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PROOF COPY 004704JBO
Investigation of morphometric parameters
for granulocytes and lymphocytes as applied
to a solution of direct and inverse light-scattering
problems
Gennady I. Ruban
National Academy of Sciences of Belarus
Stepanov Institute of Physics
Nezavisimosti Avenue 68
Minsk 220072, Belarus
E-mail: ruban@dragon.bas-net.by
Svetlana M. Kosmacheva
Natalia V. Goncharova
Ministry of Health of Belarus
Centre of Hematology and Transfusiology
Dolginovsky Avenue 160
Minsk 223059, Belarus
Dirk Van Bockstaele
Antwerpen University and Hospital
Wilrijkstraat 10
Edegem B-2650, Belgium
Valery A. Loiko
National Academy of Sciences of Belarus
Stepanov Institute of Physics
Nezavisimosti Avenue 68
Minsk 220072, Belarus
Abstract. Quantitative data on cell structure, shape, and size distri-
bution are obtained by optical measurement of normal peripheral
blood granulocytes and lymphocytes in a cell suspension. The cell
nuclei are measured in situ. The distribution laws of the cell and
nuclei sizes are estimated. The data gained are synthesized to con-
struct morphometric models of a segmented neutrophilic granulocyte
and a lymphocyte. Models of interrelation between the cell and
nucleus metric characteristics for granulocyte and lymphocyte are ob-
tained. The discovered interrelation decreases the amount of cell-
nucleus size combinations that have to be considered under simula-
tion of cell scattering patterns. It allows faster analysis of light
scattering to discriminate cells in a real-time scale. Our morphometric
data meet the requirements of scanning flow cytometry dealing with
the high rate analysis of cells in suspension. Our findings can be used
as input parameters for the solution of the direct and inverse light-
scattering problems in scanning flow cytometry, dispensing with a
costly and time-consuming immunophenotyping of the cells, as well
as in turbidimetry and nephelometry. The cell models developed can
ensure better interpretations of scattering patterns for an improvement
of discriminating capabilities of immunophenotyping-free scanning
flow cytometry.
© 2007 Society of Photo-Optical Instrumentation Engineers.
关DOI: 10.1117/1.2753466兴
Keywords: cell model; light scattering; size distribution.
Paper 06167RR received Jun. 30, 2006; revised manuscript received Mar. 7, 2007;
accepted for publication Apr. 19, 2007.
1 Introduction
Particulate matter is widely investigated by various methods
of remote optical probing. Among these methods are, for ex-
ample, turbidimetry, nephelometry, and flow cytometry. Tur-
bidimetry and nephelometry are based on light scattering by
ensembles of particles, while flow cytometry
1–3
is based on
single-particle analysis. Flow cytometry has been quickly de-
veloped within the last three decades. It is generally applied
for characterization of single cells from light scattering and
fluorescence. This technique deals with high rate analysis of
particles 共up to 5000 particles per second兲 and has numerous
applications.
1,2,4
It is widely applied for the identification of
white blood cells 共leukocytes兲 by combining light scattering
and immunophenotyping.
1
The latter is a time-consuming pro-
cess. It utilizes expensive fluorescently labeled monoclonal
antibodies. The possibility of cell injury should not be ruled
out during immunophenotyping. Therefore, for the last few
years a growing interest in the retrieval of morphological pa-
rameters of biological cells based mainly on scattering data
has been observed. In conventional flow cytometry, scattered
light is measured in two directions: forward and sideways. It
is possible to extract more information about cell characteris-
tics by measuring angular distributions of scattered light in-
tensity or polarization, which are highly sensitive to cell mor-
phology.
In some advanced experimental cytometric equipment ap-
paratus, named scanning flow cytometers, the role of scattered
radiation is extended. They measure angular dependences of
intensity and polarization of scattered light 共“fingerprints”兲
over a wide interval of scattering angles.
5,7,8
The flow cyto-
metric light-scattering patterns give new opportunities for re-
trieval of morphological parameters of biological cells and for
their discrimination. From integrated light-scattering measure-
ments, one can already distinguish two subpopulations, T8a
and T8b, within T8-positive lymphocytes, which cannot be
distinguished by ordinary histological methods.
9
Retrieval of parameters of a single particle or an ensemble
of particles by an angular pattern of scattered light is an im-
portant problem of the light-scattering theory.
10–18
To solve
that problem 共the inverse light-scattering problem兲, it is nec-
1083-3668/2007/12共4兲/1/0/$25.00 © 2007 SPIE
Address all correspondence to Gennady Ruban, Stepanov Institute of Physics,
Ave Nezavisimosti 68-Minsk, 220072, Belarus; Tel.: 375 17 2840797; Fax: 375
17 2840879; E-mail: ruban@dragon.bas-net.by
Journal of Biomedical Optics 12共4兲,1共July/August 2007兲
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essary to have optical models of the single scatterer or the
ensemble of scatterers. One of the simplest models for single
scatterer is the model of homogeneous spherical particle.
17
This simplified model can unlikely be used for the description
of light scattered by white blood cells, which we address in
this investigation, as they have complex shape and constitu-
tion. Models of leukocytes should include a number of
parameters,
17–20
in particular, refractive index
21–24
共the relative
refractive index for leukocytes is in the range of about 1.01 to
1.08兲, cell shape size, nuclear shape and size, etc. A model for
the ensemble of particles has to include the size distribution
functions as well.
