Patent Application
20230288307
This specification relates to characterizing particles using interferometric scattering optical microscopy.
The work leading to this invention has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 337969. The project leading to this application has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 841466.
Interferometric scattering optical microscopy (iSCAT) is a technique in which an imaged particle is illuminated with coherent e.g. laser light and the signal results from interference between light scattered from the particle and a reference, typically light reflected from a nearby interface. This interference can result in signals which are amplified compared with some other approaches and the technique is capable of label-free detection of even single molecules, such as biological molecules for example proteins/protein complexes.
In more detail, the iSCAT signal can be separated into three components, the reference e.g. reflected laser light, the scattered light, and the interference between the two. The reference laser light is dominant but does not carry any information, the scattered light is too weak to detect, and only the interference signal is useful.
A review of iSCAT can be found in Taylor et al. “Interferometric Scattering Microscopy: Seeing Single Nanoparticles and Molecules via Rayleigh Scattering”, Nano Letters 2019 19 (8), 4827-4835. Further background prior art can be found in: GB2552195; US2019/0195776; WO2019/110977; WO2015/059682; WO2018/189187; WO2018/047239; US2012/218629; WO2020/104814; and US2017/0248518.
One way to characterize particles using iSCAT would be to track the particles, but the interference signal can be small and tracking is hard: moving only a quarter wavelength in the depth-direction (z) can effectively make a particle disappear as the interference changes from constructive to destructive. One approach to this problem is to confine the particles in the depth direction and/or to use particle tracking techniques, but this is difficult in practice. iSCAT is typically performed on particles immobilized on a surface to stop the phase of the interference signal varying.
In one aspect there is therefore described a method of characterizing one or more particles in a fluid e.g. a liquid using interferometric scattering optical (iSCAT) microscopy. The method may comprise illuminating a region of a fluid, in particular a liquid, with illuminating light using an objective lens to generate scattered light scattered by one or more particles in the fluid, and providing reference light. In implementations the reference light and illuminating light are coherent with one another.
The method may further comprise capturing the reference light and the scattered light using the objective lens, and providing the reflected light and the scattered light to an imaging device, such that the reflected light and the scattered light interfere at the imaging device. The method may further comprise capturing a succession of images of the interference, and processing the succession of images of the interference to determine a succession of image correlation values. The succession of image correlation values may define a (gradual) decorrelation over time of, i.e. between, the captured images of the interference. The method may further comprise determining a property of the one or more particles from the succession of image correlation values defining the decorrelation over time.
The reference light may be provided by reflection from an interface. For example providing the reference light may comprise illuminating the region of the fluid through an interface, e.g. a boundary of the illuminated region, such that light is reflected from the interface to provide the reference light. Also or instead the reference light may be provided in some other way e.g. by using a portion of the illuminating light, or a source coherent with the illuminating light, as the reference. For example the illuminating light may comprise laser light and the laser light may be split into two (before or after the illuminated fluid) to generate the reference light.
In a still further arrangement excitation laser light impinges on the one or more particles from a first side of a sample of the fluid, and an optical system on a second, e.g. opposite, side of the sample collects the scattered light and the reference light i.e. the excitation laser light (which may be filtered to reduce its intensity, e.g. with a Fourier spatial filter). In such an arrangement no interface or reflection is needed (and later references to a region above a focal plane of the objective lens apply instead to a region below the focal plane).
In implementations an individual captured image may resemble noise, but by processing multiple images the correlation process is able to extract a decorrelation signal from the noise, and this can then be processed to characterize the one or more particles. This avoids the need to keep track of individual particles. Moreover the technique is able to process images in which particles are indistinguishable from noise in a single frame, where particle tracking is not possible. Thus implementations of the method are able to use iSCAT to characterize single particles, even when not apparent in the captured images. The particles may be or comprise molecules in aqueous solution, in particular biological molecules such as proteins. Further examples are given later.
The decorrelation over time may be used to determine one or more of a range of properties of the particles including their size, and/or shape, and/or number, and/or molecular weight, and/or interaction with the fluid e.g. liquid. Such a property may be determined by fitting a corresponding decorrelation function to the succession of image correlation values.
