Abstract
Hsia, C. C. W., C. J. C. Chuong, and R. L. Johnson, Jr.Red cell distortion and conceptual basis of diffusing capacity estimates: finite element analysis. J. Appl. Physiol. 83(4): 1397–1404, 1997.—To understand the effects of dynamic shape distortion of red blood cells (RBCs) as it develops under highflow conditions on the standard physiological and morphometric methods of estimating pulmonary diffusing capacity, we computed the uptake of CO across a twodimensional geometric capillary model containing a variable number of equally spaced RBCs. RBCs are circular or parachute shaped, with the same perimeter length. Total CO diffusing capacity (Dl
_{CO}) and membrane diffusing capacity (Dm
_{CO}) were calculated by a finite element method. Dl
_{CO}calculated at two levels of alveolar
 RoughtonForster technique
 morphometry
 pulmonary diffusing capacity
 membrane diffusing capacity
 random linear intercept
 capillary model
using the finite element method (FEM) (1), we previously showed that diffusive uptake of CO (Dl
_{CO}) across a geometric model of a pulmonary capillary segment is dependent on the spacing of red blood cells (RBCs), or the hematocrit, within the capillary (10). If the RBCs are assumed to be circular, the RoughtonForster (RF) technique (13) accurately recovers the conductance of the tissueplasma membrane (membrane diffusing capacity; Dm
_{CO}) at a low hematocrit but modestly overestimates Dm
_{CO} as hematocrit increases; errors arise because conductance of the membrane for CO varies with alveolar
However, under dynamic flow conditions, RBCs become distorted and assume a variety of asymmetric shapes, including parachutelike shapes (15). Such deformation of the RBC reduces shear stress and flow resistance (2, 3) but can have deleterious effects on diffusive gas exchange (17). Shape distortions might also exaggerate the conceptual errors inherent in the RF and morphometric techniques of estimating Dl _{CO}, although the magnitude of such effects has never been examined. We have utilized the geometric model and analytic approach described previously (10) to examine the effect of shape change of RBCs on the diffusive uptake of CO estimated by different methods.
METHODS
Geometric model.
The capillary model consists of a cross section (1 μm thick) through the long axis of a pulmonary capillary segment. Different numbers of RBCs are equally spaced within the capillary and are circular, as described previously (10), or parachute shaped with the same perimeter length as the circular RBCs (Fig. 1). The parachute shape of RBCs was digitized from illustrations by Skalak and Branemark (15) and Wang and Popel (17). We assume an infinite reservoir of CO in the alveolar air space. The RBCs represent infinite sinks for CO [CO partial pressure (Pco) within RBCs = 0]. The RBC component of CO uptake (1/Θ_{CO}) is modeled as a resistance to CO diffusion across a thin RBC membrane; the resistance is varied in accordance with the assumed
FEM.
We assume that the flux of CO is due entirely to tension gradients of CO driving CO diffusion into RBCs and that Pco gradients reach steady state immediately. Diffusive transport is described by the partial differential equation
Comparison with RF method.
Dl
_{CO(FEM)}calculated at 80 and 560 Torr
Comparison with morphometric method.
The geometric capillary model was subjected to standard morphometric analysis (18). Alveolarcapillary surface area and number of RBCs of the anatomic model are known. Morphometric Dm
_{CO}[Dm
_{CO(morphometry)}] was estimated using the modified method of Weibel et al. (19) and compared with Dm
_{CO(FEM)}. A grid was randomly laid over the capillary model; the distance of all intercepts of the test line with the barrier (l), from the epithelial surface to the nearest RBC membrane, was measured with a logarithmic ruler. Intercepts that do not cross both epithelial and RBC surfaces were not measured. Orientation of the grid was varied, and the measurements were repeated until at least 60 intercept lengths had been measured. The harmonic mean intercept length through the tissueplasma barrier (l
_{hb}) is given by the mean of all reciprocal intercept lengths
RESULTS
CO flux.
