Vol. 284, Issue 6, H2192-H2203, June 2003
Flow and pressure distributions in vascular networks
consisting of distensible vessels
Gary S.
Krenz1,3 and
Christopher A.
Dawson2,3,4
Departments of 1 Mathematics, Statistics, and
Computer Science and 2 Biomedical Engineering, Marquette
University, Milwaukee 53201-1881; 3 Research Service,
Zablocki Veterans Affairs Medical Center, Milwaukee 53295; and
4 Department of Physiology, Medical College of
Wisconsin, Milwaukee, Wisconsin 53226
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ABSTRACT |
We examine the influence of vessel
distensibility on the fraction of the total network flow passing
through each vessel of a model vascular network. An exact computational
methodology is developed yielding an analytic proof. For a class of
structurally heterogeneous asymmetric vascular networks, if all the
individual vessels share a common distensibility relation when the
total network flow is changed, this methodology proves that each vessel will continue to receive the same fraction of the total network flow.
This constant flow partitioning occurs despite a redistribution of
pressures, which may result in a decrease in the diameter of one and an
increase in the diameter of the other of two vessels having a common
diameter at a common pressure. This theoretical observation, taken
along with published experimental observations on pulmonary vessel
distensibilities, suggests that vessel diameter-independent distensibility in the pulmonary vasculature may be an evolutionary adaptation for preserving the spatial distribution of pulmonary blood
flow in the face of large variations in cardiac output.
flow partitioning; heterogeneity; mathematical models; nonlinear; pulmonary circulation
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INTRODUCTION |
PULMONARY CAPILLARY
PERFUSION and alveolar ventilation are adequately matched for
efficient gas exchange over a wide range of cardiac output from rest to
heavy exercise. Normally, this matching is achieved in a largely
passive manner, despite the fact that the heterogeneous and asymmetric
vascular geometry (10) results in a wide distribution of
local flows (16, 17, 22). The pulmonary arteries are also
quite distensible, as required to provide the appropriate impedance for
the right ventricle output. Even though the pulmonary arterial wall
structure varies considerably from the main pulmonary artery to the
precapillary terminal arteries (8, 44), the
distensibility, defined as the fractional change in vessel diameter per
unit change in pressure, is essentially constant and independent of
vessel diameter and vessel wall composition (2, 8, 28).
The same is true for the veins (1). We have observed that,
in model arterial (diverging flow) or venous (converging flow) treelike
structures having one common outflow or inflow pressure, respectively,
common distensibility results in the fraction of the total flow passing
through each vessel segment of the heterogeneous asymmetric tree being
constant, regardless of the total flow or the pressure at the inlet(s).
This is true, despite the fact that in such a tree the distending
pressures and, therefore, the diameters of individual vessels of
identical unstressed diameter may diverge substantially when the total
flow or inflow pressure is changed. Depending on the functional
relation between pressure and diameter, given two identical vessels
located in different parts of the tree, the diameter of one may
increase while the diameter of the other decreases in response to a
given change in total flow; yet the ratio of flows passing through the two vessels will remain the same. Thus the flow distribution, normalized to total flow, will be the same as if the vessel walls were
rigid. This perhaps counterintuitive observation led us to the
conclusion that the vessel diameter-independent distensibility of the
pulmonary blood vessels may be an adaptation that helps fix the
pulmonary flow distribution in the face of the large variations in
total pulmonary blood flow (30). This conclusion was met with some skepticism, at least in part because of the stipulation of a
common terminal outlet pressure (for an arterial tree) or inlet
pressure (for a venous tree). Thus, because the capillary inlet
pressure distribution is not known, it is not clear to what extent this
stipulation might affect the degree to which the idealized model might
reflect the behavior of the real system. In the present study, we
extend the theoretical analysis to the more general case of an entire
vascular network diverging from a single inlet and then converging to a
single outlet, with some additional observations on multiple
inlet-outlet networks.
