2.1. Phantoms

A set of five phantom materials were tested:

1.

Polystyrene microspheres in aqueous agarose gel.The sphere diameter was 100-nm at a volume fraction of 2%. The refractive indices at 488 nm were nspheres = 1.599 for spheres and nwater = 1.336 for the aqueous gel (98% water). The gel was held between a 1-mm-thick glass slide and a 120-μm-thick coverslip. One location on phantom was tested, since such gels are routinely measured in our lab.

2.

Hard polyurethane phantom.The phantom was obtained from INO, Inc., Canada, and is called Hard Biomimic phantom [

9]. See
. Three locations of the phantom were tested, but the results were very consistent for each site.

3.

Soft polyurethane phantom.The phantom was obtained from INO, Inc., Canada, and is called Soft Biomimic phantom. See . Again, three locations were tested.

4.

SpectralonTM, 99% reflectance standard.The reflectance standard was obtained from LabSphere, Inc. (New Hampshire, USA), and is now available from Pro-Lite Technology, Inc. See . One location tested.

5.

SpectralonTM, 75% reflectance standard.The reflectance standard was obtained from LabSphere, Inc. (New Hampshire, USA), and is now available from Pro-Lite Technology, Inc. See . One location tested.

2.2. Confocal reflectance microscope

The confocal reflectance scanning laser microscope (rCSLM), built in our laboratory as an inverted microscope, has been used in previous studies [

1–

3,

5,

6]. An argon-ion laser delivered ~10 mW of 488-nm wavelength to the microscope objective lens. The objective lens (NA = 0.90, water-dripping lens, LUMPlanFL, Olympus America, Melville, New York) was water-coupled to the phantoms. For the microsphere gel, the microscope was water-coupled to the coverslip.
shows the basic design. Lateral scanning was implemented by

*x*- and

*y-* galvo scanning mirrors (RS-15, Nutfield Technology Inc., Windham, New Hampshire), yielding 512 x 526 pixels of equal 0.312 μm size. Axial z-axis translation of the focus was achieved by translating the sample using a motorized scanning stage (LS50A, Applied Scientific Instrumentation, Eugene, Oregon), yielding 1-μm axial stepsizes in the axial region of interest. However, to achieve a broader axial range of imaging, the axial stepsizes were increased to 5 or 10 μm at positions above and below this central region of 1-μm stepsizes. The detection arm was a lens/pinhole/photomultiplier-tube assembly (PMT: 5773-01, Hamamatsu Photonics, Japan). Scanning and detection were controlled by a data acquisition board (6062E, National Instruments, Austin, TX) and custom software developed using Labview

^{TM}. Image reconstruction and analysis were done using MATLAB (Mathworks Inc., Natick, Massachusetts).

2.3. Raw data acquisition

shows examples of the raw images of reflectance for the phantoms, shown as log_{10}(V(z,x)) where V is the detector voltage. The abcissa, x, is the lateral position of the phantom. The ordinate, z, is the apparent depth of the focal volume equal to the difference between the focal length (FL) and the distance (h) between the objective lens and the phantom surface.

2.4. Calibration

shows the calibration of the system. The glass-water(gel) interface of the microsphere-water(gel) phantom was imaged to yield a peak voltage V_{gw} = 5.14 V. The expected reflectance from this interface was r_{gw} = ((n_{water}-n_{glass})/(n_{water}+n_{glass}))^{2} = 0.00427, where n_{glass} = 1.522. Then a calibration factor was calculated: *calib* = r_{gw}/V_{gw} = 1.204x10^{-4} [1/V]. Thereafter, any measurement V was multiplied by *calib* to yield the reflectance R, which is the fraction of light delivered by the microscope that is returned into the microscope for detection.

To check the calibration, the axial profile, R(z_{f}), for the polystyrene microsphere gel phantom was analyzed to fit the expression

As the distance height (h) of the lens above the surface of the phantom was changed, the apparent depth position of the focus varied as z = FL – h, where FL is the focal length of the lens. When h = FL, the focus is at the phantom surface. As h was decreased, z moved into the tissue. However the true position of the focus, z_{f}, increased as

where D

_{glass} is the thickness of the glass coverslip if in place (if no glass, D

_{glass} = 0). The parameter

z

_{f}/

z = tan(θ

_{1})/tan(θ

_{2}), where θ

_{1} = a sin(NA/n

_{water}) and θ

_{2} = a sin(NA/n

_{phantom}). The factor NA/n

_{phantom} is referred to here as the

*effective numerical aperture*. For the aqueous gel, the value of

z

_{f}/

z was 1.00. For the polyurethane and Spectralon

^{TM} phantoms, the value of

z

_{f}/

z was 1.20, based on an assumed value of 1.49 for n

_{phantom}. Hence, the original data versus z was converted to data versus z

_{f} before subsequent analysis.

