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Diameters and cross-sectional areas of branches in the human pulmonary arterial tree.

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THE ANATOMICAL RECORD 223~245-251(1989)
Diameters and Cross-Sectional Areas of Branches
in the Human Pulmonary Arterial Tree
The Midhurst Medical Research Institute, The Surrey Research Park, Guildford, Surrey GU2
5YU: United Kingdom (K.H.) and Department of Geography, State University of New York,
Buffalo, New York 14260 (M.J.
Measurements were made of the diameters of the three branches
meeting at each of 1,937 bifurcations in the pulmonary arterial tree, using resin
casts from two fully inflated human lungs. Cross-sectional areas of the parent branch
and of the daughter branches were calculated and plotted on a log-log plot, which
showed that mean cross-sectional area increases in a constant proportion of 1.0879
a t bifurcations. The mean value of the ratio of daughter branch diameters a t
bifurcations was 0.7849. The mean value of the exponent z in the equation flow = k
(diameter“) was found to be 2.3 k 0.1, which is equal to the optimal value for
minimizing power and metabolic costs for fully developed turbulent flow. Although
Reynolds number may exceed 2,000 in the larger branches of the pulmonary artery,
turbulent flow probably does not occur, and in the peripheral branches Reynolds
number is always low, excluding turbulent flow in these branches. This finding
seems to be incompatible with the observed value of z. A possible explanation may
be that other factors may need to be taken into account when calculating the
theoretical optimum value of z for minimum power dissipation, such as the relatively
short branches and the disturbances of flow occurring at bifurcations. Alternatively,
higher arterial diameters reduce acceleration of the blood during systole, reduce
turbulent flow, and increase the reservoir function of the larger arteries. These
higher diameters result in a lower value of z.
The lesser circulation, the vessels through which it
flows, and the changing pressures within them continue
to be of interest to pneumonologists and cardiologists
alike (Harris and Heath, 1986). Dimensions and numbers of branches in the asymmetric pulmonary arterial
tree of humans have been described by Elliott and Reid
(1965), Woldenberg et al. (1970), Singhal et al. (19731,
and Horsfield (1978). In addition, the concept of optimal
“design” (i.e., evolutionary development) in the systemic and pulmonary arterial trees has been presented
by many authors (Murray, 1926a,b; Zamir, 1976; Horsfield, 1977; Uylings, 1977; Sherman, 1981; Woldenberg
and Horsfield, 1983,1986). Pressure and flow within the
pulmonary arteries are closely related to their diameters and cross-sectional areas, so that the way in which
cross-sectional area changes down the tree is of particular interest. The purpose of this study is to measure
diameter and cross-sectional area over a wide range of
bifurcations in the pulmonary arterial tree and to relate
these measurements to various physiologic and morphometric parameters, in particular to the exponent z in
the equation flow = k x diameter’.
The branches of these trees can be classified in terms of
order, whereby branches are counted from the periphery
toward the main stem. The method of ordering that will
be used is that described by Horsfield (Horsfield and
Cumming, 1968; Horsfield, 1984). In this method two
order 1 branches meet to form a n order 2 branch, two
order 2 branches a n order 3, and so on. When branches
of differing order meet, the next order is one more than
the higher of the two meeting branches. It should be
understood that the order of the stem is determined by
the longest pathway, because order increases by 1 a t
each bifurcation. The number of branches in successive
orders down the tree increases by a constant factor, the
branching ratio Rb. Similarly, the mean diameter of
branches in successive orders up the tree also increases
by a constant factor, the diameter ratio Rd (Singhal et
al., 1973; Horsfield, 1978).
Murray (1926a) studied the trade-off between metabolic cost and power dissipation in relation to arterial
diameter. As diameter increases, resistance, and hence
work of moving the blood, fall, while the metabolic cost
of maintaining the blood and vessel wall increases because their volumes increase. Assuming laminar flow
Both the bronchial and pulmonary arterial trees in
man can be modeled in terms of a repeatedly dichotomously branching structure arising from a single main
stem, without any rejoining of branches (Horsfield and
Cumming, 1968; Singhal et al., 1973; Horsfield, 1978).
