Journal of Microscopy, Vol. 214, Pt 1 April 2004, pp. 62– 69 Received 25 June 2003; accepted 22 October 2003
Quantitative phase amplitude microscopy IV: imaging thick specimens Blackwell Publishing, Ltd.
C . J. B E L L A I R *, C . L . C U R L †, B . E . A L L M A N ‡, P. J. H A R R I S †, A . RO B E R T S *, L . M . D. D E L B R I D G E † & K . A . N U G E N T * *School of Physics, and †Department of Physiology, The University of Melbourne, Vic., 3010, Australia ‡IATIA Ltd, 2/935 Station Street, Box Hill North, Vic., 3129, Australia
Key words. Optical microscopy, phase microscopy, quantitative imaging, thick specimens, three-dimensional specimens.
Summary
1. Introduction and background
The ability to image phase distributions with high spatial resolution is a key capability of microscopy systems. Consequently, the development and use of phase microscopy has been an important aspect of microscopy research and development. Most phase microscopy is based on a form of interference. Some phase imaging techniques, such as differential interference microscopy or phase microscopy, have a low coherence requirement, which enables high-resolution imaging but in effect prevents the acquisition of quantitative phase information. These techniques are therefore used mainly for phase visualization. On the other hand, interference microscopy and holography are able to yield quantitative phase measurements but cannot offer the highest resolution. A new approach to phase microscopy, quantitative phase-amplitude microscopy (QPAM) has recently been proposed that relies on observing the manner in which intensity images change with small defocuses and using these intensity changes to recover the phase. The method is easily understood when an object is thin, meaning its thickness is much less than the depth of field of the imaging system. However, in practice, objects will not often be thin, leading to the question of what precisely is being measured when QPAM is applied to a thick object. The optical transfer function formalism previously developed uses threedimensional (3D) optical transfer functions under the Born approximation. In this paper we use the 3D optical transfer function approach of Streibl not for the analysis of 3D imaging methods, such as tomography, but rather for the problem of analysing 2D phase images of thick objects. We go on to test the theoretical predictions experimentally. The two are found to be in excellent agreement and we show that the 3D imaging properties of QPAM can be reliably predicted using the optical transfer function formalism.
A new approach to high-resolution phase microscopy has recently been proposed (Barty et al., 1998) based on propagationbased phase measurement (Teague, 1983; Gureyev et al., 1995; Paganin & Nugent, 1998). This method, called quantitative phase amplitude microscopy (QPAM), has been extensively explored for optical microscopy (Barone-Nugent et al., 2002) and also been applied to transmission electron microscopy of both materials (Bajt et al., 2000; DeGraef & Zhu, 2001) and biological samples (McMahon et al., 2002). It has also been shown to be remarkably resilient to noise in the data (Paganin et al., 2004). This technique can be described using an optical transfer function (OTF) formalism even for partially coherent illumination (Barone-Nugent et al., 2002). It is well established that the phase distribution of an object becomes visible in slightly defocused intensity images. The rate at which intensity changes with propagation is described by the so-called transport of intensity equation:
Received 25 June 2003; accepted 22 October 2003
Correspondence to: Professor Keith Nugent. Fax: +61 3 93494912; e-mail:
[email protected]
k
∂ˆ = ∇ ⋅ ( ˆ ∇Φ), ∂z
(1)
where ˆ ≡ I /〈I 〉; I is the image intensity and 〈I〉 is its spatially averaged value; k = 2π/λ; λ is the wavelength of the light; Φ is the phase in the image which, for a thin object, is given by Φ = knt, where n is the refractive index of the object and t is its thickness; and z is the propagation distance along the optical axis. Using this equation, a knowledge of the intensity and its derivative is sufficient, in many circumstances (Gureyev et al., 1995), for the unique recovery of the phase distribution even if it varies over many multiples of 2π (Allman et al., 2000). This is the fundamental idea underlying propagation-based phase recovery methods. With appropriate scaling factors, a defocus is equivalent to propagation through free-space, so defocused intensity images can be used to calculate the intensity derivative. The use of the normalized intensity in the above © 2004 The Royal Microscopical Society
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equation emphasizes that phase recovery should be independent of illuminating intensity. High-resolution microscopy requires partially coherent illumination and it has been shown that this formalism may be extended to partially coherent fields (Paganin & Nugent, 1998) with a suitable re-definition of what is meant by phase. This is the approach that is at the heart of QPAM. Previous papers in this series have developed an OTF formalism to describe the phase imaging process and demonstrated the independent measurement of amplitude and phase (Barone-Nugent et al., 2002), applied QPAM to transmission electron microscopy (McMahon et al., 2002) and examined the effects of noise on the phase retrieval process (Paganin et al., 2004). In this paper we use the OTF formalism to describe the three-dimensional (3D) imaging properties of QPAM and test our theoretical phase predictions against quantitative phase imaging experiments on a range of objects, both thick and thin, biological and inorganic.