2.5. Analysis

The behavior of R(z_{f}) depends on the parameters ρ and μ, which are described as

where Δz is the standard axial resolution, Δz = 1.4λ/NA

^{2}, where NA = sin(θ

_{1/2})n

_{phantom} with θ

_{1/2} equal to the half angle of light delivery within the phantom and n

_{phantom} is the refractive index of the phantom [

2,

3]. The value of n

_{phantom} for the polyurethane and Spectralon

^{TM} phantoms was assumed to be 1.49.

In Eq. (2), μ refers to the attenuation of light as photons move to/from the focus. When scattering is very forward directed, it is possible for photons to still reach the focus despite multiple scattering. The function

*a*(

*g*) varies from 1 to 0 as

*g* varies from 0 to 1, i.e., from isotropic scattering to forward-directed scattering. The function was determined by Monte Carlo simulations of focused light penetration to a focus at z

_{f} for varying values of μ

_{s} at a given

*g*. The change in fluence rate at the focus versus value of μ

_{s}, or F(μ

_{s}) at constant g, was fit by

Eqs. (1)–

(3) to specify the value of

*a*. Repeating for different values of

*g* yielded the function

*a*(

*g*), which can be described as [

2,

3]

The effect of absorption, μ_{a}, is negligible unless working with a very strongly absorbing material. The factor 2 accounts for the round-trip in/out path of collected photons. The factor G is a geometry factor that accounts for the extra pathlength of photons when a high NA objective lens is used. The value of G depends on the NA of the lens, and is approximated by

where value θ

_{2} is the maximum half-angle of collection at the phantom surface, which depends on the NA of the lens. The factor E

_{Gaussian}(θ) = exp(–(θ/θ

_{2})

^{2}) is a Gaussian function that describes the angular dependence of light entering the phantom. The assumption here is that the ±1/e portion of the laser beam was filling the back pupil of the objective lens and reaching the phantom. This assumption is easily modified in Eq. (6) to match a particular experimental setup. The factor T(θ) is the transport to the focus from a surface entry point at an angle θ with respect to the central z axis. Attenuation of T(θ) by tissue scattering and absorption decreases the contribution from light at larger angles of entry, which slightly decreases the average pathlength, Gz

_{f}, of photons reaching the focus. Equation (6) is more fully discussed in [

2]. In this experiment, G = 1.132.

The function *b*(*g*) describes the fraction of photons scattered within the axial Δz extent of the focus which are scattered back into the solid angle of collection of the objective lens. The function *b*(*g*) is approximated by the integral over all angles of backscatter that are within the collection angle of the objective lens:

where the scattering function p(θ) indicates the deflection of photons from their incident forward direction, π is the direct backscatter angle in radians and θ_{2} is the maximum half-angle of collection by the lens in radians. The function p(θ) was approximated by the Henyey-Greenstein function:

For the conditions of this experiment, b(g) ≈ 0.203exp(-1.716g) - 0.077exp(-0.744g), which equals 0.132 at g = 0, drops by 50% at g = 0.262 and drops by 90% at g = 0.732. Using Mie theory to generate p(θ) yields a similar *b*(*g*) as the Henyey-Greenstein function, except when the spheres are large and scattering is very forward directed (not shown).

The effective solid angle of collection by the objective lens was also dependent on the refractive index of the phantom. The θ

_{2} is the maximum angle of collection by the lens, and was used in the calculation of

*b*(

*g*) in

Eq. (3).

The functions *a*(*g*), *b*(*g*) and *G*(*NA,g*) continue to be topics of ongoing investigation.

shows an example analysis. A superficial region (5-50 μm below the surface) was used for fitting, beyond the effects of the front surface reflectance and before diffuse light begins to contaminate the signal. The noise floor due to diffuse light reflectance escaping within the solid angle eventually collected by the detector pinhole becomes important when the focus is located at depths beyond the transport mean free path, 1/(μ_{s}(1–g)). Hence, useful measurements are restricted to the superficial layer.

plots the mean μ versus mean ρ from on a log-log plot. Superimposed on this plot is a grid of iso-

*g* lines and iso-μ

_{s} lines, based on

Eqs. (3) and

(4). This grid allows interpretation of the μ and ρ data in terms of the optical properties μ

_{s} and

*g.* The experimental data point (red circle) indicates μ

_{s} = 57.7 cm

^{-1},

*g* = 0.072, μ = 130 cm

^{-1}, ρ = 9.2x10

^{-4}. Also shown is the predicted data point using Mie theory (black diamond), which has values of μ

_{s.MIE} = 58.2 cm

^{-1},

*g*_{MIE} = 0.129, μ

_{MIE} = 131 cm

^{-1}, ρ

_{MIE} = 8.2x10

^{-4}. Work continues on testing the accuracy of the first-order theory (

Eqs. (3),

(4)) and on experimental methods for preparing microsphere gels for calibration.