Vladimir V. Kulish
ASME Assoc. Mem.,
School of Mechanical & Production Engng
Nanyang Technological University
Singapore 639798
José L. Lage1
ASME Mem.,
Mechanical Engineering Department
Southern Methodist University
Dallas, TX 75275-0337
Connie C. W. Hsia
Robert L. Johnson, Jr.
Department of Internal Medicine
University of Texas-Southwestern
Medical Center
Dallas, TX 75235-9034
1
Three-dimensional, Unsteady
Simulation of Alveolar
Respiration
A novel macroscopic gas transport model, derived from fundamental engineering principles, is used to simulate the three-dimensional, unsteady respiration process within the
alveolar region of the lungs. The simulations, mimicking the single-breath technique for
measuring the lung diffusing capacity for carbon-monoxide (CO), allow the prediction of
the red blood cell (RBC) distribution effects on the lung diffusing capacity. Results, obtained through numerical simulations, unveil a strong relationship between the type of
distribution and the lung diffusing capacity. Several RBC distributions are considered,
namely: normal (random), uniform, center-cluster, and corner-cluster red cell distributions. A nondimensional correlation is obtained in terms of a geometric parameter characterizing the RBC distribution, and presented as a useful tool for predicting the RBC
distribution effect on the lung diffusing capacity. The effect of red cell movement is not
considered in the present study because CO does not equilibrate with capillary blood
within the time spent by blood in the capillary. Hence, blood flow effect on CO diffusion
is expected to be only marginal. 关DOI: 10.1115/1.1504445兴
Keywords: Gas Diffusion, Lung Diffusing Capacity, Alveolus, Transient Numerical
Simulation
Introduction
The search for a universal correlation between the lung diffusing capacity and the physical properties of each individual lung
alveolar constituent 共e.g., membranes, tissue, plasma, red blood
cells, etc.兲 has been a major research thrust, 关1– 6兴. Existing theories for estimating the lung diffusing capacity are limited by the
difficulty in characterizing the incredibly complex internal geometry 共morphology兲 of the alveolar region, including the distribution of red blood cells within the capillary bed 关7兴.
The lung diffusing capacity, as defined, is a lumped parameter
that can be estimated from relatively simple measurements 共i.e.,
measurements of inhaled gas volumetric-rate and concentration,
and of exhaled gas concentration, 关8 –10兴兲 and used to indicate
abnormalities in the respiratory diffusion process. However, it is
difficult to infer from variations in the lung diffusing capacity the
precise cause of the abnormality because the sensitivity of the
lung diffusing capacity 共a global parameter兲 to local changes in
the morphology and functionality of the lung is not well known.
There is a need for linking the local diffusion process occurring
within the alveolar region to the measurable variations in the global lung diffusing capacity parameter.
These observations provide grounds for seeking the development of suitable models that simulate locally the gas diffusion
process inside the alveolar region of the lung. Unfortunately, using the classical gas diffusion equation at the alveolus-erythrocyte
level 共referred here as the microscopic level兲 within a lung is
impractical because: 共1兲 the dimensional scale of the domain
ranges from decimeters to microns, requiring a tremendous numerical resolution, and, more importantly, 共2兲 the complex internal alveolar structure 共topology of each constituent兲 is extremely
difficult to access and to map.
A novel macroscopic model for simulating the gas diffusion
within the alveolar region of the lung, intended to overcome the
1
Corresponding author: phone 共214兲768-2361 fax 共214兲768-1473 e-mail
[email protected]
Contributed by the Bioengineering Division for publication in the JOURNAL OF
BIOMECHANICAL ENGINEERING. Manuscript received December 2000; revised
manuscript received May 2002. Associate Editor: J. B. Grotberg.
Journal of Biomechanical Engineering
scale and structure difficulties, was presented recently by Koulich
et al. 关11兴 and used for preliminary simulations of steady processes. A by-product of this model is the introduction of a macroscopic transport property of the alveolar region, called effective
diffusivity. As defined, this property depends on the internal structure of the alveolar region, bringing back the very same structurerelated problem the model originally intended to overcome.
However, it is possible to determine the equivalent effective
diffusivity of the alveolar region by simulating numerically the
single-breath CO-diffusion procedure done in the laboratory and
comparing the lung diffusing capacity obtained from the numerical results to the lung diffusing capacity obtained experimentally.
Once determined, the effective diffusivity can be used with the
three-dimensional, transient macroscopic diffusion model to investigate the effects of red cell distribution on the lung diffusing
capacity.
2
Mathematical Modeling
The macroscopic diffusion equation derived by Koulich et al.
