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1 Department of Medicine, The distribution
of red blood cells in alveolar capillaries is typically nonuniform, as
shown by intravital microscopy and in alveolar tissue fixed in situ. To
determine the effects of red cell distribution on pulmonary diffusive
gas transport, we computed the uptake of CO across a two-dimensional
geometric capillary model containing a variable number of red blood
cells. Red blood cells are spaced uniformly, randomly, or clustered
without overlap within the capillary. Total CO diffusing capacity
(DLCO)
and membrane diffusing capacity
(DmCO) are calculated by a
finite-element method. Results show that distribution of red blood
cells at a fixed hematocrit greatly affects capillary CO uptake. At any
given average capillary red cell density, the uniform distribution of
red blood cells yields the highest
DmCO and
DLCO,
whereas the clustered distribution yields the lowest values. Random
nonuniform distribution of red blood cells within a single capillary
segment reduces diffusive CO uptake by up to 30%. Nonuniform
distribution of red blood cells among separate capillary segments can
reduce diffusive CO uptake by >50%. This analysis demonstrates that
pulmonary microvascular recruitment for gas exchange does not depend
solely on the number of patent capillaries or the hematocrit; simple
redistribution of red blood cells within capillaries can potentially
account for 50% of the observed physiological recruitment of
DLCO from rest to exercise.
pulmonary diffusing capacity; membrane diffusing capacity; capillary model; finite-element analysis
IT IS KNOWN THAT PULMONARY diffusing capacity
progressively increases about two- to threefold from rest to exercise
without reaching an upper limit, even as peak exercise is approached
(2, 12). It is unclear from where these large reserves of diffusing capacity arise from rest to exercise. It is known that with exercise the increase in lung volume and the number as well as volume of patent
capillaries can augment pulmonary gas transfer (3, 18). In addition,
our group and other investigators (1, 6, 10, 11, 19) have
shown that physical properties of the red blood cell (RBC)
can also alter the diffusive process in important ways. Utilizing a
finite-element analysis and principles of heat exchange, we simulated
the diffusive uptake of CO in the lung
(DLCO)
across a geometric model of a pulmonary capillary segment and used this analysis to validate the conceptual framework underlying the
physiological and morphometric techniques of estimating
DLCO as
well as its components: membrane diffusing capacity
(DmCO) and capillary blood volume (10). We found that DmCO
and, hence,
DLCO are
sensitive to changes in the spacing of RBCs within the capillary, i.e., capillary hematocrit (10), as well as to changes in red cell shape
(11). As spacing intervals between red cell centers increase, effective
capillary surface area for diffusive gas exchange decreases; hence,
DLCO and
DmCO per unit capillary segment
length should decrease. The relationship between
DLCO of a
capillary segment and the spacing intervals between red cell centers is
such that changing spacing intervals between red cell centers has a
greater effect on
DLCO at a
low hematocrit than at a high hematocrit (10, 11). Consequently, for a
given capillary segment length containing a fixed number of RBCs, the
decrease in
DLCO caused
by an increase in the spacing between red cell centers in one region of
the capillary will not be completely compensated by a corresponding
decrease in the spacing in another region in the absence of any change in red cell numbers. The potential effect of uneven distribution of
RBCs within and among open capillaries on
DLCO has
never been examined.
Because of the above observations we ask the question: How much of the
increase in
DLCO from
rest to exercise can be explained by improving the uniformity of red
cell distribution within and among already patent pulmonary capillaries
in the absence of any structural change in the capillary bed? We
hypothesize that, for a fixed average red cell density, uneven spacing
will significantly impair diffusive gas exchange, relative to uniform
spacing. This is an important issue that is difficult to address by
physiological studies but that can be approached from a theoretical
standpoint. In the present study, we utilized the same geometric model
and finite-element analysis as described previously (10) to determine the possible magnitude by which nonuniform red cell distribution can
alter diffusive uptake of CO across the lung.
Geometric model.
