Vol. 279, Issue 6, H3047-H3057, December 2000
Form and function of fetal and neonatal pulmonary arterial
bifurcations
Stephen H.
Bennett1,
Marlowe W.
Eldridge2,
Daniel
Zaghi1,
Shaaron E.
Zaghi1,
Jay M.
Milstein1, and
Boyd W.
Goetzman1
1 Department of Pediatrics, Division of Neonatology,
University of California Davis, Davis, California 95616; and
2 Department of Pediatrics, University of Wisconsin School
of Medicine, Madison, Wisconsin 53792-4108
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ABSTRACT |
Bifurcation is a basic form of vascular
connection. It is composed of a parent vessel of diameter
d0, and two daughter vessels, d1 and d2, where
d0 > d1 
d2. Optimal values for the bifurcation area
ratio, 
= (d12 + d22)/d02,
and the junction exponent, x, in
d0x = d1x + d2x, are postulated to be
universal in nature. However, we have hypothesized that the perinatal
pulmonary arterial circulation is an exception. Arterial diameters were
measured in pulmonary vascular casts of a fetal lamb (140 days
gestation/145 days term) and a neonatal lamb (1 day old). The values
for and x were evaluated in 10,970 fetal and 846 neonatal bifurcations sampled from the proximal and intermediate
arterial regions. Mean values and confidence intervals (CI) for the
fetus were = 0.890 (0.886-0.895 CI) and x = 1.75 (1.74-1.76 CI); and for the newborn were = 0.913 (0.90-0.93 CI) and x = 1.79 (1.75-1.82 CI).
These values are significantly different from Murray's law ( > 1, x = 3) or the West-Brown-Enquist law ( = 1, x = 2). Therefore, perinatal pulmonary bifurcation design appears to be distinctive and exceptional. The decreasing cross-sectional area with branching leads to the hemodynamic
consequence of shear stress amplification. This structural organization
may be important for facilitating vascular development at low flow rates; however, it may be the origin of unstable reactivity if elevated
blood flow and pressure occurs.
pulmonary arterial morphometry; branching complexity; heterogeneity
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INTRODUCTION |
THE MAMMALIAN FETAL
PULMONARY circulation is one of the few examples in nature where
arterial vessels have made special structural adaptations in an effort
to subserve its function (23, 56). In the fetus, the
gas-exchange function of the lung is dormant (19).
Compared with an adult, fetal pulmonary arterial vessels possess an
increased wall thickness and smaller internal diameters, leading to a
very high resistance that serves to shunt blood flow away from the lung
to other organs prior to birth (54). The fetal period is
also hemodynamically unique because the pulmonary circulation requires
only a small amount of blood flow to sustain development
(69), but it experiences a large driving pressure that
would be considered hypertensive if it persisted after birth (57). At birth, the fluid-filled lung airways expand with
air, and concomitantly the pulmonary circulation experiences a 10-fold increase in blood flow as pulmonary arterial vessels dilate and capillaries are recruited to decrease pulmonary vascular resistance (64). Pulmonary arteries continue to remodel after
birth with additional increases in arterial diameter contributing to a
continuing decline in vascular resistance (54). However,
given the dramatic increase in pulmonary blood flow at birth, one
hemodynamic curiosity of the fetal pulmonary circulation is an unusual
sensitivity of pulmonary arterial vessels to endothelial injury and
vascular smooth muscle cell remodeling resulting from premature
elevations in blood flow (2, 4, 46, 52, 53). Although such
flow disturbances are not fatal to the fetus, they can lead to
postnatal complications such as persistent pulmonary hypertension and
possibly death (48, 49, 60, 61). Although the explanation
for this unusual reactivity is not presently understood, it may be
related to the complexity of a vascular network organization unique to the fetal state (17, 28, 57).
The manner in which parent and daughter vessel diameters are connected
at a bifurcation would be an elementary way in which network
organization could contribute to the unique hemodynamic properties of
the fetal pulmonary circulation (55, 76, 77). In a
bifurcation, it has been long recognized that flow travels through the
diameter of the larger parent vessel (d0)
connected to two smaller diameter daughter vessels
(d1 and d2) with flow adhering to a local power-law scaling relationship
(65)
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(1)
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where d0 > d1 d2 and
x is the junction exponent. Bifurcation form can be
summarized by three factors, the area ratio, (11), the
asymmetry ratio, 
(40, 55), and the junction exponent,
x (65). Upon defining the asymmetry ratio
between daughter vessels as = d2/d1 
1, the area
ratio, â, becomes (55)
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(2)
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According to Fig. 1, the value of
x is pivotal in controlling divergent dependencies on the
relationship between the area ratio and branching asymmetry, that is,
for x > 2, increases with increasing ; when
x = 2, is independent of ; and when x < 2, decreases with increasing .
Consequently, bifurcation form has the potential to influence its
physiological and pathological function in a variety of ways, either by
changing the distribution of flow, adjusting the pressure drop, or by
regulating hemodynamic forces, such as wall tension and shear stress,
via changes in vessel diameter and vessel wall morphology. However,
despite this potential for variation, only three basic bifurcation
designs for x have been postulated to be universal for
transport networks in nature, whereby a given design reflects an
alternative principle of economy formulated mathematically as a law.