关11兴 is based on the averaging of the diffusion equation within an
elementary representative volume 共REV兲, the building block of
the REV-domain that replaces the alveolar region shown in Fig. 1.
This REV, shown schematically in Fig. 2, contains all the constituents of the alveolar region, for instance gas region, alveolar membrane, interstitial fluid, capillary membrane, plasma, red cell
membrane, etc., except the interior of the red blood cells 共RBC’s兲.
The diffusion equation resulting from the averaging process
within an REV is
⳵具 P典
⫽D effⵜ 2 具 P 典
ot
(1)
where t is the time, 具P典 is the REV-averaged partial-pressure of the
gas being considered during the diffusion process, defined as the
average of the diffusing gas partial-pressure P within an REV of
volume V, namely
Copyright © 2002 by ASME
具 P典⫽
1
V
冕
PdV
(2)
V
OCTOBER 2002, Vol. 124 Õ 609
and D eff is the effective diffusivity of the REV, defined in 关11兴 as
兺冉
c
D eff⫽
Fig. 1 Scanning electron micrograph of the alveolar region of
the lungs „dog…, showing the architecture of alveoli. „Courtesy
of Ewald R. Weibel, University of Bern, Switzerland.…
i⫽1
冊 兺冋
c⫺1
Vi
共 D i ⫺D i⫹1 兲
D ⫹
V i
V
i⫽1
冕
A i,i⫹1
nជ i,i⫹1 ␤ i dA
册
(3)
In Eq. 共3兲, c is the total number of constituents present within the
REV, V i and D i are the volume and diffusivity of each constituent
within V, nជ i,i⫹1 is the unit vector normal to the interface with
surface area A i,i⫹1 between constituents i and i⫹1, and ␤ i is the
pseudo-diffusivity 共tensor兲 of constituent i.
Observe that once D eff is known, Eq. 共1兲 can be solved easily
within the REV-domain by applying suitable initial and boundary
conditions. Therefore, there is no need to describe the internal
structure of the alveolar region when using Eq. 共1兲, as the
structure-information of the alveolus is now embedded into the
term proportional to the area integral 共last summation term兲 of Eq.
共3兲. Unfortunately, the mathematical determination of D eff , Eq.
共3兲, requires the precise mapping of all boundaries of each constituent within the alveolar region. The mapping difficulty of modeling the gas diffusion process directly through each constituent
and across their interfaces is now inherited by the effective diffusivity of the medium, function of the structure and physical 共molecular兲 properties of each constituent within the alveolus, Eq. 共3兲.
It seems that the only alternative to avoid having to map the
topography of the alveolar region is to find a way to determine
D eff indirectly, i.e., without having to calculate the surface integral
of Eq. 共3兲. In what follows we show how the effective diffusivity
can be estimated independently of having to map the internal
structure of the alveolar region, using the lung diffusing capacity.
3
Lung Diffusing Capacity: Single-Breath Technique
The lung diffusing capacity obtained in laboratory results from
unsteady processes, such as the diffusion process during the
single-breath technique 关12兴. This technique consists of having a
gas mixture with a low concentration of CO inhaled, held for a
certain period of time 共usually ten seconds兲 and then exhaled.
During this single-breath process, CO diffuses from the gas region
of the alveolus to the RBC’s. The process is unsteady because the
potential gradient 共i.e., the CO concentration within the gas region兲 driving the diffusion varies 共decays兲 in time.
The lung diffusing capacity of the alveolar region can be obtained from the single-breath technique results using the Krogh
equation 共关12兴, p. 351兲,
具 P 典 ␯ 共 t 兲 ⫽ 具 P 典 ␯ 0 e ⫺ 共 D L P ref /V A 兲 t
(4)
where 具 P 典 ␯ is the result of volume-averaging 具P典 over the entire
REV-domain, and 具 P 典 ␯ 0 is the corresponding initial value 共i.e.,
具 P 典 ␯ at t⫽0兲. The reference pressure P ref is chosen as the total
pressure of dry gases ( P ref⫽9.51⫻104 Pa⫽713 mm Hg), and V A
is the alveolar volume equal to the inspired volume plus the residual lung volume 共a representative, normal, value is V A
⫽4,930 mlSTPD⫽4.93⫻10⫺3 m3 兲.