The pattern of CO flux over the RBC surface is shown for onehalf of an RBC in Fig. 3. The magnitude of flux is represented by the length of the vector. The distribution of flux is inhomogeneous, being more concentrated over the trailing tails of the parachuteshaped cell than over the leading surface. CO flux is low across a large portion of the RBC membrane along the infolded trailing surface. The inhomogeneity of flux distribution is more pronounced when spacing between RBCs is small and when
Diffusing capacity estimated by FEM.
Figure 4 shows total Dm
_{CO} for the capillary as well as Dm
_{CO} per RBC estimated by FEM at two levels of
For a given number of RBCs in the capillary model, Dm _{CO(FEM)}per 100μm capillary and Dm _{CO(FEM)}per RBC are lower for parachuteshaped than for circular RBCs (Fig.5); the difference diminishes as the number of RBCs increases (17% lower at 1 RBC per capillary and 8% lower at 13 RBCs per capillary). A similar pattern is seen in Dl _{CO}estimated by FEM (13% lower at 1 RBC per capillary and 6% lower at 13 RBCs per capillary).
Diffusing capacity estimated by morphometric method.
Figure 6 shows the changes in mean linear diffusion path between the epithelial surface and the RBC membrane (l _{hb}); for a given number of capillary RBCs,l _{hb} is significantly longer for parachuteshaped than for circular RBCs. Comparison of Dm _{CO} per 100μm capillary estimated by different methods is shown in Fig.7 for circular and parachuteshaped RBCs. When the harmonic barrier thickness (τ_{hb}) is used to estimate the path length for diffusion (Eq. 11 ), morphometric estimates are grossly elevated compared with corresponding estimates by FEM for both RBC shapes. Differences between FEM and morphometric estimates diminish as the number of capillary RBCs increases. Morphometric estimates range from 352% (2 cells) to 52% (12 cells) higher than corresponding estimates by FEM for circular RBCs and from 418% (2 cells) to 57% (16 cells) higher for parachuteshaped RBCs. As the number of capillary RBCs increases, morphometric overestimation of Dm _{CO}diminishes more rapidly for parachuteshaped than for circular cells. At 10 cells per 100μm capillary, overestimation of Dm _{CO} is similar for circular and parachuteshaped cells. Above 10 cells per 100μm capillary, overestimation of Dm _{CO} is slightly greater for circular cells. When values at the same number of RBCs per 100μm capillary are compared, morphometric estimates of Dm _{CO} are 5% (2 cells) and 16% (12 cells) lower for parachuteshaped than for circular cells. Similarly, morphometric estimates of Dl _{CO} are 2% (2 cell) to 13% (12 cells) lower for parachuteshaped than for circular cells.
Figure 8 shows the ratio of morphometric Dm _{CO}estimated usingl _{hb}(Eq. 12 ) to Dm _{CO}estimated by FEM. We previously showed thatl _{hb} more accurately reflects the molecular diffusion distance than does τ_{hb} (10); Eq.12 yields significantly lower estimates of Dm _{CO} and Dl _{CO} thanEq. 11 , i.e., smaller differences than estimates by FEM, particularly at low numbers of capillary RBCs. In fact, above 10 parachuteshaped RBCs per capillary, Dm _{CO(morphometry)}calculated usingl _{hb} is slightly (5–10%) below corresponding Dm _{CO}estimated by FEM. This slight underestimation disappears at 16 parachuteshaped cells per 100μm capillary when the cells are almost maximally packed.
Diffusing capacity estimated by RF method.
Deviations of Dm _{CO(RF)}from Dm _{CO(FEM)}are modest (Fig. 9). At a low hematocrit (<6 RBCs), Dm _{CO(RF)}for circular RBCs is 2% higher than corresponding Dm _{CO(FEM)}, whereas Dm _{CO(RF)}for parachute cells is 5% higher than Dm _{CO(FEM)}. As capillary hematocrit increases, errors in Dm _{CO(RF)}increase progressively for both RBC shapes to reach ∼9–13% above corresponding Dm _{CO(FEM)}.
