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Stereological Estimates of Alveolar Number and Size and Capillary Length and Surface Area in Mice Lungs.

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THE ANATOMICAL RECORD 292:113–122 (2009)
Stereological Estimates of Alveolar
Number and Size and Capillary Length
and Surface Area in Mice Lungs
Stereology and Electron Microscopy Research Laboratory and MIND Center,
University of Aarhus, Aarhus, Denmark
Department of Anatomy, Division of Electron Microscopy, University of Göttingen,
Göttingen, Germany
Institute of Anatomy, Experimental Morphology Unit, University of Bern,
Bern, Switzerland
The major function of the lung is gas exchange and depends on alveolar and capillary parameters such as surface area and volume. The number of alveoli may report on the nature of structural changes in lung parenchyma during development, illness or changing environmental factors.
We therefore developed an efficient and easily applicable stereological
design for estimating and monitoring these structural parameters in the
mouse lung. The estimation of volume fractions of different lung compartments has been carried out by point counting. A combination of cycloid
grids superimposed on vertical sections was used to estimate the capillary
surface area with isotropic test lines. Capillary length could be measured
using the harmonic mean of the surface weighted diameter. The Euler
characteristic applied in the physical fractionator with varying but known
sampling fractions (Horovitz-Thompson estimator) enabled us to estimate
alveolar number. In adult mice lungs, we obtained total values for alveolar number of 2.31 3 106 alveoli in a pair of lungs, alveolar surface area
of 82.2 cm2, capillary surface area of 124 cm2, and capillary length of 1.13 km.
All values are corrected for tissue shrinkage. With this study we present
a highly efficient combination of several design-based stereological tools
for the unbiased estimation of alveolar number and volume as well as
length, surface area, and diameter of capillaries in the mice lung. Anat
Rec, 292:113–122, 2009. Ó 2008 Wiley-Liss, Inc.
Key words: disector; Euler number; fractionator; quantitative
microscopy; vertical sections
Gas exchange is the main purpose of the lung based
on the diffusion of oxygen and carbon dioxide through
the thin air-blood barrier between the alveolar airspace
and the capillary blood in the alveolar septa. The smallest functional unit in the lung is the alveolus consisting
of its airspace and its septa carrying numerous capillaries (Caduff et al., 1986; Weibel and Taylor, 1998). Functional and structural changes in the lung can be
detected, evaluated and interpreted by monitoring the
changes in absolute number and in different parameters
of alveoli and various compartments of the lung. With
the objective of evaluating different parameters of an
Grant sponsors: European Union (Marie Curie fellowship),
Eva and Henry Frænkels Mindefond, Danish Council for Strategic Research, Lundbeck Foundation.
*Correspondence to: Jens R. Nyengaard, Stereology and Electron Microscopy Research Laboratory, University of Aarhus,
Building 1185, DK-8000 Aarhus C, Denmark.
Received 18 September 2008; Accepted 12 February 2008
DOI 10.1002/ar.20747
Published online in Wiley InterScience (www.interscience.wiley.
Fig. 1. Fixed mouse lung pair with vertical axis defined according
to the natural vertical axis.
Fig. 3. Cutting machine with agar embedded mouse lung. The lung
is embedded in agar and cut into slices of exactly the same thickness
by using this cutting machine. All slices are made parallel to the vertical axis.
Fig. 4.
Fig. 2. Sampling scheme showing all sampling stages from the
whole embedded lung to fields of view for the light microscopic evaluation of the sections. I. 1 II: The right lung (r) is rotated a random
angle around a predefined vertical axis (VA), the left lung (l) is rotated
a further 90 degrees. Both are embedded overlapping in agar and
systematically cut in slices of the same thickness parallel to the vertical axis beginning from a random point. III: Slices are placed on their
surface, grouped in two or three new blocks and embedded in paraffin. T is the slice thickness of the unprocessed tissue; MA is the
microtome advance. All cuts in the paraffin blocks are parallel to the
vertical axis. IV: Systematically, uniformly randomly chosen 2Dunbiased counting frames are observed using a light microscope on
single sections cut with a MA of 4 mm.
The agar embedded lungs cut in uniform slices.
organ, including structural quantities, it is most efficient
to do all observations in the same specimen. The efforts
for processing and observing the tissue should be kept
as low as possible.
A newly developed unbiased method for the estimation
of alveolar number in lungs based on the Euler number
has recently been published (Hyde et al., 2004; Ochs
et al., 2004). Even though described for the application
in placenta, the capillary length estimation using thin
vertical sections and the surface-weighted diameter
(Clausen et al., 2000) has not been employed in lung
research so far.
