Journal of Microscopy 187, 62-67, 1997
Stephen Ross, Richard Newton, Yu-Ming Zhou, Julian Haffegee, Mae-Wan Ho: Bioelectrodynamics Laboratory, Open University, U.K.
John P. Bolton: Physics Department, Open University, U.K.
David Knight: Bioscience Group, King Alfred's College, Winchester, U.K.
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.
Noninvasive quantitative imaging, interference colours, liquid crystals
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.
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.
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.
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,
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 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.
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).
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.
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,
R = d( -------------- - -------------- + O[S] 3) (A.4.)
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.
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.
Article first published 1997
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