In some implementations a physical size of the one or more particles is determined; the determined size may be a hydrodynamic radius (or average radius) of the one or more particles. This may be determined by fitting a decorrelation function which is dependent upon a diffusion coefficient for the one or more particles in the fluid e.g. liquid, where the diffusion coefficient depends on the (average) hydrodynamic radius.
The interface may define a boundary of the fluid e.g. the boundary of a microfluidic channel or chamber, or the boundary of a surface on which the liquid lies.
In some implementations a focal plane of the objective lens is located above the interface and the decorrelation function is substantially independent of a distance of the one or more particles from the focal plane in a direction along an optical axis of the objective lens, that is the z-direction. This can facilitate fitting the decorrelation function to the succession of image correlation values, and hence facilitates particle size determination.
The intensity of the scattered light may be used to determine particle concentration and/or a molecular weight for the one or more particles (or a total molecule weight of particles in the field of view), as the scattered light intensity depends on both of these.
Above the focal plane an intensity of the scattered light (and a pattern of the interference) depends strongly on the position of a particle in the z-direction i.e. on the distance of the one or more particles from the focal plane in the z-direction. In this region determination of particle concentration or molecular weight may have reduced accuracy. Nonetheless accuracy may be recovered by integrating over the z-direction e.g. by integrating over time such that the decorrelation is measured over a period long enough for a single particle to diffuse over the z-direction. For example this may comprise integrating over a sufficient time period for the one or more particles, on average, to diffuse a distance in the z-direction of at least a depth of field of the objective lens, or to diffuse a distance of at least half the distance between the interface and the focal plane. This dependency can also be integrated out if there is more than one particle e.g. several particles, randomly distributed in the z-direction.
In some other implementations the focal plane of the objective lens is located adjacent to or below the interface. By locating the focal plane below or at the interface the particles may be confined to the region above the focal plane. The decorrelation function may then comprise an average (of a decorrelation) over at least a region beyond the interface i.e. over a region extending in the z-direction further from the interface than the focal plane. In implementations the averaging may extend over this region a distance at least sufficiently to encompass a field of view of the imaging device in the z-direction. Optionally the decorrelation function may also be averaged over a region between the focal plane and the interface.
In implementations fitting the decorrelation function comprises identifying which of one or more basis functions best fits the succession of image correlation values. Each of the basis functions may be defined by the size of the one or more particles and may be integrated over a region further from the interface than the focal plane, e.g. a region beyond the interface, and optionally over a region between the interface and the focal plane. The precise mathematical form of the function integrated to determine a basis function may depend on what is being modelled, e.g. diffusion and particle hydrodynamic size, particle shape/orientation, and so forth.
Locating the focal plane of the objective lens above the interface places the focal plane in the bulk of the fluid, away from the interface. This effectively puts the interface out of focus, reducing the effect of scattering from the interface and increasing image contrast. Also, particularly when operating in this regime the intensity of the scattered light (later |s|2) may be used to accurately determine particle concentration, or a count of the number of particles “visible” in the particle detection region, in particular by fitting the decorrelation function (i.e. basis functions).
In implementations of the method using interferometric scattering to determine the image correlation values results in the image correlation values including a noise term representing a noise variance in the signal from the captured images (which primarily arises from the reference light). Thus fitting the decorrelation function to the succession of image correlation values may include fitting an offset representing the noise level. Fitting the decorrelation function may also include fitting an additional function representing a systematic behavior or drift of the microscopy technique, e.g. a linear additional function; and/or a background correlation may be subtracted from the succession of image correlation values.
The interference pattern generated by a particle may comprise a set of concentric rings (visible when the particle is large enough for the pattern to be clearly visible). However if the particle moves by λ/4 in the z-direction, in particular when the particle is above the focal plane of the objective lens, the path length changes by twice this—more particularly, the interfering light changes in relative phase by π—and the rings move inwards or outwards so that e.g. a central bright spot becomes a dark spot i.e. the interference pattern is inverted so that bright becomes dark, and vice-versa. During the correlation two frames may be combined by subtraction and the interference signal can take positive and negative values, so diffusion in the z-direction can result in an average zero signal. More generally diffusion in the z-direction can strongly suppress the interference signal if the frame rate is low. To mitigate this, the succession of images of the interference may be captured at a frame rate (or effective frame rate as discussed below) greater than a threshold frame rate. The threshold frame rate may be such that, on average, a particle does not diffuse in a z-direction by more than λ/4 between captured images, where is a wavelength of the coherent light and the z-direction is defined by an optical axis of the objective lens. That is, the threshold frame rate may correspond to an inverse of the (de)correlation time (which may be defined by a decay constant for the correlation). Counter-intuitively, the signal decreases approximately according to the square root of the exposure time, as does the noise, and thus a longer exposure per se may not be beneficial.