Glossary
 |
Distensibility parameter defined by D/D0 = f(P) = 1 + P (see Fig. 7)
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 |
Parameter in a general vessel segment resistance per unit length
formula
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 |
Pressure as a function of length, P = (L)
|
| µ |
Individual vessel blood viscosity; viscosity may be different in each
vessel but is assumed to be constant within a vessel as flow or
diameter is changed
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| b |
A vector in a linear system of equations: Ax = b
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| b and c |
Distensibility parameters defined by D/D0 = f(P) = b + (1  b)e cP
|
| fi |
Reference flow in vessel segment i in a vascular network
(see Fig. 1); if the vessels were rigid, with diameters fixed at geometry given at zero pressure, fi would be
flow in vessel segment i when total network inlet flow is 1 and outlet pressure is 0
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| i and j |
Vessel segment (flow through or resistance of a vessel segment) or a
node (pressure or nonlinearly transformed pressure at the inlet or
outlet of a vessel segment)
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| f(P) |
Vessel diameter-distensibility relation
|
| pi |
Reference pressure at node i in a vascular network (see Fig.
1); if vessels were rigid, having diameters fixed at geometry given at
zero pressure, pi would be pressure at node i when total network inlet flow is 1 and outlet pressure is
0
|
| nj |
Occasionally used to emphasize that subscripting refers to node number,
rather than vessel segment number
|
| r0 |
Resistance in a vessel segment if vessel diameter is fixed at geometry
given at zero pressure
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| ri |
r0 for vessel segment i in a vascular network
|
| x |
Solution to a linear system of equations: Ax = b
|
| y |
Solution to a linear system of equations: Ay = Fb
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| A |
Matrix containing conservation of flow and vessel segment pressure drop
using the usual Ohm's law (reference calculation) or the nonlinearly
transformed pressure Ohm's law (see Eq. 6)
|
| C |
Parameter in a general vessel segment resistance per unit length
formula
|
| D |
Vessel diameter, which changes with pressure (P) according the model
D(P) = D0f(P)
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| D0 |
Vessel diameter at zero pressure: D(0) = D0
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| F |
Flow in a single vessel segment or total network flow
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| Fi |
Flow in distensible vessel segment i
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| FR and FL |
Flow in right and left daughter vessels, respectively, at a bifurcation
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| L |
Length variable
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| P |
Transmural and vascular pressures, which, for simplicity, are taken as
equal
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| Pi |
Pressure at node i
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| Pin |
Pressure at the inlet of a vessel segment
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| Pout |
Pressure at the outlet of a vessel segment
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 P |
Upstream-downstream pressure drop across a vessel segment:
P = Pin Pout
|
 (P) |
Nonlinear transformation of P, e.g., for Poiseuille flow and
f(P) = 1 + P,
 . enables use of an equivalent Ohm's law for distensible vessels where upstream-downstream pressure drop for a single vessel
[(Pin) (Pout)] is directly
proportional to flow through the vessel segment; constant of
proportionality is resistance if vessel diameter is fixed at zero
pressure
|
| nj |
Nonlinear transformed pressure:
nj = (Pnj) at node
nj
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| |
Nonlinear transformed upstream-downstream pressure drop for distensible
vessels: = (Pin) (Pout)
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Fig. 1.
Vascular network consisting of 22 vessel segments. Each
vessel is assigned a unique vessel segment number, i,
ranging from 1 to 22. Filled circles denote the 18 locations or nodes
(n1-n18) where
pressures are determined. Vessel 1 is inlet
vessel.
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MODEL VASCULAR NETWORK |
The first model vascular networks we consider are those with a
single arterial inlet and a single venous outlet. To simplify the
notation, intravascular pressure at a point in the network will be
taken to be the same as transmural pressure at the point, which we will
refer to jointly as the pressure (P). We employ the standard
development for relating pressure to flow within distensible vessels
(5, 9, 13, 28, 30, 36, 53).
First, consider a single distensible vessel subjected to constant,
nonpulsatile flow. For ease of exposition, assume Poiseuille flow
within the vessel (13). If a vessel segment is modeled as
a distensible right circular cylinder and entrance effects are ignored,
the local frictional pressure drop per unit length from inlet to outlet
of a single vessel is represented as
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(1)
|
where P denotes pressure, L is vessel length, µ is
blood viscosity, F is vascular flow, and D is vessel
diameter (13). The model vessels share a common
diameter-pressure relation given by
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(2)
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We refer to Eq. 2 as the vessel distensibility
relation. For example, f(P), which has been
discussed in the pulmonary circulation literature, includes
f(P) = 1 + P and f(P) = b + (1 b)ecP (1, 2, 13,
53).