Red blood cells. Red blood cells were obtained from rats killed by exsanguination under pentobarbitone anaesthesia (100 mg kg−1). Blood was collected in tubes containing heparin (10 000 i.u. in 0.9% NaCl) and stored at room temperature. Aliquots of cells were then transferred to hypotonic (0.6% NaCl) solution and allowed to equilibrate for 10 min. An aliquot of this suspension was placed between a slide and coverslip for microscopic analysis; this is experimental specimen S4. Images of these experimental specimens are shown in Fig. 1.
2. Materials and methods
2.3. Theoretical methods
2.1. The microscope system Intensity images of experimental specimens were obtained using an inverted Zeiss Axiovert 100M microscope fitted with Zeiss LD-Achroplan objectives, ×10 NA = 0.3 and ×40 NA = 0.75 (Carl Zeiss, Karlsruhe, Germany). A piezoelectric device (Physik Instrumente, Karlsruhe, Germany) was used to move the objective to different focal positions. The intensity images were captured with a Photometrics Coolsnap fx 12-bit CCD camera (Roper Scientific, Tucson, AZ, U.S.A.) using the QPm software (Iatia Ltd, Box Hill, Australia) and detailed image analysis was performed using the IDL software package (Research Systems, Boulder, CO, U.S.A.). 2.2. Experimental test specimens A number of specimens were used to test the theoretical predictions experimentally. Mylar films. Two overlapping 3-µm-thick pieces of mylar film were positioned on a microscope slide to create experimental specimen S1, a sample containing large areas with different phase properties. Water was placed between the slide and the coverslip to reduce the amount of light scattered out of the microscope. A second experimental specimen, S2, was created in the same way using a single piece of 0.9-µm-thick mylar. Cultured airway smooth muscle cells. Airway smooth muscle (ASM) cells were obtained by collagenase and elastase digestion of airway tissue obtained from lung transplant resection patients. The resulting single cell suspension was washed in phosphate-buffered saline and then grown in phenol red-free © 2004 The Royal Microscopical Society, Journal of Microscopy, 214, 62–69
Dulbecco’s modified eagles medium (Flow Laboratories, U.K.) containing 10% fetal calf serum, l-glutamate, sodium pyruvate, non-essential amino acids and penicillin/streptomycin. The cells were incubated at 37 °C and were passaged as required by trypsinization and re-suspension at various ratios in the above media. The cells were plated on to culture dishes for at least 24 h to allow growth of the cells; this is experimental specimen S3.
Basic formalism. In an earlier paper (Barone-Nugent et al., 2002) the 3D OTF theory of Streibl (1985) was used to analyse image formation in a telecentric system under the weakly scattering (Born) approximation and assuming Kohler illumination. In that work, the phase was assumed to be recovered from the sum and difference of two slightly defocused images: 1 I + (r ) ≡ [I + δz (r ) + I −δz (r )] 2
(2a)
1 I − (r ) ≡ [I −δz (r ) − I + δz (r )] 2
(2b)
where I ±δz(r ) denotes the image obtained when the microscope is defocused by a distance ±δ z . Transfer functions were calculated for these two data sets using the Striebl formalism (Fig. 2). It was found that the phase information was completely decoupled from I+(r ). Conversely, the amplitude information was completely decoupled from I−(r ). This decoupling allows, within the approximations of the theory, the partially coherent image formation to be properly described by transfer functions. The form of the transfer functions depends on the condenser aperture setting and on the exact defocus distance so that I+(r ) and I−(r ) can be found for a range of different imaging conditions. The Born approximation assumes that a photon passing through the sample interacts only once with the sample so that a thick object may be regarded as the sum of a series of non-interacting 2D objects. The theory is therefore able to describe directly the imaging of a range of thick objects in a variety of imaging conditions. The theoretical description of the image of an object depends only on the focal position z0, the defocus distance δz
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Fig. 1. Phase images of the objects used to test experimentally the theoretical predictions of phase in different imaging conditions. (a) The two phase regions of S1 used to investigate the importance of illumination variation. (b) The edge of the mylar piece in S2 used to investigate condenser aperture settings. (c) An image of S3 showing the cytoplasmic process used as a thin sample. (d) Rat red blood cells in 0.6% NaCl solution used as thick samples.
