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Journal of Microscopy 187, 62-67, 1997.
Quantitative Image Analysis of Birefringent Biological
Material
Stephen Ross, Richard Newton, Yu-Ming Zhou, Julian
Haffegee, Mae-Wan Ho
Bioelectrodynamics Laboratory, Biology Department, Open University
Walton Hall, Milton Keynes, MK7 6AA, U.K.
John P. Bolton
Physics Department, Open University
Walton Hall, Milton Keynes, MK7 6AA, U.K.
David Knight
Bioscience Group, King Alfred's College, Sparkford Road,
Winchester S022 4NR, U.K.
Abstract
A previously described optical technique for enhancing interference
colour contrast in polarized light microscopy (Ho and Lawrence, 1993; Ho
and Saunders, 1994; Newton et al, 1995) is adapted for
quantitative image analysis. This is based on a linear relationship for
colour intensity versus molecular birefringence and degree of phase
alignment, derived using nematic liquid crystal approximations, and
verified experimentally. The image analysis is described and illustrated
on a section of the marginal rib of the dogfish egg case.
Key Words
Noninvasive quantitative imaging, interference colours, liquid crystals
Introduction
Interference colours are generated in birefringent specimens in
transmitted white light between two polarizers whose vibrational
directions are at 90. A plane-polarized light ray, on passing through the
birefringent specimen with its optic axis perpendicular to the direction
of light propagation, is split into two mutually perpendicularly vibrating
rays that propagate through the specimen at different velocities. The
retardation of the slow ray relative to the fast ray - relative
retardation measured in nanometres (nm) - generates a phase difference,
d, between the rays as they emerge
from the specimen,
d = 2Àd
(ne - no )/l(1)
where d is the thickness of the crystal, ne< and no
are the refractive indices for the extraordinary ray vibrating in
a plane parallel to the optic axis, and for the ordinary ray
vibrating perpendicular to the optic axis of the birefringent specimen,
and l is the wavelength of the light.
The slow and fast rays are recombined into a single ray as they pass
through the second polarizer, so they interfere either destructively or
constructively depending on the phase difference introduced by the
birefringent specimen. For white light with a full spectrum of wavelengths
in the visible range, from 390nm to 780nm, the phase difference varies
across the spectrum, so that a precise hue of interference colour will be
generated depending on the relative retardation (of the slow versus the
fast ray).
When the relative retardation of the specimen is below 200nm, brilliant
interference colours will not be generated unless a compensator crystal
plate of sufficient relative retardation is added in series with the
sample to bring the net retardation to within wavelengths of light in the
visible range. For weakly birefringent materials with relative
retardations less than 50nm, however, little or no colour contrast is
obtained when the compensator plate is placed with its vibrational
directions at the conventional angle of 45° from the crossed
polarizers. Instead, colour and colour contrast are greatly enhanced when
that angle is small (Ho and Lawrence, 1993; Ho and Saunders, 1994; Newton
et al, 1995). For best results over the range of weak
birefringences frequently found in biological materials, the angle was
ascertained to be 4.5° to 7.5°, and the relative retardation of
the compensator approximately 560nm - the wavelength of green/yellow
light. The same procedure was first discovered by Wright (1911) in
petrology, and described briefly again by others more recently (see Newton
et al, 1995); but it does not appear to be widely used or
developed in biological research.
We have described the applications of the technique for obtaining high
contrast colour images of live organisms which reveal their anatomical
details relatively noninvasively (Ho and Lawrence, 1993; Ho and Saunders,
1994; Newton et al, 1995). In this paper, we derive a linear
relationship for interference colour intensity versus effective
relative retardation, based on which a quantitative imaging procedure is
developed. The latter enables us to determine and plot the distribution of
retardations and angles of maximum retardation in biological specimens. In
those respects, it is complementary to the method described by Oldenbourg
and Mei (1995). We illustrate how our method is used in a section of the
marginal rib of the dogfish egg case.
Relationship of interference colour intensity versus relative
retardation and degree of coherent phase alignment
Our optical system is best analysed in terms of the intensity of light
emerging from two superposed crystal plates - the biological specimen and
the compensator full-wave plate - between crossed polarizers (see
Hartshorne and Stuart, 1970, p.302):
I /Io
= Io
- sin2(y2-
y1)
sin 2y1
cos 2y2
sin2
d1/2
+ sin2(y2
- y1)
cos 2y1
sin 2y2
sin2
d2/2
+ cos2(y2
- y1)
sin 2y1
sin 2y2
sin2(d1
+ d2)/2
(2) - sin2(y2
- y1)
sin 2y1
sin 2y2
sin2(d1
- d2)/2
where d1 and d2
are the phase differences respectively of the wave-plate and the
biological 'crystal', and y1, y2
, the angles that the vibrational directions their slow (or fast) wave
makes with the polarizer.
