SIAM J. APPL. MATH.
Vol. 62, No. 3, pp. 990–1018
c 2002 Society for Industrial and Applied Mathematics
AN ANATOMICALLY BASED MODEL OF TRANSIENT
CORONARY BLOOD FLOW IN THE HEART∗
N. P. SMITH† , A. J. PULLAN† , AND P. J. HUNTER†
Abstract. An efficient finite difference model of blood flow through the coronary vessels is developed and applied to a geometric model of the largest six generations of the coronary arterial network. By constraining the form of the velocity profile across the vessel radius, the three-dimensional
Navier–Stokes equations are reduced to one-dimensional equations governing conservation of mass
and momentum. These equations are coupled to a pressure-radius relationship characterizing the
elasticity of the vessel wall to describe the transient blood flow through a vessel segment. The two
step Lax–Wendroff finite difference method is used to numerically solve these equations. The flow
through bifurcations, where three vessel segments join, is governed by the equations of conservation of mass and momentum. The solution to these simultaneous equations is calculated using the
multidimensional Newton–Raphson method. Simulations of blood flow through a geometric model
of the coronary network are presented demonstrating physiologically realistic flow rates, washout
curves, and pressure distributions.
Key words. coronary blood flow, finite difference, mathematical model
AMS subject classification. 76Z05
PII. S0036139999355199
1. Introduction. The function of the coronary network is to supply blood continuously to meet the working requirements of cardiac tissue; this is critical to the
overall function of the heart. For this reason, there have been many experimental
and modeling studies aimed at increasing the understanding of many of the factors
influencing coronary blood flow. The mathematical studies of the coronary network
have almost exclusively used distributed or lumped parameter models [16, 21]. From
these types of models it is difficult to draw conclusions about the influence of specific
anatomical or physiological features of the system. Furthermore, the development of
lumped parameter models is limited by the difficulties in assigning values to the large
number of parameters in a complex model which have, at best, limited physiological
or anatomical significance.
By integrating axial velocity, the three-dimensional Navier–Stokes equations governing blood flow can be reduced to one dimension. One-dimensional models of vascular blood flow have previously been developed to model flow through relatively simple
network geometries [26, 36, 25]. By deriving a similar set of equations, but for transient flow, combined with a computational method which can exploit recent advances
in high performance parallel computing, it is now possible to model blood flow through
a complex geometric model. The anatomically accurate model of coupled ventricular
and coronary geometry of Smith, Pullman, and Hunter [34] provides such a geometric
representation of the large vessels in the coronary network. Using this geometry as
the foundation for the discrete blood flow model developed in this study the scope of
a lumped parameter model can be limited to the small vessel microcirculation net∗ Received by the editors April 26, 1999; accepted for publication (in revised form) August 21,
2001; published electronically February 6, 2002.
http://www.siam.org/journals/siap/62-3/35519.html
† Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland,
New Zealand ([email protected], [email protected]). This work was supported by
the awards of the Auckland Doctoral, New Zealand Vice Chancellors Committee, and Auckland
UniServices Scholarships to N. P. Smith.
990
991
AN ANATOMICALLY BASED CORONARY MODEL
works which are distributed at fine resolution throughout the myocardium. Regional
variation at a spatial scale above this resolution is dominated by the geometry of the
discrete model.
The aim of the approach presented here is to provide a modeling foundation
combining transient coronary blood flow and vascular geometry which can be used to
investigate the regional and temporal variation of blood flow through the coronary
network. Ultimately the intention is to couple this work to models of myocardial
mechanics and cellular metabolism to further quantitatively investigate blood flow
and heart disease.
2. Model development.
2.1. Blood flow equations. There are a number of fundamental assumptions
about coronary blood flow used when deriving the governing equations for the model
presented here. The studies of Perktold, Resch, and Peter [27] and Cho and Kensey [10]
indicate no significant influence of the shear thinning properties of blood in large vessels, and thus blood viscosity is assumed constant and independent of vessel radius.
While the relative size of red blood cells to vessel diameter is large, blood can be
modeled as a continuum. The distensibility of a coronary vessel wall is assumed to
dominate any effects due to the compressibility of blood.
Thus, in this study, blood is modeled as an incompressible, homogeneous, Newtonian fluid. There are typically low Reynolds numbers throughout the coronary
circulation model, with a maximum ≈ 600 in a small number of the large epicardial
−1
, and ν = 3.2mm2 .s−1 and
vessel segments (where Re= dv
ν , d ≈ 6mm, v ≈ 320mm.s
Re 31 in the majority of vessels (d ≤ 1.0mm, v ≤ 100mm.s−1 ). Thus all flows are
assumed to be laminar and also axisymmetric.
The following equations use a cylindrical coordinate system (r, θ, x) with the x
axis aligned with the local vessel axial direction. The velocity in the circumferential
direction is assumed to be zero. This removes any dependency on θ within the model.
Using the above assumptions the Navier–Stokes equations, which govern Newtonian fluid flow, reduce to [15]
(2.1)
1 ∂p
∂vx
∂vx
∂vx
+ vr
+ vx
+
=ν
∂t
∂r
∂x
ρ ∂x
and
(2.2)
∂vr
1 ∂p
∂vr
∂vr
+ vr
+ vx
+
=ν
∂t
∂r
∂x
ρ ∂r
∂ 2 vx
∂ 2 vx
1 ∂vx
+
+
2
∂r
r ∂r
∂x2
∂ 2 vr
vr
1 ∂vr
∂ 2 vr
− 2+
+
2
∂r
r ∂r
r
∂x2
.
In (2.1) and (2.2), x and vx are the axial direction and velocity, respectively, and r
and vr are the radial direction and velocity. Pressure is denoted by p, viscosity by ν,
and density by ρ. Conservation of mass is governed by
(2.3)
∂vx
1 ∂ (rvr )
+
= 0.
∂x
r ∂r
Following the dimensional analysis given by Barnard et al. [2] each quantity is
nondimensionalized such that r = Rr∗ , x = λx∗ , vx = Vo vx∗ , vr = Uo vr∗ , t = Vλo t∗ , and
p = ρVo2 p∗ , where Vo and Uo are characteristic axial and radial velocities and λ is a
o
characteristic length defined by λ = RV
Uo , where R is the characteristic inner vessel
992
N. P. SMITH, A. J. PULLAN, AND P. J. HUNTER
radius. Using these nondimensional quantities (2.1) can be written as
(2.4)
∂v ∗
∂v ∗
∂vx∗
∂p∗
λν
+ vr∗ x∗ + vx∗ x∗ +
=
2
∗
∂t
∂r
∂x
∂x∗
Vo R
∂ 2 vx∗
1 ∂v ∗
U 2 ∂ 2 v∗
+ ∗ x∗ + o2 ∗2x
∗2
∂r
r ∂r
Vo ∂x
,
and (2.2) can be rearranged to become
(2.5)
2 ∗
∗
∗
∂p∗
∂ vr
U 2 ∂vr∗
ν
1 ∂vr∗
vr∗
Uo2 ∂ 2 vr∗
∗ ∂vr
∗ ∂vr
− ∗ = o2
.
+
v
+
v
−
+
−
+
r
x
∂r
Vo ∂t∗
∂r∗
∂x∗
r∗ ∂r∗
r∗2
Vo2 ∂x∗2
RUo ∂r∗2
By assuming that the radial velocity is small compared to axial velocity, terms in
∗
U2
(2.5) multiplied by V o2 can be neglected. Thus (2.5) reduces to ∂p
∂r ∗ = 0, or pressure
o
is constant across the vessel cross-section. As r∗ is independent of x∗ , (2.3) and (2.4)
can be rearranged to be
∂ (r∗ vx∗ ) ∂ (r∗ vr∗ )
+
=0
∂x∗
∂r∗
(2.6)
and
λν ∂
∂(r∗ vx∗ ) ∂(r∗ vr∗ vx∗ ) ∂(r∗ vx∗2 ) ∂(r∗ p∗ )
+
+
+
=
2
∗
∂t∗
∂r∗
∂x∗
∂x∗
Vo R ∂r
(2.7)
r∗
∂vx∗
∂r∗
.
Introducing the quantity R∗ as the nondimensionalized inner-vessel radius and integrating (2.6) and (2.7) from r∗ = 0 to r∗ = R∗ , we get
∗
R
∗
∂
∗
∗ ∗ ∗ ∂R
∗ ∗
∗
−
[r
r
v
dr
v
]
+ [r∗ vr∗ ]R = 0
(2.8)
x
x
R
∂x∗ 0
∂x∗
and
(2.9)
∂