One design, according to Murray's law (50), postulates a
universal condition of x = 3 and > 1, based
on principles of constant wall shear stress and minimum power
(43). Another law, developed by Kurz and Sandau (41) postulates that x = 2.7 and > 1 based on the principle of constant wall tension. Alternatively,
the West-Brown-Enquist law postulates that x = 2 and
= 1, a condition leading to constant blood flow velocity
subject to an allometric relationship of blood flow delivery
and basal metabolic rate (73). In contrast, we argue that
the unusual properties of the fetal pulmonary circulation can be
accounted for by a radical conjecture that many bifurcations are
organized according to x < 2 and < 1, a
design for vascular systems that can be potentially unstable at lower
thresholds of elevated blood flow. Although the morphometric
characteristics of fetal and neonatal vessels have been well documented
(54), the form of their bifurcation design is not known.
Therefore, the purpose of this report is to measure bifurcation
diameters in the fetal and neonatal pulmonary circulation and to assess alternative design hypotheses.
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Fig. 1.
Idealized bifurcation showing the influence of daughter
vessel asymmetry, = d2/d1, on the area ratio
= (d12 + d22)/d02,
and the junction exponent, x, of the diameter power-law
relationship d0x = d1x + d2x. The latter
relationship arises when blood flow, Q, in a vessel of diameter
d, is considered to scale with diameter according to a
power-law, Q  dx (55, 62) and
continuity of flow is assumed within a bifurcation. However, when
vessels with this flow behavior are connected, the flow properties
between parent and daughter vessels scale with x. Table 1
summarizes the hemodynamic consequences for general values of
x. Optimal design conservation principles for blood flow
between parent and daughter vessels exist at certain values of
x (27): x = 2 simultaneously
minimizes drag force and power loss; x = 2.5 simultaneously minimizes vessel surface area and power loss;
x = 3 simultaneously minimizes blood volume and power
loss; and x = 4 simultaneously minimizes drag force
with flow and blood volume. Universal laws postulated to be widespread
for bifurcations in nature include Murray's law ( > 1, x = 3), leading to constant shear stress
(43), and the West-Brown-Enquist law (for larger transport
vessels, = 1 and x = 2; for smaller exchange
vessels, > 1 and x = 3) (73),
leading to constant blood flow velocity. We hypothesize that fetal
arterial bifurcations possess a design ( < 1, x < 2) in accord with its function to exacerbate
viscous hydraulic energy dissipation and permit shear stress
amplification to sustain pulmonary vascular development at low flow
rates.
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THEORY OF BIFURCATION DESIGN: ECONOMY, INTEGRATION, AND ADAPTATION |
To understand the grounds for an alternative design hypothesis, it
is instructive to recapitulate the premises upon which the universal
bifurcation design theories are based and reevaluate their
applicability to the form and function of the fetal pulmonary circulation. In general, universal theories are built upon the premise
that all vessels are optimally designed to carry flow in some sense. In
theoretical terms, optimality has been formulated using mathematical
principles of economy. Bifurcation design is hypothesized to
accommodate blood flow in the context of a cost function, which
optimizes a given measure of maximum efficiency at a minimum of
energetic cost (50, 66). However, physiological design
principles should also be expected to include factors of integration
and adaptation (71) where mathematical principles of
optimality are not necessarily the only conditions being satisfied. For
example, while vessel elements making up bifurcation demonstrate simple
behavior, their integration into bifurcations and networks may lead to
complex behavior, not predicted by the system elements themselves
(17, 71). Therefore, it is instructive to reexamine how
bifurcation complexity can modulate hemodynamic forces, especially since we now know that these forces have the capacity to modulate vessel diameter over different time scales and spatial scales (13, 15, 23). Also, universal theories, which postulate specific values for bifurcation design, predispose one to assume that
such values are fixed before birth either genetically (70) or phenotypically in combination with a given hemodynamic signal such
as shear stress (43), wall tension (41), or
blood flow velocity (73). However, a design based on
optimum economy for blood flow does not anticipate situations, such as
the fetal pulmonary circulation, where vessel form and function may be
quite different from one intended for transport vessels in general
(54, 56). Also, although universal theories of economy
predict more than one bifurcation design to be prevalent in nature,
they do not presently offer explanations for bifurcation design
variation (16, 42, 62, 75). Here, evidence of an
underlying variation may represent the functional capacity of
bifurcations to adapt their design (71) to match form and
function in response to a much broader spectrum of hemodynamic signals
than the simple ones constrained by constant shear stress, wall
tension, or blood flow velocity.
Economy.