Using Eq. (2), Newton et al (1995) plotted the intensity of
monochromatic red, green and blue light (700nm, 560nm and 450nm) as a
function of compensator plate alignment, and showed that the effect of the
small y1 angle is to reduce the
contributions of red and blue relative to the green, with the result that
the sample appears more colourful to the eye. For sample retardations (R-values)
smaller than 50nm, the small angle gives much more colour contrast.
As the wave-plate with fixed relative retardation (d2
= 2pR/l)
is kept at a constant angle with respect to the polarizers, Eq. (2) can be
simplified for small birefringences, using the approximation, siny
~y for small y
, to give the following relationships between colour intensity in the +45°
orientation of blue, Ib, and red, Ir, as a function of
relative sample retardations, Rb and Rr.
Ib = Io,b [ 0.032 + 1.8 x 10-3 Rb + 4.54 x 10-5
Rb2]
Ir = Io,r [ 0.023 + 1.1 x 10-3 Rr + 1.84 x 10-5
Rr2] (3)
The difference (DI)
between the peak intensity (+45°)
and the minimum intensity (-45°),
gives the following linear relationship:
DIb
= Io,b[
3.6 x 10-3
Rb]
DIr = I>o,r [ 2.2 x 10-3
Rr] (4)
Ignoring absorption, depolarizing scattering and dispersion effects,
these relationships are exact for mineral crystals, or crystals with
perfect molecular order. For liquid crystals, however, only the effective
relative retardation is being measured, for the light intensity
depends, not just on intrinsic molecular (and form) birefringence, but
also on the degree of coherent alignment of the individual molecules, as
expressed by the order parameter, S (see below).
For uniaxial nematic liquid crystals, which include nearly all
biological liquid crystals to first approximation, the order parameter,
S, measuring the state of alignment of the molecules along the
direction of the nematic axis, is given by,
S = <(3 cos2q
- 1)>/2 (5)
where <> denotes the average and q
is the polar angle between the direction of the nematic axis
(the average direction of alignment of the molecules) and the long axis of
the individual molecule. S varies from 0 in isotropic material to
1 when full alignment of all the molecules is attained. Isotropic material
is dark at all angles of rotation whereas fully aligned material is
maximally bright at 45° between crossed polarizers. In Appendix 1, an
expression is derived to show that, for small birefringences, the
intensity of light transmitted does indeed vary linearly with the order
parameter as well as the intrinsic birefringence. In situations where one
parameter is constant, the other can be estimated.
Experimental configuration
The measurements are made using a Prior polarizing microscopie (Prior
Scientific Instrument, Cambridge), which has been modified by Prior to
include a motorized rotating stage. The motor is driven by a Prior stage
controller: this enables the angular position of the stage to be computer
controlled, and also provides automatic focusing.
The 50 watt lamp in the microscope is connected to a Prior
computer-controlled power supply, which enables the light level to be
accurately set, and any variations in lamp brightness to be compensated
for. The compensator plate has been modified by Prior to allow manual
adjustment of the angle, while it is still inserted in the microscope.
A JVC colour CCD camera is attached to the microscope and provides the
red, green, blue (rgb) signals to the frame grabber board in the computer
(Imaging Technology CFG). The framegrabber is controlled by an image
processing program (Image Pro, Media Cybernetics), running under the
Microsoft Windows Operating System. Special software has been developed to
meet the specific requirements of our functions to be combined with the
standard Image Pro functionality to allow a wide range of image
manipulation options.
A set of mica calibration slides are used with retardations in the
range of 5 to 100nm.
Software processing
The software provides general functions to enable the user to control
the position of the stage, focusing, and the light level. The calibration
routines record the light variation as a calibrations slide is rotated
through 180. This information is stored in the computer so that it can be
used later to derive accurate values of retardations. To make quantitative
measurements from a biological sample, several steps are required. First
the sample slide is rotated on the stage from 0 to 180°, in 2 to 15°
steps. At each step, the image from the camera is sent to the disk. Next,
the centre of the image is determined either by manually, or by an
automatic scanning process. A software rotation algorithm is then used to
remove the effect of the stage's rotation on the image, so that they are
all correctly aligned. From the sequence of images, a set of 5 composite
images are then produced, in which the pixel values represent:
1. Maximum brightness during rotation
2. Minimum brightness during rotation
3. Variation in brightness during rotation (maximum minus minimum,
Max-Min)
4. Angle at which maximum brightness occurs
5. Angle at which minimum brightness occurs
The composite frames can be used to make many measurements of interest.