0 1989ALAN R. LISS, INC.
Received March 23, 1987; accepted August 8, 1988.
Address reprint requests to Dr. K. Horsfield, the Midhurst Medical
Research Institute, 30 Occam Rd., The Surrey Research Park, Guildford, Surrey GU2 5YW, U.K.
and the applicability of Poiseuille's equation, he found lated by substituting measurements of the three diameters into the equation of continuity (Suwa et al., 1963;
an optimum when
Woldenberg and Horsfield, 1983). Thus
f = kD3
fo = f1 + f2
where f is flow of blood in a branch of diameter D, and k
is a constant. For a given flow, the dissipation of energy where the subscript 0 refers to the parent branch, 1
is minimal with a fully developed laminar flow regime. refers to the major daughter branch, and 2 refers to the
With more disturbed flow regimes, more energy is dis- minor daughter branch. Therefore, substituting for f
sipated, and the optimum value of the exponent is lower from equation 2 in the equation of continuity we obtain
than 3. For fully developed turbulent flow the optimum
is 2.333 (Uylings, 1977). Thus, according to the minimum power model, the value of the exponent (z) depends
on the flow regime, and in general
Knowing the three diameters, Do, D1, and D2, a t a
f = kDZ
(2) bifurcation, equation 3 can be solved iteratively to find
the positive value of z when Do > D1 3 D2, and the
Because the observed values of z may range both above negative value of z when D1 2 D2 > Do. If D1 2 Do 2
and below 3 (Woldenberg and Horsfield, 1983) the mini- D2, then z cannot be calculated.
mum power model may not be a n appropriate explanaIn addition to the overall diameter ratio of the system,
tion for the values of z observed in the pulmonary artery we can also speak of a diameter ratio at a bifurcation
or other vascular networks. We require a more general
explanation of why flow is related to diameter by a
Rd = Do/D1
power function.
That flow in a branch is a function of its diameter is
intuitively obvious. Moving peripherally, the number of Similarly, we can define a flow ratio at a bifurcation as
branches increases while their diameters and the flow
Rf = (Do/D1)"
through each decrease. That flow should be a power
function of diameter is less obvious, but the following
argument will show why this is probable. The number and, by our previous assumption
of branches in successive orders down the tree increases
Rb E Rf
by a constant proportion,Rb. On average, the mean flow
in branches in successive orders will decrease by a similar proportion, namely 1 R b . In terms of moving up Note that Rb is the increase in the number of branches
from the periphery, mean flow in branches of successive in successive orders down the tree, and Rf is the increase
orders increases by Rb and mean diameter by Rd. Thus in mean flow in successive orders up the tree.
The ratio of daughter branch diameters, a , is a n
both flow and diameter are logarithmically related to
order and are therefore likely to be related to each other expression of asymmetry at a bifurcation with respect
to diameters, and is given by
by a power function.
Several methods have been suggested for finding the
(Y = DZ/D1 < 1.0
value of z. Direct measurement of river flow versus
width, depth, etc. to yield l/z was introduced by Leopold
In the lung the summed cross-sectional area of both
and Maddock (1953). Shoshenko et al. (1982) and Mayrovitz and Roy (1983) have derived z from direct mea- the airways and the arteries increases distally. Thus in
surements of blood flow and radius. Some predict z from
optimality equations (Murray, 1926a; Zamir, 1976; Uylings, 1977; Sherman, 1981; Roy and Woldenberg, 1982;
Woldenberg and Horsfield, 1983). Others have derived z Cancelling the constants and calling the proportional
from overall system morphometry established by order- change (usually a n increase) in cross-sectional area at a
ing (Leopold and Miller, 1956; Woldenberg et al., 1970; junction p, we get
Woldenberg, 1972; Horsfield and Thurlbeck, 1981).Murray (1927) and Hooper (1977) derived z from the slope of
p Do2 Dl2 + D2'
the regression of log diameter versus log weight of a
tree branch distal to the diameter. Horsfield and Cum- But Do = RdDl and D2 = aD1, so by substitutionwe get
ming (1968) demonstrated in human airways a linear
relation between log diameter of a branch, and log numRd2Di2 = D12 + a2Di2
ber of end branches (E) supplied by it. McBride and
Chuang (1985) noted a similar relationship, while Suwa
p Rd2 = 1 + a'
et al. (1963) found a n analogous relationship between
arterial radius and number of glomeruli in the kidney.
p=----1 a2
Woldenberg (1972) found that river width was a power
function of E. If flow is proportional to E, then flow is a
power function of diameter, and the exponent z is the
slope of log E on log diameter.