Bifurcation designs leading to postulated universal values for
x have as their foundation an underlying premise that vessel form and function in blood flow transport networks are organized on
principles of economy (50, 66, 71). Investigation into the
relationship between a vessel's diameter and the magnitude of blood
flow that it carries has had a long history of interest. James Keill,
in 1708, postulated a bifurcating branching structure to estimate the
blood flow velocity in capillaries (74). Later, according
to the review of Kurz et al. Roux, in his thesis of 1878 (40), was perhaps the first to draw attention to the fact that physical forces influence the form of bifurcations, imposing constraints on function and shape that cannot be circumvented by
genetic or regulatory means. He calculated stem-branch diameter relationships for and branching angle, and addressed x
as a fundamental parameter characterizing bifurcation form and
function. By 1901, Thoma's studies (65) in blood flow led
him to conclude that the physiological range for x was from
2-4. Hess (30), in 1913, hypothesized that in
all parts of the arterial tree there should be a vessel size where the
total energy cost of transporting blood via vessel resistance is
economized by a minimum in its relationship to its cross-sectional
area. From such a minimum, he predicted that the size of vessels must
be reduced with branching by a factor of
However, it was Murray (50) who first postulated that one
of the fundamental principles of economical physiological organization
was based on the quantitative principle of minimum work. Murray
proposed that flow Q was proportional to diameter according to a local power law, Q = kdx, where k is a
constant and x is an exponent, almost always positive. Murray demonstrated that there are two basic power "costs"
associated with moving blood through arteries. One cost component is
the pumping power of blood, that is proportional to
(Q2/D4) times length, while the
other is a power cost proportional to a metabolic cost times volume.
Murray's insight lies in recognizing that the total power is minimized
when an optimal tradeoff occurs between viscous power dissipation and
metabolic power loss. Murray demonstrated mathematically that the
optimum condition occurs when x = 3. This optimum is
distinctive because it holds for all flow rates and scales with
vascular systems of all sizes (58). Thus the elegance of
this result and the effect of the organizational principle of minimum
work on physiology cannot be underestimated. D'Arcy Wentworth Thompson
(66) emphasized quite strongly in his book On Growth
and Form that the principle of minimum work is the mechanism
"that is best possible under all circumstances." He also states
that to "believe it to be so is part of our common faith in the
perfection of Nature's handiwork" and that this mechanism should
serve as a "postulate, or methodus inveniendi, and it does not lead [the physiologist] astray" (66).
However, vascular systems do demonstrate bifurcation design deviations
from the minimum work principle (3, 16, 42, 55, 62, 75).
Furthermore, several other investigators have generalized Murray's law
and have deduced alternative minimal cost functions that could
potentially describe the diversity of bifurcation branching angles,
their area ratios, their asymmetry, and their values of x in
nature. Uylings (67) generalized the diameter
scaling relationship involving x to reflect alternative conditions of flow and concluded that the optimal value of x
ranged from x = 2.33 for turbulent conditions to
x = 3 for laminar flow. Sherman (58)
provided a generalization of Murray's optimization between vessel
volume and power that applies to the optimization of any vessel network
and does not depend upon minimizing a metabolic cost function. Zamir
(76) considered other cost functions that could be
physiologically minimized under conditions of flow Q = kd3, such as minimum lumen surface area, minimum
drag force arising from shear stress, minimum lumen volume, and minimum
power loss. Roy and Woldenberg (55) extended Zamir's
results to a generalized flow condition of Q = kdx. They found that the models which minimize a
geometric parameter, such as surface area (x = 2), or
volume (x = 3), are sensitive to variations in
x in a different way from those that minimize flow-related
parameters such as power loss due to viscous friction and shear stress.
Moreover, Woldenberg and Horsfield (75) determined that
certain ranges of x are associated with different optimality criteria: the minimization of the surface area lumen is best
represented by 0.7 < x < 2.1; minimum volume
occurs in the range 2.1 < x < 2.8; minimum power
optimization is best represented by 2.8 < x < 3.5; and the range 3.5 < x < 4 can be described
by minimum power or minimum surface area. Griffith and Edwards
(27) subsequently determined that the values of
x predicted by Zamir (76, 77) and Woldenberg
and Horsfield (75) are physiologically adjusted to the
simultaneous minimization of different optimal conditions: x = 4 minimizes both volume and drag; x = 3 either volume and power or surface and drag; x = 2.5 minimizes surface area and power loss; and x = 2 minimizes both drag and power loss. Thus, by adjusting the value of the
junction exponent x in Eq. 1, there are several
alternative ways to optimize a bifurcation to match a desired
geometric, flow, or hemodynamic signal condition.
The economy of vessel form and function extends to the organization of
bifurcations into more complex networks, but the optimal designs that
are predicted to emerge as universal tend to center on regulatory
mechanisms associated with simple hemodynamic signals, such as shear
stress, wall tension, or blood flow velocity. In vascular systems,
blood flow through arteries imparts physical forces to the vascular
wall that can be resolved into two principal components
(15). One force is pressure acting normal to the vessel
wall, which imposes a circumferential stress. The second force is shear
stress, a frictional force acting tangentially at the interface between
blood flow and the vessel (13, 14). LaBarbera
(43) argued that shear stress is a fundamental regulatory signal for which the Murray's law condition of x = 3 represents a local set point. Consequently, elementary networks with
this form of control would automatically produce a system globally minimized for minimum power dissipation during morphogenesis, growth,
and development (43, 58). The x = 3 condition appears to be selected for across divergent phyla
(42), suggesting that this design is likely convergent and
universal, representing a common end point in evolution
(68). However, in systemic mammalian arteries, the value
of x 
2.7 is observed (62), consistent with a self-organizing adjustment to conditions of constant wall tension in vascular networks, as demonstrated by Kurz and Sandau (41).