The iSCAT signal depends on a product of the reflected and scattered light intensities. Whilst the reflected light intensity may be relatively constant, it is nonetheless useful to be able to separate the reflected and scattered light intensities, for example so that the scattered light intensity can be more easily used to estimate a number or concentration of particles present. This can be achieved by determining a square root of an intensity of the images of the interference before determining the succession of image correlation values. This approximately separates the reflected and scattered light intensities into separate terms, and the reflected light intensity term then drops from the correlation.
Noise in the image correlation values derived from interference images depends linearly on the intensity but varies as the square root of a number of image frames used to determine the image correlation values. Thus when collecting images for a particular effective image frame rate it can be beneficial to use a higher frame rate, with a consequently reduced intensity signal, and then combine or integrate multiple frames. However images combined in that way should not result in a total integration time longer than the (de)correlation time. Thus in some implementations processing the succession of images of the interference may include combining (integrating) images of the interference before determining the succession of image correlation values, provided that the combined images are separated in time by no more than a characteristic time of the decorrelation e.g. a time constant of correlation described by an exponential decay.
The image correlation values may in general be determined from a real-space representation of the captured images, or from a frequency-space representation of the captured images since (providing phase is preserved) these two representations are equivalent. There are many ways of determining an image correlation value representing the correlation between two images, For example in either real-space or frequency-space an image correlation value may be determined by summing a pixel-by-pixel comparison e.g. subtraction, of the two images. The succession of image correlation values may be determined by correlating an initial image with second images chosen at successively larger time intervals. However calculated the succession of image correlation values effectively defines the interference pattern decorrelation over time.
In some implementations differences between the images may determined in real space; this can assist in attenuating a bright reflected spot which may be present in a central part of the images.
In some implementations a space-frequency transform, e.g. a Fourier transform, may be performed to transform each of the images to a frequency (Fourier) space image before determining the succession of image correlation values.
For example each frequency space image may be spatially filtered to attenuate spatial frequencies greater than a maximum expected spatial frequency. This may involve masking a region of the frequency space image outside a central area e.g. circle of the frequency space image. For example in a real space image it may be expected that there are no features spaced closer than λ/4, and the frequency space images may be spatially filtered accordingly to attenuate higher frequency noise.
After such processing the frequency space images may be transformed back to real space, or the image correlation values may be determined from the frequency space images
In some implementations the frequency space images may be used to compensate for an overall translation of the particles when the fluid e.g. liquid is flowing. This is based on the recognition that translation of an interference pattern results in just a change in phase in frequency (Fourier) space.
Thus the method may include estimating a flow rate measure of the fluid e.g. liquid, e.g. an average phase, θ, in frequency space, from a succession of the frequency space images. The flow rate measure may then be used to compensate for a flow of the fluid, for example by compensating a frequency space image, or value e.g. difference derived therefrom, for the average phase, and hence the translation, when determining an image correlation value using that frequency space image or value.
Thus estimating the flow rate measure may comprise determining a ratio of two of the frequency space images (e.g. per pixel), where the ratio defines the phase angle θ. Compensating for the flow of the fluid may then comprise adjusting a phase angle of one or more of the frequency space images. For example, a difference between two successive frequency space (Fourier) images includes a phase term from the translation (fluid flow) and a scattering term from the change in interference (decorrelation). If the scattering term changes much less than phase term, e.g. because the images are closely spaced in time, then taking a ratio of these differences can approximately cancel the scattering term to extract the phase term. The difference between two frames comprises a translation term and a diffusion term, but the former can be assumed to be constant whilst the latter is random. Thus by averaging the ratio the diffusion term may be reduced towards zero leaving the translation term.