∂t∗
0
R∗
r∗ vx∗ dr∗
∗
R
∗ ∗2 ∗ ∂R∗
∂R∗
∂
∗ ∗2 ∗
−
−
r vx R
+
r
v
dr
x
∂t∗
∂x∗ 0
∂x∗
∗
R
∗ ∗
∂p∗
λν
∗
∗ ∂vx
r
+ [r∗ vr∗ vx∗ ]R +
r∗ ∗ dr∗ =
.
∂x
Vo R∗2
∂r∗ R
0
∗
[r∗ vx∗ ]R
Since the wall is a stream surface,
(2.10)
∗
[vr∗ ]R =
∂R∗
∂R∗
∗
+
[v
]
∗
x R
∂t∗
∂x∗
or
(2.11)
∗
∂R∗ ∗ ∗2 ∂R∗
+ r vx
,
∂t∗
∂x∗
∗
[r∗ vr∗ vx∗ ]R = [r∗ vx∗ ]R
and by defining the average axial velocity to be
(2.12)
1
V = ∗2
R
∗
0
R∗
2r∗ vx∗ dr∗
AN ANATOMICALLY BASED CORONARY MODEL
993
and the nondimensionalized energy quantity α∗ as
(2.13)
1
α = ∗2 ∗2
R V
∗
0
R∗
2r∗ vx∗2 dr∗ ,
(2.8) and (2.9) can be written as
(2.14)
∗
∂(R∗2 V ∗ )
∗ ∂R
+
2R
=0
∂x∗
∂t∗
and
(2.15)
2
2
2
∗
∂(R∗ V ∗ ) ∂(α∗ R∗ V ∗ )
2λν ∗ ∂vx∗
∗2 ∂p
+
+
R
=
R
.
2
∂t∗
∂x∗
∂x∗
∂r∗ R∗
Vo R
By making the transformations R = RR∗ , α = α∗ , and V = Vo V ∗ , (2.13)
and (2.15) can be written in terms of dimensional quantities as
(2.16)
and
(2.17)
∂R
∂R R ∂V
+V
+
=0
∂t
∂x
2 ∂x
V ∂R
∂V
1 ∂p
2ν ∂vx
∂V
+ 2 (1 − α)
+ αV
+
=
.
∂t
R ∂t
∂x
ρ ∂x
R ∂r R
The above derivation eliminates vr , the radial component of velocity. However, it
requires the assumption that vx is solely a function of the radial coordinate r. This is
equivalent to specifying an axial velocity profile. Once a profile is determined, α and
the viscous term
2ν ∂vx
(2.18)
R ∂r R
can be determined in (2.17). A further simplification of the model is that this velocity
profile remains unchanged along the length of a vessel segment, which is equivalent
to setting α to be constant.
The velocity profile chosen for this model is of the form
(2.19)
vx =
r γ γ+2 V 1−
.
γ
R
Equation 2.19 defines a blunt axial velocity profile characteristic of oscillatory flow,
where γ is a constant for a particular profile. The shape of the profile defined by
(2.19) can potentially be varied between close to plug flow (most recently modeled by
Olufsen [25] using a piecewise function) and fully developed parabolic flow.
A value of γ = 9 is chosen to give a compromise fit to experimental data obtained
at different points in the cardiac cycle [15].
The form of (2.19) means that the boundary conditions of no-slip ((vx )r=R = 0)
x
and viscous axisymmetric flow ([ ∂v
∂r ]r=0 = 0) are automatically satisfied. The proγ+2
portionality constant being γ ensures that (2.12) is also satisfied. By substituting
994
N. P. SMITH, A. J. PULLAN, AND P. J. HUNTER
(2.19) into the expression that defines α, it follows that γ =
now be written in terms of α as
r 2−α
α
α−1
V 1−
(2.20)
.
vx =
2−α
R
2−α
α−1 .
Thus (2.19) can
να V
By substituting (2.20) into (2.18), the viscous term can now be calculated as −2 α−1
R2 .
Note that the case where α = 1 corresponds to a flat profile in which the no-slip
boundary condition is violated.
Equation (2.17) can be written as
(2.21)
V ∂R
∂V
1 ∂p
να V
∂V
+ 2 (1 − α)
+ αV
+
= −2
,
∂t
R ∂t
∂x
ρ ∂x
α − 1 R2
and, substituting (2.16) into (2.21), it can be further reduced to
(2.22)
∂V
V 2 ∂R 1 ∂p
∂V
να V
+ 2 (α − 1)
+
.
+ (2α − 1) V
= −2
∂t
∂x
R ∂x
ρ ∂x
α − 1 R2
Equation (2.22), along with the conservation of mass equation (2.16), provides
two equations in the following three unknowns: average velocity (V ), pressure (p),
and inner vessel radius (R). A third equation which describes the relationship between pressure and cross-sectional area must be established. This is dependent on
the mechanics of the vessel wall.
The wall of coronary blood vessels is assumed to be elastic within this model; the
transient or viscoelastic properties of the vessel wall are ignored. The approach used
has been to establish an empirical relationship between transmural pressure and the
radius (or cross-sectional area) of the form
β
R
p(R) = Go
(2.23)
−1 .
Ro
The form of (2.23) was chosen to provide a good fit to experimental pressure-radius
data with only a small number (2) of parameters. This pressure-area relationship is
similar to those proposed by Stergiopulos, Young, and Rogge [36] and Olufsen [25]
but has the advantage in the context of the current work that it can be used to derive
a steady state analytic solution. The constants Go and β, which define a particular
wall behavior, are fitted from the experimental data of Carmines, McElhaney, and
Stack [7] and extrapolated for each order of vessel using the vessel wall thickness
data of Chadwick et al. [8]. The values of Go and β for each Strahler-ordered [38]
generation of the arterial and venous vessels are included in Tables 2.1 and 2.2.
Now (2.16) and (2.22) form a set of two nonlinear, first order differential equations
relating velocity (V ), pressure (p), and radius (R) for pulsatile flow of blood, where
pressure is a function of radius via the nonlinear constitutive equation (2.23). The
system is hyperbolic, as will be shown in section 2.2. Initial conditions for solving
this set of equations are constant pressure, from which the radius is defined using
(2.23), and zero velocity. Boundary conditions are defined by prescribing pressure at
x = 0 and x = L, where L, is the segment length. The method for calculating velocity
values at the boundaries is outlined in section 2.2.
A steady state analytic solution for the flow through an elastic vessel can be derived from (2.21), (2.23), and the principle of conservation of mass. This provides a
AN ANATOMICALLY BASED CORONARY MODEL
995
Table 2.1
The parameter values for (2.23) used to model the arterial wall behavior.
Generation
11
10
9
8
7
6
Go (kPa)
7.971
7.966
10.455
11.482
13.284
14.760
β
6.469
6.473
6.617
6.717
6.894
7.051
Radius range (mm)
1.61-1.00
1.00-0.50
0.50-0.23
0.23-0.17
0.17-0.10
0.10-0.05
Table 2.2
The parameter values for (2.23) used to model the venous wall behavior.
Generation
11
10
9
8
7
6
Go (kPa)
1.462
1.498
1.996
2.256
2.762
3.251
β
4.291
4.291
4.291
4.291
4.291
4.291
Radius range (mm)
1.97-1.22
1.22-0.61
0.61-0.28
0.28-0.21
0.21-0.12
0.12-0.06
means of checking the implementation of the solution technique. By setting all transient terms to zero and substituting vessel area S = πR2 , the steady state description
of (2.21) reduces to
αV
(2.24)
να V
dV
1 dp
= −2π
+
.
dx
ρ dx
α−1 S
From the conservation of mass, given a constant flow rate Q, the velocity V =
Thus
Q
d
S
Q
dV
=α
αV
dx
S
dx
(2.25)
−αQ2 dS
.
=
S 3 dx
From the wall equation
(2.26)
p(S) = Go
S
So
β2
−1 ,
we get
(2.27)
β
Go β dS
1 dp
= 2 β S 2 −1 .
ρ dx
dx
ρSo2
Substituting (2.25) and (2.27) into (2.24) gives
(2.28)
β
να Q
−αQ2 dS
dS
2 Go β
2 −1
+
= −2π
.
β S
S 3 dx
dx
α
− 1 S2
ρSo2
Q
S.
996
N. P. SMITH, A. J. PULLAN, AND P. J. HUNTER
This can now be rearranged and integrated using the separation of variables technique
to get
−αQ2 ln S + (2.29)
β
2 Go
β
2
β
να
Qx + C,
β S 2 +2 = −2π
α−1
+ 2 ρSo2
where the constant C is calculated using the boundary condition of vessel area at the
entry point Si , where x = 0. Now an equation specifying the steady state relationship
between the vessel area S and the distance along a vessel x for a given flow rate Q
can be expressed as
β
2 Go
−αQ2 ln SSi +
β
β
ρSo2
S 2 +2 −
( β2 +2)
να
−2π α−1
Q
(2.30)
β
+2
β
2
2 Go Si
β
β
2
2 +2 ρSo
(
)
= x.
2.2. Single vessel solution procedure. The two step Lax–Wendroff finite
difference technique [28] is now applied to the equations governing flow. The first
step of the method applied to (2.16), (2.22), and (2.23) produces intermediate finite
difference representations. These are used to calculate velocity, pressure, and radius
at intermediate or half-step points. The expressions are
(2.31)
k+ 1
Vi+ 1 2
2
1
∆t 2α − 1
k
= (Vi+1 + Vi ) −
(Vi+1 + Vi ) (Vi+1 − Vi )
2
2∆x
2
+ (α − 1)
k
Vi+1 + Vi
α
− 2∆tν
,
α − 1 (Ri+1 + Ri )2
k
2
(Vi+1 + Vi )
1
(Ri+1 − Ri ) + (pi+1 − pi )
Ri+1 + Ri
ρ
(2.32)
k+ 1
Ri+ 12
2
1
∆t 1
k
= (Ri+1 + Ri ) −
(Ri+1 + Ri ) (Vi+1 − Vi )
2
2∆x 4
+
k
1
(Vi+1 + Vi ) (Ri+1 − Ri ) ,
2
(2.33)
k+ 1
pi+ 12 =
2
1
Goi+1
2