Alternatively, West et al. (73) formulated a different
branching law proposed to be universal. This law predicts that the diameters of arterial vessels posses an area-preserving design of
= 1 and x = 2, in accord with a 3/4 allometric
scaling law between basal metabolic rate and body weight, a condition
approximated by the human pulmonary circulation (34).
Lighthill (44) argued that by controlling the area ratios
of bifurcations in the systemic circulation (11) to a
value of 1 and x 2, the design
prevents unstable flow separation in larger vessels carrying flow at
large Reynolds numbers. Also, this design predicts that blood flow
velocity in arteries is constant down to exchange vessels, a condition that is also considered optimum under transport conditions
(17). Thus, although alternative universal design theories
predict common values for x in nature, it is presently
unclear as to why one single design is not preferred over another.
Integration and adaptation.
Bifurcations in nature demonstrate a wide variation of values of
x within and between different vascular systems (16,
42, 62, 75). This variation is not necessarily explained by a single cost function, and universal theories offer little explanation for such phenomena. This gap in explanation may be related to constraints imposed by premises made during the formulation of such
theories. Although bifurcation design is based mainly on premises
related to mathematical principles of economy (50, 66),
other design factors such as integrative complex hemodynamic behavior
(17, 71) and the functional capacity of real vessels to
adapt to different hemodynamic signals (23, 71) may take part in design variation.
One premise of universal design theories deals with branching
complexity and its integrating effect on the relationship between bifurcation form and function (17, 71). Classic universal theory idealizes vessel branching as symmetric (30, 51,
66), where the properties of the vessel element possess an
optimal design as the sole determinant of hemodynamic function;
however, in vascular systems such simplifications are not
necessarily true (14, 55, 62). In essence, the otherwise
simple behavior of shear stress in idealized vessels demonstrates
complex behavior when vessels are connected. It should be appreciated
from Table 1 (32, 45) that
while certain values of x do match certain hemodynamic
conditions between parent and daughter vessels as prescribed by
optimization, given values of x and can also influence the scaling of other flow conditions between parent and daughter vessels. For flow itself, x and influence only the
degree of flow partitioning between daughter vessels. However, for
other flow-related properties, such as Reynolds number, flow velocity, and shear stress, the value of x is instrumental in
determining the degree of amplification or deamplification between
parent and daughter vessels. In symmetric bifurcations, where = 1, values of x < 3 have the potential of amplifying
shear stress in the daughter vessels relative to the shear stress of
the parent vessel (11, 59). In asymmetric bifurcations,
where < 1, amplification is further accentuated along the
minor daughter pathway. Furthermore, mathematical network models
demonstrate that when asymmetry and diameter randomization are
introduced into bifurcations of an otherwise symmetric network, where
x 3, blood flow behavior in the network occurs in a
way that cannot be predicted from the behavior of the vessel
elements alone (17, 59). Along certain pathways,
fluctuations in shear stress are amplified, whereas along other
pathways shear stress is attenuated (17, 59). In addition,
as x decreases successively to values smaller than 3, the
average value of shear stress is spatially amplified to larger
magnitudes within the smaller diameter vessels of peripheral branches
(59). Such complexity phenomena in models suggest that the
integration of branching complexity in vascular systems is accompanied
by a broad spectrum of heterogeneous shear stress signals to which the
endothelium and smooth muscle must respond. Consequently, as shear
stress is a hemodynamic signal capable of modifying vessel diameter and
vessel wall function over various time scales (13), the
resulting complexity may provide a basis for explaining the tremendous
spatial heterogeneity observed in form and function in arterial vessels
(12, 24, 25, 35, 36).
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Table 1.
Bifurcation form gives rise to emergent scaling behavior for
bifurcation function stemming from the connection of vessels
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Another premise of universal theories is that bifurcation design is
fixed before birth, either genetically or in response to a predominant
hemodynamic signal acting as a selection factor of bifurcation design
(41, 42, 70-73). Consequently, for x = 3 the signal is shear stress (43), for x = 2.7 it is constant wall tension (41), and for
x = 2 it is constant blood flow velocity under
conditions of a basal metabolic rate (73). In any case, the design selection is considered significant in influencing species
survival (41, 43, 73). However, in nature, physiological systems are also selected for by their functional capacity to respond
and adapt to a variety of stimuli acting as stressors over different
time scales (23, 71). In this regard, universal theories
have not anticipated the possibility that the form and function of
vessels during development may satisfy a different physiological design
criterion than that of the adult (54). Consequently,
vessel diameter and bifurcation design adaptation at birth may
represent additional selection factors for mammalian species survival
(19, 56, 57, 60) where shear stress and wall tension are
ever-present forces regulating local physiological and pathological
responses over both short-term and long-term time scales
(14). This regulation also has a spatial component over
dimensional scales ranging from the network level down to the
individual cell nucleus (15). Under these circumstances, the biological variation of designs seen in nature can be potentially interpreted as an adaptive history of branching complexity which arises
from a broad spectrum of hemodynamic signal information (16, 42,
62).
Fetal/neonatal bifurcation design.
A basic argument for a bifurcation design radically different from
those predicted by universal design theories is based upon physiological necessity: the pulmonary circulation before birth is not
a functional transport organ expected to optimize its design according to an idealized cost function (41, 43, 73).