Again, the frequency space images may be transformed back to real space, with a correction to compensate for translation between images due to the fluid flow, and the image correlation values may be determined from the real space images; or the image correlation values may be determined from the frequency space images.
In some implementations an optical path of the coherent light to the objective lens is offset from the optical axis of the objective lens such that the coherent light illuminating the particle is at an oblique angle to the interface. However the reflected and scattered light may be captured in a direction along the optical axis of the objective lens. This can improve the contrast of the images of the interference, and also their signal-to-noise ratio.
In a related aspect there is described an interferometric scattering optical microscope system for characterizing one or more particles in a fluid e.g. a liquid. The system may comprise a particle detection region; this may have a boundary defined by an interface. In use, the particle detection region includes one or more particles in a fluid e.g. a liquid.
The system may include a source of illuminating e.g. coherent light, and a source of reference light. The reference light and illuminating light may be coherent with one another. The system may further include an objective lens to direct the illuminating light to illuminate the particle detection region such that the illuminating light is scattered by the one or more particles. The objective lens may be configured to capture the reference light and the scattered light.
The system may further comprise an imaging device, and may include an optical system to provide the reference light and the scattered light to the imaging device such that the reference light and the scattered light interfere at the imaging device.
The system may also include a processor configured to capture a succession of images of the interference. The processor may also process the succession of images of the interference to determine a succession of image correlation values. The succession of image correlation values may define a decorrelation over time of the captured images of the interference. The processor may also determine a property of the one or more particles from the succession of image correlation values defining the decorrelation over time.
The particle detection region may have a boundary defined by an interface, which serves as the source of reference light. More particularly the objective lens may be configured to direct the illuminating light to illuminate the particle detection region through the interface such that the illuminating light is reflected from the interface to generate the reference light.
The source of coherent light may be e.g. a source of laser light or light from a narrowband LED source, optionally polarized e.g. linearly polarized. An optical path between the objective lens and coherent light source may include a beam splitter to separate the returned reflected and scattered light from the incident illuminating light; in other implementations these two light paths are at different spatial positions with respect to the optical axis. The imaging device may comprise a 1D or 2D image sensor, e.g. an EMCCD (Electron Multiplying Charge Coupled Device) sensor or a fast CMOS camera. An optical element such as a lens or mirror may focus returned reflected and scattered light onto the image sensor.
The processor may comprise e.g. a general purpose computer system, mobile device, or digital signal processor under stored program control. The stored program may be provided on one or more non-transitory physical data carriers such as programmed memory, and may comprise source, object or executable code in a conventional programming language, or code for a hardware description language. Code to implement the system may be distributed between a plurality of coupled components in communication with one another.
The particle detection region may comprise an imaging region of a microfluidic channel or chamber. The interface which generates the reflected light may be an inner boundary of such a channel or chamber i.e. an interface with the fluid e.g. liquid, although in principle it could be an exterior boundary. The interface is typically flat, but in principle may be curved. In some other implementations the interface is defined by a physical surface of a transparent support, e.g. a glass coverslip, which bears a droplet of the liquid, and the particle detection region is above the physical surface of the transparent support.
If the fluid is a liquid flowing in a microchannel it may have a generally flat flow profile across a width of the channel; this may be achieved by driving flow of the liquid by using electro-osmosis. The liquid may comprise or consist essentially of water. Prior to interrogation using the interferometric scattering optical microscope system the liquid may be been filtered, separated, or otherwise treated to enhance a relative proportion of a target particle or reduce the presence of unwanted particles.
Other features of the interferometric scattering optical microscope system may correspond to those of the previously described method, i.e. the system may include sub-systems/devices to implement these features.
The particles may comprise molecules or molecular complexes, such as biological molecules/complexes, e.g. proteins, protein complexes, or antibodies. More generally the particles may have a maximum dimension which is less than a wavelength of the coherent light, or less than half this wavelength. Such particles may include metallic or other nanoparticles, colloid particles, polymer particles, viruses, exosomes and other extra-cellular vesicles, proteins, and bio-particles in general. The particles may be non-fluorescent particles i.e. label-free. In implementations the method and system may be used to detect and characterize from one to thousands or millions of particles.
