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nirn.1038

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doi: 10.1255/nirn.1038
Practical limits of spatial resolution in diffuse reflectance
NIR chemical imaging
Suzanne J. Hudak, Kenneth Haber, Gerald Sando, Linda H. Kidder and E. Neil Lewis
Malvern Instruments, Columbia, MD 21046, USA. E-mail: neil.lewis@malvern.com
Near infrared chemical
imaging (NIR-CI)
IR-CI, a technique already widely used in industry for a range of
chemical analyses, is a rapid,
non-destructive method that
combines rugged, flexible NIR spectroscopy with digital imaging.1 A single measurement enables the identification and
spatial localisation of chemical species that
can be correlated with a variety of sample
performance measurements.
Because most NIR-CI measurements are
made in diffuse reflectance, a fundamental
area of debate is the practical limitations
of high magnification optics with regard to
actual spatial resolution. Two factors impact
on spatial resolution—the diffraction limit
imposed by the wavelength of the probing
radiation, and the effective pathlength of
that radiation within the sample, a property
which is determined in part by the mode of
measurement (transmittance or reflectance).
Depending on the particular experimental
configuration, either one of these can be the
limiting factor. For NIR wavelengths, penetration of the radiation into the sample is
substantial, whereas the wavelength of light
is comparatively short. Therefore, the penetration of NIR light into the sample, not the
diffraction limit, becomes the limiting factor
for determining spatial resolution in a diffuse
reflectance imaging experiment.
While the diffraction limit is a simple
function of the wavelength of light and the
numerical aperture of the focusing optics
employed, it is less straightforward to determine the penetration depth of NIR radiation
into a sample in diffuse reflectance mode.
Although this question has been looked at
previously,2,3 this study attempts to clarify
some of the underlying issues and presents
a relatively simple model to determine a limit
for the maximum spatial resolution of diffuse
reflectance in the NIR spectral region. This
model is determined by first estimating the
effective pathlength of the diffusely-reflected
NIR radiation.
The schematic in Figure 1 shows photons
entering a sample with a finite penetration
N
6
Vol. 18 No. 6 September 2007
Incident
Photons
Diffusely Reflected
Photons
Penetration
Depth
Figure 1. Diffuse reflectance.
depth. Upon exiting the sample, the photons
will have interacted with the entire sampling
volume represented by the shaded area.
The spectral signature of the resulting NIR
radiation will be characteristic of the NIR
absorbances due to interactions with all of
the chemical species contained in this sample volume. As the contrast in NIR chemical images is based on NIR spectral signatures, it therefore implies that the depth of
penetration, which translates to a sampling
volume, will limit the highest useful lateral
magnification. In other words, for high magnifications, photons emerging from an area
on the sample surface will have interacted
with a larger volume than indicated by the
area actually imaged by a single detector.
This is observed as spectral mixing, so that
as magnification is increased, the observed
differences in adjacent spectra become
smaller, resulting in no real gain in spatial
resolution. This is true for all instrumental
implementations—array-based two-dimensional imaging, as well as line and single
point mapping.
Determining effective
pathlength as a function of
wavelength
According to Beer’s law, absorbance is
directly proportional to pathlength:
A = εcl
where A is the absorbance, ε is the extinction coefficient, c is the concentration, and l
is the pathlength.
In order to estimate the effective pathlength sampled in a typical diffuse reflect-
ance experiment, polystyrene was chemically imaged (SyNIRgi, Malvern Instruments)
in two different forms. In the first experiment,
polystyrene was measured in diffuse reflectance by finely milling the bulk material and
compressing it into a pellet of similar dimensions to that of a pharmaceutical tablet. In
the second experiment, the same sample
was measured, but an additional polystyrene window of the same material and of
known thickness was placed in the same
optical path. This has the effect of adding a
known pathlength of polystyrene to the second diffuse reflectance spectrum. The contribution of the transmittance spectrum to
the total spectrum can be calculated from
the difference of the two.
Here, we have made the assumption
that the density (concentration) is approximately the same for both the transmission
and diffuse reflectance samples. Since the
window has a known thickness of 1.01 mm,
the effective pathlength as a function of
wavelength can be calculated for the purely
diffuse reflectance measurements using the
following equation:
⎤
⎡
ADR
lDR ( λ ) = ⎢
⎥ ( λ ) × lT
⎢⎣ A TDR − ADR ⎥⎦
where lDR and lT are the diffuse reflectance
and the transmission pathlengths, respectively, and ADR and ATDR are the diffuse
reflectance and the combined transmission/
diffuse reflectance measurement absorbance values, respectively. Multiple absorption bands (spectra shown in Figure 2) were
measured relative to a baseline offset at
1310 nm and used to determine the effective pathlength as a function of wavelength.
As expected, the results, shown in Figure
3 illustrate a trend of decreasing pathlength
with increasing wavelength from ~1.7 mm
at 1210 nm to ~0.52 mm at 2360 nm.