The value of z at individual bifurcations can be calcu-
TABLE 1. Relation of flow, Reynolds number, and mean
In addition, Roy and Woldenberg(l982) have shown that
velocity to arterial branch diameter and the exponent z
at a bifurcation'
Equations 7, 8, and 9 are dimensionless. They can 14
Re a D"-l
Re 0: D1 Re 0: D2 Re a D3
therefore be used with values of a, fi, and Rd, to predict 15
v a D"-2
one variable when the other two are known.
If, flow; D, diameter of an artery; Re, Reynolds number; v, mean
Determination of z allows the calculation of flows, velocity.
velocities, and other hemodynamic parameters. For example, Reynolds number, Re, is given by
plane of focus and were corrected for magnification of x
Re = vDpIq
(10) 20. The diameters ranged from 1.0 to 0.06 mm. The third
source was new measurements, made using an eye-piece
where v is mean flow velocity across the area of a tube, scale in a dissecting microscope, of three peripheral segp = density, and TJ = viscosity. Viscosity varies, with ments broken off from different regions of a resin cast.
shear rate (Caro et al., 1978a), being about 0.03-0.04 This cast had been used to provide other material for
poise a t 1,000 s-l, and may be lower in vessels below distal data in a previous study (Horsfield, 1978).The two
0.3 mm diameter (Fahraeus and Lindqvist, 1931). We casts were made from saline-filled lungs, fully inflated
will use the value 0.03, but different values may easily via the trachea a t 25 cm H20, with resin injected into
be inserted where appropriate. For the density of blood the pulmonary artery at a pressure of about 40 cm HZO.
we take 1.0595. Mean velocity is given by flow divided
Horsfield et al. (1971) studied the form of branches a t
bifurcations and showed how the shape of the cross
by area
section changes. In the majority of cases, the cross secv = 4fID'a
(11) tion reverts to circular soon after the bifurcation; for
consistency the diameters were measured half-way along
and substitutingfor v from equation 11into equation 10 the length of each branch. In the occasional example in
we get
which the cross section was observed not to be circular
at this point, in which case it was almost always elliptical, diameter was measured in two directions a t right
4fD x 1.0595
Re =
(12) angles, and the mean value was used. However, this was
aD' x 0.03
obviously impossible for the data set obtained from the
Substitutingfor f from equation 2 into equation 12 gives photographs, and in this case there would be a small
random error, some branches being recorded as too large,
and others as too small. The effect of this procedure
Re = 44.97k D"-'
would be to increase the standard deviation of the error
of the measurement by a n unknown quantity. The error
Re a D"-l
(14) of measurement of cross-sectional area made on branches
Thus Reynolds number for a branch can be calculated of the cast was estimated from 22 repeated measurements made on selected branches on different days. For
from its diameter if k and z are known.
a branch with a mean estimate of cross-sectional area of
Mean velocity is given by
v = flarea
v a DVD~
v cx Dz-2
Thus when z = 2, v a Do; hence v is constant, and crosssectional area is constant a t the bifurcation. These relations are summarized in Table 1.
The data consist of the parent and daughter branch
diameters at pulmonary arterial bifurcations measured
on resin casts of the human pulmonary arterial tree.
The data were obtained from three sources. The first
was the data set described by Singhal et al. (1973) for
pulmonary arteries down to 0.8 mm diameter, called the
proximal zone. Diameters were obtained by measuring
a resin cast of a pulmonary arterial tree using calipers.