These and other aspects of the system will now be further described by way of example only, with reference to the accompanying figures, in which:
In the figures like elements are indicated by like reference numerals.
This specification describes techniques for characterizing particles using correlations between images captured using interferometric scattering optical microscopy (iSCAT). The techniques may be used, for example, to detect and size proteins in aqueous solution, and to determine their concentration. The proteins do not need to be labelled, and single particles as well as large concentrations can be detected.
The objective lens 112, e.g. an oil immersion objective, may be adjusted to provide generally uniform illumination in a detection region 114. The objective lens 112 may be an oil immersion objective, that is a region between the objective lens and the interface may be filled with index matched oil to reduce the intensity of oil-glass reflections and hence reduce the number of interfaces seen by the laser illumination.
In some implementations the detection region 114 may be defined by upper and lower surfaces of a chamber or channel (not shown), fabricated e.g. from glass or polymer containing a solution including the one or more particles 120. A channel/channel may be used to contain/flow the solution through the detection region; in other implementations the solution may be contained on a surface by surface tension with no chamber used.
Part of the laser illumination is reflected from an optical interface 122, e.g. a transparent support such as a cover slip or a boundary of the channel/chamber, and part is transmitted to be scattered by the particle(s) 120. The reflected light and the scattered light are captured by objective lens 112, passed back along the optical path of the illumination, directed into a separate path 124 by the beam splitter 126, and imaged. As illustrated the reflected and scattered light are focused onto an image sensor 130, e.g. a high speed camera, by imaging optics 128 such as a tube lens. The image sensor 130 may be, for example, a CMOS image sensor or an EMCCD image sensor; it may have a frame rate sufficient to capture and track movement of an imaged particle. The image sensor captures interference between the reflected and scattered light. Merely by way of example, a camera for the system may have one or more of: a field of view of around 10 μm or greater; a similar depth of field; and a pixel size of less than λ/2.
A focal plane of the objective lens 112 may be located below the optical interface 122, confining the particle(s) to a region above the focal plane. Alternatively the focal plane may be located above the optical interface 122, in which case particle(s) in a region between the focal plane and the optical interface may be characterized; this region may also extend above the focal plane. The focal plane of the objective lens 112 may be viewed, conceptually, as representing a location of the image sensor.
The iSCAT system captures images of the interference between the scattered light and reference light. In the above example the reference light comprises the reflected light generated by reflection from the optical interface 122, but reference light may be provided in other ways e.g. by splitting the beam 104 to generate a reference beam.
In some implementations the source of coherent light is linearly or circularly polarized. For example in the illustrated example the optical path from the laser to the objective lens 112 includes a quarter wave plate 109 that converts linearly polarized light from the laser into circularly polarized light (and vice-versa on the return path).
In some implementations the particle(s) 120 may be illuminated at an oblique angle, e.g. by displacing an axis of the illuminating laser beam away from an optical axis of the objective lens 112. This facilitates separating the interference (i.e. signal) from the directly reflected light (a source of noise), e.g. with a spatial filter.
In some implementations the laser 102 is a laser diode. The limited coherence length of a laser diode can reduce unwanted coherent reflections within the system. However the coherence length should be at least twice a distance from the particle to the interface. A wavelength of the laser may be in the visible; shorter wavelengths are scattered more but a sensitivity of the image sensor and transmission of the optics may reduce at shorter wavelengths.
The iSCAT system of
At step 200 the process captures a succession of interference images of a solution containing one or more particles to be characterized; each image may be termed an iSCAT signal.
The images may be captured at a frame rate sufficient that, on average, a particle does not diffuse more than
in the z-direction between successive frames, where λ is a wavelength of the illumination and the z-direction is along the optical axis of the objective lens 112. This can facilitate tracking the (de)correlation between frames. For example assuming a textbook diffusion equation a minimum frame rate may be given by fminimum=25D/λ2 where D is a diffusion coefficient for the particle(s) (in m2/s).
