Optical coherence tomography (OCT) has becomea well-established technique for obtaining high-resolution structural images of biological and othersemitransparent tissues.1,2 In recent years severalgroups have worked on expanding OCT to functionalimaging. Promising results have been reported forDoppler f low imaging,3 polarization-sensitive OCT,4
and spectroscopic OCT.5,6
Our purpose in this work is to develop a methodfor monitoring the concentration of photosensitizersin photodynamic therapy (PDT). Apart from in vivof luorescence monitoring, such methods are lacking,and the photosensitizer part of in vivo PDT dosimetryis still in its infancy.7 The need for in situ non-invasive monitoring of drug concentration rules outmethods that measure diffusion as a steady-statemolecular mobility as well as methods based on specialmeasurement or sample geometries.8 Attempts havebeen made to determine photosensitizer uptake spec-troscopically based on a photon diffusion model9 –11 re-lated to the models used for light dosimetry in PDT.12,13
However, such methods give poor spatial resolution,with a probed volume of the order of 1 cm3. OCTand confocal microscopy can provide linear resolutionthat is a thousand times better. Confocal-basedf luorescence recovery after photobleaching is a usefultechnique for f luorescence-based diffusion measure-ments.14 OCT does not rely on f luorescence and givesbetter depth penetration than confocal microscopy asa result of its higher selectivity of detected photons.
As a first step toward the goal of noninvasiveconcentration monitoring by OCT, we have tested theability of low-coherence interferometry (LCI) to mea-sure the diffusion of a PDT-related drug in 1.5% agargel. Solute diffusion in gels represents a well-studiedmodel system8,15 for which independent measurement
0146-9592/03/141215-03$15.00/0
results are available. To our knowledge our work isthe first report of measuring diffusion coefficientsby LCI.
The attenuation of light propagating in an absorbingand scattering medium is governed by the total attenu-ation coeff icient of the medium, mtl mal 1 msl, wheremal and msl are the absorption and scattering coeffi-cients of the medium, respectively, which generally arefunctions of wavelength, position, and time. Our ex-perimental system is an absorbing dye diffusing intoa scattering gel. Assuming one-dimensional diffusionand zero dye scattering, we can write
mtlz, t mt, gellz, t 1 ealCz, t , (1)
where mt, gellz, t is the attenuation coeff icient of thegel at wavelength l, Cz, t is the dye concentration asa function of depth and time, and eal is the specif icabsorption coefficient of the dye at wavelength l.
Diffusion of the dye into the gel is described by thediffusion equation
≠Cz, t≠t
=2DCz, t , (2)
where D is the diffusion coefficient. For a model inwhich the dye at time t 0 is confined to a layer ofinfinitesimal thickness at z 0, the dye concentrationis given by the one-dimensional delta-source solution16
Cz, t M
2Sp
pDtexp
µ2
z2
4Dt
∂, (3)
where M is the total amount of dye deposited on thetop surface area S of the gel at t 0.
The intensity of single-scattered light returning fromdepth z in an absorbing and scattering medium will, atthe surface of the sample, be5
Is, lz, t Is0,lRlz, texp∑
22Z z
0mtlz0, t dz0
∏, (4)
where Is0,l is the intensity incident on the sample andRlz, t is the ref lectance at depth z. After detectionand signal processing, the recorded amplitude Alz, tfor a LCI A-scan of the gel and dye sample is given by
Alz, t jrlzjGlexpΩ2
∑mt, gellz
1 eal
Z z
0Cz0, t dz0
∏æ. (5)
Alz, t is the envelope of the interference signal andis proportional to the square root of Islz, t in Eq. (4).Gl is proportional to the intensity of the light incidentin the interferometer and gain factors in the detectionsystem. For our simple model system the field ref lec-tivity of the sample, rl, is independent of z. Takingthe logarithm of Eq. (5) and defining KrGl
lnjrljGl,we get
lnAlz, t KrGl2 mt, gellz
2 eal
Z z
0Cz0, t dz0 . (6)
Equations (3) and (6) constitute the model that weuse for the interferometer signal. The diffusioncoeff icient D is determined by fitting of this model toexperimental data.
