The Non-Invasive
Measure of D-Glucose
in the Ocular Aqueous Humor using
Stimulated Raman Spectroscopy
Randall V. Tarr*
and Paul G. Steffes
School of Electrical and Computer Engineering Georgia Institute of Technology
Atlanta, GA 30332-0250
* Current address: Siemens Medical Systems Incorporated 4040 Nelson Avenue
Concord, CA 94520
I. INTRODUCTION
The efficacy of home
blood glucose monitoring in the treatment and maintenance of insulin dependent diabetic
patients is well established [1,2]. However, the necessity of drawing capillary blood for
each sample, and well as the time and cost required for each measurement limits the
frequency of such measurements to 5-10 per day, even for highly motivated patients. The
increasing availability of insulin infusion pumps, combined with the availability of
fast-acting insulin analogs, such as lispro, strongly motivates development of sensors,
either implantable or non-invasive, providing frequent or continuous blood glucose
measurement, which would be useful in patient-based blood glucose control, or in a
closed-loop automated insulin delivery system. Due to the enormous potential benefits of
tight blood glucose control in diabetic patients, many different techniques have been
tried in an attempt to non-invasively measure blood glucose concentration. The common
thread among nearly all non-invasive techniques is the use of electromagnetic radiation to
measure directly, or to infer indirectly, the blood glucose concentration. The system
developed at Georgia Tech [3] uses stimulated Raman spectroscopy to measure glucose
concentration in the aqueous ocular humor, so as to indirectly infer blood glucose
concentration.
II. APPROACH
Although many
attempts at non-invasive glucose measurement have directed electromagnetic radiation
through the skin to measure blood glucose directly, the aqueous ocular humor provides a
more promising point of measurement. The blood is a very difficult medium in which to
spectroscopically measure very small concentrations of minor constituents such as glucose,
principally because of the spectral dominance of the hemoglobin. Additionally, variations
in skin transmissivity can be problematic. The ocular aqueous humor is a spectroscopically
simpler medium, since there are fewer molecular species present which might interfere with
the measurement of glucose concentration, and the cornea is essentially transparent to
both near-infrared and visible wavelengths. Direct chemical analysis has shown that the
glucose concentration in the ocular aqueous humor is proportional to the blood glucose
concentration and tracks variations in blood glucose concentration quite promptly [4], so
that blood glucose concentration can be inferred from the measurement of the glucose
concentration in the aqueous humor.
III. USE OF STIMULATED
RAMAN SPECTROSCOPY
A. Concept
March et al. [5,6] proposed two
different approaches for the measurement of glucose in the ocular aqueous humor involving
measurement of effects of glucose on refractive index and measurement of the polarization
effects, or optical activity, of the glucose present in the aqueous humor. Unfortunately,
other refractive and optically active compounds in the aqueous humor made it nearly
impossible to unambiguously measure glucose concentration in these ways [7,8]. An
alternative approach is to use stimulated Raman spectroscopy for measurement of glucose in
the ocular aqueous humor. As in most forms of spectroscopy, the Raman spectrum of glucose
makes it spectrally differentiable from other constituents. (See Figure 1.) The use of stimulated
Raman spectroscopy makes possible detection of a single predominant feature in the Raman
spectrum using relatively low-cost fixed wavelength lasers. Additional benefits include
increased resistance to interference from luminescence and florescence, and resolution
which is limited only by the laser linewidths. (See, e.g., [9].) Sensing of ocular glucose
is accomplished by combining the two laser wavelengths and then passing them tangentially
through the front of the eye. (See Figure 2.) A detection system then measures the amount
of power translated from one wavelength to the other, which is related to glucose
concentration.
B. Sensitivity
For stimulated Raman
gain spectroscopy, the Raman-active medium is excited with a pump and a probe laser whose
frequencies are separated by a Stoke’s shift of the Raman-active medium. In this
case, the stimulated Raman effect essentially converts a portion of the pump laser power
to the proble laser wavelength. The stimulated Raman effect requires no phase matching
between the two laser beams, only that they be spatially overlapping in the Raman-active
medium [9]. When the frequency difference of the two lasers equals the vibrational
frequency of the Raman-active mode, a non-linear optical interaction occurs between the
optical field and the medium. The strength of the nonlinear optical interaction depends
upon the third order nonlinear susceptibility, c(ws), of the medium. The result of this interaction is a coherent
increase in the irradiance of the Stoke’s (probe) beam and a corresponding decrease
in the irradiance of the pump beam.
A semiclassical model
can be used to determine the gain, gs, of the Stoke’s beam in which the
electromagnetic fields are treated classically, and the medium is treated guantum
mechanically [10]. Consider the specific case of stimulated Raman gain spectroscopy, in
which the pump wave is at frequency wp and the Stoke’s wave is at frequency ws The
growth of the Stoke’s wave is governed by
(1)
where ns is the refractive index at the Stoke’s wavelength.