At step 202 the process optionally takes a square root of each image, for example by taking a square root of the intensity value of each image pixel. This can facilitate determining a level of the scattering signal (intensity), which in turn facilitates estimating properties dependent upon the scattering signal such as particle size, number, or molecular weight.
Taking the square root separates reflected (r) and scattered (s) intensity contributions to the iSCAT signal I=I0(|r|2+|r∥s|cos ϕ) where ϕ is an interference phase given by a path difference between the reference (reflected) light and the scattered light at a focal plane of the image sensor and I0 is the illumination intensity. A Taylor series expansion of √I separates the reflected and scattered contributions,
This also reduces the shot noise variance of the iSCAT signal by a factor of 4. That is, for an iSCAT signal I and a normalised noise variance σn2 the shot noise variance of the iSCAT signal σI2=Iσn2≈I0|r|2σn2 and the noise variance of the square rooted signal √I is σn2/4.
At step 204 the interference images may optionally be transformed to the spatial frequency domain, as shown in
At step 206 the process determines a succession of image correlation values e.g. from the captured image frames, or from their square roots, or from the transformed images in the spatial frequency domain. For example one way a correlation value may be determined, for two space- or frequency-domain images separated in time by Δt, is by determining a difference between the corresponding pixel values of each image and summing the result. There are many other ways of determining a measure of correlation between two images.
In some implementations the image correlation values are determined for successively increasing time intervals to obtain the succession of image correlation values. For example, as illustrated in
Then, at step 208, a decorrelation function is fitted to the succession of image correlation values to determine one or more properties of the particle(s). More particularly the decorrelation time depends on the diffusion coefficient of the particle(s), from which particle size, specifically hydrodynamic radius, can be determined.
In some implementations the decorrelation function may be fitted by fitting the succession of image correlation values to an equation describing the decorrelation. In some implementations the decorrelation function may be fitted by numerically calculating a set of basis functions, that is one or more functions using which the actual decorrelation may be modelled, and identifying one or a combination of the basis functions which best match the succession of image correlation values. Examples of both techniques are described later. The form of the basis functions may depend on the configuration of the iSCAT system and in principle could be determined by a system calibration.
The scattering signal also depends on the particle size, more particularly volume, and this in turn allows a weight or molecular weight to be estimated from an assumed particle or protein density. Also or instead a scale linking scattering signal to particle property e.g. molecular weight may be determined by calibration. The scattering signal may be determined by fitting the decorrelation function or more directly e.g. by integrating signal intensity over the scattering circle in the spatial frequency domain.
In an iSCAT system shot noise of the reflected light, more generally reference light, is a dominant source of noise. However the iSCAT signal depends on the reflected light intensity and thus naively increasing the image sensor exposure time does not improve the signal to noise ratio. When correlating two iSCAT images the noise signal is substantially the same no matter what the elapsed time between the images but the mean correlation increases as the elapsed time reduces. This recognition underlies the improved performance of iSCAT when the interference signal is used for determining a decorrelation signal from which to infer a property of the subject particle(s).
In general the correlation, g2, between two images (I, √{square root over (I)}, or frequency domain) may be defined as the normalized time average of the squared signal difference. For example in one implementation a (de)correlation between two square root images separated in time by Δt is defined as
where T denotes averaging over time and Δ√{square root over (I)} denotes a difference between the two images. Then in this example the (de)correlation between the square rooted iSCAT interference images is given by the decorrelation function
where a characteristic decorrelation time τ is defined as τ=(q2D)−1 where
and σn2/2I0 represents a noise variance of the correlation signal. Here Z denotes a distance of the particle from the focal plane of the objective lens along the z-direction, and p denotes a distance of the particle from the optical axis of the objective lens perpendicular to the z-direction.
When averaging over n frames the noise variance reduces to
which depends linearly on image sensor exposure time (∝I0) and on the square root of the number of frames. Thus noise can be reduced if image frames are combined or integrated then correlated, provided that the integration time is less than the correlation time τ (the signal to noise ratio maximum is for a single frame exposure time of less than half the correlation time).
