Patent Description:
Dynamic light scattering is a widely used method for analysing particles in which a time series of measurements of scattered light is used to determine a size or size distribution of particles. Particle characteristics are inferred from the temporal variation in the scattered light. Typically, an autocorrelation is performed on the time series of scattered light intensity, and a fit (e.g. Cumulants, CONTIN, NNLS/non-negative least squares) is performed to the autocorrelation function to determine particle characteristics. Alternatively, a fourier transform may be used to determine a power spectrum of the scattered light, and an analogous fit to the power spectrum performed to determine particle characteristics. Typically, light scattered in a single, well-defined, angle is used in a dynamic light scattering measurement.

Multi-angle dynamic light scattering (MADLS) measurements may also be performed, in which light scattered at more than one angle is used in a dynamic light scattering measurement (<NPL>).

When multiple scattering angles are used, it is necessary to determine particle characteristics that are most consistent with the scattering data obtained at each scattering angle. There is considerable room for improvement in the existing techniques for doing this.

"<NPL>) describes using a least-squares calculation that estimates weighting coefficients on the basis of the complete autocorrelation measurement.

"<NPL>) describes both combining static light scattering measurements with a few angles of dynamic light scattering data, and also recording only dynamic light scattering data at many angles and iteratively reconstructing the static light scattering data during data analysis.

"<NPL>) describes a recursive regularization method for estimating the weighting coefficients and particle size distribution from multiangle dynamic light scattering data. The algorithm distinguishing characteristics of particle size distributions and automatically chooses an inversion method from Nonnegative Tikhonov and Nonnegative Phillips-Twomey regularization algorithms.

"<NPL>) describes an iterative recursion method of estimating the weighting ratio at a second scattering angle using the particle size distribution obtained from a first scattering angle, and then estimating the weighting ratio at a third scattering angle using the particle size distributions obtained from the first and second scattering angles and the known weighting ratio at the second scattering angle, and so on.

<CIT> describes a method of determining a size distribution of a sample from a measured Taylorgram. The method comprises injecting the sample into a flow of eluent within a capillary at an injection location; flowing the sample through a length of the capillary from the injection location to a detection location; detecting a Taylorgram using a sensor at the detection location as the sample flows past the detection location; analysing the detected Taylorgram by i) determining an initial estimate for mean particle size and polydispersity index by fitting a function to the Taylorgram; ii) simulating a plurality of Taylorgrams having a different selected polydispersity index; iii) obtaining a sample of noise from the detected Taylorgram, and adding noise derived from the sample of noise to each of the simulated Taylorgrams; iv) determining an estimate polydispersity for each of the simulated Taylorgrams by fitting a function to each simulated Taylorgram; v) determining a relationship between the estimated polydispersities and the selected polydispersities; vi) using the relationship to determine a final estimate for polydispersity index from the initial estimate for polydispersity index.

Aspects of the invention are set out in the independent claims. Some optional features are set out in the dependent claims.

Embodiments of the invention will be described, purely by way of example, with reference to the accompanying drawings, in which:.

When performing a MADLS measurement with three measurement angles (θ<NUM>, θ<NUM> and θ<NUM>) the following system of linear equations must be solved (with only the relevant matrix components shown): <MAT>.

Where g(θi) is the measured correlation function at angle i, K(θi) is the instrument scattering matrix computed for angle i, and x is the particle size distribution. The aim is to determine the particle size distribution from the measured correlation functions. The coefficients a and b are scaling factors, and initial estimates for these can be calculated by extrapolation (linear or otherwise) of the correlation function to time-zero to determine the y-axis multiplier. The physical origin of a and b is the sum of the scattered intensity into each angle.

Note that the first scattering angle θ<NUM> has arbitrarily been chosen as the measurement angle to which the other measurement angles are scaled. It will be appreciated that any of the other measurement angles could alternatively be selected as having a unitary scaling factor, and the other measurement angles scaled to that angle.