Throughout the analysis, we assume that 1)
f(P) is sufficiently smooth, so when D = D0f(P) is used in
Eq. 1, the differential equation has a unique solution,
2) f(0) = 1 and
f(P) > 0 for all P, and 3) distensibility
is constant throughout the vascular network; i.e., f(P) in
Eq. 2 applies to each vessel within the vascular network. No
additional restrictions are placed on f(P). Under the above
assumptions (13, 30, 36) we can conclude the following.
Lemma 1.
If (P) denotes an antiderivative of the fourth power of
f(P), then
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(3)
|
where Pin and Pout are the inlet and
outlet pressures of the vessel, respectively, and r0 is the
vascular resistance at the zero pressure diameter
(D0).
Remark 2.
In lemma 1, individual vessel blood viscosity is part of
r0, rather than part of . Thus µ appears as a
multiplicative constant separated from . This separation allows us
to view µ as possibly being different from vessel to vessel, and µ affects only the vessel's r0.
Remark 3.
As pointed out elsewhere (30), lemma 1 is not
the most general result possible, because any resistance per unit
length formula that allows the separation of D0
and f(P) could be employed, giving rise to an appropriate
. An example of a separable local resistance per unit length
relation that describes how P changes with L would be as
follows: dP/dL = CµD
F, where F is flow in a
particular vessel segment, D is vessel segment diameter,
> 0 is fixed, and C and µ > 0 are constant within an individual vessel but might change from vessel to vessel; however, they do not change with diameter or flow. With a
distensibility relation, f(P), such that
has a unique solution: P = (L); then, as shown
elsewhere (30)
![&phgr;′(<IT>L</IT>) = −<FR><NU><IT>C</IT>&mgr;</NU><DE>{<IT>D</IT><SUB>0</SUB><IT>f</IT> [&phgr;(<IT>L</IT>)]}<SUP>&ggr;</SUP></DE></FR> F](/content/vol284/issue6/fulltext/H2192/img005.gif)
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(4)
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or
![<LIM><OP>∫</OP><LL>0</LL><UL><IT>L</IT><SUB>0</SUB></UL></LIM>{<IT>f</IT>[&phgr;(<IT>L</IT>)]}<SUP>&ggr;</SUP> &phgr;′(<IT>L</IT>) d<IT>L</IT> = <LIM><OP>∫</OP><LL>0</LL><UL><IT>L</IT><SUB>0</SUB></UL></LIM> −<FR><NU><IT>C</IT>&mgr;</NU><DE><IT>D</IT><SUP>&ggr;</SUP><SUB>0</SUB></DE></FR> Fd<IT>L</IT>](/content/vol284/issue6/fulltext/H2192/img007.gif)
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(5)
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where Pin and Pout are the inlet and
outlet pressures of the vessel segment, respectively, and and
r0 are modified from their Poiseuille flow-derived formula.
In what follows, the key observation is not dependent on whether
one uses a Poiseuille flow assumption to model vascular resistance or
selects an affine vs. an exponential relation between diameter and
pressure to model vessel distensibility. The key observation is,
instead, that should be viewed as the abstraction of pressure in a
distensible vessel, rather than actual pressure (P). With this
observation in mind, Eq. 3 can be viewed as a hemodynamic equivalent of Ohm's law, which accommodates distensibility of vessel
segments, relating resistance and flow to a pressure drop
|
(6)
|
where r0 is a fixed vessel resistance, F is flow in
the vessel, and represents the nonlinear transformed pressure
drop across the vessel segment. Rather than the usual vessel segment pressure drop, P = Pin Pout,
this "Ohm's law" relates the drop in nonlinear transformed
pressure () from inlet to outlet of the vessel segment to a fixed
reference resistance (i.e., the resistance at the zero pressure
diameter) and actual vessel flow. By ascending a single inlet-single
outlet vascular network using the standard electrical circuit analogy
and Eq. 6, one can write equations that calculate flow
fractions and nonlinear pressures for a vascular network containing
distensible vessels. Then nonlinear pressures () can be inverted to
obtain actual pressures (P).