25
That is of importance not only within the
framework of flow cytometry
26
but also beyond. Some models
of biological cells and optical scattering phenomena in whole
single cells, as well as in specific cellular organelles isolated
from cells or in situ, are considered in Refs. 17 and 27–39.
The cervical-cell model takes into account intranucleus
structure.
36,37
There are some approaches and techniques that can be
used to construct a morphometric model for leucocytes. Dif-
ferent methods are used to obtain data on cell sizes. Among
these methods are the centrifugal elutriation technique, track-
ing velocimetry, light, and electron microscopy,
40
and elec-
tronic particle volume analyzing.
41,42
The technique of centrifugal elutriation for cell size deter-
mination is based on a balance between centrifugual sedimen-
tation and centripetal flow.
43,44
By the method of centrifugal
elutriation, the authors in Ref. 44 have calculated sizes of
some human peripheral blood and bone marrow cells. Track-
ing velocimetry determines cell size by experimental deter-
mining of cell settling velocity in the natural gravity field
from microscopically obtained cell images. Human lympho-
cyte and fibrosarcoma cell size distributions were obtained by
the method of tracking velocimetry.
45
The assumptions of the
technique are: 1. the cell density does not vary from cell to
cell in a given sedimentation subpopulation, and 2. the cells
are spherical. Both techniques do not permit obtaining cell
parameters 共cell and nucleus shapes, nuclei sizes, the location
and allignment of a nucleus in the cell, etc.兲 that significantly
influence light-scattering pattern.
Light microscopy enables the size of cells on a smear or in
a suspension to be directly measured. Blood smears are very
informative for analysis of normal and pathological cells. The
data on microscopy-measured leukocytes sizes are basically
obtained from blood smears 共see Refs. 26, 46, and 47兲.Itis
worth noting that cell sizes can be distorted in smears by the
smearing and drying effect, as well as by adhesive interaction
of cells with a substrate.
Some fine details of cell morphology resolvable with elec-
tron microscopy are seemingly redundant for development of
a leukocyte model within the framework of scanning flow
cytometry today. It is worth noting that electron microscopy
methods
40,48
deal with lifeless bio-objects in high vacuum en-
vironments.
The Coulter technique
42,49,50
determines cell volume and
concentration in suspension. Data from Coulter counters are
very useful in many applications.
49
The electronic signal of
such counters is not directly related to cell volume.
45
Further-
more, the cell nuclei volumes measured by Coulter counters
may be greatly distorted with respect to the data measured by
image analysis.
51
Besides, Coulter counters do not determine
cell shape parameters that have an influence on light
scattering.
Known leukocyte size data are rather conflicting. Discrep-
ancies are seen between cell sizes obtained by examination of
smears
47,52
or suspensions,
53
as well as between data of dif-
ferent authors.
47,52
Although, as we have just described, there are vast data on
cell morphology, but a morphometric model for leucocytes
with reference to scanning flow cytometry has not been de-
veloped yet. The direct light-microscopy investigation of cells
in suspension has to be the basis to get more adequate data on
cell morphology as applied to the solution of the inverse light-
scattering problem for scanning flow cytometry.
As we wrote before, in scanning flow cytometry the im-
portance of a scattering channel is increased, owing to mea-
surement of angular structure of scattered light in a wide
range of scattering angles. It brings new opportunities for cell
discrimination and potentially can decrease the amount of
fluorochrome-labeled 共monoclonal兲 antibodies at the cell
analysis. To discriminate the cell we have to correlate its
structure and angular pattern. Remember that high rate analy-
sis is very essential to scanning flow cytometry. The recorded
angular patterns have to be analyzed rapidly in a real-time
scale 共typically on the order of miliseconds兲. That is why it is
important to find ways to improve the computer simulation.
To solve this problem, we have to narrow domains of uncer-
tainty of morphometric parameters of the cells in question, as
applied to the specified scanning flow cytometry technique.
The goal of the work is the optical microscopy investiga-
tion of morphometric characteristics of human granulocytes
and lymphocytes, at conditions corresponding to that of scan-
ning flow cytometry dealing with high rate analysis of the
cells in suspension. On the basis of the data gained by optical
microscopy, dot estimations of the distribution parameters of
granulocytes, lymphocytes, and nuclei sizes of the cells are
obtained. Probability density functions of granulocytes and
granulocyte lobes, and lymphocyte and lymphocyte nuclei are
estimated. The interrelations between the cell and nucleus
metric characteristics for granulocytes and lymphocytes are
deduced. The main attention is concentrated on granulocytes.
More detailed data on lymphocytes are published by us in
another work.
54
The obtained results are used to construct the
models of normal cells. The next step is the investigation of
pathologic cells.
There are four sections in the work. Section 1 is the intro-
duction, where the general problem is presented. It includes
our summary on the known cell morphology data and on their
applicability for morphometric model elaboration with refer-
ence to scanning flow cytometry. Section 2 is material and
methods. The object and method of our investigation are con-
sidered. Section 3 is the lymphocyte and granulocyte struc-
ture: results and discussion. Here the results of our measure-
ments for granulocytes and lymphocytes by use of optical
microscopy are described. We analized the known electron-
microscopy data with reference to the scanning flow cytom-
etry and incorporated them into the granulocyte model. The
proper references to the original publications of electron-
microscopy data are made. Section 4 has the conclusions.