DISCUSSION
The importance of capillary hematocrit in determining capillary resistance to CO diffusion has again been demonstrated, as in our previous analysis using circular RBCs. The present analysis also reveals that shape distortion of RBCs, as it develops under highflow conditions, significantly reduces diffusive uptake of CO in the lung capillaries. In addition, shape distortion of RBCs exaggerates the overestimation of Dm _{CO} caused by conceptual simplifications inherent in the RF technique. Shape distortion also exerts complex effects on the errors inherent in the morphometric technique of estimating Dm _{CO}. These effects are modulated by spacing between adjacent RBCs and are discussed below.
Hematocrit and RBC distribution.
By the classic concept of diffusive gas transfer in the alveoli, the rate of gas uptake is dependent on the diffusivity of the gas in tissue and plasma, the alveolarcapillary surface area, and the diffusion distance across the alveolarcapillaryplasma barrier. This concept does not formally consider the particulate nature of RBCs. Packaging hemoglobin within discrete RBCs retains the respiratory pigment within the vascular space and avoids the undesirable effects of hemoglobin on vascular tone. On the other hand, it leads to an inherently nonuniform distribution of hemoglobin, i.e., a mismatch of gas exchange surfaces between the RBC and the capillary endothelium. The distribution of RBCs within capillaries is a complex function of interactions among quantity, size, and deformability of RBCs, local flow dynamics, and physical properties of the capillary network. The flow and distribution of RBCs are also affected by margination and sequestration of leukocytes in capillaries (11). That static and dynamic properties of RBCs can alter diffusive gas exchange is shown by various recent reports. Geiser and Betticher (5) reported in isolated perfused rabbit lung that pulmonary diffusing capacity for O_{2}(
Deformation of RBCs.
Effects of RBC deformation on gas transport have been modeled in a single capillary by Wang and Popel (17), who reported that a change from circular to parachuteshaped RBCs decreases O_{2} flux by 26%; this shape effect is inversely related to the RBC residence time within the capillary. Betticher et al. (2) demonstrated in isolated rabbit lungs that reduced RBC deformability reduces
Errors in physiological estimate of diffusing capacity.
Our previous analysis shows that, within the geometric capillary model containing circular RBCs, Dm
_{CO(RF)}estimates are modestly higher than Dm
_{CO(FEM)}estimates at hematocrits at or above physiological level. This overestimation occurs because the RF technique assumes Dm
_{CO} to be constant regardless of
Errors in morphometric estimate of diffusing capacity.
On the other hand, previous analysis shows that estimates of Dm
_{CO} by Weibel’s morphometric technique are grossly elevated with respect to estimates by FEM when the number of capillary RBCs is low, but differences progressively diminish as the number of capillary RBCs increases. Much of the discrepancy between morphometric and FEM estimates could be attributed to an error in the stereological construct, which imposes an arbitrary factor of
1) The morphometric technique measures the distance of randomly oriented linear diffusion paths from the epithelial surface to the RBC surface, whereas FEM reveals that local Pco gradients constrain CO flux by diffusion to markedly curvilinear paths over much of the RBC surface. This curvilinearity is more pronounced for parachuteshaped than for circular RBCs and also more marked when RBCs are far apart than when they are close together. Thus approximation of diffusion distance using random linear intercepts as employed in the morphometric method yields an underestimation of true diffusion distance and an overestimation of Dm _{CO}. As more RBCs are packed into the capillary, the mean diffusion path becomes shorter and more nearly linear; thus errors in Dm _{CO} due to measured values of mean linear path length (l _{hb}) progressively diminish. 2) The morphometric method utilizes the entire available alveolarcapillary surface area in the calculation of Dm _{CO}regardless of the number of capillary RBCs. However, FEM analysis demonstrates that most of the CO flux occurs across only a small portion of the tissue membrane close to an RBC. As the number of capillary RBCs increases, the distribution of CO flux along the alveolarcapillary surface becomes more uniform; i.e., the effective alveolarcapillary surface available for diffusive gas exchange increases and approaches that estimated by morphometry. Thus morphometry grossly overestimates Dm _{CO} at a low hematocrit, and errors diminish progressively as capillary hematocrit increases. 