The capillary model consists of a cross-section (thickness 1 µm)
through the long axis of a pulmonary capillary segment (length 90 µm). The capillary segment was divided into 12 equal pockets; each
pocket can accommodate one circular-shaped RBC (diameter 7.5 µm). One
to twelve RBCs are placed single-file within the capillary in different
distributions: uniformly spaced (equal distance between neighboring red
cell centers), randomly spaced (unequal distance between neighboring
red cell centers), or clustered (distance between adjacent RBC centers = 7.5 µm and lateral RBC membranes touch one another) (Fig.
1). Adjacent RBCs do not overlap. Dimensions and constants employed (7, 9, 15) are listed in Table
1.
ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX
INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX
METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX
View larger version (24K):
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Fig. 1.
Geometric model of a pulmonary capillary segment divided into 12 pockets and containing 4 red blood cells (RBCs) arranged in a uniform
distribution (equal spacing between red cell centers), a random
nonuniform distribution (unequal spacing between red cell centers), and
a clustered distribution (adjacent red cell membranes touch but do not
overlap). See Table 1 for dimensions.
Table 1.
Dimensions and constants of capillary model
CO, where
CO is specific rate of CO
uptake by RBCs), is modeled as a resistance to CO
diffusion across a thin red cell membrane; the resistance is varied in
accordance with the assumed alveolar
O2 tension (in mmHg) to accurately
mimic the values of CO measured
by Holland et al. (9) in dog RBCs at 39°C
|
(1) |
|
(2) |
is Bunsen solubility coefficient in lung tissue and plasma
(mmHg
1);
DCO is diffusion
coefficient (µm2/s);
PCO is in mmHg; and 
is
gradient operator (=
i · 
/x + j · /y + k · /z)
(µm1).
The boundary conditions are that
PCO = 1 mmHg in the alveolar
phase 5 µm above the air-tissue interface and
PCO = 0 mmHg at the inner
membrane surface of the RBCs. Because RBCs may not be equally spaced
and, hence, the capillary model may not be symmetrical, we analyzed the
entire capillary segment for each asymmetric distribution. The model is
divided into 3,724 connecting quadrilateral and triangular elements and
3,805 nodal points, each with its own respective diffusion properties
in air, tissue, and plasma (Fig.
2, A
and B). Through this discretization
process, Eq. 2 is transformed into
3,381 simultaneous algebraic equations (excluding boundary constraints)
from which the PCO at each nodal
point can be solved in the same manner as described previously (10).
The matrix equation has the form
|
(3) |
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(4) |
PCO/n
denotes PCO gradients evaluated
along the normal direction from a constant
PCO surface. The total CO flow, equivalent to
DLCO of
each typical region, is obtained by summing the flow along the boundary
surface of air-tissue barrier for all the elements
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(5) |
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(6) |
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RESULTS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX
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Effect of RBC distribution on diffusing capacity
within a 90-µm capillary. The pattern of CO flux over
the red cell surface is shown in examples of a uniform, nonuniform
random, and clustered distribution in Fig.
3,
A-C.
The magnitude of flux is represented by the length of the vector.
Figures 4 and 5 show the
effect of red cell distribution on
DmCO and
DLCO,
respectively. At a given red cell density, the uniform distribution
yields the highest values, whereas the clustered distribution yields
the lowest. Other nonuniform distributions yield intermediate results.
The magnitude of the effect of cell distribution on
DLCO is
shown in Fig. 6. A simple rearrangement of
RBCs at a given red cell density within a single capillary segment can
change DLCO
by up to 33%. Red cell distribution has the greatest effect on
DLCO when
there are between three and seven cells per capillary segment, corresponding to a hematocrit of 18-43%.
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Estimation of DLCO
for capillaries with uneven RBC spacing.
The estimation of
DLCO by
finite-element analysis in a capillary with uniformly spaced RBCs is
simplified by symmetry. Because of the symmetry of the model and flux
distributions, finite-element analysis is required on only one quadrant
of an RBC to describe the entire capillary. Using the results obtained
assuming uniform spacing of RBC at different hematocrits, we calculated
the relationship of
DLCO per
unit capillary segment length between red cell centers and spacing
intervals, as shown in Fig. 7.
|
1 · mmHg1.
For the same number of RBCs with uniform spacing of 22.5 µm, DLCO would
be given by (4 × 22.5 × 0.0260) = 2.34 µm3 · min1 · mmHg1.