For example, the retardation values can be derived from the Max-Min image,
and the angle of orientation from the angle at which maximum brightness
occurs. The measured angle at which the minimum occurs is much more
subject to noise than that of the maximum, and is mainly used as a check,
as it should be 90o
from the angle at which the maximum occurs.
The computer software records the image in colour using the 3 image
planes, rgb (red, green and blue). Measurements of retardation and
orientation are made on a specific plane, normally red or blue, as these
show the greatest variation in intensity.
Quantitative image analysis
Quantitative image analysis is developed based on the linear
relationship described in Equation (4). Colour intensities are measured
with a ccd colour camera in red, green and blue. The experimental values
will deviate from the theoretical as the camera detects a bandwidth for
each colour, and the spectrum of the microscope light will deviate from
the 'white' ideal. This can be corrected for in the theoretical curve, but
is unnecessary if the measurement system is calibrated. Thus,
provided the response of the camera (and video board) is linear, colour
intensity values can be linearly transformed into retardances. (In cases
where the response departs from nonlinearity, the values are curve-fitted
to include higher order terms and reasonable estimates of retardances can
still be obtained.)
The output of the camera is digitized by a PC video acquisition card
and stored on the computer's disk. Calibration is achieved with the set of
standard mica plates of known retardations to correct for variations
associated with different microscope objectives and levels of lighting
necessary to produce good images. The retardations of the standards were
independently checked by measurements using a Sénarmont compensator
on a Zeiss Universal microscope fitted with a mercury vapour light source
and interference filter to isolate the green line at 546nm. For each
standard, the +45o
blue and red colour intensities were determined by automated stepwise
rotation at 5o
intervals for 180o.
The calibration curve of DI
versus retardation should cover the range of retardations under
investigation. Calibration is repeated for each set of recording
conditions so that different recordings can be compared where required.
The orientation of the sample is determined with respect to some reference
axis, such as the vibrational direction of the polarizer or the analyser.
The sample is then rotated as for the standards and the colour intensities
recorded.
Samples to be compared are recorded under the same conditions, as are
the calibration curves. For time-lapse studies carried out over long
periods, the stability of the colour response is monitored by the
background values, which, in our case are found to remain constant to
within +5%. The main causes of fluctuation are the light level and
the focus (especially where long time-lapse sequences are involved). These
are minimized by feedback voltage control to the microscope lamp, and
autofocus interfacing between the computer and the microscope, both of
which are built into the software.
Molecular orientations in collagen ribbons of dogfish egg case
Interference colour imaging was used to determine molecular
orientations in ribbons of collagen in the marginal rib of the dogfish egg
case extruded from the extrusion dies in the nidamental gland. The image
of the section, the plot of retardation and angle of orientation along a
transect, a two dimensional orientation graph indicated by vectors, and a
grey-tone image giving relative retardations over the whole area of
interest are shown in Figure 1. The results of this investigation,
reported in detail elsewhere (Knight et al, 1996) suggest that
rheological interaction with nematic liquid crystallization are
responsible for producing the bent nematic arrangement seen in this
material. This interpretation is consistent with observations from low
angle X-ray diffraction, bright field microscopy, electron microscopy and
confocal microscopy performed on the same material (Knight et al,
1996).
The quantitative imaging technique described in this paper is unique in
that it gives detailed anatomical information as well as physical
information concerning birefringence, orientation and degree of coherent
phase ordering of the molecules making up the tissues. For relatively
'inert' materials and sections, such as the dogfish egg case and human
cornea (see Newton et al, 1996), it complements measurements
obtained with other techniques such as X-ray diffraction and electron
microscopy. It is especially suitable for characterizing liquid
crystalline mesophases of polymers and other biomimetic materials. Because
of its noninvasive/ nondestructive nature, it can yield information on
rapid changes in liquid crystalline mesophases in living organisms and
cells that cannot be obtained by any other means (see Ho et al,
1996), and is equally suitable for following fast dynamics of liquid
crystalline phase transitions in vitro.
We have shown that, for small birefringences, the intensity of light
transmitted varies linearly with the degree of alignment as well as
intrinsic birefringence. In situations where one parameter is constant,
the other can be estimated. In the development of Drosophila larva
body wall musculature, for example, it is known that the molecular
constituents of muscle are randomly arranged when first formed, to become
organized into regular myofilament bundles typical of muscle some time
afterwards (Abmayr, et al, 1995). We have observed a rapid
increase in colour intensity in time-lapse images of the maturing Drosophila
larva which may be correlated with the increase in coherent alignment of
the molecular constituents during the condensation of the body wall
musculature (Ho et al, 1996).