Equation 共4兲 can be derived from a lumped-capacity approach
to the diffusion process inside the alveolar region. Consider the
alveolar region as a closed impermeable system of volume V A ,
filled with a gas having a certain concentration of CO at time
equal to zero. The volumetric rate of CO, V̇ CO , being depleted
共sink兲 from inside this system 共known as gas-uptake兲, expressed
in terms of the lung diffusing capacity D L , assumed constant, is
V̇ CO 共 t 兲 ⫽D L 共 具 P 典 ␯ 共 t 兲 ⫺ 具 P 典 h 兲
(5)
where 具 P 典 h is the average partial-pressure of CO inside all the
RBC’s, considered equal to zero for the case of CO diffusion.
Now, the species concentration conservation law requires
Fig. 2 Schematic representation of the alveolar region: „a… microscopic level, with all individual constituents; „b… intermediate level, as a representative elementary volume, REV, domain.
610 Õ Vol. 124, OCTOBER 2002
ṁ CO 共 t 兲 ⫽⫺V A
d 具 c 典 ␯共 t 兲
dt
(6)
Transactions of the ASME
where ṁ CO (t) is the CO mass flow rate 共mass gas uptake兲 and
R D eff⫽
具 c 典 ␯ is the time-dependent volume-averaged CO concentration.
Equations 共5兲 and 共6兲 can be combined by setting
V̇ CO 共 t 兲 ⫽
ṁ CO 具 t 典
c ref
(7)
V A d 关 具 c 典 ␯ 共 t 兲兴
c ref
dt
(8)
The last term of Eq. 共8兲 can be re-written in terms of the partialpressure of CO, using the ideal gas law (c⫽M P/RT) for an isothermal process, then
D L 具 P 典 ␯ 共 t 兲 ⫽⫺
V A d 具 P 典 ␯共 t 兲
P ref
dt
(9)
Solving Eq. 共9兲, from time t⫽0 to t,
具 P 典 ␯ 共 t 兲 ⫽ 具 P 典 ␯ 0 e ⫺ 共 D L P ref /V A 兲 t
(10)
When the reference pressure P ref is chosen as the total pressure of
dry gases, Eq. 共10兲 becomes identical to the Krogh equation, Eq.
共4兲.
Observe how the solution of Eq. 共9兲 leads to an exponential
decay of 具 P 典 ␯ with time. The lung diffusing capacity can be obtained easily from the partial-pressure results by rearranging Eq.
共10兲,
D L⫽
冋
具 P 典 ␯0
VA
ln
P reft
具 P 典 ␯共 t 兲
册
(11)
During clinical single-breath tests, Eq. 共11兲 is used for determining the lung diffusing capacity by measuring the gas partialpressure in the inhaled mixture 具 P 典 ␯ 0 , the total test time t and the
gas partial-pressure in the exhaled mixture 具 P 典 ␯ (t).
It is useful to determine a condition for the validity of the
lumped capacity approach 共and the validity of the Krogh equation兲 by considering the definition of the lung diffusing capacity
as equivalent to the definition of mass transfer coefficient from
Newton’s Law of cooling. In this context, one has
ṁ CO 共 t 兲 ⫽c refD L 共 具 P 典 ␯ 共 t 兲 ⫺ 具 P 典 h 兲 ⫽
共 具 P 典 ␯共 t 兲 ⫺ 具 P 典 h 兲
R DL
(12)
where the equivalent convective-like resistance R D L is simply
R DL⫽
1
c refD L
(13)
⬙ 共CO mass rate per unit of area兲 can be
Now, the mass flux ṁ CO
obtained by calculating the mass crossing all the RBC surfaces 共of
total area A兲 in terms of the effective diffusivity of the alveolar
region D eff and local partial-pressure gradient ⵜ 具 P 典 , i.e.,
⬙ 共 t 兲 ⫽⫺D eff
ṁ CO
冉 冊
M
ⵜ具 P典兩
RT
(14)
In a scaling sense, Eq. 共14兲 can be re-written as
ṁ CO 共 t 兲 ⬃AD eff
冉 冊冉
M
RT
具 P 典 ␯共 t 兲 ⫺ 具 P 典 h
LK
冊
⫽
具 P 典 ␯共 t 兲 ⫺ 具 P 典 h
R D eff
(15)
⬙ ⫽ṁ CO /A, and L K is the representative diffusion
where ṁ CO
length 共scale兲 in the volume-averaged domain where 具 P 典 ␯ is calculated. Observe that the diffusion resistance R D eff is, from Eq.