+ Goi 
k+ 1
2Ri+ 12
2
Roi+1 + Roi
β

 − 1
.
Final or full-step values are calculated with the following equations. These have
been obtained by applying the second step of the Lax–Wendroff method again to
AN ANATOMICALLY BASED CORONARY MODEL
(2.16), (2.22), and (2.23).
Vik+1
=
Vik
=
Rik
(2.34)
(2.35)
(2.36)
Rik+1
pk+1
i
997
∆t 2α − 1 Vi+ 12 + Vi− 12
Vi+ 12 − Vi− 12
−
∆x
2
2
Vi+ 12 + Vi− 12
+ (α − 1)
Ri+ 12 − Ri− 12
Ri+ 12 + Ri− 12
k+ 12
1
pi+ 12 − pi− 12
+
ρ

k+ 12
1
1
V
+
V
α 
i+ 2
i− 2

− 4∆tν
,

2 
α−1
Ri+ 12 + Ri− 12
∆t 1 Ri+ 12 + Ri− 12
Vi+ 12 − Vi− 12
−
∆x 4
k+ 12
1
V 1 + Vi− 12
Ri+ 12 − Ri− 21
,
+
2 i+ 2


β
k+1
R
i
= Goi 
− 1 .
Roi
The finite difference scheme outlined above provides equations relating the values
of pki , Rik , Vik for i = 1, N , where N is the total number of grid points defined on an
, Rik+1 , Vik+1 for i = 2, N − 1. Initial values
arterial segment, to the values of pk+1
i
= 0, Rik=0 = Ro , Vik=0 = 0.
for each grid point i when k = 0 are set such that pk=0
i
A boundary condition scheme is needed to calculate the end values pk+1
, Rik+1 , Vik+1
i
k+1
k+1
k+1
and pN , RN , VN .
If (2.23) is substituted into (2.22) to eliminate pressure, then, in combination with
(2.16), it can be written in a quasi-linear matrix form as
(2.37)
(2α − 1) V
1 0 ∂ V
+
R
0 1 ∂t R
2
2 (α − 1) VR +
2
V
βGo Rβ−1
ρRoβ
−2να V ∂ V
= α−1 R2 ,
0
∂x R
which can be cast in the general form
(2.38)
∂u
∂u
+ A(u, x)
= z(u, x).
∂t
∂x
For the purposes of examining the properties of the characteristic directions, we now
define x as a function of t, i.e. x = x(t), and u as the vector (V, R). Then
(2.39)
∂u
∂t
∂u (x(t), t)
∂u ∂u dx
=
+
.
∂t
∂t
∂x dt
Comparing (2.38) and (2.39) shows that if ∂x
∂t I = A, then along the path x(t),
= z (u, x(t)). Therefore, the value of u(x, t) is dependent only on its initial value
998
N. P. SMITH, A. J. PULLAN, AND P. J. HUNTER
and on the integration of ∂u
∂t along the path x(t). Thus x(t) defines the characteristic
paths along which information propagates in (x,t) space. The slope of x(t) or the
characteristic directions λ = dx
dt are thus calculated from A (u, x) = λI or
2
βGo Rβ−1 (2α − 1) V − λ 2 (α − 1) VR + ρR
β
o
(2.40)
= 0.
R
V −λ
2
Thus from (2.40) there are two characteristic directions specified by
1
βGo Rβ 2
2
λ = αV ± α (α − 1) V +
.
2ρRoβ
(2.41)
As α is always greater than one, there are two real values of λ, and thus (2.37)
represents a hyperbolic system. At each end of a vessel segment there is one characteristic path sloping toward the interior grid points i = 2, N − 1 [32]. Thus these
interior points are affected at the next time step by only two boundary values, one at
each end of the segment. These boundary values are generally chosen to be a pressure.
Using an elastic wall equation, radius is simply a function of pressure. However, a
method for calculating the velocity at the two segment endpoints needs to be found.
At the beginning of the vessel segment,
the
difference representation for the system
of equations centered at the point 32 , k + 12 is
(2.42)
V1k+1 + V2k+1 − V1k − V2k +
∆t
(2α − 1) V V2k+1 + V2k − V1k+1 − V1k
∆x
2
V
k+1
+ 2 (α − 1)
R2 + R2k − R1k+1 − R1k
R
1 k+1
k+1
k
k
p
+ p2 − p1 − p1
+
ρ 2
= −4∆tν
R1k+1
(2.43)
+
R2k+1
−
R1k
−
R2k
V
α
,
α − 1 (R )2
∆t 1 k+1
R V2
+
+ V2k − V1k+1 − V1k
∆x 2
k+1
k+1
k
k
+ V R2 + R2 − R1 − R1 = 0,
k+1
k+1
where V , R , and P refer to the points V3/2
, R3/2
, and pk+1
3/2 , respectively. Using an
elastic vessel wall equation, R1k+1 can be calculated directly from pk+1
. This means
1
that R2k+1 or V2k+1 can be eliminated, which reduces the error associated with including variables from outside the domain of dependence defined by the characteristic
directions.
Rearranging (2.43) such that
(2.44)
V2k+1 =
2∆x k
(R + R2k − R1k+1 − R2k+1 )
∆tR 1
2V
− (R2k+1 + R2k − R1k+1 − R1k ) + V1k+1 + V1k+1 − V2k
R
AN ANATOMICALLY BASED CORONARY MODEL
999
and now substituting (2.44) into (2.42) to eliminate V2k+1 and collecting terms produce
the following expression relating the boundary value of velocity V1k+1 to quantities
from the k and k + 1 time step which have been previously calculated:
(V )2 k+1
∆t
2α
R2 + R2k − R1k+1 − R1k
V1k+1 = V2k +
2∆x
R
1 k+1
k
−
−
p
p2 + pk2 − pk+1
1
1
ρ
(2.45)
V
α
− 2∆tν
α − 1 (R )2
k+1
1 ∆x
R1 + R2k+1 − R1k − R2k
+ + 2αV
R
∆t
2V − Rk+1 − R2k ,
R
where pk+1
is positive and is the prescribed variable and R1k+1 is obtained directly in
1
k+1
terms of p1 from
(2.46)
R1k+1 = Ro1
β1
pk+1
1
+1
.
Go1
To ensure the correct implementation of the single vessel finite difference scheme,
the steady state analytic function can be plotted and compared to the results obtained
from the numerical code. The sample problem is a single 50mm vessel segment which
has an unstressed radius of 1mm, an initial pressure of 5.6kPa (42 mmHg), and initial
velocity set to zero. From 0s to 0.1s pressure at the inflow is linearly increased to
10.6kPa (80 mmHg) and then held constant until a steady state solution is reached.
The relative fluid density is set at 1.05, viscosity ν at 3.2mm2 s, the wall elasticity
constants Go and β are 21.2kPa (158 mmHg) and 2, respectively, and the flow profile
parameter α is 1.1. The time step ∆t is 0.1ms, and the space step ∆x is 0.926mm. The
comparison is done using the flow rate Q and the entry radius calculated numerically
for a single vessel once a steady state solution has been reached. The steady state
solution for this comparison was defined as the solution at time t such that root mean
squared change in the velocity field at the set of difference points along the vessel
is less than 0.001% for the last 1000 time steps. The numerical and analytic steady
state solutions were compared for increasing values of β. This parameter is chosen
because it has the largest effect on pressure distribution and therefore also on radius
values. Figure 2.1 shows an extremely good match between the analytic solution of
(2.30) and the numerical results for all values of wall stiffness. The lower values of
wall stiffness show an increasingly nonlinear relationship between radius and distance
along the vessel because a more compliant wall has a larger change in radius for a given
−2ναV
pressure drop. The change in radius also effects the viscous term (α−1)R
2 , which is
dependent on radius and velocity and thus is lower at the left-hand end, where radius
∂p
is therefore also lower, and
is higher and velocity lower. The rate of pressure drop ∂x
the steady state solution thus shows an increasing pressure gradient down the vessel.
Increasing β is a movement toward the limiting case, where the vessel is completely
rigid (i.e., β = ∞) and where the relationship between radius and distance will be
constant.
1000
N. P. SMITH, A. J. PULLAN, AND P. J. HUNTER
Comparision of Analytic and Numerical Solutions
1.5
radius (mm)
1.4
1.3
1
numeric 1
analytic 1
numeric 2
analytic 2
numeric 3
analytic 3
numeric 4
analytic 4
1.2
2
3
4
1.1
1
0
10
20
30
40
distance along the vessel (mm)
50
60
70
Fig. 2.1. Comparison of the steady state numerical solution of radius values at grid points along
the vessel with the analytic solution calculated using (2.30). The β values used for each solution are
labelled beside each plot.
2.3. Bifurcation model. The equations governing fluid flow at bifurcations
within a network now need to be introduced in order to model blood flow through
branching structures such as the coronary network. Previously published one-dimensional bifurcation models have simulated flow through less complicated networks [31]
or focused on predicting changes in pressure and flow patterns in large vessels [37].
The following scheme has been developed as a computationally efficient way to couple
the boundary conditions of the individual vessel segments while ensuring conservation
of mass and momentum across each bifurcation. Furthermore, this new algorithm is
straightforward to parallelize and provides a way of calculating the pressure at each of
the grid points which surround a bifurcation once the single vessel scheme calculations
are complete.
The bifurcation model is based around approximating the junction as three short
elastic tubes. The tubes are assumed to be sufficiently short such that the velocity
along them is constant, and thus they are parallel-sided, and losses due to fluid viscosity are negligible. No fluid is assumed to be stored within the junction. The finite
difference grid point at the end of the parent vessel entering the junction is denoted
as a1 , and the point proximal to a1 is denoted as a2 . The grid points at the beginning
of the daughter vessels are labelled as b1 and c1 and the distal points as b2 and c2 ,
respectively. Fa1 , Fb1 , and Fc1 are the flows through each junction segment, and Po
is the pressure at the junction center.
Conservation of mass through the junction is thus governed by
(2.47)
− Fbk+1
− Fck+1
= 0.
Fak+1
1
1
1
The conservation of momentum for tube a is governed by the resultant axial force
being equal to the rate of change of momentum of fluid in a segment length la radius
Ra , i.e.,
∂ ρla πRa2 Va
2
πRa (pa − po ) =
(2.48)
.
∂t
Similar expressions can be written for tubes b and c. These are expanded using a
AN ANATOMICALLY BASED CORONARY MODEL
1001
central difference representation about the (k + 12 ) time step to give
2La k+1
Fa1 − Fak1 ,
∆t
2Lb k+1
Fb1 − Fbk1 ,
=
∆t
2Lc k+1
Fc1 − Fck1 ,
=
∆t
(2.49)
k
k+1
− pko =
pk+1
a1 + pa1 − po
(2.50)
k
pk+1
+ pko − pk+1
o
b 1 − pb 1
(2.51)
k
pk+1
+ pko − pk+1
o
c1 − pc1
where F = πR2 V and
La =
(2.52)
ρla
,
πRa2
Lb =
ρlb
,
πRb2
Lc =
ρlc
.
πRc2
For all calculations presented in this work, La , Lb , and Lc are set equal to 1 ×
10−9 . By adding (2.49)–(2.51) and then enforcing the conservation of mass (2.47), an
can be constructed in terms of pressures at the ends of the vessel
equation for pk+1
o
segments at time steps k and k + 1, namely
pk+1
o
(2.53)
=
−pko
+
k
pk+1
a1 +pa1
La
k
pk+1
pk+1 +pk
b1 +pb1
+ c1 Lc c1
Lb
1
1
1
La + Lb + Lc
+
.
By combining (2.16) and (2.23), assuming R = 0 and using the chain rule to write
∂R
dR ∂p
∂F
∂R
2 ∂V
∂t as dp ∂t and ∂x as 2πV R ∂x + πR ∂x , we get
1 ∂F dp
∂p
+
= 0.
∂t
2πR ∂x dR
(2.54)
Expanding this for each vessel segment using a central difference representation at
the k + 12 time step gives
∆t k+1
− Fak1 =0,
Fa2 + Fak2 − Fak+1
1
∆x
∆t k+1
k+1
k+1
k
k
Fb2 + Fbk2 − Fbk+1
− Fbk1 =0,
pb 1 + p b 2 − p b 1 − p b 2 − A b
1
∆x
∆t k+1
k+1
k
k
F
pk+1
+ Fck2 − Fck+1
− Fck1 =0,
c1 + pc2 − pc1 − pc2 − Ac
1
∆x c2
k+1
k
k
pk+1
a1 + pa2 − pa1 − pa2 − Aa
(2.55)
(2.56)
(2.57)
where