Bryant (referenced above) discloses deriving a and b from the count rate, and the use of a linear solver, but this method is prone to error if the count rate is not stationary over time. Cummins (<NPL>) discloses using a non-linear solver to determine a, b and x. Applying a non-linear solver in this way is problematic, because there are a large number of coefficients (in x) to be determined and there is a significant possibility that the solution will not converge reliably.

In embodiments of the present invention, a nested approach is used, in which a non-linear solver (such as Nelder-Mead simplex, Levenberg-Marquardt, Gauss-Newton or another method) is used to iterate an estimate for a and b, and within each iteration, a linear solver (e.g. non-negative least squares) is used to determine a best fit for x and consequently determine a residual error. This is exemplified in the following pseudocode:
<IMG>.

This nested technique allows a fast and robust linear method (e.g. NNLS) to be applied to the linear problem of solving the particle distribution. Coefficients a and b are variable scalars that require a non-linear approach, and a non-linear solver is therefore appropriate for this. This approach is preferable over tackling the entire problem with a non-linear solver (the approach taken by Cummins). Embodiments therefore provide more robust, faster to execute, solutions for MADLS. This in turn enables a larger number of size classes (i.e. a longer vector x), so that MADLS can be used to span a greater size range than in the literature, or to provide a particle size distribution with a greater resolution than in the prior art.

This method can be generalised to apply to the following set of equations, for n number of scattering angles: <MAT>.

With αi corresponding with the scaling coefficient for angle i.

Conventionally, the first scattering (i = <NUM>) angle may have a scaling coefficient defined as <NUM> (α<NUM> = <NUM>). As mentioned above, this is merely an arbitrary convention: in alternative embodiments the scaling may be performed relative to any of the measurements <NUM> to n. It will also be appreciated that the order of concatenation of measurement results is also arbitrary, and any of the measurements may be placed first. For ease of notation, the scaling mentioned herein is denoted as relative to the first measurement, but this does not imply a limitation.

The generalised pseudo code therefore becomes:
<IMG>.

<FIG> schematically illustrates a method of determining a particle size distribution according to an embodiment. At step <NUM>, a time series of scattering intensity is measured (or determined from a time history of photon arrival times) at each scattering angle (e.g. sequentially, or in some embodiments, simultaneously). At step <NUM>, correlation functions are determined for each scattering angle from the respective time series of scattering intensity (e.g. using a correlator). At step <NUM>, the equation (<NUM>) is solved to determine a particle size distribution x.

The process of solving equation (<NUM>) to determine the particle size distribution comprises:.

When performing a multi-angle DLS measurement, the data that is measured at each angle must be representative of the same sample. In most measurements, it can be assumed that the sample will not change over the duration of the measurement. If Typically, because of the relatively high cost of suitable detectors for DLS (which typically employs a photon counting detector such as an avalanche photodiode) the scattering measurements at each angle are taken sequentially, with an optical path to a single detector cycled between each detection angle. Additionally, due to the field of view of the detector, different measurement angles sample a subtly different scattering volume, even if the volume centres are coincident. Under these measurement circumstances, a transient contaminant (such as dust, or filter debris) will tend to contaminate a single scattering angle. This will adversely affect the result, since it will not be possible to find a common solution that satisfies the measurement data (because one of the measurements is not representative of the sample).

It is possible to accommodate the presence of noise at a single angle during the fitting process by including terms (specific to the noise type) in the solution that depend each only on a single angle. The particle size distribution result should remain common across all angles. Noise usually manifests as a slowly decaying contribution to the correlogram caused by dust at one measurement angle. However, noise types of any form can be accounted for if they are suspected to exist.

As already discussed above, the regular system of linear equations is of the form g = Kx, where g is the measured correlation function, K is the instrument scattering matrix, and x is the particle size distribution (neglecting the scaling factor coefficients for simplicity of notation).