共15兲,
Journal of Biomechanical Engineering
(16)
For the lumped capacity approach to be valid, one would expect
R D L to be larger than R D eff. Using Eqs. 共13兲 and 共16兲, the criterion
for valid lumped capacity approach is
where c ref is a constant, reference gas concentration value. Using
Eq. 共5兲 and 共6兲, Eq. 共7兲 becomes
D L 具 P 典 ␯ 共 t 兲 ⫽⫺
L K RT
M AD eff
c refRT L K D L
LK DL
⫽ P ref
⬍1
M
A D eff
A D eff
(17)
Equation 共17兲 is important for providing an engineering estimate of the validity of the Krogh equation, Eq. 共4兲. We will show
later that this aspect translates into a limit on the time-duration of
the single-breath test for the results to be consistent.
4
Transient Simulations: Normal Distribution
To use the single-breath test results it is necessary, then, to
model the diffusion process as occurring during the experimental
measurement of D L . Unfortunately, this means that the transient
term of the macroscopic model Eq. 共1兲 must be retained. The
model equation becomes dependent on D eff , a quantity not known
a priori.
However, Eq. 共1兲 can be used with a guessed D eff value, in a
numerical simulation mimicking the single-breath test. As a first
guess, the D eff value obtained from the steady-state analysis of
Koulich et al. 关11兴 共independent of D eff兲 can be used with Eq. 共1兲
to determine the time-evolution of the volume-averaged CO
partial-pressure 具 P 典 ␯ . Once the results are obtained, a lung diffusing capacity D L can be found from Eq. 共11兲.
This D L value is not expected to match the lung diffusing capacity found experimentally, because the initial D eff value 共obtained from the steady-state results兲 was just an approximation to
the correct value. Therefore, a new guess for D eff is used to simulate again the diffusion process. The new results yield a new value
of D L to be compare against the experimental value. A predictorcorrector iteration scheme can then be used to fine-tune the D eff
value used in Eq. 共1兲 until the expected 共measurable兲 D L value is
obtained from the numerical results. Resulting from this procedure is the correct D eff value.
Three-dimensional, transient numerical simulations of alveolar
CO diffusion, mimicking the single-breath test, is performed by
discretizing Eq. 共1兲 using finite differences 共central-difference兲,
and solving the algebraic equations within a representative alveolar cubic domain.
The initial condition for the numerical simulations is 具 P 典 (0)
⫽ 具 P 典 0 ⫽133.3 Pa 共⫽ 1 Torr兲 everywhere within the cubic domain, except at the RBC locations where the partial-pressure of
CO is always equal to zero, i.e., 具 P 典 h⫽0. The domain boundary is
set as impermeable to gas diffusion, therefore, ⳵ 具 P 典 / ⳵ n⫽0 at the
boundary, where n is the coordinate along the direction normal to
the boundary. The red cell density, defined as the ratio of RBC
total volume to the alveolar volume, is ␳ ⫽0.034, a value consistent with 45 percent hematocrit 共alveolar region兲. The RBC sites
are distributed randomly 共normal distribution兲 within the REVdomain. Additional details of the numerical procedure can be
found in 关13兴.
The estimated effective diffusivity found numerically by Koulich et al. 关11兴 from the steady-state simulations, D eff⫽9.4
⫻10⫺8 m2 /s, is used in Eq. 共1兲 as a first guess to simulate the
transient process within the alveolar region. After several iterations, it is found that D eff⫽2.68⫻10⫺7 m2 /s yields results from
which D L , obtained from Eq. 共11兲 using only the value of 具 P 典 ␯
from the numerical simulation at t⫽10 s, matches to within one
percent the value assumed as representative of normal RBC distribution, i.e., D L ⫽17 mlCO /mmHg min⫽2.13⫻10⫺9 m3 /sPa.
The corresponding time-evolution of 具 P 典 ␯ , obtained from the
numerical simulations with D eff⫽2.68⫻10⫺7 m2 /s, is shown on
Fig. 3 in terms of ␥, where
OCTOBER 2002, Vol. 124 Õ 611
冋
␥ ⫽ln
具 P 典 ␯0
具 P 典 ␯共 t 兲
册
(18)
The results shown in Fig. 3 indicate that the time decay of the
volume-averaged CO partial-pressure 具 P 典 ␯ approximately follows
an exponential curve, as anticipated by Eq. 共10兲.
With D eff known, the characteristic time of the diffusion process modeled by the REV-averaged equation, Eq. 共1兲, can be obtained and compared to the characteristic time of the process as
modeled by Krogh’s equation, Eq. 共4兲.