By this comparison, efficiency of CO uptake would be reduced 16.5% by
the nonuniform RBC distribution.
On the other hand, if significant interaction between CO fluxes
generated on either side of unequally spaced RBCs develops, errors may
result in this comparison. To determine the magnitude of this potential
error, we carried out finite-element analysis on 12 sets each of
representative random distributions of three, six, and nine RBCs in a
90-µm capillary at O2 tensions
of 80 and 560 Torr (a total of 72 comparisons). The resultant
comparison (Fig. 8) of
DLCO,
obtained by direct analysis of nonuniformly distributed RBCs and
corresponding indirect estimates made by using the data in Fig. 7,
demonstrates a correlation so close to the line of identity that we can
neglect this potential source of error. Hence, all of our subsequent
analyses of the effects of nonuniform red cell spacing on diffusive
uptake of CO are based on data from Fig. 7.
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX
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Summary of findings. The present theoretical analysis provides the first quantitative evidence that uneven distribution of RBCs at a fixed hematocrit or red cell density significantly affects diffusive uptake of CO in pulmonary capillaries. For a given red cell density, uniform spacing of RBCs is associated with a higher DmCO and DLCO than is nonuniform spacing; clustering of cells is associated with the lowest DmCO and DLCO. The magnitude of this effect is influenced by the average red cell density, being greatest at a density equivalent to a capillary hematocrit of 18%. Simple rearrangement of RBCs within a single capillary segment can change DLCO by up to 33%. For a given red cell number, improvement in the uniformity of RBC distribution among five separate patent capillary segments can potentially increase DLCO by over 50%. In the physiological range of capillary hematocrit ranging from 25 to 43% (equivalent to between 25 and 35 RBCs, respectively, distributed among five 90-µm capillaries), improvement in red cell distribution from a clustered to uniform pattern could increase DLCO by 30-50%.
Effect of red cell characteristics on diffusing capacity. The particulate nature of RBCs leads to an inherently nonuniform distribution of hemoglobin that creates a mismatch between the gas-exchange surfaces of the red cell and the septal tissue. Distribution of RBCs within capillaries is determined by various factors, including the physical properties of RBCs and the capillary network, local flow dynamics, as well as the sequestration of capillary leukocytes (14). Changes in the static and dynamic properties of RBCs can alter diffusive gas exchange in important ways. For instance, in isolated perfused rabbit lung, diffusing capacity for O2 (DLO2) is lower when the lung is perfused with red cell suspensions than with hemoglobin solutions (6), suggesting that a uniform distribution of hemoglobin facilitates O2 uptake. A decrease in the deformability of RBCs in isolated rabbit lungs reduces DLO2 (1). This effect may be due to a thicker unstirred layer around undeformed RBCs flowing at low velocities; the unstirred layer is thinner around deformed RBCs flowing at higher speed because of better mixing. Alternatively, the loss of red cell deformability may also lead to a nonuniform distribution of capillary flow resistance, resulting in nonuniform regional hematocrits (17). The deleterious effect of deformation on diffusive uptake may be offset by the simultaneous improvement in hydrodynamics of the deformed cells, which might lead to greater homogeneity in the distribution of capillary RBCs. These opposing effects highlight the complex structure-function interaction between the red cell and the capillary network, complicating the interpretation of physiological data.
Wang and Popel (19) simulated the deformation of RBCs from a circular to parachute shape and reported that deformation results in a 26% decline of O2 flux, the effect being inversely related to the capillary transit time of the red cell. We reached a similar conclusion by finite-element analysis of CO flux in a geometric model of the pulmonary capillary containing circular or parachute-shaped RBCs (11). We found that the lower DLCO associated with parachute-shaped RBCs is due to a more heterogeneous distribution of CO flux over the red cell surface, and the effect is greater as capillary red cell density is reduced. Thus, in addition to capillary red cell volume, the mean spacing between adjacent RBCs, the geometry of the RBC, and regional red cell distribution are all important factors that affect DLCO.