Here some remarks on the obtained results are made.
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2 Materials and Methods
Cells of fresh peripheral blood were investigated. The blood
was obtained by venipuncture from apparently healthy men
and women ages 20 to 46. Heparin 共20 units per
1ml of
blood兲 was utilized as an anticoagulating agent. The blood
was diluted in physiologic saline
共pH=7.4兲 at a ratio of two
to one.
Leukocytes were isolated by a density gradient technique.
To obtain mononuclear cells 共MNCs兲 or granulocytes in ac-
cordance with the last technique,
3ml of Ficoll-Verografin
共Sigma, USA兲 was placed in a tube. Ficoll-Verografin density
was 1.097 and
1.077 g/cm
3
for granulocyte and lymphocyte
separation, respectively. The blood 共4to
5ml兲 was layered
over the Ficoll-Verografin cushion. The tube was centrifuged
at
460 g at +5°C for 60 to 80 min. During centrifugation, the
cells are separated at phase interface, according to their
density.
The isolated fractions of the cells were resuspended in a
phosphate buffered 共physiological兲 saline 共PBS, pH 7.4兲
supplemented with 1% heat inactivated pooled AB共IV兲 serum
from five healthy individuals. The resuspended cells were pel-
leted by centrifugation at
460 g at +5°C for 20 min. The cell
pellets were then washed twice by resuspending in phosphate
buffered saline 共PBS兲 with the AB共IV兲 serum and centrifuga-
tion at
210 g at +5°C for 10 min.
Viability of the isolated cells was determined by a standard
technique with the help of trypan blue stain 共Sigma, USA兲.
The viability of isolated MNCs and granulocytes being tested
with 500 to 600 cells for each individual was ranged from 95
to 100%. The viable and fixed cells were studied. The cells
were fixed with 2% solution of paraformaldehyde 共Sigma,
USA兲 in PBS at
+4°C in the dark for 20 min, washed in the
PBS.
Light microscopy was used to investigation the cell sus-
pensions. For that purpose they were hermetically sandwiched
between an object-plate and cover-slip, which makes up a
“microcuvette.” We used the Leica DMLB2 microscope in the
bright field, fluorescence, and differential interference contrast
共DIC兲 modes as per the microscope manufacturer’s instruc-
tions. An oil immersion objective was used with magnifica-
tion
100⫻ and numerical aperture 1.25. Optical micrographs
were made by the microscope-mounted digital camera Leica
DC 150 with a resolution of
⬃5 MPixels. The granulocyte
micrographs in Fig. 1 are displayed as an example. The Leica
image processing software IM 1000 has been exploited to
analyze 2-D images of the cells. Size calibration was made
with a Leica test object.
The mononuclear suspensions included lymphocytes and
monocytes. We recognized and excluded monocytes via mi-
croscopically visual differences in cell morphology. We dis-
tinguished B lymphocytes from the mononuclear cells by the
fluorescently labeled 共with phycoerythrin兲 monoclonal anti-
bodies directed against surface antigen CD19.
3 Granulocyte and Lymphocyte Structure:
Results and Discussion
In this section we consider lymphocyte and granulocyte
shapes and sizes and estimate size distribution laws for granu-
locytes, their lobes, lymphocytes, and their nuclei. Correla-
tions between cell and nucleus metric characteristics for
granulocytes and lymphocytes are described.
To construct the adequate model, it is desirable to measure
cells having the least possible disturbance. So initially, viable
cells were studied. However, the process of cell image record-
ing is time consuming. That is why it is preferable to deal
with fixed cells. The fixation can change cell morphological
characteristics. To estimate the influence of fixation, we com-
pared fixed and viable cells for some individuals. We detected
differences of mean sizes of viable and fixed cells for the
same individuals. The differences were small. They did not
exceed 1% 共for lymphocytes兲 and3to5%共for granulocytes兲.
For example, mean sizes of viable and fixed lymphocytes
were 6.65 and
6.70
m, respectively 共individual 4兲; mean
sizes of viable and fixed granulocytes was 9.24 and
9.52
m,
respectively 共individual 11兲. That is why we do not distin-
guish the results for viable and fixed cells described next.
3.1 Lymphocyte and Granulocyte Shape
Some researchers consider lymphocytes as perfect spherical
particles, while others consider them as nonspherical
particles.
40,55–57
Our image analysis of the cells has shown that
lymphocytes and their nuclei basically have an ellipsoidal
shape or a shape close to the spherical one. The lymphocyte
nucleus commonly has ellipsoidal or round shapes and some-
times more complex ones. Detailed lymphocyte morphomet-
ric data are presented in our article.
54
Some contours of granulocytes, drawn with Leica image
management software IM1000, are displayed in Fig. 2. Con-
tours of granulocytes are less various in shape compared to
lymphocyte
54
ones. Our measurements and analysis indicate
that the shape of granulocytes is commonly slightly elongated
共ellipsoidal兲 or round. Sometimes we observe granulocytes of
Fig. 1 Images of 共a兲 segmented 共neutrophilic兲 granulocyte and 共b兲 stab
共neutrophilic兲 granulocyte with C-like nucleus shape. Differential in-
terference contrast mode.