and the relative aperture of the condenser, ξ ≡ NAcon/NAobj. The theoretical phase predictions under different imaging conditions are quantitatively tested in section 3. Image simulation. We regard a thick object as the sum of a set of 2D slices arrayed along the optical axis of the microscope. In general, light scattered by one object slice will be further scattered by the additional object slices between it and the microscope. This multiple-scattering process is inherently non-linear and renders quantitative image interpretation very difficult. The Born approximation assumes that subsequent scattering is negligible and so an image of such a thick object can be regarded as the incoherent sum of the appropriately (de)focused images of each slice. The properties of the image of a thick object were simulated in this way. Three different theoretical objects were simulated: (i) a thin stripe, (ii) a thin circular disc and (iii) a 3D ellipsoid. These were intended to mimic the properties of the experimental specimen described in section 2.2. It is possible to calculate intensity images of the theoretical objects under a variety of imaging conditions by using the 3D OTF formalism to find I+(r ) and I−(r ) from the specified phase and amplitude information. The slightly defocused images I ±δz(r ) and I − δz(r ) can then be calculated. If δz is set to zero, a focused image may also be calculated. The observed phase can be recovered using these three intensity images.
3. Experimental results 3.1. Effect of intensity variations It is predicted that the recovered phase should be independent of the illuminating intensity (section 1). This prediction was tested by varying the current through the microscope condenser lamp and taking a series of intensity images of specimen S1. A region of S1 where the mylar pieces overlapped was chosen so the average phase across two different phase regions could be compared (Fig. 1a). The phase was then recovered at different illumination intensities. The phase in one part of the image was plotted against the mean intensity value (in arbitrary units) in the in-focus image (Fig. 3). It can be seen that the phase varies with a standard deviation of 5% between mean intensity values of 500 and 2500 units. The accuracy of the recovered phase was seen to deteriorate at the extremes of the intensity range. We attribute this to the effect of non-linearities in the system. For example, saturation in parts of the image will begin to have an effect at the higher intensity levels. The phase recovery procedure depends on measuring small differences between two images and this will suffer from digitization errors at lower intensities. However, these results indicate that the recovered phase values are independent of the illumination intensity provided the detector does not suffer from either saturation or digitization errors during data acquisition. © 2004 The Royal Microscopical Society, Journal of Microscopy, 214, 62– 69
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Fig. 3. Plot of the phase variation with changing microscope illumination intensity for S1 (Fig. 2a). A region of relatively constant phase values is clearly visible between image mean intensities of 500 and 2500 units. Phase value variations outside this region are caused by non-linearities in the data acquisition process.
The image contrast was estimated using the root mean square visibility, VRMS, as a measure of the contrast between the positive and negatively defocused images: 1
I (i, j) − I (i, j) 2 2 − δz , VRMS = ∑ +δz ROI I +δz (i, j) + I −δz (i, j) Fig. 2. Three-dimensional OTFs showing (a) the intensity and (b) the phase response of the imaging system at different frequencies when ξ = 0.8.