With application to MADLS, the equation becomes (with only the relevant matrix components shown): <MAT>.

In prior art analyses, the vector x comprises a set of N scalar values, each scalar value defining an intensity of scattering from a particular range of particle sizes (or bin). In embodiments of the present invention, the vector x includes at least one additional value that is included to accommodate the presence of contaminants at one or more scattering angle. For n = <NUM>, corresponding with three measurement angles, and a particle size distribution comprising N size bins, the vector x may take the following form: [x<NUM>, x<NUM>, x<NUM>,. , xN, n<NUM>, n<NUM>, n<NUM>], where x<NUM> to xN are the scattering intensities corresponding with each size bin, and the terms n<NUM> to n<NUM> are noise intensities corresponding with each of the three scattering angles.

The matrix <MAT> has the following form: columns in K are computed as the expected instrument response according to each element in x. Columns at indexes according to elements x<NUM> to xN may be calculated theoretically for each correlator lag-time, τ, and angle, θ. Columns at indexes n<NUM> to n<NUM> take the form of a computed noise contribution, at each angle, based on an assumption about the characteristics of the noise.

In an example embodiment, the noise contribution is assumed to take the form of large transient particles in the scattering volume. For example, the model used to estimate the noise contribution columns of K may mimic that used to compute the expected instrument response for a size bin, but use a fixed particle diameter of <NUM> microns at the particular scattering angle i, and for each correlator lag-time, τ: <MAT> Where:.

Since the noise at each angle is not related to the noise at any other measurement angle (for a sequential multi-angle measurement), elements are zero at angles other than for which the noise is considered. If large material is present at just one angle, the solution residual will minimise if intensity is assigned to the noise bin during fitting. The solver will not compromise the particle size distribution fit result by addition of spurious particles.

In an example, a mixture of polystyrene latex spheres of diameter <NUM> and <NUM> was prepared in an aqueous dispersion. Two methods were used to fit a particle size distribution to the same instrument data. The first method did not assume any single-angle noise. <FIG> shows the resulting particle size distribution, which comprises a peak <NUM> at <NUM>, a peak <NUM> at <NUM> and a peak <NUM>, corresponding with a contaminant, at <NUM> microns. The fit residual has a magnitude of <NUM>. 6e-<NUM> (L<NUM>-norm).

<FIG> shows the vector x calculated in accordance with an embodiment, in which a single-angle noise contribution is assumed to be present to a greater or lesser degree at each scattering angle. The result is a particle size distribution that includes a peak <NUM> at bins corresponding with <NUM>, a peak <NUM> at bins corresponding with <NUM>, negligible contaminant at <NUM> microns and significant single-noise contribution <NUM> in the noise bins that are at the end of the vector x. In this case the noise contribution was present in back-scatter but not in either of side-scatter or forward scatter, as illustrated in <FIG>, which shows the intensity assigned to contaminants at each of the three scattering angles of this measurement. The fit residual has a magnitude of <NUM>. 8e-<NUM> (L<NUM>-norm) - smaller than achieved without consideration of a single-angle noise contribution.

Although the foregoing example has illustrated the application of this technique to three scattering angles, it will be understood that fewer or more scattering angles may be used in accordance with alternative embodiments.

In some embodiments, the same principles can be applied to measurements that are not taken at different scattering angles (but are taken at different times, with at least some, or all, at the same scattering angle), so that a particle size distribution can be determined from the ensemble data that discounts scattering contributions from contaminants. This approach will be effective in removing scattering contributions from contaminants that are not present in every measurement, and which are well approximated by the model used to simulate the scattering contribution from the contaminants.

Because the MADLS problem (and more generally, the DLS problem) is ill-conditioned, regularisation may be used to bias the solution against fitting to noise and to enforce some predefined property of the result. The system of linear equations g = Kx becomes (with only the relevant matrix components shown): <MAT>.

