µ α Q L
π
Divided by 5D (0) and let ε = D(L)-D(0)
4
 2[D(L)-D(0)] 128 µ α Q L
( D(L)-D(0))1 +
=
4
D(0)
π
D
( 0)


Q D(L)-D(0) = α [P(L)-P(0)]
 2α [P(L)-P(0)] 128 µ α Q L
∴α [P(L)-P(0)]1 +
=
4
D(0)
D
π
( 0)


128 µ Q L
2α
2
∆P ) =
i.e. ∆P(1 +
(21)
4
π D (0)
D0
∴ ∆P =
- D 0 + D02 + 8 α D0 ∆Pp
⇒ ∆Pn =
4α
- D n + Dn2 + 8 α n Dn ∆Ppn
(22) Newtonian flow
4α n
Non − Newtonian blood flow

π (α ( P − P ) + D
k1 + k 2
4Q
dP 128 
−
=
4
*
dz
π
α (P − P ) + D
*
[
]
*
2



Q (23)
Normalized pressure drop v.s. log compliance (LCCA)
9. Integration of Theory &
Experiment
Interaction of anatomy, elasticity, vasoactivity, tissue/vessel interaction,
analysis & experiment
10. Concluding Remarks
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Mathematical model of coronary circulation
can be constructed based on
– physical laws
– measured data on anatomy & elasticity of blood
vessels
– muscle/vessel interaction & vasoactivity
– Rheology of blood
– Boundary conditions
10. Concluding Remarks
(cont’d)
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Model of normal hearts and
pathological state can be studied by
changing model parameters
Building model based on continuum
mechanics and using measured
geometric & elasticity data
Ad hoc hypotheses are kept to
minimum!