3) The error in morphometry caused by underestimation of molecular diffusion distance due to linear approximation of a curvilinear diffusion path is counterbalanced in parachuteshaped cells by another error arising from a biased sampling distribution over the RBC surface. This source of error is related to RBC geometry and the probability that some portions of the infolded perimeter of the parachuteshaped cell are preferentially sampled by a randomly oriented line, particularly as RBC spacing diminishes (Fig. 10). The probability of sampling any given point along the infolded perimeter of a parachuteshaped RBC by a randomly oriented line through a given point on the epithelial surface (point a) varies from a finite value (regions 1 and3) to zero (region 2), even though these regions subtend the same angle. Because of the concentration gradient of CO and the axial symmetry of the capillary segment, according to FEM most of the CO flux acrosspoint a will reachregion 3 of the RBC, whereas a random linear intercept from point a toregion 1 in fact violates physical laws by running against the local Pco gradient. Therefore, by the random linear intercept method, a significant portion of the infolded RBC perimeter closest to point a is undersampled, whereas the regions farthest frompoint a are oversampled. The net result of this sampling bias is an overestimation of mean diffusion path length over the infolded surface of the RBC, leading to an underestimation of Dm _{CO}. As the number of capillary RBCs increases, this sampling bias increases. However, beyond a certain closeness of RBC packing (16 cells per capillary), this bias disappears, because lateral surfaces of adjacent RBCs become relatively hidden and inaccessible to linear sampling from the epithelial surface. Hence, the apparent mean barrier thickness again decreases (Fig. 6) and morphometric Dm _{CO}abruptly increases (Fig. 7). This sampling bias arises for parachuteshaped but not circular RBCs, because parachuteshaped cells lack full rotational symmetry. We would expect a similar sampling bias to occur in other asymmetric shapes assumed by RBCs.
Limitations of FEM.
As pointed out previously (10), our model is not meant to reproduce reality but, rather, to provide a uniform framework and an independent analytic technique that could be utilized to explore the conceptual basis of our understanding of the pulmonary diffusion process and to reconcile differences between current physiological and morphometric methods of estimating pulmonary diffusing capacity. This stylized capillary model is twodimensional and static; no motion of RBCs is implied. The selection of cell shapes is necessarily arbitrary, since RBCs, in fact, can assume numerous irregular shapes during capillary transit. However, the circular and parachute shapes are representative of a symmetric and an asymmetric configuration, respectively. Furthermore, the parachute is a shape seen in perfused capillaries under direct observation. The boundary conditions are also arbitrary, but variations would not have altered our general conclusions. The primary variable examined in this study is Dm
_{CO} and the potential sources of error in its estimation by the RF and the morphometric methods; we assumed that in vitro measurements of Θ_{CO} at different levels of
We conclude from finite element analysis that shape distortion of the RBCs as develops under highflow conditions alters the distribution of CO flux across the RBC surface and reduces the diffusive uptake of CO. Distortion of RBCs exaggerates conceptual errors in the RF and the morphometric technique of estimating diffusing capacity via different mechanisms. Errors in the RF technique arise from the same source regardless of RBC shape and are most sensitive to changes in RBC spacing in the physiological range of hematocrits. The various sources of error in the morphometric technique exert opposing effects on the estimate of Dm _{CO}; their net effect is most sensitive to changes in RBC spacing when the capillary hematocrit is low. In vivo, the unfavorable effect of RBC shape distortion on diffusive gas uptake may be mitigated by its favorable effect on hydrodynamics and the distribution of capillary RBC flow.
Acknowledgments
This project was supported by National Heart, Lung, and Blood Institute Grants R01HL40070, R01HL45716, and RO1HL46185. C. C. W. Hsia was supported by an Established Investigator Award from the American Heart Association.
Footnotes

Address for reprint requests: C. C. W. Hsia, Dept. of Medicine, University of Texas Southwestern Medical Center, 5323 Harry Hines Blvd., Dallas, TX 752359034.

Parts of this work have been published in abstract form (FASEB J. 10: A362, 1996).
 Copyright © 1997 the American Physiological Society