Assuming plane wave solutions to the wave equation, the electric field at the Stoke’s
frequency, ws is
(2)
where 
is
a vector constant and ks is the propagation constant. It has been
determined that for isotropic media, such as liquids, there are only 21 non-zero elements
in the third order susceptibility tensor and only three of these are independent [11]. For
a liquid medium illuminated by two spatially overlapping fields with parallel polarization
of the vector fields, the susceptibility tensor reduces to a scalar quantity [121. Thus
the third order polarization of the medium at the Stoke’s frequency, 
, is
(3)
where Ep represents the vector field at the pump frequency wp, Es represents the Stoke’s field at frequency ws and c(3)(ws) is the
third order susceptibility constant of the medium. Substituting for Es & Ps, in
equation (1) and time averaging gives
(4)
Solving for the
propagation constant, ks,
(5)
where 
is the
pump laser irradiance. Using the binomial theorem leads to the approximate relation
(6)
In the vicinity of a
resonance the susceptibility can be broken down into its component parts,
is
the real part of the resonant contribution, 
is the imaginary part of
the resonant contribution, and cNR is a term accounting for the nonresonant
background [12]. Using this relation for 
and defining 
the expression for ks,
can be separated into its real and imaginary parts
Now this expression
for ks, can be substituted back into equation (2)
to yield
(8)
Now the irradiance of
the Stoke’s field, Is(z), is of the form
(9)
where Is(0) is the initial irradiance of the Stoke’s wave and
the stimulated Raman gain, gs, is given by
(10)
From a quantum
mechanical treatment of a liquid medium, the imaginary part of the third order
susceptibility constant 
can be expressed in terms of the spontaneous Raman
scattering cross section, 
and the Raman transition line width G as
[12]
(11)
where D is the difference in probability of occupation of two vibrational
levels, N is the density of molecules, and hmks is Planck’s constant in mks units. Substituting into the
expression for the stimulated Raman gain yields
(12)
and at the resonance
peak wp-ws=w0 which leads to a peak stimulated Raman gain of
(13)
Unfortunately, no
measurement of the nonlinear susceptibility of D-glucose has been reported. However, an
estimate of the imaginary part of the third order susceptibility for D-glucose may be
obtained from the spontaneous Raman spectra of a 30% D-glucose solution shown in Figure 1.
From Equation (11) the imaginary part of the third order susceptibility may be determined
from the spontaneous Raman cross section, 
, and the linewidth, G for
the 518 cm-1 resonance of D-glucose. The spontaneous
Raman cross section for the 518 cm-1
resonance can be estimated from the spontaneous Raman cross section of the 3655 cm-1 resonance of water which has been measured to be (9±2) x 10-30 CM2/sr.mol [13]. From Figure 1 it is evident that the linewidths of
these two resonance peaks differ. To account for this difference between the linewidths of
these two resonance peaks, the peak Raman scattering cross section for each resonance is
determined. Assuming a Lorentzian shape for each Raman line, the peak Raman scattering
cross section, 
can be related to the spontaneous (integrated) Raman cross section by [13].
(14)
where G is the resonance linewidth in Hertz. By fitting a Lorentzian line
shape to the 3655 cm-1 resonance of water, the linewidth in
wavenumbers, is found to be approximately 200 cm-1 and the corresponding peak Raman scattering cross section, is 3 x 10-32 cm3/sr.mol.
from Equation (14). From the spontaneous Raman spectrum for the D-glucose solution-shown
in Figure 1, the peak Raman scattering cross section for the 518 cm-1 resonance of D-glucose is
(15)
Abundance
normalization of the peak Raman cross section yields
(16)
The linewidth of an
equivalent Lorentzian line which has been fit to the 518 cm-1 resonance Raman cross section for the 518 cm-1 resonance of D-glucose is calculated to be
(17)
The imaginary part of
the third order susceptibility for the 518 cm-1
resonance of D-glucose may be estimated by using the calculated value for the spontaneous
Raman scattering cross section and the linewidth with Equation (11). Specifically,
substituting the following values:
and the value for the
Raman scattering cross section into Equation (11) yields a value for the imaginary part of
the third order susceptibility for D-glucose,
of
(18)
Figure 3 shows the stimulated Raman signal gain vs. the D-glucose
concentration with various laser beam diameters. These gain curves have been calculated
using Equation (10) with a pump laser power of 10mW and a Stoke’s wavelength of .85
mm. The smallest D-glucose concentration used in these calculations, h=.0001, is smaller than the minimum D-glucose concentration
present in the ocular aqueous humor [6, 7]. The optical power level of 10mW is
approximately equal to the optical power levels used in our hardware implementation shown
in Figure 4. Values on the gain curves can be related to a stimulated Raman signal
amplitude by
(19)
where z@1 cm in the ocular aqueous humor [4]. Upon substitution of the
appropriate values for D-glucose into Equation (13), we have
(20)
where h is the glucose concentration and r
represents the laser beam radius in centimeters. From these gain curves it is evident that
a very small beam diameter is necessary to produce stimulated Raman signal levels which
are measurable in the presence of noise. The noise level is determined by the amplitude
noise present on the probe laser and by the shot noise produced during photodetection.