In this study we aimed at developing an efficient and
practical design-based combination of stereological tools
for the estimation of various parameters of lung structure in mice. This includes total alveolar number, sur-
Fig. 5. Estimation of section sampling fraction ssf and shrunken
tissue thickness T* by exhaustive sectioning of one block. The paraffin
block containing the lung slice is seen edge-on and the lines x, y, and
z mark points in the slice where optically prominent parts of the tissue
can be well observed and followed throughout the whole thickness of
the slice. As all incomplete sections of one block are mounted on
slides one has to identify those that precisely represent the appear-
ance and disappearance of one well-defined point in the tissue. In our
example, these are sections nx1 and nx2, ny1 and ny2 and nz1 and nz2
showing the appearance and disappearance of tissue at the points x,
y, and z, respectively. The number of sections N in-between them, say
the difference n2–n1, represents the slice thickness T* of the shrunken
slice and enables us to determine the exact section sampling fraction,
ssf, of an arbitrary number of sections per block.
face area, and volume as well as capillary surface area,
whereas the estimation of capillary length is a modelbased method. The sampling and counting steps are
described in sufficient detail to allow researchers to
adopt these methods to study the lung phenotype in
gene manipulated mice models using appropriate
design-based stereological methods.
lungs were embedded overlapping to achieve a unimodal
area distribution on the systematic set of sections
(Ogbuihi and Cruz-Orive, 1990).
Lungs were systematically cut in slices starting at a
uniform random point. We used a cutting machine and
chose a slice thickness of 1.7 mm, providing about seven
slices from each pair of lungs (see Figs. 3 and 4).
Sampling was done by using a multistage physical
fractionator (Gundersen, 1986) with a varying sampling
fraction being a general Horovitz-Thompson estimator
that enabled us to estimate the alveolar number disregarding shrinkage.
The first sampling level is the cutting and sampling of
blocks, the block (slice) sampling fraction (bsf) which is 1
because all blocks are sampled. The second sampling step
is the section sampling fraction (SSF). One complete section is taken from each block. It is not necessary to section
all the blocks exhaustively to estimate the average section
sampling fraction. We embedded the agar slices in two to
three different paraffin blocks and sectioned a systematic,
uniformly random fraction of these blocks exhaustively,
which yielded an average number of sections N per block
representing the block thickness (see Fig. 5).
All sections from the very beginning of the tissue in the
block until the first complete profile are mounted on
slides. At the end all sections from the last complete profiles until the very end of the tissue are also mounted.
These are stained with Mayer’s haematoxylin. In the part
where every section covers the whole area of the slice,
sections are taken for the purpose of our estimations.
On the haematoxylin stained serial sections, one fixed
point in the section is defined and followed through the
tissue from the very first time of its appearance until it
disappears. During embedding one has to make sure that
the agar slice really contains tissue throughout its whole
thickness and on all—or at least a well-known part—of
the area seen on the upper surface. The number of
sections N between the appearance and disappearance of
this very point in the tissue represents the shrunken
slice thickness T* in a dimensionless way (see Fig. 5).
We used female mice of the strain CL 57 B6 that were
purchased from Bomholtgaard (Ry, Denmark). The study
has been approved by the Danish Animal Inspectorate.
Five mature animals were taken from different litters,
three of them were 46- and two were 69-days old. The
animals were weighed and anaesthetised by injecting
3 mL per kg body weight Mebumal (50 mg/mL pentobarbital) intraperitoneally. The trachea was cannulated
through a midline cervical incision while the animal was
still breathing. A bilateral pneumothorax was produced
by puncturing the diaphragm from its abdominal surface. We infused 4% phosphate-buffered formaldehyde
pH 5 7 into the trachea at a transpulmonary pressure
of 20 cm H2O. The trachea was ligated when the flow
ceased (Weibel, 1984). Lungs were removed from the
thorax and then placed in fresh fixative at 48C for
another 4 hr to complete fixation while intrapulmonary
pressure was still maintained.
The fixed pair of lungs was separated and carefully
dried on its outer surface. We defined the vertical axis
according to the natural vertical orientation of the lungs
in vivo and all further cuttings were made parallel to
the vertical axis (see Figs. 1 and 2).
To avoid artificial edges the lungs were rotated as a
whole to get slices that still possess their natural borders, that is, are totally surrounded by pleura. The angle
of rotation for the right lung around the vertical axis
was randomly chosen and the left lung (see Fig. 2) was
rotated 908 with respect to the right lung. Both lungs
were embedded in agar in one block. If possible both
Fig. 6. Sectioning of the block. Sections are only taken in that part
of the block were one can make sure that all slices may contribute full
intersections. In the figure this part is defined by the thick lines. Inside
this range, taking sections in different heights may influence whether a
lung slice is (e.g., slice 4 in sect. b) or is not (e.g., slice 4 in sect. a)
contributing tissue to the section.
Taking one complete section out of every slice we
know that the sampling fraction at this moment is 1/N.