The second source consisted of the photographs made for
the study of the intermediate zone of the same cast
(Singhal et al., 1973). Measurements of diameter were
made a t dichotomies where all the branches lay in the
35.33 mm' the standard deviation (SD) of the measurement was 1.138 mm', and the coefficient of variation
(SDImean x 100) was 3.2%. For a branch with a mean
estimate of cross-sectional area of 2.197 mm2, the SD of
the measurement was 0.2457 mm2, and the coefficient
of variation was 11.2%. The error in the estimated value
of z depends on the errors of three different diameter
measurements in a rather complicated way, as suggested by equation 3, and the coefficient of variation for
this has not been calculated. However, Horsfield and
Thurlbeck (1981)showed that the calculated value of z
is very sensitive to errors in the measurement of diameters, and may therefore be subject to considerable variability, which is shown by the wide range of values for z
which we found, from 0.4 to 14.5. In such circumstances
it is essential to make a large number of estimates in
order that the mean value shall be a good estimate of
the true value.
The three data files were merged and a computer
program written to calculate z using equation 3, alpha,
and cross-sectional areas at each bifurcation. Equation
3 was solved by iteratively substituting values of z with
decreasing increments, until a value was found that
gave a n error of less than 0.0001. The equation for the
- 2.4
-5 1.6
0 .
0 .
Fig. 1. Grouped data for plot of log sum area of daughter branches
against log sum area of parent branch at 1,937 arterial bifurcations.
Data grouped at log intervals of 0.2. 0 , 1-10 data points; 0,
data points;
31 -100 data points; 0 , more than 100 data points.
regression of y = log summed daughter areas versus
x = log parent area was calculated to see whether there
was any change in the ratio of daughter areas to parent
area as a function of the size of the parent branch.
Logarithms were used because the data are log normally
distributed, and a bivariate regression of untransformed
data would be invalid.
Figure 1 shows grouped data for y = log sum area of
daughter branches against x = log area of parent. Mean
values and the regression equation were calculated from
the data before grouping; the equation is y = 0.0367
1.0003x, r = 0.993, mean x = -0.2601, mean y =
-0.2235 and n = 1,937.
The slope is not significantly different from 1.0, and
we therefore consider it reasonable to interpret the line
as being parallel to the line of identity. It also lies above
the line of identity. These findings imply that mean
cross-sectional area a t a bifurcation increases in constant proportion, independently of the size of the parent
branch. Given mean x, mean y, and a slope of 1.0, the
intercept can easily be calculated. It is 0.0366, and its
antilog is the area ratio, p = 1.0879. Table 2 gives the
95% confidence limits for the regression line and the
corresponding values of fl calculated from the antilogs of
the given values of y and x. The range is wide (0.648 to
1.831) but differs little with varying values of x.
Table 3 shows the range, mean values, standard deviation, and 95% confidence limits of the means of CY and
log z, along with the corresponding values of z. Mean log
z was used because the value of z showed a log normal
distribution. Although both CY and z have wide ranges,
the 95% confidence limits of the means are fairly narrow
because of the large number of observations. Neither z
nor CY was found to correlate with parent branch diameter. Mean diameter ratio can be calculated from equation 8, and substituting 0.7849 for a! and 1.0879 for 0,
we get Rd = 1.2188.
Considering a hypothetical idealized bifurcation in
which the system average values of a and Rd apply, the
value of z may be calculated. Let Do = 1.0; then D1= 1/
Rd = 0.8205. Similarly D2 = D ~ C=Y0.6440. From equation 3
+ 0.6440" = 1.0
which, when solved iteratively, gives z = 2.2925. Alternatively, the same answer can be obtained from equation 9. This answer compares with a mean value of z
from the data of 2.3779, and a geometric mean of 2.1951.
Thus the three estimates of mean z fall approximately
in the range 2.3 k 0.1.
Considering flow a t this same bifurcation, then from
equation 2, f0 a 1.0", f i 0: 0.8205", and f2 a 0.6440". For
unit flow in the parent branch, and using the idealized
value of z = 2.925, we calculate the proportion of flow to
each daughter branch
TABLE 2. Confidence limits (95%)of the regression line of
log sum daughter areas versus log parent area (Fig. 1) and the
corresponding values of p'
Best estimate
x = log
of y = log sum
parent area daughter areas
limits of y
Corresponding values of
= (antilog y)/(antilog x)
1.0902 (0.6491-1.8310)
1.0877 (0.6489-1.8238)
1.0859 (0.6480-1.8239)
area ratio at a bifurcation.