Instead, the gas exchange organ is dormant, subsisting via a small and sufficient blood flow for its development (69). Also, it
is important to emphasize that a very narrow range of flows delivered to the fetal pulmonary circulation is necessary to ensure a successful adaptation at birth (2, 47, 52), because beyond an
undetermined low-flow threshold pulmonary arterial vessels change their
structure and function adversely (4, 7-10,
46-48). Hence, if such a design is radically different from
an adult, fetal bifurcations must possess the functional capacity to
adapt to another design after birth to accommodate to the dramatically
different hemodynamic conditions. Therefore, just as the function of
the fetal pulmonary circulation prior to birth appears contrary to that
which is intended for general optimal vascular transport networks in
nature (41, 43, 73), it is now reasonable to justify that
that the form of fetal pulmonary bifurcations are likewise exceptional:
instead of 1 and 2 x 3, fetal
bifurcations are hypothesized to satisfy an alternative condition of
â< 1 and x < 2.
A fetal bifurcation design of < 1 and x < 2 leads to several hemodynamic consequences, most of which are economical
only for the fetal state. One consequence of lower values of
x is that the magnitude of the principal resistance within
the pulmonary circulation is enhanced (16, 17). This
design is spatially expanded to other vessels, thereby increasing the
loci of viscous hydraulic energy dissipation. A spatially expansive
increase in resistance represents a plausible design condition for a
high-resistance shunt, intended to divert blood flow to other organs
(56). In addition, with < 1, a continuously
branching cascade of decreasing cross-sectional area leads to shear
stress amplification (6). As shear stress demonstrates a
spectrum of actions on the endothelium and smooth muscle over different
magnitudes and time thresholds (14), it is plausible that
this design would be instrumental in influencing pulmonary vascular
development at very low flow rates (69). Not only would
this design place fetal distal arterial shear rates closer to their
postbirth values, it would thereby reduce the likelihood of endothelial
damage near the gas exchange region when the surge in blood flow occurs
at birth (29). Also, at birth, when blood flow increases,
such a network design would be effective in a more rapid transduction
of shear stress and its effect on increasing arterial diameter via
flow-dependent endothelial mechanisms (1, 19, 20).
However, it must be emphasized that this design confers an
organizational form of structural reactivity (23) that is
potentially unstable, whereby excessive blood flow to the fetal
pulmonary circulation above a low-flow threshold could induce
peripheral arteriolar endothelial dysfunction secondary to a shear
stress injury (7-10, 21, 48, 52, 53).
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METHODS |
Cast preparation.
The diameters of the extant fetal and neonatal pulmonary circulation
were evaluated via lung casts. Under an approved animal use protocol at
the University of California Davis, a pregnant ewe with a fetus of 140 days gestation, along with a 1-day-old newborn lamb, were euthanized by
an overdose of pentothal. A thorachotomy was performed, the trachea was
clamped, and the lungs were removed en bloc. A 4.5-French cannula was
inserted into the pulmonary artery. Fetal airways remained filled with
amniotic fluid. The newborn airways were expanded using saline
under a hydrostatic pressure gradient of 20 cmH2O via a
tracheal cannula. The pulmonary arterial circulation was washed free of
blood by saline perfusion. Methyl methacrylate plastic (Coe Tray
Plastic; GC America, Chicago, IL) was then injected into the pulmonary
arterial system slowly via a syringe over a 2-min period. The setting
time of the plastic is 15 min, with dramatic increases in viscosity
preventing plastic infusion by 7 min. The plastic was allowed to
polymerize overnight, whereupon the lung tissue was macerated in a 20%
KOH bath for 3-5 days. The remaining tissue was washed away gently
with distilled water, and the cast was allowed to dry.
Diameter measurement and calculations.
Bifurcation diameters were measured using a video micrometer. The video
micrometer consisted of a Macintosh IIci computer equipped with a Data
Translation model DT-2255 image frame-grabber board connected to a
Zeiss stereomicroscope fitted with a Panasonic model WW1500X video
camera. Images were acquired and analyzed via a program Object Image
developed by Norbert Vischer (http://simon.bio.uva.nl). Object Image is
an extended version of the program, NIH Image (http://rsb.info.nih.gov/nih-image), allowing diameter measurements and
derived calculations to be recorded into a database. To measure diameters, the three-dimensional branching aspect of each cast was
broken into pieces to facilitate placement of bifurcations onto a
plane. Assuming vessels were circular, vessel diameter was calculated
as the average of two diameter measurements per vessel segment. Vessel
diameters were measured at bifurcations, consisting of a parent vessel,
d0, and two daughter vessels,
d1 and d2, where
d0 > d1 d2. From bifurcation diameters, the diameter junction exponent, x, of the equation
(d1/d0)x + (d2/d0)x = 1 was solved by iteration (34), while the area ratio,
, was calculated according to Eq. 2.
Statistical analysis.
We assumed that the bifurcations sampled from the lung casts were taken
from a dichotomously branching network with self-similar (17, 39,
70-72) and statistically self-similar branching properties (63). Self-similarity and statistical self-similarity are
postulates that establish a null hypothesis of universal network design
for all bifurcations (41, 43, 73) that can be tested via
analysis of variance methods using bifurcation sampling techniques
(16, 17). Our null hypothesis is that if the design
properties of lung branching are postulated to adhere to fixed
universal laws following the principle of self-similarity (17,
71-73), then the averages of and x should be
identical, in a statistical sense, at all levels of branching. The
additional postulate of statistical self-similarity (63)
assumes that the variances of and x are also equal and
independent of branching level, even if alternative methods are used to
categorize vessel diameters into topologically related groups
(17, 39), such as by ranks (17), generations,
or orders (33).