To illustrate the computational methodology proposed above,
consider the vascular network in Fig. 1.
First, we examine the reference case, where reference inlet flow is 1, reference outlet pressure is 0, and reference resistances are
r1, ... , r22. For this reference setting,
vessel segment i would have flow fi (i = 1, ... , 22), and the vascular network would
experience pressures p2, ... , p18; e.g.,
p2 is the pressure at node 2 and p18
is the pressure at node 18. Then, in the reference case,
conservation of flow at node 18 in Fig. 1 would imply
f1 = 1, whereas at node 17, conservation of
flow would yield f1 = f2 + f5 or, equivalently, f1 + f2 + f5 = 0. Similarly, at node
5, f12 + f13 = f18,
which one should write as f12 + f13 f18 = 0. At node 7,
f6 + f12 = 0. The upstream pressure
at node 2 is p2 = f22r22, or f22r22 p2 = 0. The two distinct pathways to node
15 would yield f10r10 + p11 p15 = 0, as well as, e.g.,
f11r11 + p12 p15 = 0. Overall, the 17 (conservation of) flow
equations and 22 upstream pressure equations in the reference setting
result in a system
|
(7)
|
where x = [f1,
f2, ... , f22, p2,
p3, ... , p18]', b = [1,
0, 0, ... , 0]', the symbol ' denotes vector transpose, and the
matrix A (see APPENDIX A) captures the left-hand sides of
the equations corresponding to conservation of flow and pressure calculations.
We now turn to the distensible vessel flow and pressure
calculations for the vascular network depicted in Fig. 1. Lowercase subscripted variables refer to the previous reference setting calculation; i.e., fi,
ri, and
pnj are reference flow
(fraction), reference resistance, and reference pressure, respectively,
whereas corresponding uppercase variables denote the distensible vessel value.
Remark 4.
Because antiderivatives differ by at most an additive constant, we may
select such that (Pn1) = 0 at pressure Pn1. This is
equivalent to selecting the nonlinear outlet pressure to be the zero
baseline, simplifying notation and computations without affecting the
generality of the results. We do not require that
Pn1 = 0.
For any nonzero total inlet flow (F), the flow in the ith
distensible vessel segment will be denoted Fi.
Suppose the ith distensible vessel segment is between nodes
nk and nj. We will show
that 1) Fi = Ffi and 2)
Pnj is obtained from the nonlinear
equation
|
(8)
|
The same can be shown for Pnk.
To see this, Eq. 6 is used as an equivalent Ohm's law. The
matrix equation arising from conservation of flow and nonlinear transformed pressure drop is
|
(9)
|
where y = [F1,
F2, ... , F22, 2,
3, ... , 18]'. The matrix A is
exactly the matrix obtained in the reference calculation. Uniqueness of
the solution of Eq. 7 implies that the system of equations
that determines the flows (Fi) and nonlinear pressures (nj) is equivalent to
|
(10)
|
where y = Fx are related through
Fi = Ffi
(i = 1, ... , 22) and
nj (nj = 2, ... , 18) are given by Eq. 8.
With this example of calculation methodology in mind, we can now
state the main result. Suppose each vessel segment in a vascular network is assigned a unique number i.
Theorem 5.
Suppose that, in an arbitrary single inlet-single outlet vascular
network, 1) every vessel segment has the same distensibility relation, D/D0 = f(P)
(each vessel segment, however, may have a different
D0), 2) µ, although it may be
different in each vessel segment, remains constant within a vessel
segment as flow or diameter changes, and 3) up to a
multiplicative constant, every vessel segment has the same separable
local resistance per unit length relation. Then, for each vessel
segment i in the vascular network, there exists a unique
constant fi, independent of F, such that
|
(11)
|
relating the flow in the vessel (Fi) to the
nonzero total inlet flow (F).
This result follows directly from the calculation methodology
using conservation of flow and nonlinear pressure () and relating the reference calculations to the distensible vessel calculations.
Corollary 6.