D. Plane Wave
Propagation through the Ocular Aqueous Humor
In order to determine an optical path
laterally through the ocular aqueous humor, a plane wave analysis has been performed. The
eye is approximated by a sphere with a radius of curvature, r =7.42mm, comparable to that
of the cornea. The optical beam is approximated by a cylindrically symmetric array (a
bundle) of filament rays (plane waves). This constructions allows the intensity to vary
across the wavefront and the curvature of the dielectric interface to be modeled. The
optical path length through the sphere is constrained to be ~1 cm which is comparable to
that of the ocular aqueous human [14]. Figure 5 illustrates the path through the sphere by
a filament ray at the center of the bundle.
A computer model
has been constructed to simulate plane wave propagation through the ocular aqueous humor.
The program traces multiple filament rays which comprise an optical beam through a sphere
with a radius of curvature equal to 7.42 mm [14] which approximates the curvature of the
cornea. Figure 6 shows a cross section of the optical beam which illustrates the
quantization of an optical beam into multiple filament rays. The initial power
distribution among the filaments within the beam can be adjusted to model various
intensity profiles, such as Gaussian. Several simulations have been performed with
different beam diameters and initial intensity profiles. Table 1 gives the initial power
distributions among the filaments for both a uniform intensity profile and a Gaussian
intensity profile. During the computer simulation the optical power levels are monitored
in the regions shown in Figure 7. From these simulations it is evident that even though a
collimated beam is incident upon the sphere, the optical beam is focused to a point within
the sphere before exiting on the opposite side. This focusing effect is useful since the
simulated Raman signal is directly related to the irradiance of the optical beam within
the Raman-active medium. From Table 2 a comparison of the optical power levels in the defined regions may be made
for various optical beam diameters and initial intensity profiles.
E. Laser Power
Constraints
The total laser power
used in this implementation of the stimulated Raman effect is limited by the maximum
permissible exposure to the eye. The energy in the near-infrared wavelength band is highly
absorbed by the retina, but not by other structures within the eye [15]. Table 3 gives the
maximum permissible exposures for direct intrabeam viewing with various exposure times,
specifically for optical power at 800 nm wavelength. From the analysis in the previous
section with 15 mW incident power approximately 3 mW remains in the sphere. For exposure
times greater than ~0.3 sec, this energy exceeds the maximum permissible exposure levels
in Table 3. However, from the geometry of the opttical system, the optical power which
remains within the aqueous humor does not reach the retina since it is blocked by the
iris. Therefore, it may be permissible to safely exceed the established standards for the
maximum permissible exposure without ocular damage.
IV. EXPERIMENTS
Two test systems were developed to demonstrate the feasibility of
this measurement approach. The first was a fiber-optics based system wherein both the
filtering and combining system for the laser beams from the two solid-state laser sources,
and the detector system, were constructed from relatively low-cost fiber optic components,
with the test cell, representing the eye, being the only region of unguided wave
propagation. The second system was a free space optical system, which provided better
isolation between the pump and probe wavelengths in the detection system and avoided any
spurious Raman activity in the fibers. (See Figure 4.) With relatively low power
solid-state lasers, (probe power: 50 mW pump power: 100 mW) it has been possible to detect
a 0.5% concentration of glucose over a 10 cm long path in an aqueous solution. (See figure
8.) Since the sensitivity of such systems is related to the power and stability of the
lasers used, in addition to the spatial compactness of the beams, the use of higher
stability, higher power semiconductor lasers with more compact and uniform spatial
beamwidths should make possible measurement at the 0.01% level or better over paths of
less than 1 cm.
V. CONCLUSION
A technique has been
devised to non-invasively measure the D-glucose concentration in the ocular aqueous humor.
This approach is radically different from conventional invasive blood glucose measurement
techniques. A major benefit of this technique is that no wavelength tuning is required,
which allows the use of economical fixed wavelength semiconductor lasers. Using the
limited resources available, this concept has been experimentally demonstrated by the
construction and evaluation of two independent in-vitro measurement systems.
However, the sensitivity of the experimental systems must be improved in order to measure
the extremely small D-glucose concentrations present in the aqueous humor. In addition,
theoretical studies have been performed to determine the feasibility of the stimulated
Raman spectroscopy technique for in-vivo measurements in the ocular aqueous humor.
The results from these theoretical studies indicate that the stimulated Raman signal
amplitude levels produced from the very low D-Glucose concentrations present in the
aqueous humor should be detectable using solid-state electro-optical components.
VI. REFERENCES
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P.G. Steffes, “Non-invasive blood glucose measurement system and method using
stimulated Raman spectroscopy,” United States Patent #5,243,983 - September
14, 1993.
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