k+ 1
2
N− 1
2
2R
β Goa1 + Goa2  Ro
a1 +Roa2
Aa =
(2.58)
β−1

k+ 1 2πRN −21 Roa1 + Roa2
2
k+1
from (2.55)
with similar expressions for Ab and Ac . Using (2.49) to eliminate Fa1
produces the following expression:
(2.59)
pk+1
a1 =
k
k+1
pk+1
a1 + pa2 + pa2 +
Aa ∆t
∆x
Fak+1
+ Fak2 + 2Fak1 −
2
1+
Aa La ∆t2
2∆x
∆t
2La
pka1 − pk+1
− pko
o
.
1002
N. P. SMITH, A. J. PULLAN, AND P. J. HUNTER
This relates the pressure at the end of vessel segment a at time k + 1 to quantities at
interior points which have been previously calculated at time steps k and k + 12 using
k+1
the general scheme already presented. Expressions for pk+1
b1 and pc1 like (2.59), which
1
k+1
, and Fck+1
, can
consist only of information known at times k, k + 2 and Fa2 , Fbk+1
2
2
k+1
k+1
k+1
be obtained. Once pa1 , pb1 , and pc1 have been calculated, the variables for radius
and velocities are obtained from the boundary scheme ((2.42) and (2.43)). The use
of points Fak+1
, Fbk+1
, and Fck+1
introduces a small error because information lying
2
2
2
outside the domain of dependence has now been included in the solution. This error
is demonstrated by calculating the conservation of mass Fa1 − Fb1 − Fc1 for a junction
which from (2.47) should be zero. The introduced error means that this expression
quickly grows to be larger than the total flow and thus needs to be controlled.
Starting from the pressures calculated in the explicit scheme above, the error
is then reduced using a Newton–Raphson iterative scheme. The scheme seeks to
simultaneously satisfy the nonlinear system formed from (2.47) and (2.49)–(2.51),
i.e.,
(2.60)
(2.61)
(2.62)
−2 2 La Fak1 + Lb Fbk1 + pkb − pka ,
La Fak+1
+ Lb Fbk+1
=
1
1
∆t
∆t
−2 2 k+1
k+1
k+1
k+1
La Fa1 + Lc Fc1
La Fak1 + Lc Fck1 + pkc − pka ,
pa1 − pc1 −
=
∆t
∆t
k+1
k+1
Fak+1
−
F
−
F
=0.
c1
b1
1
k+1
pk+1
a1 − pb1 −
All terms in (2.60)–(2.62) either are constants that are calculated at time k or are
, Pbk+1
, and Pck+1
. Thus (2.60)–(2.62) can be approximated with
functions of Pak+1
1
1
1
a first order Taylor series which is rearranged to form a matrix equation from which
the Newton step s = (s1 , s2 , s3 ) can be calculated:


k+1
∂Fbk+1
2Lb
2La ∂Fa1
1
−1 − ∆t
0

1 − ∆t
 

∂pk+1
∂pk+1
a1
b1


k+1
k+1  s1

∂F
∂F
a1
c1
2Lc
 s2 
1 − 2La
0
−1 − ∆t


∆t
∂pk+1
∂pk+1
 s3

a1
c1
k+1


k+1
k+1
∂Fb1
∂Fa1
∂Fc1
(2.63)


−
−
k+1
k+1
k+1
∂pa1
∂pb1
∂pc1
2L k+1
 2La k+1

k
k
k
k+1
k
b
− Fa1 + ∆t Fb1 − Fb1 + pk+1
b1 + pb1 − pa1 − pa1
∆t Fa1
k+1
k+1
k
k+1
k 
a
c
=  2L
− Fak1 + 2L
− Fck1 + pk+1
c1 + pc1 − pa1 − pa1 ,
∆t Fa1
∆t Fc1
k+1
k+1
k+1
Fc1 + Fb1 − Fa1
where
∂Fak+1
1
∂pk+1
a1
∂Fak+1
1
(2.64)
∂Vak+1
1
∂pk+1
a1
∂pk+1
a1
2 ∂Vak+1
∂Rak+1
1
1
= π Rak+1
+ 2π Rak+1
Vak+1
;
1
1
1
k+1
∂pa1
∂pk+1
a1
is calculated from differentiating (2.45) with respect to pk+1
a1 :
(2.65)
∂Vak+1
1
∂pk+1
a1
is calculated using the chain rule