According to their tissue distribution in the agar at the
ends of the organ, the first and the last lung slice may
or may not provide a tissue section (Fig. 6). In the pilot
study, it was noted for all slices at which depth (section
number) a complete tissue profile was always seen. The
maximal, first complete section number is then sampled
from all slices, with its following look-up section, and
most of the blocks are not sectioned further.
For the alveolar counting we employed a stain with
orcein and a nominal disector height of 4 mm, that is,
the disector is two consecutive sections. This seems
rather small given an expected alveolar diameter of
about 45 mm (Faffe et al., 2002). However, we are
actually not counting alveoli themselves, meaning the
whole alveolus as a particle, but instead counting the alveolar openings. When we keep track of the appearance
or disappearance of alveolar openings we need a much
smaller disector height.
Examining disectors under the microscope 2D
unbiased counting frames are sampled in a known and
predetermined distance from each other (Gundersen,
1977). Knowing the frame area and the step length
between the counting frames we calculate the final area
in which the counting was performed on each section
and can accordingly derive the area sampling fraction
(asf) from this.
We estimated lung volume with the Cavalieri principle
(Michel and Cruz-Orive, 1988) on the embedded and cut
tissue and had to correct for shrinkage afterwards to
achieve values for unshrunken lung volumes. With a
1.253 Olympus objective (Olympus, PlanApo 0.04) and
at a final magnification of 23.83 a point grid was superimposed on the sections and thus area of the section
estimated. Having taken the sections in a fixed distance
from the tissue surface we know the distance between
the sections will be equal to the shrunken slice thickness. Under these circumstances we are able to calculate
lung volume after shrinkage by the Cavalieri principle:
a X
V ðlung; shrÞ ¼ T PðlungÞ
T* being the slice thickness after shrinkage, P(lung)
the number of points hitting the shrunken lung tissue
and a/p the area per test point of the counting grid.
Alveolar Number Estimation by Means
of the Euler Number
Before we can count anything we have to get an idea
of what we actually want to count, for alveoli as air
spaces have no precise spatial definition and border that
would make them countable in ordinary sections (Burri
et al., 1974; Blanco et al., 1989). The disector offers the
opportunity to solve these problems as it is a 3D probe
sensitive to particle of which Euler number is just an
especially simple case (Gundersen, 1986). Still it is necessary to define a feature that can unmistakably be recognized and counted in a disector and is characteristic
for alveoli and only for them. Alveolar openings are such
a feature (Hyde et al., 2004; Ochs et al., 2004), unique of
the alveolus, but the question remains, how to count
them? Alveolar openings are characterized by bundles of
elastic fibres outlining them, visible at the very edge of
the alveolar septum on the section (Burri et al., 1974).
Inside an acinus one can imagine these interconnected
elastic fibres as a net in whose meshes the alveoli are
fixed. The very basis for quantitations in a strict mathematical sense is additivity. In a net, the feature that can
be determined additively is connectivity.
The mathematical idea underlying connectivity and
the instrument that enables us to measure it in an
object by specific counting rules is the Euler number
(Hadwiger, 1959). The Euler characteristic of an object
provides information about the connections of complicated 3D structures. Connections in their very meaning
are never ever isolated or separated from the structure
in question.
For a structure of a certain dimension the Euler number is defined as the sum of significant topological
changes of the Euler number in a probe of one dimension less swept through the structure disregarding its
orientation. For the estimation of the Euler number of
the original structure, it is necessary to define changes
of the Euler number in the probe (Fig. 7).
In a two-dimensional structure a segment can appear
as a point in the reference section of the appropriate
probe when it is not visible in the look-up section and it
is then counted as 11. We refer to this as a new Island.
If one line in the reference section splits into two different new ones in the look-up section this is counted as
21. We refer to this as a new Bridge. Bridges and
Islands are significant topological changes and contribute to the Euler number of the 2D object (Fig. 7).
For the alveolar number estimation we use, as aforementioned, the abstraction of a 2D elastic fibre net of alveolar openings or the net of alveolar wall ridges. This
2D net is expanded in a 3D space. Consequently, we are
never able to make statements about it using just isolated, single 2D sections. The disector is the instrument
which can sample counting events by sweeping a 2D
plane through a 3D space (Gundersen, 1986).
According to the situation in a 2D object one can
observe certain significant topological changes between
the reference section and the look-up section in a 3D
object as well.
When there is a septal segment in the reference section that has disappeared in the look-up section this is
referred to as an Island. When the look-up section shows
two (septal) lines in direct neighborhood that form one
uninterrupted line in the reference section this is called
a Bridge (Fig. 8).
v is the disector contribution to the Euler number v3
of the 2D net in 3D space and is defined as:
v ¼ #ðislandsÞ #ðbridgesÞ
dxdy ;
asf ¼
where a(frame) is the frame area of the
counting frame and dx and dy are the step lengths
between counting frames.