Fig. 2 Contours of granulocytes.
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ovoid-like and other shapes. The mean ratio of the major and
minor granulocyte axes 共maximum and minimum linear sizes兲
measured for one of the individuals 共individual 6兲 is 1.08
共standard deviation
=0.06兲. This value shows that granulo-
cytes, as well as lymphocytes, are elongated cells. Note that
slightly elongated and rounded granulocytes are indicated in
the known cell micrographs
40
and figures
46
as well.
According to our estimates, about 90% of granulocyte nu-
clei are segmented 共lobulated兲. The lobes have different
shapes. Sometimes they can be modeled as ellipsoidal-like
one. The lobes are more elongated than the granulocyte. For
one of the considered individuals, the mean ratio of the major
and minor lobe axes is 1.2
共
=0.1兲. Nuclei in stab 共neutro-
philic兲 granulocytes have C-like, S-like, and other shapes.
Granulocyte surfaces can expose some features.
55
Some-
times we observed a knobby contour of granulocytes by opti-
cal microscope. As it was shown by electron microscopy, the
surface of neutrophil granulocytes can be wrinkled.
58
Granu-
locyte nucleus surfaces can also have some features. For ex-
ample, neutrophil nuclei can be folded.
56
As it is well known, lymphocytes and granulocytes are
surrounded by a plasma membrane with a nanometer-scale
thickness. Cell membranes are composed mostly of lipids and
proteins with refractive indexes in the range of 1.46 to 1.54
共see Ref. 59 and references therein兲. Backscattered light cal-
culated for particles with a thin shell is strongly sensitive to
shell thickness and to shell refractive index, as one can see
from Fig. 3 共our original data are obtained using a code by
Babenko, Astafyeva, and Kuzmin
25
兲 and data published in
Ref. 59. Thus the cell membrane should be taken into account
in the construction of detailed cell morphometric and optical
models, with reference to scanning flow cytometry, when
measurement of light intensity scattered in the backward
hemisphere is available.
3.2 Size Data for Lymphocytes and Granulocytes
Size of a cell is one of its most important biological and
optical parameters. Some results of our measurements for size
distributions of lymphocytes and lymphocyte nuclei are
shown in Figs. 4 and 5, respectively. Here the maximal linear
sizes 共major axes兲 of lymphocytes and lymphocyte nuclei are
presented. In Figs. 4 and 5, the relative frequency
p
i
is plotted
on the
y axis. The relative frequency p
i
= f
i
/n, where f
i
is the
observed absolute frequency corresponding to the
i’th interval
of sizes 共
i=1,2...k, where k is the quantity of the intervals
and
n is the amount of sampling兲. Maximum linear size is on
the
x axis. The range of the sizes, the sampling average size,
and the standard deviation for lymphocytes of some individu-
als are presented in Table 1. These values for lymphocyte
nuclei of two individuals are as follows: 4.7 to 8.9, 6.4, and
0.7
m 共individual 1兲: and 4.8 to 8.3, 6.3, and 0.5
m 共indi-
vidual 14兲, respectively. Our measurements demonstrate that a
nucleus within a lymphocyte is generally eccentric and
off-oriented.
54,60
The lymphocyte nucleus is sometimes situ-
ated near the cell membrane. Applying this to the problem of
light scattering by the cell results in the appearance of high-
order modes in the expansion of the cell internal fields.
18
Comparison of our data on lymphocyte size with data from
Coulter counters
45
shows good agreement between the results.
In this investigation we mostly pay main attention to
granulocytes. The range of sizes, the sampling average size,
and the standard deviation for granulocytes are presented in
Table 2. The mean size of granulocytes is larger than the one
Fig. 3 Calculated angular dependences of light-scattering intensity for
the single coated 共line 1兲 and uncoated 共line 2兲 particles with diam-
eter 7.5
m and refractive index 1.37. Refractive index of particle
surroundings is 1.35. Shell thickness of coated particles is 10 nm, and
shell refractive index is 1.52. Wavelength of the incident light is
0.63
m.
Fig. 4 Histogram of the maximal linear sizes for lymphocytes. Indi-
vidual 1, n=324.
Fig. 5 Histogram of the maximal linear sizes for lymphocyte nuclei.
Individual 26, n=474.
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of lymphocytes, as can be seen from Tables 1 and 2. The size
ranges of the granulocytes and lymphocytes overlap. Histo-
grams of the granulocyte size 共the maximal cell size兲 distri-
bution as well as the ratio distribution of the maximal and
minimal sizes of granulocytes for one individual are presented
in Figs. 6 and 7, respectively.
Granulocyte nuclei are frequently lobulated. The nucleus
lobes are connected by thin filaments of chromatin.
56
We ob-
served adjoined, overlapped, and separated nucleus lobes. For
the individuals that we examined, the number of nuclei lob-
ules in granulocyte set is in the range 2 to 6, commonly 3 to
4. Granulocyte subsets are characterized by other researchers
as follows
61
: neutrophil granulocytes contain 3 to 5 lobules,
basophils contain 2 to 3 lobes, and eosinophils contain 2 lobes
commonly. The range of the maximal linear sizes, the sam-
pling average size, and the standard deviation for single
nucleus lobes in our measurements are 2.6 to 6.6, 4.35, and
0.65
m, respectively. These values for minimal linear size
共minor axis兲 of nucleus lobes are 2.4 to 5.7, 3.6, and
0.55
m, respectively 共individual 9兲. Total projection area of
the granulocyte lobes 共of the 2-D cell images兲 is in the range
of 21 to
49
m
2
; the mean total area is 33
m
2
with standard
deviation
5.5
m
2
共individual 9兲. Histograms of the measured
size distributions of nuclei lobes are displayed in Figs. 8–10.