Under the same suppositions as for theorem 5, if
FR denotes the flow in one daughter vessel at a bifurcation
and FL denotes the flow in the other daughter vessel, then
FR/FL is the same for every nonzero total inlet
flow (F) through the vascular network.
To provide a concrete numerical example, we employ an f(P)
that exhibits autoregulatory-like behavior to the vascular network in
Fig. 1. This example was chosen, despite the focus of the introduction on the pulmonary circulation in which passive mechanics dominates, because the example tends to be a rather severe challenge to one's intuition. In a network experiencing autoregulatory behavior, it is
easy to see that, for two vessels having the same
D0, a change in total flow can result in an
increase in diameter of one and a decrease in diameter of the other,
but perhaps not so obvious is that the fraction of flow through each
vessel will remain the same. One such autoregulatory distensibility
relation is f(P) = a1 + 
, where a1 = 0.9, b1 = 0.25, b2 = 1, c1 = 0.25, and
c2 = 0.25 (30). A graph of
f(P) vs. P is given in Fig. 2.
This f(P) provides a maximum diameter of 131%
D0 but falls to only 90%
D0 as pressure increases.
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Fig. 2.
An autoregulatory-like distensibility relation. Vertical
axis is distensibility D/D0 = f(P); horizontal axis is pressure (P). Peak distension
occurs at P ~ 3.6.
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The autoregulatory f(P) was applied to the 22-vessel
vascular network in Fig. 1, where vessels in the network are numbered i = 1, ... , 22 and nodes are denoted
n1, ... , n18, and
vascular network outlet pressure
(Pn1) was set to zero. Employing the
vessel-numbering scheme, ri denotes the
reference resistance that would result at the diameter corresponding to zero pressure in vessel segment i. With the assumption of
Poiseuille flow, (P) would be an antiderivative of the fourth power
of f(P). Figure 3 is one such
antiderivative, with the appropriate additive constant such that
(0) = 0. For the vascular network in Fig. 1, we
set r1 = 1/8, r2 = r21 = 1/2, r3 = r4 = r5 = r18 = r19 = r20 = 1, r6 = r8 = r10 = r12 = r13 = r15 = r16 = r17 = 2, r7 = r9 = r11 = r14 = 3, and r22 = 1/16. Particular
reference resistances would correspond to a known vessel geometry at
zero pressure. However, because flow division depends solely on the
reference resistances, it is sufficient to select the resistances.
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Fig. 3.
Antiderivative for 4th power of f(P) in
Fig. 2. Vertical axis, abstraction of the concept of pressure in
distensible vessels [(P)]; horizontal axis, actual pressure (P).
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Solving the reference case Eq. 7 and numerically inverting
the nonlinear expression Eq. 8 provide the pressures
Pnj at the nodes of Fig. 1. The
pressures, as a function of total vascular network flow, are given in
Fig. 4. At any fixed total vascular
network flow, a vertical line would cut through the various pressure
curves, which, along with the zero outlet pressure, would depict the
complete nodal pressure distribution in the network. At a total network
flow of 20, the pressure at node 3 is P3 = 4.90, at node 4 the pressure is P4 = 2.82, and, in regard to further increases in flow and, therefore, pressures,
vessel 1 has become a rigid tube with diameter ~90% of
D0. In Fig. 5, the
pressure drop across each individual vessel in Fig. 1 is plotted as a
function of individual vessel flow. If the vessels were rigid, each
plot would be a straight line. Although each vessel has the same
distensibility relation, the nonlinear interaction of distensibility,
pressure, and vascular network connective structure is readily apparent in the nonlinear appearance of the curves and, more importantly, in the
differences in concavities of the curves, and yet every vessel segment
in the vascular network is receiving a constant fraction of the total
vascular network flow over the entire range of total network flow.
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Fig. 4.
Pressure-flow relation for individual vessels for
vascular network in Fig. 1. Horizontal axis, total vascular network
flow (F). Individual vessel flow is a constant fraction of total
vascular network flow. Vertical axis, pressure (P) at
nodes.
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Fig. 5.
Individual pressure drop vs. flow for each vessel in
network in Fig. 1. Horizontal axis, individual vessel flow; vertical
axis, pressure difference between nodes at each end of vessel segment.