TABLE 3. Mean, range, and standard deviation of a and log z, with 95%
confidence limits of the means'
log z
limits of the mean
(2.1518-2.2408 1
'Values in parentheses are the corresponding values of z.
TABLE 4. Morphometric parameters in the human
pulmonary arterial tree
fo = 1.0
f i = 0.6354
f2 = 0.3646
It will be seen that fo = fl + f2, a s required by continuity. The ratio of flow in the daughter branches is f2/f1 =
0.5739. Mean flow velocity v is given by fkross-sectional
area, and using the idealized values for f and D given
above, we obtain vz/vl = 0.9314.
We have already shown that the ratio fi/fo s 1/Rb, so
that the flow ratio, Rf, is given by fo/f1 = Rb. Since
How obtained
Regression equation from data
Mean value from data
Calculated from (Y and
Arithmetic mean from data
Geometric mean from data
Calculated from Rd and a
Calculated from Rd
Calculated from RdZ
area ratio at a bifurcation; a,diameter ratio of daughter branches;
z,exponent relating flow to diameter, where flow a DZ;Rd, diameter
ratio; Rb, branching ratio; Rf, flow ratio; w, order of main pulmonary
Rb z Rf
(Do/D1)" = RdZ
and given the values of z = 2.2925 and Rd = 1.2188,
then Rb G Rf = 1.574 (Woldenberg, 1972; Horsfield and
Thurlbeck, 1981).
The number of orders in the arterial tree can be estimated from the diameter ratio, thus
D1 x RdW-'
where w is the order of the main pulmonary artery.
Given Dw = 30 mm, the mean diameter of a n order 1
branch, D1 = 0.013 mm, and Rd = 1.2188, then w = 40.
Note that the number of generations (counting bifurcations down from the main pulmonary artery) is equal to
the number of orders on the longest pathway. Table 4
gives a summary of the values obtained for the more
important morphometric parameters in the pulmonary
arterial tree.
In order to calculate Reynolds number from equation
13, the value of k is required. This of course depends on
the total flow (cardiac output), and as a n example this
will be taken to be 5.0 literdmin (83.33 ml/s) in the main
pulmonary artery of 3.0 cm diameter. From equation 2
k = f/D" = 83.33/3.02.2925
k = 6.7143
Substituting for k from equation 19 into equation 13
Re = 44.97 x 6.7143 Dzpl
Re = 302 D1.2925
Table 5 gives some examples of Re calculated from equation 20. Since viscosity is reduced in vessels less than
0.3 mm in diameter, perhaps by as much a s 50% (Fahraeus and Lindqvist, 1931), calculated Re for such vessels may be too high by the same factor. These
calculations were based on the assumption of steady
flow throughout the cardiac cycle, but mean systolic or
peak systolic flow could be used instead as required.
The diameter data were obtained from two pulmonary
artery casts that were measured in three different ways,
so it was reassuring that the log area data showed a
linear correlation with r = 0.993. Apart from a few
trichotomies, all the available data down to 0.8 mm
diameter were utilized. In the case of the photographs
and the distal cast, samples were taken from three separate sites in the lung. Within these samples, all available data were utilized. Thus sampling bias could only
arise from the choice of sites for each of the two sets of
three samples. These sites were deliberately taken from
widely separate parts of the casts, and no indication of
the results they might give could be obtained until after
they had been measured. We therefore do not consider it
likely that sampling bias was a serious problem. We
consider a n ordinary least squares regression to be a
reasonable method to use when r2 = 0.986. Furthermore, because the measurement errors in y (log summed
daughter areas) were assumed to be about twice the
errors in x (log parent area), the regression line would
give a good estimate of the functional relation (Davies
and Goldsmith, 1980).
More important is the fact that the data were obtained
from measurements of casts. With the lung inflated a t
25 cm H2O and the resin injected at a pressure of about
40 cm HzO, the vessels are probably near or a t their
maximum diameters, and thus may well have been
smaller during life. The data must therefore be taken as
applying to maximum arterial diameters. However, if
vessels maintain the same relative dimensions that they
have a t lower pressures, then the estimated values of z,
diameter ratio, and area ratio would still be applicable,
although k would be too small.