Before proceeding, however, it is very important to recognize
that some features of the microscopic diffusion process are ‘‘averaged away’’ when modeled using the REV-averaged equation,
Eq. 共1兲, or the Krogh equation, Eq. 共4兲. This is an intrinsic drawback feature of any averaging technique, a penalty necessary for
scaling-up the microscopic transport process. The results obtained
in terms of 具P典 are REV-averaged results; therefore, they are a
representation 共in an average sense兲 of the microscopic process,
such as the local gas exchange around a single RBC. The REVaveraged results are obtained at an intermediate scaling level: one
step above the microscopic level 共individual RBC’s, membranes,
etc.兲, and one level below the macroscopic level 共the entire alveolar region of volume V A 兲. Results evolving from the Krogh equation, on the other hand, are obtained by considering the diffusion
process at the macroscopic level. Understanding the distinction
between all these levels is fundamental for identifying the correct
length- and time-scales ruling the processes.
The characteristic time of the diffusion process modeled by the
REV-averaged equation, Eq. 共1兲, is L R2 /D eff , where L R is the characteristic diffusion length. This length, estimated as the distance
between RBC sites in the REV-domain, is of the order of 10⫺3 m,
for RBC density equal to 0.034 and normal distribution. Hence,
using D eff⬃10⫺7 m2 /s, L R2 /D eff⬃10 s. On the other hand, the
characteristic time for the diffusion process modeled by the Krogh
equation, Eq. 共4兲, is V A /(D L P ref), typically equal to 24.4 s 共using
V A ⫽4.93⫻10⫺3 m3 , D L ⫽2.13⫻10⫺9 m3 /sPa, and P ref⫽9.51
⫻104 Pa兲. Both characteristic times being of the same order-ofmagnitude verifies the consistency of the REV approach.
It is important now to re-visit Eq. 共17兲, the criterion for the
validity of Eq. 共11兲, by using order-of-magnitude estimates for all
the parameters involved, namely:
P ref⬃105 Pa,
DL
⬃10⫺9 m3 /sPa, D eff⬃10⫺7 m2 /s. The only remaining parameter,
the ratio L K /A, is estimated as (V A ) 1/3/ 关 ␳ V A (6/␦ ) 兴
⬃(10⫺3 ) 1/3/ 关 0.034(10⫺3 )(6/12⫻10⫺6 ) 兴 ⬃10⫺3 m⫺1 , where ␦ is
the representative RBC diameter. The left side of the inequality
shown in Eq. 共17兲 is then found as equal to 1.0. This result indicates that the lumped capacity model, from which the Krogh
equation can be derived, is in the threshold of not being valid.
This conclusion is corroborated by the deviation of the curve ␥
versus t from the straight dashed-line also shown in Fig. 3.
The short time used in Fig. 3 共about 10 s兲, mimicking the duration of a typical clinical single-breath test, masks the deviation
of the REV-averaged model results from the lumped capacity
model results 共straight dashed line shown in Fig. 3兲. If values of
具 P 典 ␯ (t) at times different than 10 s were used in Eq. 共11兲, we
would obtain different values for D L 共because 具 P 典 ␯ (t) does not
follow precisely the expected exponential decay兲 for each time
chosen. We would observe D L behaving as a time-dependent
quantity, invalidating a fundamental hypothesis behind Eq. 共11兲,
that of constant D L . This is a very important aspect because it
highlights a fundamental discrepancy between the Krogh equation, which is strictly valid only if the gas partial-pressure within
the domain is uniform during the entire test, and the REVaveraged equation, which allows for spatial variation of gas
partial-pressure. The discrepancy between the results of the two
models becomes more evident if the time range of Fig. 3 is increased beyond ten seconds.
612 Õ Vol. 124, OCTOBER 2002
Fig. 3 Time evolution of ␥ Äln†ŠP‹␯0 ÕŠP‹␯„t…‡, for ␳ Ä3.4% and
normal „random… RBC distribution. The dashed-line, showing
the results from Eq. „10… using D L Ä2.13Ã10À9 m3 ÕsPa, depict
the deviation to Krogh’s equation.
5
Other Red Cell Distributions
Observe that the RBC distribution within the alveolar domain
does not alter the effective diffusivity of the domain 共as long as
the RBC density is the same兲, but it does affect the lung diffusing
capacity. This is because the effective diffusivity, as defined in Eq.
共3兲, does not depend on where the RBC sites are located within
the REV-domain, as they are simply internal boundaries for the
diffusion process modeled by Eq. 共1兲. Consequently, other RBC
distributions can be simulated numerically once the effective diffusivity of the alveolar region is determined for a certain RBC
density, as long as the red cell density is kept the same.