In subsequent experiments, the illumination level was chosen to avoid these extremes and the fluctuations of the recovered phase seen in Fig. 3 will be used to estimate the experimental uncertainties. 3.2. Effect of condenser aperture variation It has been shown that it is possible to recover the phase of a microscope image even when the illumination is partially coherent (Barone-Nugent et al., 2002), although in the extreme limit of incoherent illumination at the object – equivalent to a self-luminous or fluorescent object – it is not possible to make such a measurement (Paganin & Nugent, 1998). It is therefore to be expected that the variation with defocus reduces as the coherence is decreased. It has also been shown elsewhere (Barone-Nugent et al., 2002) that the phase measurement is largely independent of the condenser setting; however, it is anticipated that the contrast, and therefore the signal-to-noise ratio, in the data will degrade with increasing condenser aperture. On the other hand, the image resolution will increase with condenser aperture (Born & Wolf, 1995). © 2004 The Royal Microscopical Society, Journal of Microscopy, 214, 62–69
(3)
where ROI indicates that the sum is to be taken over a specified region of interest, I +δz is the positively defocused intensity image and I –δz is the negatively defocused intensity image. As the coherence of the illumination is reduced, the contrast between the two defocus images will reduce and it is expected that this measure should tend to zero. The edge of the mylar piece in sample S2, as indicated in Fig. 1, was imaged as a function of ξ ≡ NAcon/NAobj and the experimental results are shown in Fig. 4. This sample was chosen as the edge of the mylar provided sharp contrast changes between positively and negatively defocused intensity images. The solid line in this figure is the prediction of our theoretical model, which was normalized to produce the best fit to the data. It can be seen that the agreement between the theory and the experiment is good. By this measure, the phase signal is strongest in the range 0.4 < ξ < 0.6. However, the phase signal is still substantial out past ξ = 1, indicating that a range of condenser settings can be used with little degradation of the phase image. In particular, the manufacturer-recommended setting corresponding to ξ = 0.8 allows good phase recovery. 3.3. Imaging thin objects The depth of field of a microscope is a measure of its ability to resolve separate points along the optical axis – to an excellent
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Fig. 4. Plot showing the contrast in phase images of S2 (Fig. 2b), measured using the root mean square visibility, as a function of ξ. The contrast peaks for 0.4 < ξ < 0.6; however, there is significant contrast at all measured values of ξ.
approximation, all objects within the depth of field will appear correctly focused in the image plane and the object will be indistinguishable from one that is perfectly thin. The depth of field depends on the numerical aperture of the objective, NA, and on the wavelength of the light, λ, forming the image. We use the depth of field definition: Dz =
λ 1 − (NA) 2 . (NA) 2
(4)
With this definition, the NA = 0.3 objective used here to image specimen S1 has a depth of field of approximately 6 µm, which exceeds the 3 µm thickness of the mylar in S1. We are therefore able to consider this sample as effectively perfectly thin. A through-focus series of intensity images of specimen S1 was obtained with mid-range intensity and ξ = 0.8. A section of the image was selected where the mylar pieces were slightly separated, making the phase object the space between the mylar pieces. Phase values were obtained as a function of the dimensionless quantity δz/Dz and the results are shown in Fig. 5. The experimental results show the phase values remaining constant for a range of defocus distances before gradually decreasing. The separation of the mylar pieces was measured to have a width of 21 ± 1 µm and so the measurements were compared with theoretical predictions for a strip with lateral width of 20 µm. The experimental uncertainties were estimated based on the known effects of intensity fluctuations over the time taken to record the intensity images and the precision of the piezoelectric device. The theoretical prediction was fitted to the experimental data at small defocus distances by scaling the theoretical prediction to the experimental phase values. There is good agreement between the experimental and the
Fig. 5. Plot of phase value variation with increasing δz/Dz in S1. The solid line respresents the theoretical prediction fitted to the experimental data (). The graph shows good agreement between the theoretical prediction and the experimental data.