An implicit assumption has been that the values of z
and k are the same in the three branches forming a
bifurcation, and as the daughter branches become the
parent branches of the next bifurcations, it is also implied that the values are constant all the way down the
tree. Alternatively, both k and z might vary together in
such a way that equation 2 holds true for each individual bifurcation. We plotted the value of z calculated at
each bifurcation against the diameter of the parent
branch and found no change in mean z with parent
branch diameter. Thus there is no evidence of a systematic change in z down the tree. However, considerable
local variations in the value of z were observed. If at
junction (i) the diameters of the three branches and the
flow in the parent branch are known, and equation 3
TABLE 5. Diameter of arterial branches and Reynolds
number calculated from equation 20 with cardiac output of
5.0 litedmin
Diameter (cm)
Reynolds number
holds exactly (except when D1 2 Do > Dz), then k(i) and
the flows in the daughters can be found. Should a daughter of junction (i) be also the parent of junction (j), then
the values of z(j) and k(j) can similarly be calculated.
Thus, if f(c)is the flow in the common branch, then
[k(j)/k(i)] Dz(i)-z(i) = 1
dissipation during laminar flow, when z = 3, other factors that are optimized a t higher diameters are also
operative, with the combined result that z has a lower
observed value of 2.3.
Part of this work was supported by Collaborative Research Grant 376184 from the Senior Scientists Program
(21) of N.A.T.O.
Equation 2 is only true in a statistical sense for the tree
as a whole; there can never be perfect agreement at each
individual junction, if only because of the errors in the
The optimum value of z, calculated by Uylings (1977)
to minimize the total of metabolic power losses (proportional to volume) plus frictional power losses for fully
turbulent flow, is 2.333. A similar calculation by Murray (1926a) gave the optimum value for z for fully developed laminar flow as 3.0. Our mean values therefore fall
on the lower limit of the predicted range of 2.3-3.0.
Turbulent flow probably does occur in the proximal aorta
(Caro et al., 1978b), in which Reynolds number exceeds
2,000. However, even though the main pulmonary artery has the same flow and a similar diameter, turbulent
flow probably does not occur in the pulmonary artery
because it has more elastic
(Caro et al., 1 9 7 8 ~partly
walls. As Table 5 indicates, Re falls off rapidly in the
smaller branches, and turbulent flow in these is out of
the question. The finding of such a relatively low value
of z throughout the pulmonary arterial tree is therefore
unexpected, although low values have previously been
noted (Woldenberg and Horsfield, 1983).Optimum values of z for flow in a branching tubular structure with
short segments has not yet been calculated, and these
features might help to account for the low value. Alternatively, z may be optimized for other variables in addition to minimum power dissipation.
A value of z lower than 3 suggests the possibility that,
for equation 3 to hold true, diameters are greater than
those appropriatefor minimum power loss with laminar
flow. This suggestion raises the question of what might
be the explanation €or such a n increase in the arterial
diameters. First, a n increase in diameter lowers mean
velocity and hence reduces the work of accelerating the
blood during systole, a factor that we have ignored in
our calculations. Second, the reduced velocity diminishes the likelihood of turbulent flow, which dissipates
more energy than laminar flow. Third, the larger vessels have a reservoir function, which, along with the
elasticity of their walls, enables much of the blood ejected
during systole to be accommodated, and then forwarded
during diastole, thus distributingflow more equally over
the cardiac cycle. Finally, because we used a pressure of
40 cm water for the resin when making the casts, the
arterial diameters would have been a t or near their
Murray (1926a) showed that a t the optimum for minimum power dissipation with laminar flow, when z = 3,
the work of overcoming friction during arterial blood
flow is half the metabolic cost of the arterial blood volume. When, as a result of the requirement for increased
arterial diameters, and hence volumes, the optimum
moves towards more metabolic work, the value of z
decreases. Thus, in summary, while the pulmonary arterial tree may be optimized partly for minimum power
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sectional, diameter, branches, pulmonaria, area, cross, tree, human, arterial
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