Consequently we get the final estimator equation:
P ðTi vi Þ
ðvi Þ
dx dy X
v3 ¼ 1 v
MA aðframeÞ
This is a ratio unbiased estimate
P of the varying SSF.
It is therefore important that (1) vi 100, so CE(vi) <
0.1; and (2) enough slices are used when Ti is quite
The Euler number of a 3D object can in a more structural way be defined as:
v3 ¼#ðisolated partsÞ #ðextra connectionsÞ
þ #ðinternal cavitiesÞ
Fig. 7. Changes in the Euler number in a 2D object, observed from
opposite directions. If we consider the object to be a net we can
clearly see it has four holes. Estimating the number of holes, that is,
connectivity, by the Euler number we should obtain the same result. A
line is taken as a one-dimensional probe and for reasons of efficiency
swept through the object from opposite directions. Applying the
counting rules for Euler number the topological changes in the probe
are summed up. From this we may derive the connectivity of the
object which corresponds to the number of holes. In total nine bridges
(B) and three islands (I) are sampled contributing each by 21 or 11,
respectively, to the Euler number. Dividing the sum by two, because
we count in two directions, the Euler number equals (3 2 9)/2 5 23.
Inserting this value in the connectivity equation Eq. (7) we obtain: connectivity 5 # holes 5 # isolated objects – Euler number 5 12(23) 5
4. Our final result is in accordance with what common sense tells us,
that there are four holes in the observed object.
Counting in disectors and for reasons of efficiency we
always count in both directions in the reference and the
look-up sections and then divide our estimate by two.
#ðislandsÞ #ðbridgesÞ
In this fractionator design the Euler number v3 of the
3D structure is calculated as follows:
v3 ¼
v is summed over all disectors in all sections
and F is the aggregate sampling fraction:
F ¼ bsf ssf asf
where bsf is the block sampling fraction, ssf the slice
sampling fraction and asf the area sampling fraction.
bsf 51; because we sample all lung blocks.
ssf ¼ PMA
; where MA is the constant microtome
PðTi vi Þ
ðvi Þ
advance, Ti is the shrunken slice thickness of slice i,
and vi is the Euler number in slice i.
Trying to apply this equation to our example of the 2D
elastic fibre net we can obviously exclude the number of
internal cavities.
As connectivity W is the number of extra connections
we may transform the equation as follows:
W ¼ #ðisolated partsÞ v3
When alveolar number in terms of connectivity is estimated for the whole lung one has to keep track of the
number of isolated parts. Isolated parts mean the number of separate 2D elastic fibre nets in one pair of lungs
which equals the number of acini.
Thus we finally derive the alveolar number N(alv):
NðalvÞ ¼ W ¼ #ðaciniÞ v3
NðalvÞ ¼ W ¼ #ðaciniÞ
2PðT v Þ
Pi i
ðvi Þ
dx dy
#ðislandsÞ #ðbridgesÞ7
5 ð10Þ
MA aðframeÞ
How this equation really can reflect the alveolar number becomes more graphic if one realizes that gaining or
loosing one extra connection exactly reflects the existence or nonexistence of one hole or mesh in the net,
that is, one alveolus in the acinus.
There remains one unknown factor in Eq. (10) and
that is the number of acini. The pulmonary acinus can
be defined as the sum of all respiratory bronchioles (nonexistent in mice lungs) and alveolar ducts with appendant alveoli peripheral to a terminal bronchiole. The
transition from a terminal bronchiole to a respiratory
bronchiole is marked by the change from a purely conductive airway with high cuboidal epithelium into respiratory airspace outlined with the typical flat epithelium
that is crucial for gas exchange (Rodriguez et al., 1987;
Randell et al., 1989; Ten Have-Opbroek, 1991). Randell
et al. (1989) found about 5,000 acini in rat lungs using
serial reconstructions. We cannot transfer this value to
mice lungs offhand, but assuming a similar dimension a
number of a few thousand would not significantly influence the alveolar number which is in the range of millions in mice.
Fig. 8. Disector image of orcein stained lung tissue in the CAST
system. The reference section and the look-up section are corresponding areas on two adjacent sections that show significant structural changes concerning the Euler number estimation. Counting can
be performed in both directions taking each section as reference and
look-up section. Bridges, B, are counted where two alveolar wall end
points are interconnected in the reference section but appear to be in-
dependent in the look-up section. The counting frame determines the
counting area where significant changes situated inside the frame or
touching the inclusion line (thinner black line on the top and to the
right) have to be counted while changes touching the exclusion line
(thick black line on the bottom and to the left) have to be ignored
(thick arrow).
Immunohistochemical Staining of Capillaries
Dell Monitor and had the CAST vers. 2.158 stereology
software (Visiopharm, Hørsholm, Denmark) installed.