We categorized parent diameters of bifurcations into independent
"bifurcation levels" (3), so that vascular systems
with different distributions of parent diameters could ultimately be related to a canonical self-similar branching model (3,
17) that satisfies the requisite assumptions for an ANOVA
(78). The data set consisted of parent diameter,
d0, area ratio, , and junction exponent,
x, which were derived from the parent vessel's corresponding daughter vessels. The resulting data set was then sorted
from the largest parent diameter down to the smallest diameter, by
levels k = 0, 1...n categorized according
to Rd
k (d0/D0) > Rd(k+1). Here,
d0 is a parent diameter of a bifurcation in the
sorted list, D0 is the diameter of the main
pulmonary artery, and Rd was an average diameter
ratio (Rd = d0/d1) computed from the
pool of bifurcations and was set to 1.315. The ranking
procedure can be thought of as yielding an equivalent asymmetric
branching network, whose branching properties are summarized by a
diameter ratio, Rd, of the major daughter
pathway (parent diameter to largest daughter diameter) (17,
32). The ranking procedure results in a nonoverlapping range of
parent diameters for each bifurcation level, where each level is
assumed to be statistically independent from another, and where the
transformed values [atan and atan x (arctangents,
expressed in radians)] possess a statistically self-similar
Gaussian distribution with bifurcation level (63). The
two-factor ANOVA was evaluated on arctangent transformed values of and x using Statview 4.5. Results are reported as mean
values along with their associated 95% confidence intervals (CI) for transformed (atan and atan x) and their
inverse-transformed variables [i.e., = tan(atan
)]. Non-overlapping confidence intervals for and
x within bifurcation levels were considered to represent
statistically significant difference at a P < 0.05 level of significance.
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RESULTS |
Diameter measurements were made in 10,970 fetal and 846 neonatal
pulmonary arterial bifurcations. Figure 2
summarizes how the numbers of bifurcations, along with their parent
diameters, vary with bifurcation level relative to the main pulmonary
artery. The range of diameters include the proximal and distal zones of the pulmonary arterial circulation. In the case of the fetal pulmonary cast, some parent vessels proximal to capillaries were sampled. The
Fig. 2, left, indicates that the numbers of bifurcations
within a level increase toward smaller diameter vessels as a result of the ranking procedure. In general, from the main pulmonary artery (level 0), down to the smallest diameter compared
(level 13), newborn vessels were ~1.8 times the diameter
of fetal vessels. Figure 3 shows the
histograms for the pooled fetal and neonatal estimates for transformed
(left) and transformed x (right)
compared with their Gaussian fit. Figure
4 illustrates the relationship among ,
, and x in the fetal and neonatal state summarized as a
bivariate distribution (55). The averages for were
nonnormally distributed, with the fetal mean values = 0.62 ± 0.23 SD (arithmetic), 0.59 (geometric), and 0.56 (harmonic), while the newborn mean values were = 0.56 ± 0.21 SD (arithmetic), 0.51 (geometric), and 0.46 (harmonic). Figure 4
demonstrates that fetal and neonatal vessel daughter diameters
in the lamb possess a locus of design concentrated on x < 2 and < 1, where many bifurcations with the same design
are found in the adult human pulmonary arterial tree (55).
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Fig. 2.
Numbers of bifurcations (left) along with
their corresponding average parent diameter (right)
expressed as a function of bifurcation level (k = 0, 1, 2...n) derived from when bifurcations are categorized
according to the parent diameter scaling relationship
Rdk (d0/D0) > Rd(k+1).
Rd is an average diameter ratio
(Rd = d0/d1) calculated from
all bifurcations set to Rd = 1.315. Level
k = 0 is main pulmonary arterial bifurcation.
Left: Nk bifurcations/level.
Right: corresponding average diameter vs. bifurcation level.
Newborn vessels are ~1.8 times larger than fetal vessels at all
bifurcation levels studied. ANOVA was performed on levels 0-13,
but additional numbers of bifurcations were sampled in the fetus to
levels 14-19.
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Fig. 3.
Population histograms of pooled estimates (N = 11,816 bifurcations) for arctangent (atan) transformed
values of {left:  = 0.891 [0.672-1.171 confidence interval (CI)]} and
x [right:  =0.891
(1.22-2.72 CI)]: estimates for x are closer to
Gaussian distribution than estimates for . Gray areas mark regions
of "fetal" bifurcation design ( < 1 and x < 2).
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Fig. 4.
Area ratio () and asymmetry ratio () of fetus
(left) and newborn (right) expressed as a
bivariate frequency distribution. Gray scales represent the numbers of
bifurcations within a particular range of , , and x.
Average  : fetus = 0.62 ± 0.23 SD; newborn = 0.56 ± 0.21 SD. The equation of the lines is given by Eq. 2 in text.