Acknowledgment
This research was supported by an EPSRC-LINK grant to M.W.H. and
funding from King Alfred College (D.P.K.). We thank Lyndon Davies for his
interest and support in coordinating the LINK grant, Prior Scientific for
supplying the Prior polarizing microscope and accessories and Data Cell
for the imaging software for which our image analysis is an extension. We
thank Dr. D.J. Goldstein of Sheffield University for advice on measuring
birefringence, and for lending us the quarter-wave plate for 546nm. Thanks
are also due to Michael Lawrence for help with microscopy and some of the
video-recordings, to Mike Dodd for advice on statistics, to Phil Bland and
Mike Stewart for loan of microscopes, Ian Wood of Hitachi for loan of a
ccd colour camera. Helpful comments were provided by two anonymous
referees.
References
Abmayr, S.M., Erickson, M.S. and Bour, B.A. (1995). Embryonic
development of the larval body wall musculature of Drosophila
melanogaster. TIG 11, 153-159.
De Jeu, W.H. (1978). The dielectric permettivity of liquid crystals. In
Liquid Crystals (L. Liebert, ed.) p. 109-145, Academic Press, New
York.
Hartshorne, N.H. and Stuart, A. (1970). Crystals and the Polarizing
Microscope, Edward Arnold, London.
Ho, M.W. and Lawrence, M. (1993). Interference colour vital imaging: A
novel noninvasive microscopic technique. Microscopy and Analysis September,
26.
Ho, M.W. , Haffegee, J., Newton, R., Ross, S., Zhou, Y.M. and Bolton,
J.S. (1996). Organisms as polyphasic liquid crystals.Bioelectrochemistry
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Knight, D.P., Hu, S.W., Gathercole, L.J., Rusaouën-Innocent, M.,
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Scyliorhinus canicula; a novel lyotropic liquid crystalline
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Appendix 1
The relationship between the light intensity and the order parameter
The light intensity differences between +45° and -45° orientation of the sample are given by Equation (4),
DIb = Io,b[ 3.6 x 10-3 Rb]
DIr = Io,r [ 2.2 x 10-3 Rr] (A.1)
where Rb,r, =d (ve1 - ve2 );Rb,r are the retardations of the sample for blue and red; e1 and e2 are small dielectric constants parallel and perpendicular to the optic axis of the biological sample, and d is the thickness of the sample.
In the mean field approximation (De Jeu, 1978),
e1 =1+ 4 p NrhF/M ( a + 2/3 ø S + (Fm2/ 3KBT) (1-(1-3 cos2q) S) (A.2)
e2 =1+ 4 p NrhF/M ( a + 2/3 ø S + (Fm2/ 3KBT) (1+1/2(1-3 cos2q) S)
where r is density, h =3 e/(2e+1), e is the mean dielectric constant, F =1/(1- af),
f = 4p Nr( 2e -2)/3M(2e+1), a is the mean polarization, m is the dipole moment, M is the molecular weight, S is the order parameter, ø is the polarizability anisotropy. Equation (A.2) can be simplified:
e1 =1+ A (a + b +B S)
(A.3)
e2 =1+ A (a + b + CS)
where A, B, C and b are constants unrelated to S.
Equation (A.1) can be simplified in the case of small retardations, where e1 and e2 are small, that means both A(x+B S) and A(x+CS) are much smaller than 1. Substituting the expressions of dielectric constants (A.3) into the expression for retardation (A.1) and expanding it, gives,
 0.75ABS0.09375A2B2S2
R = d( -------------- - -------------- + O[S] 3) (A.4.)
(1+Ax)1/2(1+Ax)3/2
Effective retardation, R, and hence the intensity of light transmitted, is approximately linearly related to S, because the second order term, S2 is much smaller than the first order term, and higher order terms, O[S] 3, are even smaller.
Legends
Figure 1. (a) Interference colour image of section of dogfish egg capsule marginal rib cut at 15o to the long axis, where the plane of the lamellae curves. The line marking a transect is 400m in length. (b) Plots of relative retardation and angle of orientation along the transect indicated in (a). (c) Two-dimensional plot of orientation of collagen fibres overlaid on the image. The length of individual vectors is proportional to the relative retardation. (d) Grey tone plot of relative retardation over the area of interest in the image. Note the loss of resolution between adjacent orange and blue ribbons.
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