To avoid edge effects in our estimations in the CAST
we used rather large counting frames although this
would somehow affect overall sampling efficiency, but
due to the fact that we expect distribution of alveoli in
the lung to be rather homogeneous it should not turn
out to be a problem for the overall sampling efficiency.
The point counting of all volume densities and the
counting for alveolar number and surface area estimation were performed using a 203 objective (NA 0.70) at
a final magnification of 4733. The estimation of capillary surface area and length required a 403 objective
(NA 0.75) at a final magnification of 13703.
Adjacent sections were mounted on one slide to
improve the efficiency of the counting process. For the
Euler number counting procedure in the physical disector the CAST provided, after the delineation of the two
sections, a fast and precise automatic switch between
corresponding fields of view on the two sections. For every lung we aimed at a number of counting events of
about 100 for alveolar number and between 150 and 200
for all other counted parameters. The system chose
fields of view by systematic, uniformly random sampling.
Anti-Aquaporin-1 (AQP-1) was used for the staining of
lung capillaries. AQP-1 water channels are known to be
expressed in mammalian lung microvascular endothelium (King et al., 1996; Effros et al., 1997; Bai et al.,
1999). In lung, AQP-1 is not expressed in alveolar type I
cells but is in rare occasions and small amounts detectable on type II cells (King et al., 1996; Effros et al., 1997).
As type II cells are in their morphology distinctively different from endothelial cells we can probably trust the
immunohistochemical identification of capillaries.
Immunohistochemical Treatment
Peroxidase is blocked with 30% H2O2 and methanol
and afterwards the sections are microwave treated at
450 W in citrate buffer three times for 5 min. Incubation
with AQP-1 (Chemicon; rabbit anti-AQP-1; polyclonal) is
performed at a dilution of 1:500 for 1 hr at 378C. Then
the sections are treated with the secondary antibody
(anti-rabbit IgG; Vector; dilution 1:100) at room temperature for 2 hr, stained with DAB and counterstained
with Mayer’s haematoxylin.
Microscopical Evaluation
All observations were made with a modified Olympus
BH2 microscope equipped with a MT12 microcator and
ND281 readout (Heidenhain, Germany), a motorised
specimen stage (Märzhäuser, Germany) and an object rotator (Olympus, Denmark). A CCD camcorder (JAI-2040,
Protec, Japan) connected to a PC was mounted on the top
of the microscope. The computer was connected to a 2100
Point Counting
In the observed lung tissue we collected data about
the changes in the distribution of volume fractions on
different subtypes of tissue. Airspace in the parenchyma
and airspace of the conductive airways down to terminal
bronchioles were point counted separately.
Fully aware of the fact that we possibly cannot unambiguously distinguish between air space in alveoli and airspace in alveolar ducts we still regarded it necessary to
make this difference in order to be able to calculate some
value of mean alveolar volume for reasons of comparison
and to get an idea of how the distribution of airspace in
those two compartments changes. Thus, we defined alveolar airspace as the area seen on the section which is surrounded by alveolar septa and an imaginary straight line
between the free edges of the alveolar opening.
We decided to distinguish between points hitting
parenchymal tissue and those hitting nonparenchymal
tissue. Parenchymal tissue includes all tissue that could
be identified as belonging to alveolar septa, including
capillaries, epithelial, and interstitial cells and extra cellular matrix. Nonparenchymal tissue is defined as all
tissue not being part of alveolar septa, that is, the large
connective tissue septa, walls of larger vessels and conductive airways with endothelial, epithelial, interstitial,
muscle, and cartilage cells and their surrounding connective tissue. Vessels above a diameter of 10-mm make
up one separate compartment as well.
Surface Area of the Gas Exchange Tissue
For the estimation of surface area, S, one can usually
count intersections of structure boundaries on the section with straight 3D isotropic test lines of a superimposed counting grid. Alveolar surface area is estimated
on the sections as follows:
SðalvÞ ¼
I ðalvÞ
2 l P
ð1 Shrinkage3D Þ3 VðunshrÞ
where I(alv) is the number of intersections between test
lines and alveolar septa, l/p is the length per test point
on the counting grid and P(lung) is the number of points
hitting lung tissue, V(unshr) the volume of the
unshrunken lung and Shrinkage3D is the volume shrinkage, see Eq. (16). With this combination of vertical sections and cycloids it is possible to create IUR test lines.
Capillary Length and Surface Area
Capillary length and surface area were observed on
single thin (2 mm) sections. Surface area of the capillaries was estimated according to the estimation of alveolar surface area by counting intersections of cycloids
with the capillary walls on systematically, uniformly,
randomly chosen fields of view (Fig. 9). The equation is
identical to the alveolar surface area estimation Eq. (11)
except that I(alv) is exchanged with I(cap), which is the
number of intersections between cycloids and capillary
For the estimation of capillary length we superimpose
a cycloid counting grid on the section and measure the
diameter of every capillary profile that is intersected by
a cycloid test line. If the capillary is hit twice by the test
line its diameter is sampled and recorded twice as well.