=
V
2
∆t 
2α
2∆x
R

∂Rak+1
1
∂pk+1
a1
1
1
− + ρ
R
−∆x
+ 2αV
∆t
∂Rak+1
1
∂pk+1
a1
−
2V ∂Rak+1
1
;
R ∂pk+1
a1
AN ANATOMICALLY BASED CORONARY MODEL
and, finally,
k+1
∂Ra
1
∂pk+1
a1
∂Fbk+1
1
∂pk+1
b
1
is obtained by differentiating (2.46) with respect to pk+1
a1 :
∂Rak+1
1
(2.66)
∂pk+1
a1
and
∂Fck+1
1
∂pk+1
c1
1003
Ro
=
βGo
( β1 −1)
pk+1
a1
+1
.
Go
are derived in a similar manner.
The Newton–Raphson iterations are now performed by solving (2.63) and updating the pressures via
(2.67)
 k+1new   k+1old   
pa1
pa1
s1
pk+1new  = pk+1old  + s2  .
b1
b1
new
old
s3
pk+1
pk+1
c1
c1
The process is then repeated calculating a new vector s = [s1 , s2 , s3 ]t with each set of
updated pressure values until the process has converged to within a specified tolerance.
The initial starting solution as calculated using (2.59) is sufficiently close to satisfying
(2.60)–(2.62) such that the scheme converges under all flow conditions tested.
A converged solution is usually achieved in less than three iterations. This method
proved to be an extremely efficient and stable way to achieve a solution that satisfied
the governing conservation equation at bifurcation points within the finite difference
grid.
2.4. Microcirculation model. The microcirculation network formed by the
arterioles, capillaries, and venules, defined for the purposes of this work as vessels
with radii less than 100µm, is both topologically and functionally different from the
network of large conduit vessels. At this spatial scale, blood increasingly can no longer
be considered a homogeneous Newtonian fluid. The flow properties are strongly influenced by the individual red blood cells it contains in suspension [30]. This affects fluid
viscosity (known as the Fahraeus effect), flow profiles, and distribution of flow at bifurcations [29]. Thus the equations used to model flow through the larger vessels in this
study are no longer valid. The large number of microcirculation networks connecting
each small artery to a small vein also makes the method of discretely modeling individual vessel segments for each microcirculation network computationally prohibitive.
To overcome these problems, a lumped parameter model of microcirculation is developed based on the intramyocardial pump of Spaan, Breuls, and Laird [35]. This is
used to reproduce the observed flow responses to arteriole and venule pressure of an
anatomically based model combining nonlinear resistive and capacitive elements. This
is a computationally efficient way of reproducing experimentally observed behavior
while maintaining some of the fundamental physics of the problem. The five element
lumped parameter model is shown schematically in Figure 2.2. Ra , Rc , and Rv are
arterial, capillary, and venule resistances, respectively, and C1 and C2 represent the
1004
N. P. SMITH, A. J. PULLAN, AND P. J. HUNTER
Ra
Rv
Rc
C1
C2
Arterioles
Capillaries
Venules
Fig. 2.2. Schematic of the lumped parameter microcirculation model.
proximal and distal capacitances of the microcirculation model.
(2.68)
(2.69)
(2.70)
(2.71)
(2.72)
pa − p 1
,
Ra
p1 − p 2
Fc =
,
Rc
p2 − p v
,
Fv =
Rv
dp1
C1
= Fa − F c ,
dt
dp2
= Fc − F v .
C2
dt
Fa =
(2.68)–(2.70) govern the flow through the resistive elements Ra , Rc , and Rv ,
and the fluid volume stored in the capacitive elements of the microcirculation bed
is described in (2.71) and (2.72). Using a central difference approximation of (2.71)
and (2.72) about the k + 12 time step and a forward difference approximation of
(2.68)–(2.70) about the k + 1 time step produces the following set of simultaneous
equations:
− pk+1
pk+1
a
1
,
Rak+1
pk+1 − pk+1
= 1 k+12 ,
Rc
− pk+1
pk+1
= 2 k+1v ,
Rv
(2.73)
Fak+1 =
(2.74)
Fck+1
(2.75)
Fvk+1
1005
AN ANATOMICALLY BASED CORONARY MODEL
(2.76)
(2.77)
pk+1
− pk1
1
= Fak+1 + Fak − Fck+1 − Fck ,
∆t
k+1
pk+1 − pk2
C2 + C2k 2
= Fck+1 + Fck − Fvk+1 − Fvk .
∆t
C1k+1 + C1k
To couple the lumped parameter model to the arterial/venous network finite difference model requires finding the pressures Pa and Pv which simultaneously satisfy
the conservation equations in both models. This is done by calculating the simultaneous solution to (2.73)–(2.77) and the boundary condition (2.45)–(2.46).
Defining the variables Ψ1 and Ψ2 to be functions of pressures pa and pv and using
(2.73)–(2.75) to eliminate p1 , p2 , and Rc , we can write (2.76) and (2.77) as
(2.78)
+ Fak+1 + Fak − Fck+1 − Fck
Ψ1 (pa , pv ) ≡Fak+1 Rak+1 − pk+1
a
+ pka − Fak Rak = 0,
(2.79)
Ψ2 (pa , pv ) ≡Fvk+1 Rvk+1 + pk+1
− Fck+1 + Fck − Fvk+1 − Fvk
v
− pkv − Rvk Fvk = 0,
∆t
C1k+1 + C1k
∆t
+ C2k
C2k+1
where
(2.80)
Fck+1 =
− Fak+1 Rak+1 − Fvk+1 Rvk+1 − pk+1
pk+1
a
v
.
Rck+1
The equations governing the variation of Ra and C1 , fitted from an anatomically
based model described below, are cast solely in terms of pa . Likewise, Rv and C2
are expressed solely in terms of pv . The boundary condition (2.45) can be used to
express Fa as a function of pa , and, likewise, Fv can be expressed solely as a function
of pv using (2.46). The equations for Fa and Fv in terms of pa and pv are the same
as those used for the bifurcation model. Thus, using (2.80), Ψ1 and Ψ2 are also solely
functions of pa and pv . The simultaneous solution to (2.78) and (2.79) is found by
iteratively applying the Newton–Raphson method. The system of equations solved at
each iteration is


∂Ψ1
∂Ψ1 −1
  ∂pk+1
∂pk+1
v
Ψ1
∆pa

 a
(2.81)
= −
.