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Figure 5 illustrates the distribution of
and x as a function of bifurcation level. For
bifurcation levels 0-13, ANOVA demonstrated a slight
but significant difference between the fetal and neonatal group alone:
for [F = 6.51, P = 0.011: fetus mean = 0.890 (0.87-0.90 CI); newborn mean = 0.913 (0.90-0.93 CI)], or for x [F = 1.812, P = 0.173: fetus mean x = 1.75 (1.74-1.76 CI); newborn mean x = 1.79 (1.75-1.82 CI)]. ANOVA showed a significant difference between
levels (0-13) for both lungs [for (F = 18.17, P < 0.0001) and x (F = 13.79, P < 0.0001)], indicating that and x,
are different between bifurcation levels. In addition, ANOVA
demonstrated a significant interaction between group (fetus vs.
newborn) and bifurcation level (0-13) for both (F = 174.98, P < 0.0001) and x (F = 2.98, P < 0.0002), indicating that the fetus and
newborn manifest differences in design in branching toward smaller
peripheral diameter vessels. In both the fetal and neonatal lungs
studied, the larger diameter parent vessels (levels
1-7) were consistent with the West-Brown-Enquist law of
= 1 and x = 2. However, beyond level
7, bifurcations followed a fetal pattern of < 1 and
x < 2, both decreasing with increasing branching level
away from the main pulmonary artery. Figure 5 reports additional
numbers of smaller bifurcations in the fetus beyond level
13, down to a bifurcation level of 19, where from Fig.
2, the parent diameters were ~30 µm, consistent with vessels
proximal to immature capillaries. As Fig. 5 indicates, the bifurcation
design in the distal zone reaches a minimum and x at a
bifurcation level of 15, just beyond the range of available
measurements in neonatal lamb.
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Fig. 5.
Distribution of and x as a function of bifurcation
level. Left: transformed scale of . Right:
transformed scale of x used in ANOVA. Gray area demarcates
fetal design values for and x. Bifurcation levels
11-13 are common to both fetus and newborn where ( < 1, x < 2). However, newborn exhibits more levels of
fetal design, but the fetus studied showed more dramatic change in
design with increasing levels. Overall, proximal levels show
bifurcation design behavior consistent with West-Brown-Enquist law
( = 1, x = 2), but both systems manifest
"fetal-like" bifurcation designs ( < 1, x < 2) in distal bifurcation levels that are heterogeneous with
branching.
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DISCUSSION |
The null hypothesis anticipated that fetal vessel connectivity was
optimized, either for constant shear stress, constant wall tension, or
constant flow velocity, as postulated by universal laws predicted by
biological network theory. Such a hypothesis stipulates that
bifurcation design criteria satisfy 1 and 2 x 3 for all levels of branching. Under these
circumstances, bifurcation design in the fetal, neonatal, and adult
state would be identical and homogeneous at all levels of branching.
Consequently, differences in the fetal and neonatal state could be
attributed solely to vessel geometry alone, concordant with a simple
diameter increase due to a vasodilation mechanism after birth
(19). However, the results of this study support the
alternative hypothesis that the form of bifurcation design in small
arterial vessels of the fetal and neonatal pulmonary circulation is
distinctive, with < 1 and x < 2. Furthermore, this bifurcation design is not homogeneous with a uniform
distribution with branching but instead shows a serial spatial
heterogeneity. These results implicate an alternative form of network
organization at the bifurcation and network level different from that
previously supposed by universal network theories that assume
a fixed homogeneous adult arterial bifurcation design of 1 and 2.0 < x < 2.8 for the several adult species
that have been studied (16, 34). The configuration of
fetal bifurcations confirms a hypothesis made by Hopkins
(31) that the perinatal pulmonary circulation dissipates
hydraulic energy in a manner that is quite different than other stages
of pulmonary growth and development and different organ systems
(31, 37). However, whereas Hopkins addressed a particular
mechanism related to pulsatile power dissipation in the viscoelastic
vessel wall, this study identifies an underlying organizational form of
structural reactivity (23) that was not previously
anticipated by any previous theory of bifurcation design (41, 43,
73). This form of reactivity is embodied by the way vessels are
interconnected and would suggest that the fetal and neonatal
configuration must remodel substantially during postnatal
development by changes in bifurcation complexity to expand the
range of flows permissible without injury (34, 55). While
the overall average values for , , and x are much higher in the adult ( 0.8, 1.0, x 2.3), fetal-like bifurcation designs appear with
relatively high frequency (55). In this regard, our Fig. 4
should be compared with figure 2 of Roy and Woldenberg
(55) where several bifurcations in the adult show a
similar pattern of design as our fetal and neonatal lamb bifurcations, with < 1, x < 2, and 0.3-0.4. The appearance of excessive numbers of fetal-like
bifurcation designs in the adult may therefore be suggestive of a
remodeling process due to a prenatal injury (26) or may
indicate an incomplete adaptive remodeling process during growth and
development. In either case, remnants of fetal bifurcation design in
the pulmonary arterial circulation may play a role in predisposing some
individuals to endothelial injury at elevated blood flow
(18).
The acceptance of these results and implications must be viewed
carefully in light of the present study limitations related to casting
methods. One important limitation with casting methods (62) is that arterial diameters are not observed directly
(16) and rely on an apparent diameter filled by the
hydraulic conditions of a high-viscosity polymer (16).