To measure the capillary diameter an imaginary line
perpendicular to the direction of the largest profile
dimension is drawn and measured at its greatest width.
The calculated mean over all measured diameters is an
estimate of the surface-weighted mean diameter of the
Fig. 9. A cycloid grid superimposed on an image of alveolar septa.
The sections are stained with AQP- 1 to make the identification more
reliable, which is not visible in this black and white image. Every time
a cycloid intersects the boundary of a capillary wall, it is registered
and the capillary diameter is measured. If a capillary is hit one time
(thin arrow) or two times (thick arrow) by the cycloids the capillary diameter is measured once or twice, respectively.
capillaries in our sample. As this surface weighted mean
diameter, dS(cap), is not equal to the ordinary lengthweighted mean capillary diameter, d(cap), we cannot
simply derive length from the equation length 5 surface
area/diameter p, but we have to convert our surfaceweighted mean diameter, dS(cap) to the ordinary lengthweighted diameter, d(cap), by computing the harmonic
mean, dS (cap) (Clausen et al., 2000):
dðcapÞ ¼ dS ðcapÞ ¼
di ðcapÞ
here di(cap) is every diameter that is sampled with
cycloids in the fields of view and mi is the total number
of times cycloid test lines intersect the boundaries of
sampled capillary profiles (Fig. 9).
The total length of the capillaries in a lung is then:
SV ðcap=lungÞ
LðcapÞ ¼
ð1 Shrinkage3D Þ3 VðunshrÞ
ðcapÞ p
here SV(cap/lung) is the estimated surface area density
of capillaries and dðcapÞ
the estimated mean lengthweighted diameter. Both Eqs. (12) and (13) are modelbased estimators and require the capillary cross sections
to be circular. If capillaries are collapsed, Eq. (12) pro
vides an underestimate of dðcapÞ
and LðcapÞ is overestimated using Eq. (13).
Shrinkage Measurements
As mentioned earlier we had to embed the sections
into paraffin because of the immunohistochemical treatment. This makes of course the problem of tissue deformation even more prominent because paraffin embedded
tissue is usually affected by pronounced tissue deforma-
tion. Where shrinkage is no problem for number estimation, because cardinality is unaffected by deformation,
estimations of length, surface and volume must be corrected for shrinkage to preserve unbiasedness (DorphPetersen et al., 2001).
Performing exhaustive cutting with a calibrated
microtome we have the possibility of estimating shrinkage on these sections. Knowing the exact thickness of
the slices made in the cutting machine and having exact
knowledge about the microtome advance we can compute the thickness of the shrunken tissue and the linear
shrinkage that occurred during the histological processing (see Fig. 5).
For the determination of the sampling fraction the average number of sections N one slice provides has already been identified. Multiplying the number of sections with the exact microtome advance we calculate the
thickness of one slice after embedding and tissue deformation. Thickness T* after shrinkage:
T ¼ N MA
where N is the number of sections one slice provides on
average and MA is the microtome advance.
The difference between the original slab thickness and
the value after cutting is the measured linear shrinkage,
Shrinkage (linear):
Shrinkage ðlinearÞ ¼
where T is the original thickness of a slice, T* is the
thickness after shrinkage.
Assuming isotropic shrinkage, we can estimate volume
shrinkage Shrinkage3D:
Shrinkage3D ¼ 1 ð16Þ
The volume of the unshrunken lung tissue is:
V ðunshrÞ ¼
V ðshrÞ
mm3 (0.22) [77 mm3, 130 mm3] making up 31% (0.08)
[29%, 35%] of total lung volume. Every pair of lungs
comprised on average 2.31 106 alveoli (0.23) [1.68 106,
2.96 106]. Left and right lung showed in accordance
with their volume shares different absolute numbers of
alveoli. The left lung contained 740 103 alveoli (0.23)
[550 103, 940 103] which corresponds to a relative
share in total alveolar number of 32% (0.10) [0.29%,
0.36%]. With a number 1.57 106 alveoli (0.25) [1.13 106,
2.1 106] the right lung consisted of more than twice as
many alveoli, namely 68% (0.05) [0.66%, 0.71%] of the
total amount. The observed lungs showed an alveolar
number density N(alv) per mm3 parenchyma of 9.55 103
alv/mm3 (0.31) [7.25 103 alv/mm3, 14.8 103 alv/mm3]
in the unshrunken lung with a mean alveolar volume of
59.5 103 mm3 (0.26) [34.5 103 mm3, 75.0 103 mm3].