Numbers (1-22) indicate vessel segments corresponding
to plot.
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To demonstrate the alternative condition, i.e., the effect of changing
total flow on individual vessel flows in networks with varying vessel
distensibility, we carried out the simulations depicted in Fig.
6. The columns in Fig. 6 exemplify the
impact of diameter-dependent distensibility for the network in Fig. 1. For Fig. 6, the vessel distensibility relation was taken as
D/D0 = 1 + P, which is
more consistent with pulmonary vessels (1, 2, 13, 53).
With vessel segments having a fixed length-to-diameter ratio,
Poiseuille flow, and constant µ for each vessel, the reference resistances translate into an initial P = 0 geometry. Starting with the same P = 0 resistances as in Fig. 4 and, hence, an
implied P = 0 geometry, the left column in Fig. 6 shows
the hemodynamic consequences of a particular choice of increasing
with decreasing vessel size ( for each vessel was taken to be
proportional to its D
, with ranging
from 0.0625 to 3). In the right column of Fig. 6, decreases with decreasing vessel size ( for each vessel was taken to
be proportional to its D
with the
diameters the same as for the left column, ranging from
16 to 0.33). The middle column shows the results with
constant = 0.57, chosen to provide overall network pressure
drops in the same range as the left and right columns. The individual segment flow reversal on increasing total flow in the left column is the result of inversion of the
order of the pressures at nodes 6, 10, and 14,
i.e., P6 > P10 > P14, which occurs at a total network flow of 11.3.
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Fig. 6.
Hemodynamic calculations for network topology in Fig. 1 using
distensibility relation D/D0 = 1 + P. Each column represents a different choice of : increases with decreasing vessel size (left), is the
same for all vessels (middle), and decreases with
decreasing vessel size (right). From top to
bottom: pressure at 18 nodes as a function of total network
flow, vessel segment flow as a function of total network flow, fraction
of total flow in a vessel segment as a function of total network flow,
and pressure drop across a vessel segment as a function of vessel
segment flow. Negative values indicate retrograde flow.
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DISCUSSION |
The primary observation of this study is that if a heterogeneous
asymmetric vascular network (having the stated properties) consists of
blood vessels, each of which has the same distensibility relation,
despite potentially wide variations in pressures within vessels that
have a common diameter at a given pressure, the flow distribution
within the network will be unaffected by changes in total network flow
and the accompanying redistribution of pressures. A similar observation
was made previously for diverging (arterial) and converging (venous)
trees, with the restriction that there was a common outlet or inlet
pressure, respectively. Because there is little available experimental
information on capillary pressure distributions (24), it
is unknown how damaging this restriction might be to the relevance of
the theorem to any real vascular system. Extension to the single
inlet-single outlet network helps address this question to the extent
that an arterial (or venous) tree can now be considered a part of
a network for which there is somewhere downstream (or upstream) a
common pressure and in which distributed arterial outlet or venous
inlet pressure would be the normal condition for a heterogeneous
asymmetric network.
As indicated in the introduction, this study was motivated by an
attempt to understand the significance of observations indicating that
the distensibility of the pulmonary arteries (Fig.
7) and veins (1) is
virtually independent of vessel size. The network model does not
completely resolve the question, because the specific assumption
invoked is that the arteries, capillaries, and veins have the same
distensibility relation. Over the physiological range of pulmonary
pressures, that assumption appears to be reasonable for the pulmonary
arteries and veins (1, 2, 26). Whether the same can be
said for the capillaries is not so clear; in part, it depends on
whether the capillaries are viewed as cylinders (21, 47),
which distend with a uniform increase in diameter, or as a punctuated
sheet, wherein distension is only orthogonal to the alveolar surface
(13, 14, 47), or somewhere between these extremes. For a
cylindrical capillary, the geometric component of the resistance would
involve the fifth power of the diameter, as in the cylindrical arteries
and veins, whereas the sheet resistance involves the fourth power of
the dimension orthogonal to the alveolar surface
(13). On the other hand, the available data suggest that the capillary distensibility is at least within the same order of
magnitude as the arteries and veins, with values of capillary distensibility (defined as the fractional change in the vessel dimension orthogonal to the alveolar surface per unit change in pressure), obtainable from the literature, ranging from ~0.023/mmHg for the dog lung (40) to ~0.07/mmHg for cat and dog
lungs (13, 15).