The curve of
The scattering intensity s2/4, optionally with the noise level (at Δt=0) subtracted, depends on the size (volume) of the particle(s) as the scattering cross-section of a single particle is given by
ϵp is the permittivity of the particle, ϵm is the permittivity of the medium (solution) and V is the particle volume. For a particular particle type e.g. a protein, a particle density and permittivity may be assumed. The particle volume then allows a measure of the total particle “weight”, and an estimate of a particle molecular weight if the number of particles is known. Alternatively a number or concentration of the particles may be estimated if the particle molecular weight (and permittivity) is known.
In another approach the scattering intensity s2/4 may be calibrated using known numbers of particles of different molecular weights to establish a (linear) calibration curve. This can then be used to estimate the molecular weight or number/concentration of particles in a solution to be characterized by the system.
In some implementations a value for the characteristic decorrelation time τ, and hence for the diffusion coefficient D, may be found by fitting the decorrelation function to the data i.e. to the correlation values. In one approach this involves calculating a set of curves of the type shown in
In some implementations the dependence on Z (and ρ) is integrated out, especially where Z>0. Physically this corresponds to an assumption that multiple particles are present with a distribution of values of Z (and ρ), and/or to integration over a time period over which the particle(s) move in Z. For example the integration may be over a time period sufficient to allow a particle to diffuse in Z a distance equal to the depth of field of the objective lens; or to diffuse in Z at least half the distance between the interface and the focal plane.
For example, in some implementations a basis function is given by:
where Zf defines a location of the interface 122, a negative value indicating that the interface is below the focal plane (i.e. nearer to the illuminating source than the focal plane), ρmax is a value which may be chosen e.g. dependent on the horizontal field of view of the objective lens, and θ is defined by ρ and Z (as shown in
Not shown in
The basis function curves cease to be significantly dependent on Zf when the interface 122 is around 20 μm or more below the focal plane i.e. Zf≤−20 μm. In this regime, i.e. where the focal plane of the objective lens lies within the bulk of the fluid, the scattering intensity is also a reliable metric. An added benefit is that the interface 122 may not be in focus, reducing unwanted scattering from this surface.
Some implementations of the system may be used to characterize particles in flow, by separating a component of the signal due to the flow, most easily done by processing in the spatial frequency domain.
In the Fourier domain a translation is equivalent to a phase difference and thus for a translation (Δx, Δy) the Fourier transform of fn+1(x, y)=fn(x+Δx,y+Δy) is Fn+1(ϵ, η)=exp(iθ)Fn(ϵ, η) where θ≡2π(ϵΔx+ηΔy). As before taking the square root of the iSCAT signal separates the reflected and scattered intensity and the Fourier transform of frame n can be written as F{√{square root over (I)}n}≈Fr+Fs,n. The Fourier transform of frame n+1, Fs,n+1 is related to Fs,n by a difference in the scattering pattern δFs,n arising from diffusion of the particle, and by a translation, Fs,n+1=exp (iθ)Fs,n+δFs,n+1. The difference used to compute the interferometric iSCAT signal is therefore
Δn+1=Fs,n+1−Fs,n=(exp (iθ)−1)Fs,n+δFs,n+1
and a similar forward calculation gives
Δn=Fs,n−Fs,n−1=(exp (−iθ)((exp (iθ)−1Fs,n+δFs,n).
where θ=0 corresponds to no flow. The ratio Δn+1/Δn may be taken as constant between pairs of frames to detect the offset by assuming δF to be small so that δn+1/Δn≈exp(iθ). This can be used to calculate a value for θ, and hence the translation between frames and flow rate. Further, when θ is known, a difference ΔΔn dependent upon δF terms (which can be evaluated more easily than F) can be determined as follows:
ΔΔn=Δn+1−exp (iθ)Δn=δFs,n+1−δFs,n
Since ΔΔn is a sum of Gaussian random variables with the same variance, ΔΔn may be used as a correlation value as previously described, the contribution due to the flow having been removed.
Many alternatives will occur to the skilled person. The invention is not limited to the described embodiments and encompasses modifications apparent to those skilled in the art lying within the scope of the claims appended hereto.
| Number | Date | Country | Kind |
|---|---|---|---|
| 2010411.3 | Jul 2020 | GB | national |
| Filing Document | Filing Date | Country | Kind |
|---|---|---|---|
| PCT/EP2021/068359 | 7/2/2021 | WO |