Although the resulting diameters are presumed to mimic the extant
vessel morphology, the polymerization process is known to be subject to
diameter distortion (62). Casting results in a vessel
distention error, reported to scale allometrically with vessel radius,
where the absolute error in diameter measurement propagates to the
smallest vessels. Suwa and colleagues (62)
estimated that for vessels from casts in the range 3,000 µm down to
10 µm, the scaling follows the power-law r = 1.369r00.9609 where r is
the radius of resin fixed vessels of the arterial cast and
r0 is the corresponding radius of Formalin-fixed
vessels. Consequently, for the smallest 30-µm parent diameter fetal
vessels observed in this study, this relationship predicts that they
may be over-distended by a factor estimated to be up to 25%. However, it must be emphasized that we looked only at diameters within a
bifurcation using measurements that normalized their values. Thus the
error rates due to distension should be distributed equally among the
vessels within a bifurcation. A greater possible source of error
contributing to the uncertainty in and x arises from the
violation of our assumption of cylindrical vessel shape from our method
of measuring diameters (38). If such diameter errors were
present and random, then they were compensated for in using a large
number of bifurcations at different levels of branching. Despite these
limitations, casting methods, utilized widely in the adult pulmonary
circulation, lead to consistent findings for bifurcation design for
several species and correlate well with direct visualization techniques
(16). However, in using our bifurcation sampling
technique, the connective branching topology of the entire pulmonary
arterial tree was lost, so we cannot compare the mean and variance
properties of our categorization process using bifurcation levels with
other topologically based methods, such as Weibel generations,
Horsfield ordering, or Strahler ordering. Consequently, the underlying
variance of x using bifurcation levels (3) may
not necessarily yield the same probability distribution properties of
other ordering methods (33, 38). Likewise, as with other
casting studies, a large number of sample lungs taken from the
population are difficult to achieve, given the lack of an automated
imaging procedure and the labor-intensive process of characterizing
thousands of diameter measurements per lung. As a result, this study
was limited to a single fetal and neonatal sample from sheep.
Therefore, we presently do not know at what time after birth
bifurcation design takes on adult values or whether our results reflect
an average pattern of design adaptation at birth representative of
different mammalian species. In any case, it must be emphasized that
fetal and newborn sheep are widely used experimental models of human
pulmonary vascular development and disease process (19, 21, 22,
46-48, 52, 53, 56, 57) for which morphometric data of this
type are lacking. Consequently, the results of this study suggest that
the fetal design can persist in the immediate neonatal period and that
a fetal bifurcation design implicates a possible structural origin of
hypertensive disease (23) based upon branching complexity
(17). Another limitation of this study was that it was
only possible to sample sufficient numbers in the proximal and
intermediate regions of the pulmonary circulation. Thus the smaller
distal arterial vessels, infant capillaries, and veins are excluded
from analysis. However, our morphometric results, although limited in
scope, predict radical differences in the perinatal organizational
state in the pulmonary circulation that can be verified independently
by hemodynamic methods that depend explicitly upon the
theoretical consequences of bifurcation design. Such methods include,
but are not limited to, the slope of the pressure-flow curve
(16), the cumulative distribution of resistance as a
function of vascular volume (39), and the broad-band wave
reflection interpretation of vascular input impedance (5,
6).
In view of these limitations, the results of this study emphasize an
elementary concept of branching complexity: individual vessels, with an
assumed known form and function, may manifest quite different emergent
scaling behavior when connected (17). This study
identifies an example in nature where bifurcations are not necessarily
endowed with a design based on theoretical principles of economy
(43, 66, 73). We must emphasize that the fetal and
neonatal pulmonary circulation are not necessarily the only examples
where bifurcation values for < 1 form elements of
branching design in arterial trees (3, 11, 55, 75). However, in the case of the fetal and newborn pulmonary arterial system, this connective behavior provides a new perspective to explain
hemodynamic phenomena underlying the process of adaptation or
maladaptation during pulmonary vascular growth and development. In
effect, branching design may originate from a state far removed from a
theoretical one optimized for minimum energy dissipation, as proposed
by Thompson (66), and still be economical to the form and
function of the fetal pulmonary vascular state. However, to survive
after birth, pulmonary arterial bifurcations must possess the
functional capacity to alter their design radically in response to a
broad spectrum of hemodynamic conditions (71). Whether such behavior is present in other mammalian fetal and neonatal pulmonary vascular systems, and what their properties are in the general population, requires further investigation.
| |
ACKNOWLEDGEMENTS |
We thank Jim Jones for illuminating discussions on optimality. We
also thank an anonymous reviewer for suggesting that fetal bifurcation
design acts to precondition fetal distal arteries to endothelial shear
rates closer to their postbirth values.
| |
FOOTNOTES |
This work was made possible by a grant provided by the Children's
Miracle Telethon Network.
Address for reprint requests and other correspondence: S. H. Bennett, Division of Neonatology, Dept. of Pediatrics, Univ. of
California Davis, Davis, CA 95616 (E-mail: shbennett{at}ucdavis.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.
Received 18 May 2000; accepted in final form 27 July 2000.
| |
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TOP
ABSTRACT
INTRODUCTION