In the above, g may represent a matrix of n correlation functions (i.e. <MAT>), each corresponding with a measurement taken at a different time and/or at a different scattering angle, and K may represent a matrix <MAT>.

Typically, the regularisation coefficient, γ, takes the form of a scalar value to enforce more or less regularisation, dependant on the magnitude of γ, with larger γ resulting in more regularisation. The regularisation matrix, Γ, can take several forms. When performing DLS, it is often desired to enforce smoothness in the result as particle size distributions are believed to be mostly continuous. Alternatively, the solution norm may be biased toward zero if the particle size distribution is believed to be monomodal.

The regularisation term, γΓ, forms part of the residual to be minimised: <MAT>.

In this example embodiment, the matrix Γ may act as a low pass operator to penalise curvature in the solution, x. When measuring the particle size distribution across a large dynamic range of size bins, it is not always possible to realise a regularisation coefficient that suitably penalises curvature at small particle size and large particle size simultaneously. This is not only in part due to the relative separation of neighbouring size bins (since these are typically log-spaced) but also the differing particle characteristics at small and large particle size.

According to embodiments, this problem is solved by using a vector of regularisation coefficients γ, such that the regularisation coefficient is dependent on the particle size.

This allows more regularisation to be enforced at large sizes where particle size classes are more widely spaced and we wish to prevent a spiky solution. Conversely, this allows less regularisation at small size where we wish to employ more resolving capability. In this way, the highest possible resolution may be maintained across a large size range - <NUM> to <NUM>.

In order to illustrate this, an example DLS measurement will be simulated. The simulated example comprises four separate particle components:.

This represents a two-component mixture of protein monomers in the presence of a small number of large particulates (the large particulates are similar by scattering intensity, because this scales with the sixth power of particle diameter). A white noise contribution of <NUM>% was added to the simulated auto-correlation functions. Shown in <FIG> is the resultant particle size distribution, derived using the MADLS method with a vector of regularisation coefficients. All peaks are resolvable, with peaks <NUM> to <NUM> respectively corresponding with peak ID <NUM> to <NUM> in the above table.

The vector of regularisation coefficients, γ, employed in the above analysis varied with particle size in a way that is linear in log space, according to the function: <MAT>.

In this example, m=<NUM> and c=<NUM> (with x measured in nm), with the result that the regularisation coefficient varied as shown in <FIG>. Other functions of particle sized may be used to determine the regularisation vector (e.g. a non-log function, a polynomial etc).

The results in <FIG> can be contrasted with results obtained with a scalar regularisation coefficient (γ = <NUM>), which are shown in <FIG>. In <FIG> the large particle contribution is over-resolved because of noise on the simulated measurement.

Increasing the scalar regularisation coefficient (γ = <NUM>), does not remedy this problem, because this causes the small particle contribution to be poorly resolved, as shown in <FIG>. The applicant has found that using a vector regularisation that varies with particle size to provide improved measurement accuracy in the particle size distribution.

It is possible to determine an appropriate regularisation coefficient (e.g. automatically). One method described in the literature is the L-curve method (<NPL>). According to this method, the optimal regularisation coefficient is that which minimises both the residual norm (∥Kx - g∥) and the regularisation norm (∥Γx∥). Plotting the residual norm against the regularisation norm across a range of regularisation coefficient typically results in an L-shaped curve, the corner of which represents the optimum regularisation coefficient.

This approach can be adapted to determine an optimum regularisation vector, γ. In the example above, in which a linear log function is used to determine the regularisation vector, the intercept, c, and the gradient m, can be iterated, and multiple residual norm vs regularisation norm (L-curves) can be plotted. Following this, an appropriate regularisation intercept and gradient pair can be deduced. In principle, a similar analysis can be applied to compare any functions for determining an appropriate regularisation vector. Such an analysis can be applied automatically by a processor/instrument, based on measurement data (simulated or actual) that is representative of a particular use-case for an instrument. Alternatively, a regularisation vector (or vectors) can be determined that is generally appropriate for a particular customer requirement. In some embodiments, the user may be able to select between alternative regularisation approaches (e.g. scalar, first vector (low gradient), second vector (high gradient), etc).