theoretical results, indicating that the theory is able to describe the imaging of thin objects properly. Figure 5 also shows that a large range of defocus distances can be used to retrieve the phase information. The exact extent of this range appears to depend on the lateral width of the object being imaged. In order to explore the effect of recovering images that are improperly focused, a through-focal series of images of ASM cells was taken using the ×10, NA = 0.3 objective. The cytoplasmic processes of ASM cells have thickness around 5 µm, as determined using confocal slicing techniques, and so may be considered thin in these imaging conditions. A section of the image containing a long cytoplasmic process with a relatively constant lateral width of 20 ± 1 µm was selected for analysis, and phase images were acquired for constant δz as a function of z0/Dz. The results are shown in Fig. 6. Here the phase values are symmetrical around the z0 = 0 plane where the ASM cell is correctly in focus. The phase values remain relatively constant for a small distance around this plane and then drop off quickly. The object structure was modelled as a thin strip of constant refractive index for the theoretical predictions. The results of the prediction are also shown in Fig. 6 and can be seen to be in good agreement with the experimental data. These results show that improperly focused objects will contribute to phase images for a range of z0 positions which depend on the lateral width of the object. 3.4. Imaging thick objects The results shown in section 3.3 show that the OTF formalism is a good way to describe the imaging of thin objects. As the theory considers thick objects as the sum of a set of thin slices, © 2004 The Royal Microscopical Society, Journal of Microscopy, 214, 62– 69
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Fig. 6. Plot of phase value variation as a function of z0/Dz at a fixed defocus distance for a cytoplasmic process in S3 (Fig. 2c). The z0 = 0 plane corresponds to the cytoplasmic process being correctly focused with all parts of the process lying within the depth of field of the microscope. Here again there is clearly good agreement between the experimental data () and the theoretical prediction (solid line).
we can now compare theoretical phase predictions of thick objects with experimental data. Phase images of rat red blood cells, sample S4, were obtained from a through-focal series of intensity images using the ×40, NA = 0.75 objective (Fig. 1d). Red blood cells in a hypotonic solution are assumed to be spherical in shape with diameters of 5–7 µm. In these imaging conditions the depth of field of the objective was approximately 0.7 µm so the red blood cells could not be regarded as thin, and intensity images of these cells contain some out-of-focus artefacts. Images were obtained as a function of δz/Dz with the microscope focused on the mid-plane of the cells, and the phase was measured at the centre of the cell. The results are shown in Fig. 7 where, as before, the experimental uncertainties were estimated using the statistical fluctuations observed elsewhere and the precision of the piezoelectric device. The same trends are observed in Fig. 7 as in Fig. 5, the constancy of the phase values for a range of defocus distances followed by a gradual decrease as the defocus distance increases. Figure 7 also shows the theoretical prediction for an ellipsoidal object of the same size as the red blood cell, and the agreement between the theoretical prediction and the experimental data is found to be excellent. This demonstrates that a range of defocus distances can be used for thick samples without losing phase information. The defocus distance was then fixed and phase images were acquired as a function of z0/Dz to investigate the importance of focus when imaging thick objects. The experimentally obtained phase values at the centre of the cell were compared with the theoretical prediction for an ellipsoidal object with the same dimensions as the red blood cells (Fig. 8). The graph shows a © 2004 The Royal Microscopical Society, Journal of Microscopy, 214, 62–69
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Fig. 7. Plot of phase value variation with increasing δz/Dz for a rat red blood cell in S4 (Fig. 2d). As the thickness of the rat red blood cell is much greater than the depth of field of the microscope, out-of-focus portions of the cell are contributing to the phase. The experimental data () are in good agreement with the theoretical prediction (solid line), confirming that the OTF formalism can be used for thick objects.
Fig. 8. Plot of phase value variation as a function of z0 /Dz for a rat red blood cell (Fig. 2d). Here the z0 /Dz = 0 plane corresponds to the mid-plane of the cell. For planes around the mid-plane there is excellent agreement between the experimental data () and the theoretical prediction (solid line).