The equivalent lung diffusing capacity for each RBC distribution can be calculated from Eq. 共11兲 using the numerical results,
and compared to the base lung diffusing capacity, D L-n
⫽17 mlCO /(min mmHg), of normal 共random兲 red cell distribution.
Initially, a uniform distribution is considered in which the
RBC’s are uniformly distributed in spaced within the domain. The
resulting time-evolution of the partial-pressure for this case is
shown in Fig. 4, next to the result for the normal RBC distribution.
The transient numerical simulation yields, from Eq. 共11兲 also
with t⫽10 s, the value D L-u ⫽17.9 mlCO /(min mmHg) for the
uniform distribution. In comparison with the lung diffusing capacity obtained from the normal RBC distribution, the D L value from
the uniform distribution is slightly higher. In fact, it would have
been easy to anticipate the higher lung diffusing capacity of the
Fig. 4 Time evolution of ␥ Äln†ŠP‹␯0 ÕŠP‹␯„t…‡, for ␳ Ä3.4%: comparison of normal „random… and uniform RBC distributions
Transactions of the ASME
Fig. 5 Schematic representation of corner- and center-cluster
RBC distributions. The cubes represent the clustered RBC
sites where Š P ‹ is always equal to zero
uniform distribution because this distribution maximizes the volume of influence of each red cell within the domain, minimizing
the competition for gas, which is characteristic of red cell clustering. The very small increase in D L 共approximately five percent兲
reveals that the normal 共random兲 RBC distribution yields a configuration similar to the configuration for optimum diffusion process obtained with uniformly distributed RBC’s.
To investigate the red cell clustering effect 共reported in 关11兴 for
the case of steady diffusion兲, two other RBC distributions are
investigated. They are the center-clustering and the cornerclustering distributions, obtained by clustering the RBC’s around
the center of the domain or adjacent to one of the corners of the
domain, respectively, as shown schematically in Fig. 5.
The time-evolution of the diffusion process is shown in Fig. 6
for the case of center-cluster distribution. The RBC’s are clustered
inside the centered cube shown in the top-left frame, where the
partial-pressure is always zero ( 具 P 典 ⫽0.0 Torr). Each frame, corresponding to a certain time during the single-breath test, shows
Fig. 6 Time-evolution of isoconcentration surface Š P ‹Ä0.1 Torr, for the case
of RBC’s clustered in the center of the domain. As the CO is progressively
consumed by the RBC’s in the cluster, the isoconcentration surface expands
toward the boundaries of the domain where the gas concentration is higher.
Journal of Biomechanical Engineering
OCTOBER 2002, Vol. 124 Õ 613
Fig. 7 Time-evolution of isoconcentration surface Š P ‹Ä0.1 Torr, for the case
of RBC’s clustered in the lower corner of the domain. As the CO is progressively consumed by the RBC’s in the cluster, the isoconcentration surface expands toward the boundaries of the domain where the gas concentration is
higher.
the isosurface corresponding to 具 P 典 ⫽0.1 Torr 共this value is ten
percent of the initial partial-pressure value兲. Observe how the isosurface 具 P 典 ⫽0.1 Torr moves from the RBC cluster 共at the center
of the domain兲 towards the boundaries of the domain where the
partial-pressure is initially higher 共i.e., 具 P 典 ⫽1.0 Torr兲. This evolution indicates the directional depletion of CO within the domain,
from low 共center兲 to high 共boundaries兲 partial-pressure 共or concentration兲 regions.
The corresponding evolution of the corner-cluster distribution is
shown in Fig. 7. In this case, the depleting front evolves from the
corner of the domain, where the RBC’s are clustered, toward the
opposite corner further away from the gas-sink region.
The time-evolution of the average partial-pressure for these
cases is shown in Fig. 8. Corresponding values of lung diffusing
capacity, obtained from Eq. 共11兲 with t⫽10 s, are: D L-cec
⫽1.79 mlCO /(min mmHg) and D L-coc ⫽0.99 mlCO(min mmHg),
for center- and corner-cluster distributions, respectively. The two
cluster results show radical reduction in D L when compared to the
uniform and normal RBC distributions. This is expected because
the RBC’s inside the cluster are shielded by the RBC’s on the
614 Õ Vol. 124, OCTOBER 2002
Fig. 8 Time evolution of ␥ Äln†ŠP‹␯0 ÕŠP‹␯„t…‡: comparison of results from center-cluster and corner-cluster RBC distributions,
for ␳ Ä3.4%.
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Table 1 Normalized lung diffusing capacity D L , RBC
distribution-centering d, and RBC distribution-radius r. D L - n is
the lung diffusing capacity of normal RBC distribution; s is the
half-length of the diagonal of the alveolar domain.