Compare our data to the ratios of the cell to nucleus size
for lymphocytes and granulocytes. We characterize the size of
a granulocyte nucleus by its effective diameter. It is deter-
mined as the diameter of the disk having the same area as the
total area of the granulocyte lobes. The mean ratio of the
major axis of lymphocyte to its nucleus axis is about 1.2. This
means that in the majority of cells, the nucleus and lympho-
cyte sizes are comparable. The mean ratio of the granulocyte
size 共maximal linear size of the cell兲 to the 共previously deter-
mined effective diameter of the granulocyte lobes is about 1.6
versus 1.2 for the proper lymphocyte value.
As it is known, granulocyte cytoplasm holds granules.
61
They have sizes close to the limit of the light microscope
resolution. Published data of electron-microscopy studies are
contradictory. Electron-microscopy micrographs commonly
display elongated granules.
61,62
Analyzing the micrographs of
eosinophil granules, as published in Refs. 61 and 62, we find
that the ranges of ratios of maximal and minimal sizes are 1.2
to 1.7 and 1.2 to 2.5, respectively. According to the results of
Ref. 56, the eosinophil granules have spherical shapes; baso-
phil granules are roughly spherical or are seldomly, irregular.
Granules are different in organization. For example, spe-
cific eosinophil granules are heterogeneous: they include a
peripheral matrix and inner core. This core is an elongated
Table 1 Lymphocyte size data.
Individual
Range of
lymphocyte
size,
m
Mean value of
lymphocyte
size,
m
Standard
deviation,
m
Number
of cells
2 6.1 to 8.8 7.4 0.5 809
4 5.2 to 8.0 6.6 0.4 324
5 6.3 to 8.6 7.5 0.4 138
23 6.4 to 9.8 7.9 0.6 223
24 6.7 to 9.0 7.7 0.6 168
25 6.3 to 10.1 8.2 0.8 124
Table 2 Granulocyte size data.
Individual
Range of
granulocyte
size,
m
Mean value of
granulocyte
size,
m
Standard
deviation,
m
Number
of cells
6 8.0 to 11.7 9.6 0.5 3040
7 7.9 to 12.1 9.7 0.7 583
8 8.1 to 12.0 9.6 0.6 697
9 7.9 to 11.2 9.5 0.6 225
10 7.3 to 12.1 9.4 0.7 187
11 7.6 to 13.2 9.2 0.7 316
12 7.8 to 13.0 9.8 0.9 118
15 7.9 to 12.4 9.4 0.7 326
Fig. 6 Histogram of the maximal linear sizes for granulocytes. Indi-
vidual 6, n=3040.
Fig. 7 Ratio distribution of the maximal linear size to the minimal one
for granulocytes. Individual 6, n=112.
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crystalloid.
61,63
Other eosinophil granules are homogeneous
共coreless兲.
56
Granules in granulocyte subsets are various in size. The
neutrophil granulocyte cytoplasm holds specific granules with
a size of about
0.2
m 共80 to 90% of the total number of
granules兲 and nonspecific ones with a size of about
0.4
m
共10 to 20%兲. Basophil granulocytes contain specific granules
with sizes 0.5 to
1.2
m. Sizes of specific granules in eosi-
nophil granulocytes are 0.6 to
1
m. The previous data are
obtained by electron microscopy.
61
The lower limit of specific
eosinophil granules measured by the time-resolved patch-
dampt capacitance technique is 0.45 to
0.5
m.
64
The surface
area of eosinophil granules is about
0.7
m
2
.
64
Some comments on granule packing and ordering are as
follows. Granules occupy a noticeable part of the cytoplasm,
as one can see from electron micrographs.
61
This visual im-
pression is verified by numerical data. For example, the area
of granulation in eosinophils is 87%.
26
The number of grains
in the cytoplasm of neutrophil granulocytes is 50 to 200. Re-
ciprocal disposition and orientation of elongated granules is
not fully random, as seen from the electron micrographs of
granulocytes.
61,62
We analyze the described electron-microscopy data on
granulocyte granules with reference to scanning flow cytom-
etry. First, the previously mentioned high value of area filling
of the cytoplasm by granules 共0.87兲 implies that at first ap-
proximation, when constructing a model for granulocyte mor-
phology, intracellular constituents other than granules 共and
nucleus兲 can be neglected. Second, we deduce that granulo-
cyte granules are an ensemble of densely packed particles in
terms of optics of scattering media. Granule disposition and
orientation are partially ordered. Dense packing gives rise to
overirradiation and near-field effects.
65,66
Ordered disposition
and orientation give rise to interference of light scattered by
granules. These facts should be taken into account when se-
lecting a proper method to solve direct and inverse light-
scattering problems. Third, the size of granulocyte granules is
less than or comparable with the wavelength of visible light.