Converting effective
pathlength to sampling
volume
The next step is to correlate the effective
pathlength of the NIR radiation to a sampling volume. A random walk model is used
a r ti c le s
1
Diffuse Reflectance + Transmission
Diffuse Reflectance
Absorbance
0.8
0.6
0.4
0.2
0
1200
1400
1600
1800
2000
2200
2400
Wavelength (nm)
Figure 2. NIR absorption spectra of polystyrene.
to estimate the volume sampled by a typical photon.4 This model assumes that an
object, here a photon, travels a set distance
and then changes directions randomly. This
process is repeated many times before the
photon ultimately leaves the sample and is
measured by the detector.
The following equation is used to calculate the standard deviation, σ, of the displacement of the photon from the origin:
σ2 =
t 2
ε
δt
where t is the time elapsed since the start of
the random walk, ε is the size of each step
taken before a change in direction, and δt
is the time elapsed between two successive steps. In this model, the step size is
assumed to be equal to the wavelength of
light. This is likely to be an underestimation
of the scattering length, leading to an overestimation for the highest attainable spatial resolution predicted resolution > actual
resolution).
The pathlength calculated from Beer’s
law and the transmission results along with
the speed of light provide the value for t,
the time it takes the photon to enter, interact and leave the sample. The wavelength
of light, along with the speed of light provide
the value for δt, as discussed earlier.
For the polystyrene “tablet” the calculated σ values vary as a function of wavelength from 46 µm at 1210 nm to 31 µm at
2360 nm. Since these numbers correspond
to a sampling radius from which ~68% of the
measured spectral intensity imaged onto a
single detector pixel originates, it therefore
suggests that the upper limit of the spatial
Figure 3. Effective diffuse reflectance pathlength as a function of wavelength.
resolution is a sphere with minimum diameters between 92 µm and 62 µm.
The Nyquist criterion states that the sampling interval should be at least two measurements (2 pixels) for each spatial resolution
element probed. Therefore, using the lower
limit of 62 µm, it implies that an appropriate sampling frequency (magnification) for a
NIR chemical imaging system should not be
significantly less than 31 µm.
Determining effective
pathlength in a
pharmaceutical tablet
Pharmaceutical products such as tablets
are typically imaged in diffuse reflectance
and in order to estimate the spatial resolution attainable for this type of sample, a similar experiment was repeated. In this case,
the polystyrene tablet was replaced with a
commercial aspirin tablet. The rationale for
this was that both polystyrene and aspirin
contain similar aromatic groups, and for
the purposes of these calculations we have
assumed that the extinction coefficients
and “concentrations” for the two samples
are the same. For polystyrene, the overtone
of the aromatic CH stretch is observed at
1680 nm compared to 1660 nm for aspirin.
The transmission experiment absorbance
value for the polystyrene plastic was calculated by taking the difference between the
Figure 4. Aspirin (red) and Aspirin + polystyrene (black) NIR spectra.
Vol. 18 No. 6 September 2007
7
ar ti c l es
absorbance values at 1680 nm in the diffuse reflectance and the diffuse reflectance/
transmission experiments. The calculated
absorbance intensity values at 1680 nm
for polystyrene and at 1660 nm for aspirin
are 0.54 and 0.42, respectively. Assuming
ε to be the same, an effective pathlength of
780 µm is calculated for aspirin at 1660 nm,
resulting in a predicted sampling volume
equivalent to a sphere of 72 µm in diameter.
Within the limits of the assumptions made
in this article, the results are similar to those
obtained for the polystyrene samples.
Effect on spatial resolution
What this means in terms of practical spatial
resolution is that, depending on wavelength,
~70% of the photons exiting the sample at
any given point in a “typical” diffuse reflectance measurement will have interacted with
8
Vol. 18 No. 6 September 2007
a spherical region with a diameter of ~90–
60 µm. Several factors such as the selected
scattering lengths and the use of a single
standard deviation suggests that mixture
spectra will be observed over areas and
volumes significantly larger than this, and
spatial resolution is degraded accordingly.
Further, while optics can be constructed
resulting in magnifications of 10 µm/pixel
or less, the usefulness of this is questionable due to the optical scattering effects
described above. Increasing the magnification beyond the inherent spatial resolution
only accomplishes a decrease in the area
sampled in any given time. The values calculated in this study are relatively consistent
with previous estimates.2,3
References
1. E.N. Lewis, J.W. Schoppelrei, E. Lee and L.H.
Kidder, “NIR Chemical Imaging as a Process
Analytical Tool”, in Process Analytical Technology, Ed by K. Bakeev. Blackwell Publishing, Ch. 7, p. 187 (2005).
2. F.C. Clarke, S.V. Hammond, R.D. Jei and
A.C. Moffat, “Determination of the Information
Depth and Sample Size for the Analysis of
Pharmaceutical Materials Using Reflectance
Near-Infrared Microscopy”, Appl. Spectrosc.
56(11), 1475 (2002).
3. R. Spragg, T. Locke, R. Hoult and J. Sellors,
“Microscopic Chemical Imaging in the NIR
Region”, in Proceedings of the 11th International Conference NIR Near Infrared Spectroscopy, Ed by A.M.C. Davies and A. Garrido-Varo. NIR Publications, Chichester, UK
(2003).
4. B.D. Hughes, Random Walks and Random Environments. Oxford University Press
(1996).
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