Distribution:
Uniform
Normal
Cluster:
Center
Corner
Fig. 9 Transient diffusion: comparison of uniform, normal,
center-cluster, and corner-cluster red cell distributions, for
␳ Ä3.4%.
periphery of the cluster, becoming less effective sinks of CO. This
shielding effect was observed also during the simulations of the
steady-state model, as reported by 关11兴, and in the twodimensional capillary simulations performed by Hsia et al. 关7兴.
Somewhat surprising is the strong effect of cluster-location on
D L . The corner-clustering configuration yields a low diffusion
performance 共lower than the center-clustering performance兲 because the RBC cluster is placed adjacent to the corner-boundary
of the domain. That is, three out of the six cluster boundaries are
left without access to CO. Therefore, one would expect that
D L-cec /2⬃D L-coc , a prediction confirmed approximately by the
numerical results.
A summary of the results is presented in Fig. 9, where the time
evolution of the average partial-pressure for all RBC distributions
is presented. Obviously, one can expect the lung diffusing capacity to vary between D L-cec and D L-coc if the same cluster is placed
anywhere else in the domain. Moreover, it is believed that the
corner cluster provides the worst possible RBC distribution, yielding a lower-bound value for the lung diffusing capacity. Assuming
the D L value for uniform RBC distribution as an upper bound
value, the effect of RBC distribution on D L then spreads itself
within the range 0.99 to 17.9 mlCO /(min mmHg).
Trying to quantify the location-versus-distance effect of RBC
distribution, we devised two geometrical parameters to help characterize each distribution. One of these parameters is the
distribution-centering d representing the distance between the
geometrical center of the RBC distribution, defined by the coordinates (X c ,Y c ,Z c ), and the center of the domain at (X 0 ,Y 0 ,Z 0 ).
The coordinates of the geometrical center of the RBC distribution
are found from the equation
共 X c ,Y c ,Z c 兲 ⫽
1
N
N
兺 共 x ,y ,z 兲
i⫽1
i
i
(19)
i
D L /D L-n
d/s
r/s
1.05
1.00
0.05
0.03
0.47
0.48
0.11
0.06
0.15
0.55
0.27
0.21
the diffusion process is less effective. A small r-value indicates the
RBC’s are close together, maybe forming a cluster, also leading
to a less effective diffusion configuration. Hence, d and r are
in principle two good candidates for quantifying the sensitivity
of the lung diffusing capacity to the three-dimensional RBC
distribution.
Values of d and r normalized with respect to the diagonal halflength of the domain, and values of D L for each RBC distribution
normalized by the normal D L value are summarized in Table 1.
Observe that the uniform and center-clustering distributions are
not perfectly centered in the domain because the number of RBC’s
necessary to reach 3.4 percent RBC density does not allow a
perfectly symmetric distribution in relation to the center of the
three-dimensional REV-domain. The distortion caused by this effect, however, is minor.
Notice from Table 1 that the normal 共random兲 and uniform
distributions have very similar d and r, and that is why these two
distributions yield very similar D L . The d and r values for the two
cluster distributions indicate that d, varying by 270 percent, is not
very influential on the value of D L , varying only 29 percent.
Based on these observations, the normalized lung diffusion values presented in Table 1 are plotted, versus the normalized radius
of distribution r/s, in Fig. 10. The continuous line is a polynomial
curve-fit, namely
冉冊 冉冊 冉冊
DL
r
r
⫽⫺1.6 ⫹19
D L-n
s
s
2
⫺68
r
s
3
⫹93
r
s
4
(23)
The deviation between numerical results and the curve-fit Eq.
共23兲 is attributed to parameter d, not accounted for in Eq. 共23兲.
7
Summary and Conclusions
A three-dimensional, unsteady diffusion model is used to simulate the gas exchange process in the alveolar region of the lungs.
By mimicking the single-breath experimental technique, the results of the numerical simulation lead to an effective diffusivity
value equivalent to the measured lung diffusing capacity.