Such particles have a larger ratio of sideways and backward
light-scattering intensity compared to particles larger than a
wavelength. Therefore, angular light-scattering patterns by a
granulocyte should have distinctive features as compared to
an agranulocyte of a proper size, e.g., a monocyte and a large
lymphocyte. Fourth, the size of the majority of neutrophil
granulocyte granules is approximately 3 to 6 times smaller
than that of eosinophil and basophil granulocytes. Thereafter,
the angular scattering pattern of light by neutrophil granulo-
cytes can expose distinctive features as compared with the
one for eosinophil and basophil granulocytes. Fifth, having
analyzed the previously listed data, it is reasonable to con-
clude that distinctive polarization features of scattered light
correspond to the elongated and partially ordered eosinophil
granules with crystalloid cores. These features can be used to
distinguish eosinophils.
A major subset of granulocytes is neutrophils. The mor-
phometric model of a neutrophilic granulocyte with lobulated
nucleus is shown schematically in Fig. 11.
Some comparisons of our results with the data of Coulter
measurements and smear measurements for the size of granu-
locytes are indicated next. The average neutrophilic granulo-
cyte volume as measured with a Coulter counter is
370 fL.
53
The corresponding neutrophil diameter is then 8.9
m for an
equivolume sphere. The mean size of
9.58
m of granulo-
cytes, as we observed, is slightly larger inasmuch as the
granulocytes in our measurements include neutrophils, eosi-
Fig. 8 Histogram of the maximal linear sizes for granulocyte lobes.
Individual 9, n=462.
Fig. 9 Histogram of the minimal linear sizes of granulocyte lobes.
Individual 9, n=213.
Fig. 10 Total-area distribution of granulocyte lobes cross-sections. In-
dividual 9, n=236.
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nophils, and basophils. The last two are larger in size than
neutrophils.
52
Moreover, a granulocyte actually is a somewhat
oblong ellipsoid rather than a perfect sphere. So our size data
agree well with the literature data for lymphocytes and granu-
locytes in suspension. We measured size ranges of granulo-
cytes in suspension. The lower and upper limits of the ranges
for different individuals are 7.3 and
13.2
m, respectively. It
worthwhile to note that the sizes of granulocytes in a smear
共12 to
17
m兲
47,52
and in suspension 共7.3 to 13.2
m兲 are
noticeably different. The lower limit
共12
m兲 of the
granulocyte-size ranges in a smear is close to the upper limit
of the ranges in suspension
共13.2
m兲.
Our measurements and calculations have shown that there
is a positive correlation between areas of granulocyte lobes
and of the granulocyte itself. It is statistically highly signifi-
cant, the probability
P⬍0.0001. The probability P is deter-
mined as follows:
P= P
N
共兩r兩 ⱖr
0
兲, where r is the correlation
coefficient, and
r
0
is the sampling correlation coefficient.
67
Values of correlation coefficients for two individuals are indi-
cated in Table 3. Constants
A
g
and B
g
of the linear regression
equation
共y = A
g
+B
g
x兲 of the lobe area y on granulocyte area
x are presented in Table 3 as well. The experimental data of
the lobes and granulocyte areas are displayed in Fig. 12. A
linear correlation dependence of sizes for T-lymphocyte and
its nucleus
54
and B-lymphocyte and its nucleus 共Table 4兲 are
also revealed. Constants
A and B of the linear regression
equation
共y = A + Bx兲 of nucleus size 共major nucleus axis y兲 on
lymphocyte size 共major lymphocyte axis
x兲 for B-cells are
presented in Table 4. Correlation coefficients between sizes of
nucleus and B-lymphocyte for some individuals are indicated
in the table as well.
The discovered correlations of metric characteristics for
the cells and their nuclei simplify simulations of scattering
patterns. Actually, the correlations reduce a set of cell-nucleus
size combinations to the statistically admissible subset of the
combinations. Just this subset has to be taken into consider-
ation under simulation of cell scattering patterns. It permits
faster analysis of light scattering to discriminate cells in a
real-time scale.
3.3 Estimation of a Size Distribution Law
In this subitem, the distribution density of granulocytes and
lymphocytes is estimated. To promote a hypothesis for distri-
bution law
f共x兲 of the cell sizes, we 1. calculate the dot evalu-
ation of distribution parameters of granulocytes, and 2. con-
struct a histogram of size distribution of these cells. Then 3.
we check the hypothesis by an evaluation of the probability
density function in accordance with criterion
2
.
3.3.1 Estimation by Dot Parameters
For an initial assessment of the distribution law, it is expedi-
ent to use a simple method. To make a conclusion about the
distribution law, the scheme of sampling dot evaluations ob-
tained by a method of moments is used.
67
According to this
Fig. 11 Scheme of lobulated neutrophil granulocyte.
Table 3 Sampling correlation coefficient and constants A
g
and B
g
of the linear regression equation of
granulocyte lobe area on granulocyte area.
Individual
Correlation
coefficient
A
g
,
m
2
Standard
deviation
of A
g
,
m
2
B
g
Standard
deviation
of B
g
Standard
deviation
of lobe
area,
m
2
Number
of cells
9 0.65 1 0.2 0.5 0.04 4.3 193
16 0.41 6.3 0.3 0.25 0.03 3.5 285
Fig. 12 Scatter plot 共points兲 and linear regression 共solid line兲 of the
projection area of granulocytes and total projection area of the granu-
locyte lobes. Individual 9, n=193.