∆pv
Ψ2
 ∂Ψ2
∂Ψ2 
∂pk+1
a
∂pk+1
v
pa and pv are then incremented to pa + ∆pa and pv + ∆pv . The derivatives in (2.81)
are calculated from
(2.82)
k+1
∂Ψ1
∆t
∂Fak+1 k+1 ∂Rak+1 k+1
∂Fa
∂Fck+1
=
−
1
+
R
+
F
+
−
a
k+1
k+1
k+1 a
k+1
k+1
k+1
∂pa
∂pa
∂pa
∂pa
∂pa
C1 + C1k
∂C1k+1
−∆t
+ Fak+1 + Fak − Fck+1 − Fck ,
2
k+1
C1k+1 + C1k ∂pa
1006
N. P. SMITH, A. J. PULLAN, AND P. J. HUNTER
(2.83)
∂Ψ1
∂Fck+1
−∆t
= k+1
,
k+1
∂pv
C1 + C1k ∂pk+1
v
(2.84)
∂Ψ2
∂Fck+1
−∆t
=
,
∂pk+1
C2k+1 + C2k ∂pk+1
a
a
k+1
∂Ψ2
∆t
∂Fvk+1 k+1 ∂Rvk+1 k+1
∂Fc
∂Fvk+1
=1 +
Rv +
F
−
−
k+1
k+1
k+1 v
k+1
k+1
k+1
∂pa
∂pv
∂pv
∂pv
∂pv
C2 + C2k
(2.85)
k+1
∂C2
−∆t
− Fck+1 + Fck − Fvk+1 − Fvk ,
2
k+1
C2k+1 + C2k ∂pv
(2.86)
k+1
k+1
∂Fck+1
k+1 ∂Ra
k+1 ∂Fa
Rck+1
=
1
−
F
−
R
a
a
k+1
k+1
∂pk+1
∂p
∂p
a
a
a
k+1
k+1
2
k+1 k+1
k+1 k+1
k+1 ∂Rc
/ Rck+1 ,
− p a − Fa R a − Fv R v − p v
k+1
∂pa
(2.87)
k+1
k+1
∂Fck+1
k+1 ∂Rv
k+1 ∂Fv
Rck+1
=
−1
−
F
−
R
v
v
∂pk+1
∂pk+1
∂pk+1
v
v
v
k+1
2
k+1 k+1
k+1 k+1
k+1 ∂Rc
/ Rck+1 .
− pk+1
−
F
R
−
F
R
−
p
a
a
a
v
v
v
k+1
∂pv
Because of the elasticity of the vessel segments, the value of the resistive and
capacitive components are dependent on blood pressure. The relation between the
value of each component and pressure is determined using an anatomically based
microcirculation model. Using the data of Kassab et al. [19, 20, 18], the topology
of the arterioles, capillary, and venule networks is reconstructed. The relationships
proposed by Chadwick et al. [8] are used to define the values and variation in vessel
distensibility, collapsibility, and recruitment with blood pressure. To account for the
shear thinning properties of blood flow in small vessels, the empirical relationship
developed by Pries et al. [29] is used to determine the value of blood viscosity as a
function of vessel size. The flow through each of the arteriole, capillary, and venule
networks is initially calculated for a given proximal and distal pressure using the
explicit technique proposed by Mayer [22] assuming fully developed Poiseuille flow.
From the pressure values in each vessel segment, the values of viscosity, radius, and
recruitment are then updated, and the process is repeated until a converged solution
is obtained. Using this method, the anatomically based model can be used to estimate
Ra , Rc , Rv , C1 , and C2 for any given pressure values. However, while accurate to
the morphology and accounting for many of the physical phenomena, they are too
computationally expensive to include in the whole organ model. For computational
efficiency, empirical relationships are fitted to the more detailed model results to
provide a more efficient way of evaluating the resistance and compliance values. Each
of the relationships is modelled using a rational polynomial of the form
(2.88)
R=
N1 + N2 q + N3 s + N4 qs + N5 q 2 + N6 s2
,
D1 + D2 q + D3 s + D4 qs + D5 q 2 + D6 s2
AN ANATOMICALLY BASED CORONARY MODEL
1007
Table 2.3
The variables associated with x and y for each component in the lumped parameter model and
the RMS error in fitting (2.88) to the physically based model.
Variable
Ra
Rc
Rv
C1
C2
Units
kPa.s.ml−1
kPa.s.ml−1
kPa.s.ml−1
ml.kPa−1
ml.kPa−1
q
Pa
P1
Pv
Pa
Pv
s
Fa
P2
Fv
P1
P2
RMS error
5.157
13.371
9.056
3.123e-05
3.615e-06
Table 2.4
The fitted nominator values for (2.88) for each component in the lumped parameter model.
Variable
Ra
Rc
Rv
C1
C2
N1
0.00051137
1.0
0.0034779
3.0104e-05
0.00020578
N2
-7.3677e-05
-0.11236
-0.00066418
3.2133e-06
1.3136e-05
N3
0.0035752
-0.083349
-0.0047076
-1.2763e-05
-4.428e-05
N4
-0.0021702
0.0055689
-0.0125
-8.6688e-07
-2.3643e-06
N5
4.2085e-06
0.0082348
7.4907e-05
6.9337e-08
5.2479e-06
N6
1.0
0.0069464
1.0
3.3129e-06
8.2425e-06
Table 2.5
The fitted denominator values for (2.88) for each component in the lumped parameter model.
Variable
Ra
Rc
Rv
C1
C2
D1
3.1192e-07
0.00043352
7.6075e-06
1.0
1.0
D2
-1.0483e-08
1.9563e-05
3.276e-07
0.069804
0.001729
D3
-3.17e-05
3.2255e-05
-0.00046939
-0.74405
-0.36117
D4
-5.0848e-07
2.4147e-06
-2.8796e-05
-0.023532
-0.0062574
D5
2.2473e-09
9.9149e-06
2.1416e-07
0.0015796
0.0443551
D6
0.0010545
1.2144e-05
0.0027147
0.16945
0.05926
where q and s are the input variables, Nn are the polynomial coefficients in the
numerator, and Dn are the coefficients in the denominator of (2.88). The values of
the constants Nn and Dn are calculated using the minimax fitting technique [28]
which seeks to minimize the maximum deviation between the two models.
The form of (2.88) provides an adequate number of degrees of freedom such that a
close approximation to the physical model can be achieved. The value and derivatives
of (2.88) are also computationally inexpensive to evaluate, and the Newton–Raphson
method converges quickly to the solution of (2.78) and (2.79). Table 2.3 presents the
variables q and s used in (2.88) to fit each component relationship and the root mean
squared (RMS) error in each fit. Tables 2.4 and 2.5 present the values of Nn and
Dn for each fit. The q and s variables used to fit Ra and Rv are the inlet pressure
and the flow rather than the inlet pressure and the outlet pressure. This is done to
avoid having to form an implicit relationship. Using the boundary conditions (2.45)
and (2.46) from the finite difference scheme, we can express Fa solely as a function of
pa . Thus Ra = Ra (pa , Fa (pa )), and
(2.89)
dRa
∂Ra
∂Ra dFa
=
+
.
dpa
∂pa
∂Fa dpa
If Ra is fitted with pa and p1 , then Ra (pa , p1 ) = Ra (pa , pa − Ra Fa ), which defines
Ra implicitly and thus becomes a more expensive relationship to evaluate. Similarly,
pv and Fv are used for fitting Rv . Once the values of Ra and Rv have been determined,
1008
N. P. SMITH, A. J. PULLAN, AND P. J. HUNTER
p1 and p2 can be calculated from
(2.90)
p 1 = p a − R a Fa
and
(2.91)
p 2 = p v + R v Fv .
Now the values of C1 , C2 , and Rc can be evaluated. The derivatives
c
and ∂R
∂pv are given by
(2.92)
dC1
∂C1
∂C1
=
+
dpa
∂pa
∂p1
(2.93)
dC2
∂C2
∂C2
=
+
dpv
∂pv
∂p2
(2.94)
∂Rc
∂Rc
=
∂pa
∂p1
(2.95)
∂Rc
∂Rc
=
∂pv
∂p2
1 − Ra
1 + Rv
1 − Ra
1 + Rv
∂Fa
∂Ra
− Fa
∂pa
∂pa
∂Fv
∂Rv
+ Fv
∂pv
∂pV
∂Fa
∂Ra
− Fa
∂pa
∂pa
∂Fv
∂Rv
+ Fv
∂pv
∂pv
dC1 dC2 ∂Rc
dpa , dpv , ∂pa ,
,
,
,
.
Each of the five elements of the lumped parameter model is now cast in a form such
that (2.78) and (2.79) can be solved using the numerical scheme outlined above.
3. Results. A finite difference grid is generated on the finite element geometric
model of the coronary vessels of Smith, Pullan, and Hunter [34]. Grid points are
spaced equally in the ξ space of each element. The number of grid points contained
in each element is such that ∆x is approximately equal throughout the finite difference
grid.
The transient coronary blood flow equations are solved using this finite difference
grid with the kinematic viscosity ν set at 3.2mm2 .s−1 [24]. The space step ∆x is set
at approximately 0.926mm such that each vessel element contains at least three grid
points. The time step ∆t is set at 0.1ms such that the finite difference scheme is both
converged and stable for this value of ∆x. Initially the pressure and velocity at each
grid point are set to zero, and thus initial radius values are equal to the unstressed
radius values Ro . Pressure at the arterial inflow is linearly increased from 0kPa (0
mmHg) to 12.63kPa (95 mmHg) (pressure at the beginning of diastole) over 0.3s and
then is held constant until a steady state flow solution is reached. Pressure at the
venous outflow is held constant at 0kPa (0 mmHg).
The change in pressure through time for the full arterial and venous tree is shown
on the six generation coronary mesh in Figures 3.1 and 3.2.
These results show that the numerical scheme is performing stably with conservation of mass and momentum and continuity of pressure across all bifurcation points.
The delay in flow transmission through the microcirculation network is because of the
capacitance in the lumped parameter model.
The high vessel compliance at zero transmural pressure means that the vessel
wave speed is relatively slow. Thus large pressure gradients are set up as the pressure
AN ANATOMICALLY BASED CORONARY MODEL
(a)
(b)
(c)
(d)
(e)
(f)
1009
Fig. 3.1. The calculated blood pressure solutions in the coronary network through time. The
surface of the ventricular endocardium of the heart model is shaded red, while the coronary network
embedded in the myocardium is exposed. The shaded vessels indicate arterial (left) and venous
(right) pressures at 100ms (a), (b), 200ms (c), (d), and 300ms (e), (f).
1010
N. P. SMITH, A. J. PULLAN, AND P. J. HUNTER
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 3.2. The calculated blood pressure solutions in the coronary network through time. The
surface of the ventricular endocardium of the heart model is shaded red, while the coronary network
embedded in the myocardium is exposed. The shaded vessels indicate arterial (left) and venous
(right) pressures at 400ms (a), (b), 500ms (c), (d), and 600ms (e), (f).
1011
AN ANATOMICALLY BASED CORONARY MODEL
at arterial inflow is quickly increased. These pressure gradients create large arterial
velocities shifting fluid from the larger vessels into the smaller vessels. Once the
pressure wave has reached the high resistance microcirculation network, the coronary
slosh phenomenon is demonstrated [13], where the velocities in the larger vessels
are reduced by the high pressures in the smaller vessels created by the increased
myocardial fluid volume. As pressure at the arterial inflow continues to increase until