small range of z0 planes, centred on z0 = 0, returning a constant phase value with the phase decreasing quickly outside this region. The agreement between theory and experiment is acceptable, although not as good as elsewhere, possibly owing to irregularities in the cell dimension and shape or because aberrations in the optical system have not been taken into account. These results from thick objects show the OTF formalism is able to describe phase imaging of thick objects properly and
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that it is possible to obtain quantitatively accurate phase images of thick objects. Conclusions Comparisons of experimental results with theoretical predictions in section 3 show that the 3D OTF theory properly describes the imaging of thick objects. For any object of specified thickness and lateral width we can use this theory to predict whether an accurate phase image can be recovered from a particular combination of δz and z0. Given the experimental confirmation of the accuracy of the theory, it is now possible to develop some guidelines for accurate phase measurement. As the theory is a good description of the imaging process we can use it to develop guidelines relating the dimensions of an object to those δz and z0 combinations that should produce accurate phase values. To obtain these rules, images of 3D oblate spheroids were simulated. These objects were circular when viewed along the optical axis, and elliptical with ellipticity e ≡ T/W when viewed transversely to the optical axis, where W is the maximum diameter perpendicular to the optical axis and T is the maximum dimension along it. A series of theoretical predictions were made for a range of thicknesses and diameters using δz = Dz (see Fig. 7) with the microscope focused on the mid-plane (z0 = 0) of the object. These predictions generated the family of curves shown in Fig. 9. The phase behaviour was found to depend only on the ellipticity of the object. This can also be seen in our earlier results (Figs 5–8). In general, and consistent with the data in Fig. 7, phase recovery was accurate provided e ≤ 1. If an object is thicker than it is wide, the phase is no longer accurate because of the effect of poorly measured phase values in out-
of-focus planes (Fig. 6). The NA of the microscope is approximately unity, which implies that the image of a point on the furthest object plane will be defocused to a disc of size equal to the object when e ∼ 1. We conjecture that, at this aspect ratio, some object planes will be completely contaminated by out-offocus artefacts so that no accurate phase recovery is possible. In the case of the red blood cells imaged in section 3.4, W/ Dz ≈ 8. Assuming the cells are spherical, e = 1 and T/Dz ≈ 8, the relationship between the phase values and the optical thickness of the red blood cells can be found in Fig. 9. The theory predicts that the quantitative values in phase images of red blood cells would be 70% of the actual optical thickness. This relatively poor result is to be expected for objects that are as thick as they are wide. When the same set of predictions are made for different defocus distances and with the microscope focused on different planes, another guideline emerges linking object shape to δz and z0. As a general rule, it was observed that the microscope should be focused such that z0 ≤ (Dz/e) − 2δz. In simple terms, good phase recovery requires objects to be wider than they are thick, and in this case the microscope should be focused on the centre of the object to within a couple of depths of field. If these conditions are satisfied, the resulting phase measurement corresponds to that produced by the full optical thickness of the object. This work has primarily concentrated on structurally simple objects symmetrical around the mid plane and we have shown that it is possible to obtain phase images corresponding to the phase accumulated through an entire 3D object. We have found that OTFs can be used to model the phase imaging process and that the predictions are in good agreement with experiment. In addition, we have developed guidelines to assist in the selection of appropriate defocus distances and focal positions. Although these guidelines were based on very simple samples, we believe that they will provide a useful guide to the practitioner in the imaging of more complex objects. For very complex objects, the full OTF formalism should still apply and so permit a full understanding of the imaging process. Acknowledgements We would like to acknowledge the support of the Australian Research Council and IATIA Ltd. The quantitative phase microscopy system is being marketed by IATIA and K.A.N. has a financial interest in that company. We would also like to acknowledge the generous assistance of Alastair Stewart and Trudi Harris in providing ASM cultures for analysis. References
Fig. 9. Plot of phase variation as a function of object thickness, T/Dz, for different object widths ranging from 1 to 10. The phase is constant and accurate for thicknesses less than the width, leading to a general rule that phase images of thick objects are only quantative if e > 1.
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McMahon, P.J., Barone-Nugent, E.D., Allman, B.E. & Nugent, K.A. (2002) Quantitative phase amplitude microscopy II. Differential interference contrast imaging for biological TEM. J. Microsc. 206, 204 –208. Paganin, D., Barty, A., McMahon, P.J. & Nugent, K.A. (2004) Quantitative phase-amplitude microscopy III: the effects of noise. J. Microsc. 214, 58 – 68. Paganin, D. & Nugent, K.A. (1998) Non-interferometric phase imaging with partially coherent light. Phys. Rev. Lett. 80, 2586 –2589. Streibl, N. (1985) Three-dimensional imaging by a microscope. J. Opt. Soc. Am. A2, 121–127. Teague, M.R. (1983) Deterministic phase retrieval: a Green’s function solution. J. Opt. Soc. Am. 73, 1431–1441.