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scheme, the sampling values are calculated for the factors of
asymmetry
A and excess E, and their standard deviations S
A
and S
E
.
Then it is believed that the empirical law is consistent with
the hypothetical one, provided that the following inequalities
hold:
兩A − M共A兲兩 ⬍ 3S
A
, 共1兲
兩E − M共E兲兩 ⬍ 3S
E
, 共2兲
where M共A兲 and M共E 兲 are the mathematical expectations of
the
A and E values.
As a hypothesis, let us consider the normal law of distri-
bution of the experimental data. For the normal distribution
we have:
M共A兲= M共E 兲=0. Calculation of the values A, E, S
A
,
and
S
E
shows that inequalities in Eqs. 共1兲 and 共2兲 are carried
out for the individuals. Hence, the hypothesis about normal
distribution is accepted. It is worth noting that the inequalities
in Eqs. 共1兲 and 共2兲 have empirical basis, and the obtained
conclusion has to be considered as the preliminary one. Fur-
ther analysis is carried out next.
3.3.2 Estimation by Histogram
For construction of the histogram, the range of the measured
values of cell sizes is divided into the intervals of width
⌬
i
.
Then the absolute frequency
f
i
, the relative frequency p
i
= f
i
/n, and the normalized relative frequency f
i⌬
= f
i
/共n⌬
i
兲,
corresponding to the intervals, are counted up 共
n is the
amount of sampling兲. On the basis of that data, the histogram
of granulocyte size distribution is constructed 共Fig. 6兲.
The obtained sampling estimations of the distribution pa-
rameters and the constructed histogram allow putting forward
a hypothesis
H
0
of the normal distribution law. A detailed
check of that hypothesis is carried out in the following.
3.3.3 Estimation by the Chi-square Criterion
To test the identity of the distribution density of the sample
data, and the hypothetical distribution density, the
2
criterion
is used by the following scheme.
67
1. A hypothesis H
0
f共x兲= f
0
共x ,
兲 is put forward. Here f共x兲
is a distribution law of the random value represented by the
sample
兵x
j
其, j=1...n. f
0
共x ,
兲 is the model distribution law
characterized by a vector of parameters
=关
1
...
m
兴.
2. The range of the measured sizes is divided into
k group-
ing intervals of width
⌬
i
with X
i
as the right boundary of an
i’th interval, i=1...k. Absolute frequency f
i
appropriate to
the
i’th interval is counted up over the sample 兵x
j
其. Some
intervals on the tail areas of the distribution are combined to
provide for statistical representativeness of frequency
f
i
共in
that case
i=1...k
0
, k
0
⬍k兲, and an expected probability P
i
=⌽共Z
i+1
兲−⌽共Z
i
兲 is calculated. Here the function
⌽共Z兲 =
冕
−⬁
z
exp共− t
2
兲dt, 共3兲
is the distribution function of the standardized normal random
value
Z with zero mean Z
m
=0 and unit variance
z
2
=1. The
expected absolute frequency
F
i
= P
i
n 共i =1...k
0
兲.
3. Checking on the hypothesis includes three steps.
Step 1. A critical point
␣
2
is determined for a significance
level
␣
and the degrees of freedom
=k
0
−1−m. Under a
statistical treatment of the results of medical investigations, it
is often agreed that
␣
=0.01. The values of
␣
2
are tabulated
共see Ref. 67兲.
Step 2. A value of criterion
2
is calculated.
2
=
兺
i
k
0
i
2
=
兺
i
k
0
共F
i
− f
i
兲
2
/F
i
. 共4兲
Calculation of the
2
-criterion value is elaborated for one in-
dividual in Table 5.
Step 3. Comparing the criterion
2
value with the
␣
2
value, we accept or reject the hypothesis H
0
on the signifi-
cance level
␣
. The domain of acceptance of the hypothesis is
determined by an inequality
2
ⱕ
␣
2
.
The considered scheme is used to analyze measurement
results. The previous inequality is fulfilled for all considered
individuals. For example, for individual 6, the value of
2
=23.34 共for granulocytes兲. It is less than the value of the
critical point for this individual
共
␣
2
=26.21兲. Hence, on the
significance level
␣
=0.01, we do not have ground to reject
hypothesis
H
0
of the normal distribution law
f共x兲 =1/关共2
兲
1/2
兴exp − 共x − x
m
兲
2
/2
2
. 共 5兲
For the experimental data grouped in Table 5 共individual 6兲,
x
m
=9.58
m, and
=0.52
m.
The values of criterion
2
are close to the proper critical
points
␣
2
for the considered individuals. That is why it is
reasonable to examine other distribution laws. Assuming nor-
mal distribution of logarithms of the measured values, we
Table 4 Constants A and B of the linear regression equation of nucleus size on lymphocyte size for
B-cells.
Individual
A,
m
Standard
deviation
of A,
m B
Standard
deviation
of B
Standard
deviation
of nucleus
size,
m
Correlation
coefficient between
nucleus and
lymphocyte sizes
30 1.9 0.06 0.5 0.15 0.6 0.61
31 2.7 0.75 0.5 0.1 0.5 0.62
32 3.6 0.7 0.4 0.1 0.4 0.55
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found that
2
values are noticeably smaller than the proper
␣
2
values 共for example, for individual 6, criterion
2
=11.11兲 It means that the log-normal law
f共x兲 =1/关共2
兲
1/2
x
兴exp共− 关ln共x/x
m
兲兴
2
/2
2
兲, 共6兲
describes the experimental data better than the normal one.