0.3s, the reverse pressure gradient is overcome, and flow at the arterial inflow increases
again. When inflow pressure stops increasing, the same slosh phenomenon can be seen
again although with smaller changes in velocity.
The steady state solution is reached after 1.2s when the venous outflow equals 99%
of the arterial inflow. Figure 3.3 shows steady state pressure as a function of vessel
size comparable with those reported by Defily et al. [11] for a vasodilated coronary
network. The total steady state flow of 20ml.s−1 for the 200cm3 ventricular model
also compares well with the range of 0.075ml.s−1 g−1 to 0.15ml.s−1 g−1 reported by
Chilian [9] for similar arterial venous pressure difference. This indicates that the
global hemodynamics are modeling coronary blood flow in the absence of contraction
effectively.
Statistical Distribution of Pressures with Segment Radius
arterial pressure
venous pressure
Defily et. al.
14
pressure (kPa)
12
10
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
radius (mm)
1.2
1.4
1.6
1.8
Fig. 3.3. The mean, upper, and lower quartiles of segment pressure grouped by segment radius.
Flows through the myocardium have been reported to be spatially heterogeneous
[4, 5]. Relative dispersion (RD) is used to measure flow heterogeneity for spatial
resolution or tissue sample volume and is defined as the standard deviation of flow σf
divided by the mean myocardial flow µf into each sample measured as flow per ml
tissue, i.e.,
(3.1)
RD =
σf
.
µf
For a fractal distribution in space RD scales with tissue sample size v, according to
1012
N. P. SMITH, A. J. PULLAN, AND P. J. HUNTER
Probability Density Functions of Normalized Flow
0.2
volume=0.2073
volume=0.4144
volume=0.8289
volume=1.6579
volume=3.3156
density function of flows
0.18
0.16
0.14
ml
ml
ml
ml
ml
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
relative flow
3
3.5
4
Fig. 3.4. The probability density functions of relative flow ( ffi for different sample sizes,
av
where fi is the flow into a given tissue sample and fav is the average flow for that sample size).
Plot of Normalized RD as a Function of Normalized Tissue Volume
calculated values
fitted relationship
0.4
RD
log( RD
)
o
0.3
0.2
0.1
0
-3
-2.5
-2
Fig. 3.5. The log log plot of RD
-1.5
RD
RDo
-1
log( VVo )
-0.5
0
versus normalised sample volume
0.5
V
Vo
1
and the least squares
RD
= (1 − D) log VV . RDo is calculated at Vo equal to 3.3156ml, and
linear fit of these points log RD
o
o
D (the fractal dimension) is fitted as 1.124 within a 68% confidence interval of ±0.0128. The linear
relationship indicates that a fractal model may be appropriate.
AN ANATOMICALLY BASED CORONARY MODEL
1013
[4],
(3.2)
RD = RDo
v
vo
1−D
,
where RDo is the RD measured at reference volume vo and D is the fractal dimension.
The RD is calculated by first refining the heart host mesh along lines of constant ξ
into equal volume blocks. Regional flow into each block is then determined by adding
the flows from terminal arterial segments contained within the block.
Figure 3.4 shows the flow probability density functions for average tissue sample
volumes of 3315.6mm3 , 1657.9mm3 , 828.9mm3 , 414.4mm3 , and 207.3mm3 , where each
volume has an RD value of 1.17, 1.34, 1.41, 1.67, and 1.72, respectively.
The regional variation in flow is determined using only six of the eleven generations
of the coronary arterial network. The RD calculated is significantly higher than the
experimentally observed value; Bassingthwaighte, Liebovitch, and West [5] report an
RD of 0.135 for 1.0g tissue samples compared with the calculated RD of 1.41 for
a similar (0.8289mm3 ) volume size. The discrepancy exists because the number of
terminal arterioles feeding into a given volume is greatly reduced from that of a full
coronary network. Thus, at a given tissue sample size, the calculated RD is much more
susceptible to variation between blocks of tissue. This indicates that to model the
whole organ transport properties of the coronary network, regional distribution below
the level currently included in the six generation model may need to be accounted
for.
The relationship between RD and tissue sample size can still be calculated and
fitted to (3.2) to test the fractal behavior of the model. The log log plot of RD verses
tissue sample size in Figure 3.5 shows a near linear relationship indicating that a
fractal model with a fractal dimension D = 1.124 may be appropriate.
Literature exists detailing experiments in which tracer wash-out from the heart
has been measured [3, 33]. These provide a useful comparison between experimental
data and results from a flow simulation using the full coronary mesh which have been
run until a steady state is achieved.
When a unit impulse of a flow-limited tracer is applied at the inflow to the coronary network, the normalized dilution time curve measured at the venous outflow is
equivalent to the probability density function of transit times h(t) [6]. Bassingthwaighte and Beard [3] have contended that the tail of such a wash-out curve can be
fitted as a power law in the form of
(3.3)
h(t) = AαD tαD −1 .
For our model, h(t) was calculated from the steady state flow results. Each flow path
was determined by tracing back proximally from a terminal arterial segment up to
the inflow/outflow segment in the tree. The transit times ttransi for each grid point
i are determined by calculating the sum of the Euclidean distance from the current
grid point to the proximal and to the distal grid points and then dividing this value
by twice the velocity value (Vi ) at the grid point, i.e.,
xi − xp 2 + xi − xd 2
ttransi =
(3.4)
,
2Vi
where xi , xp , and xd are the Cartesian positions of the current, proximal, and distal
grid points. These grid point transit times were summed for both venous and arterial
1014
N. P. SMITH, A. J. PULLAN, AND P. J. HUNTER
Relative tracer concentration
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
time (s)
2.5
3
3.5
4
Fig. 3.6. Tracer outflow concentration as a function of time after a unit step in concentration
at time 0s.
velocities for all grid points along a given flow path to find that path’s total transit
time tp . h(t) is plotted by summing all flow along paths which have transit times
δt
within a specific time interval (t − δt
2 , t + 2 ) divided by the total flow.
Figure 3.6 shows the outflow concentration function O(t) in response to an inflow
with a unit step in concentration. This outflow concentration curve O(t) is calculated
as a cumulative frequency graph of transit times for each flow path corresponding to
a terminal grid point i weighted by the fraction of total flow through that terminal
segment, i.e.,
(3.5)
O(t) =
ttransi ≤t
πRi (Vi )2
,
Ftot
where Ftot is the total inflow (which is equal to the outflow in steady state) and Ri
and Vi are the radius and velocity through the terminal grid point i.
Figure 3.7 plots the normalized dilution time curve h(t), which can also be thought
of as the derivative of the curve in Figure 3.6 with respect to time. As transit time
through the microcirculation bed is not considered, the delay in the response in outflow
concentration is much smaller than in reported experiments. The fundamental form
should not, however, be different.
The tracer is assumed to be flow limited, and thus no diffusion or dispersion of the
concentration is modeled. The lack of diffusion, combined with the aliasing effects
of measuring discrete transit times using a small time interval, probably accounts
for the scatter shown in Figure 3.7. However, Beard and Bassingthwaighte [6] have
shown that the power law form of the tail of h(t) is unaffected by the diffusion of
concentration.
The tail of h(t) is plotted on a log log scale in Figure 3.5, and the data is fitted with
a linear function using the Marquardt–Levenberg least squares regression algorithm.
1015
AN ANATOMICALLY BASED CORONARY MODEL
0.5
0.45
0.4
0.35
h(t)
0.3
0.25
0.2
0.15
0.1
0.05
0
0
2
4
6
time (s)
8
10
12
14
Fig. 3.7. The normalized outflow dilution curve as a function of time after a unit step in
concentration at time 0s.
The linear fit has an χ2 value of 8.76, which is within the 99% confidence interval for
the sample size. From (3.3) A was fitted with 0.753 within a 68.3% confidence interval
of 0.0928. More significantly, αD was fitted with −2.98 within a 68.3% confidence
interval of 0.123, which compares well with the value of −3.0 reported by Beard and
Bassingthwaighte [6] and Bassingthwaighte and Beard [3]. This shows that the model
simulates the observed tracer wash-out process and indicates that it may indeed be
approximately fractal.
4. Summary and model limitations. The major contribution of this study
is the development and application of a blood flow model that accounts for much
of the fundamental physiology of vascular blood flow while retaining the efficiency
required for implementation on large vascular networks. A lumped parameter model
of the microcirculation is coupled with the terminal segments of the discrete model
to simulate flow in vessels with radius less than 100µm. Below 100µm, many of the
fundamental physical properties of the models are altered, and a discrete model of
the network becomes computationally prohibitive. Using a geometric representation
of the coronary network, blood flow simulations have been performed and compared
with experimental data to verify the coronary model. The major results from these