With this effective diffusivity, a quantity independent of the RBC
positioning within the domain, several RBC distributions are
investigated.
where (xi ,yi ,zi) are the coordinates of each red cell, and N is the
total number of red cells in the domain. Hence, the distance to the
center of the domain is simply,
d⫽ 关共 X c ⫺X 0 兲 2 ⫹ 共 Z c ⫺Z 0 兲 2 ⫹ 共 Y c ⫺Y 0 兲 2 兴 1/2
(20)
The other geometrical parameter characterizing the RBC distribution is the effective radius of the distribution, r,
r⫽ 共 r 2x ⫹r 2y ⫹r z2 兲 1/2
(21)
with,
共 r x ,r y ,r z 兲 ⫽
1
N
N
兺 关 兩 共 X ⫺x 兲兩 , 兩 共 Y ⫺y 兲兩 , 兩 共 Z ⫺z 兲兩 兴
i⫽1
c
i
c
i
c
i
(22)
A large d-value indicates the RBC distribution is far from the
center of the REV-domain, hence, close to the boundaries where
Journal of Biomechanical Engineering
Fig. 10 Normalized lung diffusion coefficient versus normalized distribution radius, ␳ Ä3.4%
OCTOBER 2002, Vol. 124 Õ 615
The equivalent lung diffusing capacity is obtained, for each
RBC distribution, from the numerical results using the Krogh
equation. An analytical criterion for estimating the validity of the
single-breath lung diffusing capacity results obtained from the
Krogh equation is derived by considering the lumped-capacity
model. The criterion indicates that the Krogh equation, when used
for tests that are 10 seconds or longer, is only marginally valid.
This prediction supports the numerical results indicating a deviation from the lumped-capacity model.
Results for normal 共random兲, uniform, center-cluster, and
corner-cluster RBC distributions suggest a strong influence of
RBC distribution over the lung diffusing capacity. The introduction of two descriptive geometrical parameters, i.e., distributioncenter and distribution-radius, are helpful in characterizing the
RBC distributions. A polynomial correlation is proposed for estimating the lung diffusing capacity as function of the distributionradius, believed to be the predominant parameter. Although independent of the distribution-center parameter, this simplified
correlation yields reasonably good predictions as compared to the
numerical results.
The arbitrarily chosen RBC distribution type and location are in
line with our objective to investigate the effects of different RBC
distributions. Only a detailed mapping of the alveolar region can
identify the REV size and the actual RBC distribution. Nevertheless, our study covers the two possible extreme cases: cornercluster and uniform RBC distribution.
The lung diffusing capacity, although clinically measurable, is
affected by the RBC distribution within the alveolar region, therefore it must depend also on the domain used for the theoretical
analysis.
It is concluded that the three-dimensional distribution of the red
cells, characterized by the distribution radius, has a fundamental
impact on the lung diffusing capacity. This conclusion indicates
that the Roughton-Foster lung diffusing capacity model, used in
physiology and morphometric analysis, must be re-interpreted in
light of the red cell distribution effect within the alveolar region.
Nomenclature
A ⫽ area, m2
c ⫽ total number of constituent present in the alveolar
region
c ref ⫽ reference gas concentration, kg/m3
d ⫽ distribution-centering or distance from RBC distribution center to center of domain, m
D L ⫽ lung diffusing capacity, m3/共Pa s兲
D eff ⫽ effective diffusivity, Eq. 共3兲, m2/s
L K ⫽ representative diffusion length in the Krogh’s domain, m
L R ⫽ representative diffusion length in the REV domain,
m
ṁ CO ⫽ CO mass flow rate 共mass gas uptake兲, kg/s
M ⫽ gas molecular mass, kg/kg-mol
N ⫽ total number of RBC’s
nជ ⫽ unit vector
P ⫽ gas partial-pressure, Pa
P ref ⫽ reference gas partial-pressure, Pa
具P典 ⫽ REV-averaged partial-pressure, Pa
具 P 典 ␯ ⫽ volume-averaged 具P典 within computational domain,
Pa
r ⫽ distribution-radius or effective radius, m
R ⫽ universal gas constant, Pa m3/K kg-mol
616 Õ Vol. 124, OCTOBER 2002
R DL
R D eff
s
t
T
V
VA
V̇ CO
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
equivalent convection resistance, Pa s/kg
equivalent diffusion resistance, Pa s/kg
diagonal half-length, m
time, s
temperature, K
volume, m3
alveolar volume, m3
CO volumetric flow rate 共volumetric gas uptake兲,
m3/s
x, y, z ⫽ Cartesian coordinates, m
X, Y, Z ⫽ coordinates, m
Greek Symbol
␤ ⫽
␦ ⫽
␥ ⫽
Subscript
c ⫽
cec ⫽
coc ⫽
i ⫽
n ⫽
o ⫽
u ⫽
0 ⫽
pseudo-diffusivity, m2/s
RBC diameter, m
nondimensional gas partial-pressure, Eq. 共18兲
center
center-cluster distribution
corner-cluster distribution
counter
normal
domain center
uniform
initial, at time zero
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