Surface area of total alveolar airspace was 82.2 cm2
(0.17) [63.3 cm2, 101 cm2] in unshrunken tissue, which
resulted in a mean surface area of one alveolus of 3,620
mm2 (0.14) [2,870 mm2, 4,260 mm2]. The observed lungs
comprised a volume of alveolar airspaces of 138 mm3
(0.29) [102 mm3, 205 mm3], alveolar duct airspace of
37.0 mm3 (0.30) [22.9 mm3, 48.1 mm3] and alveolar
septal tissue of 72.6 mm3 (0.21) [50.1 mm3, 90.3 mm3].
We obtained a capillary surface area corrected for
shrinkage that was 124 cm2 (0.13) [109 cm2, 143 cm2].
Capillaries in these mice lungs had a length of 1.13 km
(0.13) [0.98 km, 1.35 km].
We present a way of effectively combining different
stereological estimators for the evaluation of lung alveoli
and capillaries in mice. We used the whole lungs of five
adult female mice from different litters. Female mice
were chosen because they are used in a bigger study of
lung development. We expect that female mice have
smaller lungs than male mice and that growth of female
lungs will reach a plateau in contrast to male mice.
Results are shown as mean (CV 5 SD/mean) and
[min; max]. The evaluation of the error variance, coefficient of error (CE), has for the Cavalieri principle and
alveolar number estimate been estimated from the equations provided in Gundersen et al. (1999). The CE’s for
the ratio estimates of alveolar surface area and capillary
surface area and length were estimated from Kroustrup
and Gundersen (1983).
Mean body weight of the mice was 20.6 g (0.13) [18.3 g,
23.8 g]. Mean lung volume estimated by the Cavalieri
principle and corrected for volume shrinkage of 35% was
307 mm3 (0.17) [264 mm3, 395 mm3]. The right lung
comprised four lobes and made up 69% (0.04) [65%,
71%] of total lung volume with an absolute volume of
210 mm3 (0.15) [187 mm3, 264 mm3]. The left lung consisted of only one lung lobe with a mean volume of 97
Tissue Processing
Lungs were fixed by instillation fixation in situ to
allow evenly expansion and unfolding of the delicate
interalveolar walls and to minimize the time between
death and complete fixation of the tissue, avoiding post
mortal tissue variances (Randell et al., 1989; Gil, 1990).
Instillation via the airways is the method of choice for
all investigations on lung tissue except for the occasions
whenever the preservation of the physiologically normal
status of the airspace side of alveolar septa is required,
what would be the case in, for example, research on
intraalveolar surfactant or intraalveolar pulmonary
oedema (Fehrenbach and Ochs, 1998).
We decided to apply the Cavalieri principle instead of
fluid displacement for the estimation of lung volume
(Michel and Cruz-Orive, 1988) because the volume estimated by this method is closest to the volume all further
observations are made in (Yan et al., 2003). Yan et al.,
measured 13%–25% lower lung volume in guinea pigs
and dogs with the Cavalieri method than with the fluid
displacement and concluded that VCav measured on lung
slices after relaxation of pressure more precisely repre-
sents the state of the tissue to be used for subsequent
morphometric analysis (Yan et al., 2003).
We are aware of the fact, that a 4 hr fixation in 2%
formaldehyde does not prevent tissue deformation due to
elastic recoil after releasing pressure when cutting the
slabs. Oldmixon et al. found a 30%–40% lower airspace
dimension after releasing airway pressure and only fixation in glutaraldehyde and dehydration by perfusing the
lung with ethanol would prevent tissue from shrinkage
(Oldmixon et al., 1985). As we need to use antibodies for
the estimations of capillaries we are not able to apply
this fixation technique. Neither could we embed the tissue in methacrylate instead of paraffin for quality
improvement of the thin sections, because plastic would
as well prevent the use of antibodies. Knowing at least
two circumstances which produce shrinkage in our samples: fixation and the embedding in paraffin, estimation
and monitoring of the magnitude of shrinkage and afterwards correction of our results for shrinkage are inevitable to get hold of reasonably reliable results for length,
surface and volume estimations, as these parameters
are affected by shrinkage in contrast to number (Yan
et al., 2003). However, our shrinkage correction is less
than perfect because we assume isotropic shrinkage.
Tissue Sampling
The fractionator sampling by Gundersen (2002) enables us to estimate alveolar number without any regard
to shrinkage. Very simple and unbiased in itself its efficiency in lung tissue is increased by the relative homogeneous particle density of alveoli. Moreover, the
sampled number of particles is rather high and the size
of the pieces that are sampled on each sampling step
only varies negligibly. The alveolar number estimation
can be carried out on sections using a combination of
the disector and the Cavalieri principle for lung volume
estimation as well (Ochs et al., 2004). Still, this method
may produce biased results and overestimate alveolar
number as far as shrinkage is not monitored and corrected for.
Mice are commonly used as models to study the mechanisms involved in lung development and disease by
manipulating their well-known genome. The mouse is a
very small mammal and presents good opportunities for
a rigorous sampling design. On the level of block sampling we can still include all blocks and achieve a block
sampling fraction, bsf, of 1. On the level of section sampling it is then enough to take only one section per slice
for further observations and to survey it in about fifty
fields of view.