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Fig. 7.
Pulmonary arterial distensibilities () from 6 species
obtained from 26 studies of vessels of various internal diameters
(D), where is defined by D(P) = D(0) + P/D(0).
Values were reported as such in cited reference or estimated from data
provided in the reference. Transmural pressure (P) range was generally
0-30 mmHg. Dashed line, representing = 0.02/mmHg,
appears to reflect central tendency of data reasonably
well.
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Cox (8) was apparently the first to point out that the
mechanical properties of the pulmonary arterial vessel walls are essentially independent of vessel diameter and the composition of the
individual vessel walls defined by relative amounts of connective
tissue and smooth muscle. This is reiterated by the compilation of data
in Fig. 7, updated from the findings reported by al-Tinawi et al.
(2), wherein the vessel diameter independence of the
distensibility coefficient is reflected by the fact that the data
from several studies obtained using various methods can be correlated
by a virtually constant value of over several orders of magnitude
in D0 from the main pulmonary artery to terminal arterioles and represent many more orders of magnitude in individual vessel segment resistances. Despite the diameter independence, there
is, in fact, variability in the individual values within a given
diameter range, even between studies on the same species. The reasons
for this are not clear but may reflect sensitivity to some aspect(s) of
study conditions that has not been systematically identified. Thus it
seems probable that the variability within a given diameter range in
Fig. 7 is greater than would be expected within any particular
individual lung. However, the objective of the analysis is not to
provide an argument that the distensibility is constant. Rather, it
points out that limits on the distribution of individual vessel
distensibilities would be a logical result of evolutionary pressure to
maintain gas exchange efficiency (i.e., the ventilation-perfusion
distribution) over a wide range of cardiac output.
Determination of the impact of the various obvious differences between
the model and the real system (e.g., pulsatile flow and gravity
effects) will probably require numerical simulations beyond the scope
of present study. Thus, even having generalized the model to encompass
an entire network, it remains idealized. This allows for the analytic
approach to understanding the model behavior, and we believe that the
observations provide a reference point for understanding the
implications of vascular network design in a sense similar to other
idealizations, including "Poiseuille's law," "sheet flow," the
"fifth power law" (13, 53), "Murray's law"
(32), and others (31, 36, 43, 49).
It may also be useful to reiterate that the theorem presented
here is not dependent on the assumption of Poiseuille flow. Rather, in
the deviation of Eq. 9, the existence of a and the reference r0 is what is needed, where the reference
r0 might be thought of as the resistance that would exist
if the vessel diameters were fixed at their zero pressure values. This
is accomplished if, up to a multiplicative constant, each vessel in the
vascular network has the same local normalized resistance per unit
length expression. Although the results allow for blood viscosity being different in each vessel, the restriction of constant viscosity within
a vessel may be viewed as more limiting. However, changes in viscosity
within any single vessel segment due to physiologically reasonable flow
or diameter changes are small (29).
Although our primary goal was to examine the potential for fixed flow
partitioning within a heterogeneous asymmetric vascular network, the
methodology can be employed to determine flows within multiple
input-multiple outlet vascular networks where each vessel experiences
the same distensibility relation (see APPENDIX B). It is
further clear that reference flow distribution calculations apply to
the distensible vessel case under any of the following conditions:
1) a multiple inlet-single outlet vascular network, where
the inlet flows may increase or decrease, but inlet flows are delivered
in a fixed ratio, 2) a single inlet-multiple outlet vascular
network, where all outlet pressures are fixed at the same value, and
3) a multiple inlet-multiple outlet vascular network, where
the inlet flows may increase or decrease but the inlet flows are
delivered in a fixed ratio and all outlet pressures are fixed at the
same value. In each case, the individual vessel segment flows
throughout the network would follow the constant partitioning results
described above.
The observation that distensibility of the pulmonary arteries and veins
is diameter independent over several orders of magnitude in vessel
diameter may reflect a design feature that takes advantage of the
observations described above. A structure with vessels sharing a common
distensibility should result in a stabilizing effect on the impact of
changing cardiac output on the pulmonary capillary flow distribution
without requiring an elaborate controlling mechanism (28).