<FIG> shows a set of L-curves <NUM>-<NUM> respectively corresponding with values of gradient, m, in equation (<NUM>) of <NUM> to <NUM> (in increments of <NUM>), with each curve <NUM>-<NUM> having a series of values for intercept, c, incremented between <NUM> and <NUM>. Low gradient with moderate intercept are shown by <FIG> to be optimal (e.g. similar to the parameters used for <FIG>).

<FIG> illustrates an instrument according to an example embodiment, comprising a light source <NUM>, sample holder <NUM>, detector <NUM>, processor <NUM> and optional display <NUM>.

The light source <NUM> may be a laser (or an LED) and illuminates a sample on or in the sample holder <NUM> with a light beam. The sample comprises a fluid in which particles are suspended, and the light beam is scattered by the particles to create scattered light. The scattered light is detected by the detector <NUM>, which may comprise a photon counting detector such as an APD. Suitable collection optics may be provided to collect light scattered at a particular scattering angle (such as a non-invasive backscatter, or NIBS, arrangement that is employed in the Zetasizer Nano from Malvern Panalytical Ltd). The collection optics may be configured to allow a single detector to receive light scattered at different angles (e.g. using optical fibres and an optical switch).

The detector <NUM> provides measurement data (e.g. a sequence of photon arrival times) to the processor <NUM>, which may be configured to determine a scattering intensity over time. The processor determines a correlation function from the measurement data, for use in determining particle size in accordance with dynamic light scattering principles. Specifically, the processor <NUM> is configured to perform at least one of the methods described herein.

The processor <NUM> is optionally configured to output a particle size distribution to a display <NUM>, which displays the result to a user. The processor <NUM> may comprise a system on chip/module, a general purpose personal computer, or a server. The processor <NUM> may be co-located with the detector <NUM>, but this may not always be the case. In some embodiments the processor <NUM> may be part of a server, to which measurement results are communicated (e.g. by a further computing device).

Claim 1:
A method of determining particle size distribution from dynamic light scattering data, comprising:
obtaining measured correlation functions g(θ<NUM>) to g(θn); and
solving an equation of the form: <MAT>
wherein:
g(θi) is the measured correlation function for measurement time i, corresponding with a scattering angle θi, K(θi) is the instrument scattering matrix computed for the angle θi, x is the particle size distribution and αi is the scaling coefficient for measurement time i (with α<NUM> = <NUM>);
the vector x takes the form: [x<NUM>, ..., xN, n<NUM>, ... , nn], where x<NUM> to xN are the scattering intensities corresponding with each size bin, and the terms n<NUM> to nn are noise intensities corresponding with each of the measurement times or angles ; and
columns in K are computed as the expected instrument response according to each element in x, with columns in K at indexes according to elements x<NUM> to xN calculated for each correlator lag-time,τ and angle, θ, and columns at indexes n<NUM> to nn calculated as a computed noise contribution for each correlator lag-time,τ, at each angle, θ, based on an assumption about the characteristics of the noise;
wherein the computed noise contribution is based on the expected instrument response to a large particle of at least <NUM> microns in diameter in a scattering volume of the instrument; and
wherein the computed noise contribution is determined according to: <MAT> where:
g<NUM>(τ) is the instrument-measured field autocorrelation functions at lag time , τ; q is the scattering wave vector ( <MAT>); n<NUM> is the dispersant refractive index; λ is the vacuum wavelength; θi is the scattering angle; Dt is the translational diffusion coefficient ( <MAT>); kB is the Boltzmann constant; T is the absolute temperature; η is the dispersant viscosity; and d is the assumed large particle hydrodynamic diameter.