The combination of the disector as 3D sampling probe
and the Euler number as a counting principle allows
unbiased estimation of the alveolar number. On a trial
run, we adjusted the superimposed counting grids with
the different probes and the size of the counting frame
to the lung size. This small effort makes it possible to
estimate lung volume and surface area densities parallel
to the Euler number count in the same sampling session, thus avoiding a considerable loss of time by
repeated delineations of tissue and sampling of fields of
view. Whereas alveolar number in terms of Euler number was always estimated in both the reference and the
look-up section, we could skip the look-up section for the
estimation of all other parameters.
Alveolar Number and Size
Weibel and Gomez developed a formula to estimate
the number of alveoli in lungs assuming a particular
shape and size distribution of alveoli that enter the calculation as K the size distribution coefficient and b the
shape coefficient for alveoli (Weibel, 1984). This has for
a long time been of great use in lung research but nowadays falls behind in comparison with the new unbiased
stereological tools.
The mean chord length Lm obtained from single histological sections as an index of alveolar size has been a
common way of deriving the total number of alveoli from
alveolar size and total lung volume (Tomkeieff, 1945;
Weibel and Gomez, 1962). However, this method entails
bias as alveolar airspace and airspace of alveolar ducts
are not distinguishable on single sections and therefore
mean chord length would always be an intermediate
value of Lm in acinar ducts and alveoli (Hansen and
Ampaya, 1974; Blanco et al., 1989). Moreover the ratio
of airspace in ducts and alveolar airspace can vary and
shows different responses to experimental conditions. In
addition, any estimation of size in single sections is
influenced by size, shape, and distribution of alveoli.
Weibel and Gomez developed a formula to estimate
the number of alveoli in lungs assuming a particular
shape and size distribution of alveoli that enter the calculation as K the size distribution coefficient and b the
shape coefficient for alveoli (Weibel and Gomez, 1962;
Weibel, 1984). This has for a long time been of great use
in lung research but nowadays falls behind in comparison with the new unbiased stereological tools.
Blanco et al. (1989) first used the selector method
(Cruz-Orive, 1987) on stacks of serial sections to determine alveolar volume in rat lungs and indirectly derived
alveolar number from mean alveolar volume, alveolar
airspace density and lung volume (Blanco et al., 1989;
Massaro and Massaro, 1996). The method is fraught
with nearly the same problems as the Lm-based estimator. Mercer et al. (1994) first used the disector for alveolar number estimation but the identification of one single alveolus was only possible by employing the rather
time consuming observation of serial sections and three
dimensional reconstruction as well (Mercer et al., 1987)
and could not be ensured to be unambiguous because
the definition of alveoli for the counting procedure was
The first approaches combining the disector and the
Euler number was made by Hyde et al. (2004) and Ochs
et al. (2004) for Rhesus monkey and rat and human
lung, respectively.
The mean total number of alveoli of 2.3 106 we estimated for mice lungs is about in the same order of magnitude as the values Mercer et al. found in their study
(1994). Those mice lungs comprised 4.2 106 alveoli, but
the animals weighed about 1.5 times as much as ours
and had a 2.7 times larger lung volume. The lung volume estimated by Mercer et al. is uncorrected for
shrinkage, but shrinkage should not be too pronounced
as the lungs were embedded in epoxy resin.
Capillary Length
If one wants to apply the described method of capillary length estimation one should meet four major
requirements: (1) The surface area has to be obtained
from vertical sections using a cycloid counting grid. (2)
The diameter has to be measured according to the
explained procedure as a surface-weighted diameter. (3)
To perform all further calculations the surface-weighted
diameter has to be transformed to the ordinary lengthweighted diameter by calculating its harmonic mean. (4)
The length estimator requires that the capillaries are
circular. With our way of lung fixation via the airways
we may not entirely fulfil the fourth requirement. It is
therefore usually an advantage to choose isotropic sections so the length estimation can be performed with a
single profile count.
We have demonstrated the reliable and efficient application of a combination of unbiased stereological tools
for the estimation of various parameters of lung structure. The estimation of alveolar number by the means of
Euler number in a fractionator design is an easily applicable tool. The combination with the capillary length
estimation on vertical sections using the surface
weighted diameter provides us with a method that renders an easy, fast and essentially unbiased way of monitoring structural changes in lung tissue under whatever
experimental circumstances possible. The approach presented here should allow the wide application of designbased stereology for the quantitative lung phenotype
analysis of mice.
The authors thank the skillful technical assistance of
Maj-Britt Lundorf and Anette Larsen as well as the linguistic assistance of Karin Kristensen.
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estimates, length, alveolar, area, mice, lung, surface, size, number, capillary, stereological
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