Red cell distribution and recruitment of diffusing capacity. From rest to exercise there is a nearly twofold increase of DLCO in a linear relationship with respect to cardiac output (2, 13). Up to peak exercise, there is no evidence of an upper limit being reached in this relationship, indicating the existence of large physiological reserves in diffusing capacity. It is believed that recruitment of diffusing capacity reserves occurs because the increased pulmonary perfusion opens previously collapsed capillaries and distends patent capillaries, resulting in a higher pulmonary capillary blood volume as well as a larger red cell-endothelial interface for diffusion. In highly aerobic animals that possess a large splenic reservoir of blood, i.e, horses and dogs, autotransfusion by splenic contraction is an additional factor that further augments DLCO on exercise by increasing total blood volume and hematocrit (20). Theoretical analyses suggest that an increase in capillary hematocrit from 10 to 50% can potentially increase DLCO and DLO2 more than threefold (5, 10). Although it has been suggested that a more uniform regional red cell distribution may also contribute to increasing DLCO during exercise, this effect has not been previously demonstrated. At rest, red cell distribution among pulmonary capillaries is markedly heterogeneous; there is a wide range of transit times through the capillary bed (8, 16). As perfusion increases, both the mean and the relative dispersion of red cell transit time distribution decrease, suggesting a more homogeneous red cell distribution, which mitigates the decline in mean red cell transit time and maintains a minimum transit time just above the theoretical threshold required for complete saturation with O2 (16). The improved homogeneity of red cell distribution can potentially augment DLCO by another 30-50% without further change in the number of perfused capillaries.
We thus conclude that changes in the distribution pattern of capillary RBCs can account for a large component of the recruitment in DLCO observed physiologically from rest to exercise.
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APPENDIX |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX
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How a Random Distribution of Cells Is Defined Within a Single Capillary
We define a random distribution of red cells within and among capillaries the same as the spatial distribution of particles in statistical mechanics. Space is subdivided into small pockets or cells; assumptions are made about the relative probability of a pocket being filled, whether particles and pockets are distinguishable, and how many particles can exist in the same pocket. We assume the following.1) A 90-µm capillary is subdivided into 12 pockets, the width of each is the diameter of a red cell (7.5 µm). The capillary is circularized to eliminate discontinuities at the ends.
2) Each pocket has an equal probability of being occupied.
3) RBCs are indistinguishable from one another.
4) Pockets are distinguishable from one another only by occupancy or lack of occupancy.
5) No more than one RBC can occupy a pocket at a given time.
6) Multiple capillaries are distinguishable only if occupancy patterns are different.
The random distribution of a given number of RBCs (q) in n pockets follows a Fermi-Dirac statistic (4). The number of possible ways (N) that q cells can be arranged in 12 pockets is given by
|
(A1) |
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(A2) |
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How a Random Distribution of Cells Is Defined Among Multiple Capillaries
The random distribution of q indistinguishable cells among n capillaries follows Bose-Einstein statistics (4), with the added restriction that no more than 12 RBCs can be present in a given capillary. One can generate by hand or by computer all the possible combinations of cell occupancy; Table 2 gives an example of q = 30 cells and n = 5 capillaries.
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In Table 2, the number of permutations (P) of each combination (i) is calculated as follows
|
(A3) |
i Pi.
Note that the completely uniform distribution of six cells each in all
five capillaries (combination 252)
has the lowest probability. An example of the random probability
distribution for 30 cells in 5 capillaries is shown in Fig.
11.
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ACKNOWLEDGEMENTS |
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The authors gratefully acknowledge the statistical assistance of Dr. William H. Frawley of Academic Computing Services, University of Texas Southwestern Medical Center.
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FOOTNOTES |
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This project was supported by National Heart, Lung, and Blood Institute Grants R01-HL-40070, R01-HL-54060, and R01-HL-45716. C. C. W. Hsia was supported by an Established Investigator Award from the American Heart Association. Parts of this work have been published in abstract form in FASEB J. 12: A498, 1998.
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. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: C. C. W. Hsia, Dept. of Medicine, Univ. of Texas Southwestern Medical Center, 5323 Harry Hines Blvd., Dallas, TX 75235-9034 (E-mail: Connie.Hsia{at}emailswmed.edu).
Received 10 November 1998; accepted in final form 3 February 1999.
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REFERENCES |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX
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