simulations show (1) realistic pressure distributions through arterial and venous vessel
segments, (2) a fractal relationship between spatial heterogeneity of blood flow and
tissue sample size, and (3) a power law behavior for the tails of simulated wash-out
curves.
The major assumptions included in the blood flow model concern the pressure
radius relationship, the radial velocity profile, and pressure loss at vessel bifurcations. The effect of vascular smooth muscle is not included in the pressure radius
relationship, and thus vessels are assumed to be in a maximally dilated state. This
may approximate ischaemic conditions, where coronary flow reserve is substantially
1016
N. P. SMITH, A. J. PULLAN, AND P. J. HUNTER
reduced. However, since regional flows show marked differences between dilated and
auto regulated states [1], it will be important to include vasoregulatory mechanisms
to further investigate many clinically relevant phenomena using this model. A second assumption inherent in (2.23) is that the vessel wall behavior is purely elastic.
While the study of Giezeman et al. [12] indicates that an elastic model is suitable, the
viscoelastic properties of the vessel wall may also have an effect [14].
Recently, Kajiya et al. [17] have reported blunt velocity profiles in epicardial
vessels typical of the form defined by (2.20). However, in some instances, these profiles
were not axisymmetric, and their form changed throughout the cycle developing a
temporary “M” shape during early and/or mid-diastole. The significance of a phasic
change in the form of the velocity profile is as yet undetermined. However, such an
“M”-shaped profile could be accounted for by acceleration of the vessel wall relative
to blood flow due to deformation of the heart wall. Moore et al. [23] have produced a
preliminary model which indicates that blood vessel movement may have a significant
effect on coronary blood flow. The effect of vessel movement would be incorporated
into the current model by determining the flow profile constant α as a function of
vessel movement relative to blood flow velocity.
While losses due to viscous and flow separation effects at bifurcations have been
ignored on the basis of low Reynolds numbers in coronary blood flow, such losses may
become more significant when investigating pathologies such as vessel stenosis. The
work of Stettler, Niederer, and Anliker [37] provides potential directions to incorporate
these effects in the current model.
The determination of the effect of vessel movement and heart contraction point
clearly to the next step in the development of the coronary blood flow model presented
in this study. This is the inclusion of pressure exerted by the contraction of the
heart on the coronary vessels in the blood flow model. Coupling heart contraction to
coronary blood flow will allow the magnitude and regional and temporal variation of
systolic flow impediment to be investigated.
Acknowledgment. The authors would like to acknowledge the detailed and
constructive comments of the journal referees.
REFERENCES
[1] R. J. Bache and F. R. Cobb, Effect of maximal coronary vasodilation on transmural myocardial perfusion during tachycardia in the awake dog, Circ. Res., 41 (1977), pp. 648–653.
[2] A. C. L. Barnard, W. A. Hunt, W. P. Timlake, and E. Varley, A theory of fulid flow in
compliant tubes, Biophys. J., 6 (1966), pp. 717–724.
[3] J. B. Bassingthwaighte and D. A. Beard, Fractal o-labelled water washout from the heart,
Circ. Res., 77 (1995), pp. 1212–1221.
[4] J. B. Bassingthwaighte, R. B. King, and S. A. Roger, Fractal nature of regional myocardial
blood flow heterogeneity, Circ. Res., 65 (1989), pp. 578–590.
[5] J. B. Bassingthwaighte, L. S. Liebovitch, and B. J. West, Fractal Physiology, Oxford
University Press, Oxford, UK, 1994.
[6] D. A. Beard and J. B. Bassingthwaighte, The fractal nature of myocardial blood flow
emerges from a whole-organ model of arterial network, J. Vasc. Res., 37 (2000), pp. 282–
296.
[7] D. V. Carmines, J. H. McElhaney, and R. Stack, A piece-wise non-linear elastic stress
expression of human and pig coronary arteries tested in vitro, J. Biomech., 24 (1991), pp.
899–906.
[8] R. S. Chadwick, A. Tedgul, J. B. Michel, J. Ohayon, and B. I. Levy, Phasic regional
myocardial inflow and outflow: Comparison of theory and experiments, Am. J. Physiol.,
258 (1990), pp. 1687–1698.
AN ANATOMICALLY BASED CORONARY MODEL
1017
[9] W. Chilian, Microvascular pressures and resistances in the left ventricular subepicardium and
subendocardiun, Circ. Res., 69 (1991), pp. 561–570.
[10] Y. I. Cho and R. Kensey, Effects of the non-Newtonian viscosity of blood flows in a diseased
arterial vessel, Biorheology, 28 (1991), pp. 241–262.
[11] D. V. Defily, L. Kuo, M. J. Davis, and W. M. Chilian, Segmental Distribution and Control
of Coronary Microvascular Resistance, Springer-Verlag, New York, 1993, pp. 270–282.
[12] M. J. M. M. Giezeman, E. VanBavel, C. A. Grimbergeb, and J. A. E. Spaan, Compliance
of isolated porcine coronary small arteries and coronary pressure-flow relations, Am. J.
Physiol., 267 (1994), pp. 1190–1198.
[13] M. Goto, A. Kimura, K. Tsujioka, and F. Kajiya, Characteristics of coronary artery inflow
and its significance in coronary pathophysiology, Biorheology, 30 (1993), pp. 323–331.
[14] K. Hayashi, Experimental approaches on measuring the mechanical properties and constitutive
laws of arterial walls, J. Biomed. Engrg., 115 (1993), pp. 481–488.
[15] P. J. Hunter, Numerical Solution of Arterial Blood Flow, Master’s thesis, University of Auckland, Auckland, New Zealand, 1972.
[16] R. M. Judd, D. A. Redberg, and R. E. Mates, Diastolic coronary resistance and capacitance
are independent of duration of diastole, Am. J. Physiol., 260 (1991), pp. 943–952.
[17] F. Kajiya, S. Matsuoka, Y. Ogasawara, O. Hiramatsu, S. Kanazawa, Y. Wada, S.
Tadaoka, K. Tsujioka, T. Fujiwara, and M. Zamir, Velocity profiles and phasic flow
patterns in the non-stenotic human left anterior descending coronary artery during cardiac
surgery, Cardiovasc. Res., 27 (1993), pp. 845–850.
[18] G. S. Kassab, D. H. Lin, and Y. C. Fung, Morphometry of pig coronary venous system, Am.
J. Physiol., 267 (1994), pp. 2100–2113.
[19] G. S. Kassab, C. A. Rider, N. J. Tang, and Y. C. Fung, Morphometry of pig coronary
arterial trees, Am. J. Physiol., 265 (1993), pp. 350–365.
[20] G. S. Kassab, C. A. Rider, N. J. Tang, and Y. C. Fung, Topology and dimensions of pig
coronary capilary network, Am. J. Physiol., 267 (1994), pp. 319–325.
[21] J. Lee, D. E. Chambers, S. Akizuki, and J. M. Downey, The role of vascular capcitance in
coronary arteries, Circ. Res., 55 (1984), pp. 751–762.
[22] S. Mayer, On the pressure and flow-rate distributions in tree-like and arterial-veneous networks, Bull. Math. Biol., 58 (1996), pp. 753–785.
[23] J. E. Moore, N. Guggenheim, A. Delfino, P. Doriot, P. Dorsaz, W. Rutishauser, and
J. Meister, Preliminary analysis of the effects of blood vessel movement on blood flow
patterns in the coronary arteries, J. Biomed. Engrg., 116 (1994), pp. 302–306.
[24] W. W. Nichols and M. F. O’Rourke, McDonalds Blood Flow in Arteries: Theoretic, Experimental and Clinical Principles, 3rd ed., Hodder and Stoughton, London, 1990, pp.
85–87.
[25] M. S. Olufsen, Structured tree outflow condition for blood flow in larger systemic arteries,
Am. J. Physiol., 276 (1999), pp. 257–268.
[26] T. J. Pedley, The Fluid Mechanics of Large Blood Vessels, Cambridge University Press,
Cambridge, UK, 1980, pp. 317–318.
[27] K. Perktold, M. Resch, and R. O. Peter, Three-dimensional numerical analysis of the
pulsatile flow and wall shear stress in the carotid artery bifurcation, J. Biomech., 24 (1991),
pp. 409–420.
[28] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes
in FORTRAN. The Art of Scientific Computing, 2nd ed., Cambridge University Press,
Cambridge, UK, 1992.
[29] A. R. Pries, K. Ley, M. Claasen, and P. Gaehtgens, Red cell distribution at microvascular
bifurcations, Microvasc. Res., 38 (1989), pp. 81–101.
[30] A. R. Pries, D. Neuhaus, and P. Gaehtgens, Blood viscosity in tube flow: Dependence on
diameter and hematocrit, Am. J. Physiol., 263 (1992), pp. 1770–1778.
[31] J. Raines, M. Jaffrin, and A. H. Shapiro, A computer simulation of arterial dynamics in
the human leg, J. Biomech., 7 (1972), pp. 77–91.
[32] R. D. Richtmyer and K. W. Morton, Difference Methods for Initial Value Problems, 2nd
ed., Wiley-Interscience, New York, 1967.
[33] S. Sideman and R. Beyar, Imaging Measurements and Analysis of the Heart, Hemisphere,
New York, 1992, pp. 230–235.
[34] N. P. Smith, A. J. Pullan, and P. J. Hunter, The generation of an anatomically accurate
geometric coronary model, Ann. Biomed. Engrg., 28 (2000), pp. 14–25.
[35] J. A. E. Spaan, N. P. W. Breuls, and J. D. Laird, Diastolic-systolic coronary flow differences
are caused by intramyocardial pump action in the anesthetised dog, Circ. Res., 49 (1981),
pp. 584–593.
1018
N. P. SMITH, A. J. PULLAN, AND P. J. HUNTER
[36] N. Stergiopulos, D. F. Young, and T. R. Rogge, Computer simulation of arterial flow with
applications to arterial and aortic stenoses, J. Biomech., 25 (1992), pp. 1477–1488.
[37] J. C. Stettler, P. Niederer, and M. Anliker, Theoretical analysis of arterial hemodynamics
including the influence of bifurcations, Ann. Biomed. Engrg., 9 (1981), pp. 145–164.
[38] A. N. Strahler, Quantitative analysis of watershed geomorphology, Am. Geophys. Union
Trans., 38 (1957), pp. 913–920.