Parameters
x
m
and
of the log-normal distribution law for
granulocytes, granulocyte lobes, lymphocytes, and lympho-
cyte nuclei are displayed in Table 6 for some individuals.
The graphics of normal and log-normal distributions and
experimental data are displayed in Fig. 13 for one individual.
The experimental data are taken from the histogram of Fig. 6.
Figure 13 shows that both distributions agree well with the
experimental data and the log-normal law describes the mea-
surement results better.
4 Conclusions
The peripheral blood granulocytes and lymphocytes of
healthy adult individuals are investigated by methods of spe-
cialized light microscopy. The morphometric parameters of
the cells in suspension are measured and analyzed. The cell
structures are characterized as applied to a problem of cell
discrimination in the frame of polarizing scanning flow
cytometry.
Size distribution parameters of the cells and nuclei are sta-
tistically estimated. The size probability density functions of
cells and their nuclei are obtained on the base of the estimated
parameters. The interrelations between the cell and nucleus
metric characteristics are deduced for granulocytes and lym-
phocytes to provide the simplification of the computer simu-
lation and the reduction of the running time of cell discrimi-
nation algorithms. The obtained data are processed and
generalized to construct the morphometric models of a seg-
mented neutrophilic granulocyte and a lymphocyte. Our mor-
phometric data meet the requirements of flow cytometry deal-
ing with the cells in suspension and real-time computations.
The findings of our investigation can be used as input param-
eters to solve the direct and inverse light-scattering problems
in scanning flow cytometry, dispensing with costly and time-
consuming cell labeling by monoclonal antibodies and fluo-
rochromes, as well as in turbidimetry and nephelometry.
The obtained results are related to cells of healthy indi-
viduals. Consideration of pathologic cells is the next step of
the investigation. The cell models developed in the present
work can contribute to theoretical grounding of cell discrimi-
nation and diagnosis methods. These models make it possible
to establish relationships between the cell structure and the
angular pattern of scattered light to determine dominant and
secondary origins of light scattering by a cell. This can pro-
vide background for better biological and medical interpreta-
Table 5 The data to calculate the
2
criterion and to test the hypoth-
esis of normal granulocyte size distribution 共individual 6兲.
iX
i
fi Z
i
⌽共Z
i
兲 P
i
F
i
i
2
1 8.20 12 −2.6545 0.0040 0.0040 12.0735 0.0004
2 8.41 13 −2.2495 0.0122 0.0083 25.1398 5.8622
3 8.62 52 −1.8445 0.0326 0.0203 61.7630 1.5433
4 8.83 124 −1.4394 0.0750 0.0425 129.0594 0.1983
5 9.04 257 −1.0344 0.1505 0.0755 229.3805 3.3256
6 9.25 376 −0.6294 0.2645 0.1141 346.7701 2.4638
7 9.46 465 −0.2244 0.4112 0.1467 445.9143 0.8169
8 9.67 493 0.1806 0.5717 0.1604 487.7392 0.0567
9 9.88 433 0.5856 0.7209 0.1493 453.7889 0.9524
10 10.09 334 0.9906 0.8391 0.1181 359.1265 1.7580
11 10.30 233 1.3956 0.9186 0.0795 241.7494 0.3167
12 10.51 122 1.8006 0.9641 0.0455 138.4213 1.9481
13 10.72 76 2.2057 0.9863 0.0222 67.4136 1.0936
14 10.93 30 2.6107 0.9955 0.0092 27.9246 0.1542
15 11.14 20 3.0157 0.9987 0.0045 13.7358 2.8568
Table 6 Parameters x
m
and
of log-normal size distribution calcu-
lated by criteria
2
for granulocytes, granulocyte lobes, lymphocytes,
and lymphocyte nuclei.
Individual Cell
x
m
,
m
Number
of cells
6 Granulocyte 9.56 0.054 3040
7 Granulocyte 9.70 0.067 583
8 Granulocyte 9.63 0.063 697
9 Granulocyte lobes 4.31 0.15 462
1 Lymphocyte 7.78 0.056 406
2 Lymphocyte 7.40 0.065 809
14 Lymphocyte 7.67 0.079 1089
14 Lymphocyte nuclei 6.30 0.077 1089
Fig. 13 The granulocyte size distribution f共x兲. The dashed line is the
normal distribution and the solid line is the log-normal distribution.
Points f
i⌬
indicate experimental data. Individual 6, n=3040.
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tions of the scattering patterns, to improve discriminating and
diagnostic capabilities of immunophenotyping-free scanning
flow cytometry, as well as for further adaptation of cell mod-
els to the real-time computations.
Acknowledgments
This research was supported by the Programme of Basic Re-
search of Belarus “Modern technologies in medicine”, and
partially sponsored by NATO’s Scientific Affairs Division in
the framework of the Science for Peace Programme, Project
977976. We express our gratitude to Olga Gritsai for taking
part in the cell image recordings. The authors are thankful to
Alfons Hoekstra, Alexei Gruzdev, and Valeri Maltsev for
fruitful discussions.
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