When the distensibility is not constant throughout the network, in
particular, when it is diameter dependent, the fraction of total flow
within any one branch of the network may diverge from the initial flow
distribution, and even reversal of flow in some segments is possible
(Fig. 6). Some observations of the effects of changing cardiac output
on the pulmonary flow distribution (4, 23, 46) appear to
be generally consistent with a nearly constant flow distribution.
These observations may have implications for the function of
diseased lungs that are somewhat analogous to the effect of the distribution of airway mechanics on the breathing frequency dependence of the distribution of ventilation. That is, in diseased lungs having
an abnormally broad distribution of time constants among respiratory
units, the increase in breathing frequency generally accompanying an
increase in total ventilation results in a redistribution of the
fraction of the total ventilation received by a given respiratory unit
(41). Likewise, increasing cardiac output in a lung with a
disease extended distribution in individual vessel distensibilities and
would tend to result in a redistribution of blood flow. Little information is available regarding any changes in the longitudinal or
parallel distributions of distensibilities of vessels that might occur
as the result of pulmonary vascular remodeling in pulmonary diseases.
| |
APPENDIX A |
The connective structure of the single inlet-single outlet
vascular network and the explicit choice of the separable local resistance per unit length relation determines the matrix A in Eq. 7. For the example vascular network in Fig. 1, A is
given by
where
and
| |
APPENDIX B |
Provided that sufficient and appropriate boundary
conditions are given so that they uniquely specify the reference
calculation, a linear system of equations can be created to determine
the resulting flows and pressure throughout the vascular network
that contains distensible vessels. For example, consider the vascular
network in Fig. 8, where each vessel has
the same distensibility relation f(P) and, up to a
multiplicative constant, each vessel has the same separable local
resistance per unit length relation, which gives rise to some .
View larger version (9K):
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|
Fig. 8.
Vessel network with several inputs. Each vessel is
assigned a unique vessel segment number, 1-6. Filled circles
denote the 6 locations or nodes,
n1-n6, where
pressures are determined.
|
|
We denote the direction of individual vessel flow in Fig. 8 to be from
a higher node number to a lower node number; e.g., the flow in
vessel 4, F4, is considered directed from
n3 to n2. Furthermore,
suppose we set pressures at n1,
n4, and n6 such that (Pn1) is the smallest of the
terminal pressures. We define
nj = (Pnj) (Pn1), for j = 1, ... , 6. Conservation of flow and calculation of
nonlinear pressures, using Eq. 6,
gives
which can be summarized as a system of linear equations
Ay = b with A a 12-by-12 matrix,
y = [F1, ... , F6,
1, ... , 6]', b = [0, 0, 0, 0, 0, 0, V1, 0, 0, 0, V2, 0]', where V1 and
V2 denote known values from the vascular network
pressure boundary conditions, and ri denotes fixed reference resistances. Although the flows can repartition as
pressures are changed, flow calculation, as well as the nonlinear pressures , is a linear problem. Nonlinearity comes into play only 1) for setting V1 and
V2 and 2) if one needs to invert
to obtain the actual pressures. This example also illustrates the necessary modifications if one does not select to have a zero baseline.
In this setting, if, say, V2 is always a fixed
constant multiple of V1, then each vessel in the
network in Fig. 8 will experience a fixed fraction of the total network flow.
| |
ACKNOWLEDGEMENTS |
This research was supported by National Heart, Lung, and Blood
Institute Grant HL-19298 and the Department of Veterans Affairs.
| |
FOOTNOTES |
Address for reprint requests and other correspondence:
G. S. Krenz, Dept. of Mathematics, Statistics, and Computer
Science, Marquette University, Milwaukee, WI 53201-1881 (E-mail:
gary.krenz{at}marquette.edu).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
First published January 30, 2003;10.1152/ajpheart.00762.2002
Received 30 August 2002; accepted in final form 20 January 2003.
| |
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Am J Physiol Heart Circ Physiol 284(6):H2192-H2203
TOP
ABSTRACT
INTRODUCTION
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