For the measurements we use a bulk Michelsoninterferometer, as shown in Fig. 1, with wavelengthmultiplexing of two superluminescent diodes. Thecenter wavelengths are 675 and 805 nm, with 0.4 and0.8 mW of power and 10- and 18-nm spectral FWHMs,respectively. Demultiplexing is carried out by a ruledgold grating, and the amplitude of the interferometersignal is obtained at both wavelengths. The systemhas a dynamic range of 90 dB.
We have studied the diffusion of aluminumphthalocyanine tetrasulfonate chloride dye (AlPcS834,Porphyrin Products, Inc.) in 1.5% agar gel. The dye’smolecular weight is MW 895.19 Da, and its absorp-tion coeff icient is 3 orders of magnitude larger at675 nm than at 805 nm. With our interferometerwe measured the attenuation coefficient at 675 nmof 1.5% agar without dye to be mt, gel675 0.06 mm21,giving a mean free path for light–agar interactions of1.7 cm. This is three times the maximum scanningdepth of our instrument and ensures that we can usea single-scattering model for light propagation andattenuation. Backscattering from the agar gel alonegives a LCI signal 25 dB above the noise level ofour instrument. The experiments were performedon phantoms with constant scattering, and only thesignal at the 675-nm wavelength was used to de-termine the diffusion coeff icient. On more realistic
tissue phantoms, e.g., with depth-dependent scatter-ing, we will need to use the signal recorded at 805 nmto correct for scattering.
Agar gels were prepared in 10-mm-deep cylindricalcuvettes with an inner diameter of 18 mm. A con-trolled amount of dissolved dye was deposited onto thegel as evenly as possible. The cuvette was then cov-ered with a glass slide and placed in the sample arm ofthe interferometer. The LCI amplitude was recordedas a function of depth and time for the wavelength of675 nm for ten agar samples. For each sample, mea-surement started immediately after deposition of thedye, and a series of 40 A-scans was acquired over aperiod of 2.5 min. The sample was displaced 10 mmtransversely between A-scans to allow speckle averag-ing. To study the time development of the diffusion,we also acquired data series from the same positionon the sample at four later time points. Thus a totalof f ive data series were recorded during 40 min afterdye deposition for each of the ten agar samples. Thedepth information was converted from optical depthto geometrical depth z with n 1.34 as the refrac-tive index. Figure 2(a) shows images of the logarith-mic amplitude for one sample plotted as a functionof depth and time, with t and z in the plot corre-sponding to t and z in Eq. (3). The black areas in theimage correspond to time periods in which no data wereacquired. Shortly after deposition the backscatteredintensity falls rapidly with imaging depth because of ahigh concentration of dye near the gel–glass interface.Later diffusion of the dye into the gel results in a lowerconcentration and a slower decrease in backscatteredintensity with depth.
To obtain speckle-averaged data suitable for modelfitting, we averaged each scan series transversely andtook it to represent the average time ti of the scanseries. The model in Eqs. (3) and (6) was fitted tothe logarithm of the averaged amplitude at 675 nm,with D, M , and K as fitting parameters. Parameterfitting was done in Matlab with a least-squaresalgorithm. Figure 2(b) shows the experimental data(jagged curves) along with the fitted model (smoothcurves) for one agar sample. Results from tensamples gave D 2.5 6 0.2 3 10210 m2s as the 95%
Fig. 1. Setup for the wavelength-multiplexed interferome-ter. BS, beam splitter; PC, personal computer.
Fig. 2. Experimental data for one of the ten samples.(a) Gray-scale image showing the logarithmic ampli-tude of f ive series of A-scans recorded over the same0.4 mm 3 1.0 mm cross-sectional area of the sample. Theabscissa represents the time t after dye deposition, and theordinate axis is the geometrical depth in the sample, wherez 0 is the agar–glass interface. (b) Speckle-averagedamplitude as a function of depth for the f ive images in(a). The smooth curves represent the least-squares-f ittedmodel of Eq. (6). The ref lection from the agar–glassinterface seen as an area of high ref lectivity at z 0 in(a) was removed from the data before parameter fitting.The mean time for each of the images is t1 (bottom curve)to t5 (top curve), and the data sample rate in depth is1340 samplesmm.
confidence interval for the diffusion coeff icient. Thisresult is in agreement with our preliminary measure-ments,17 which gave D 2.9 6 0.6 3 10210 m2s, themean values not being statistically different at a 5%significance level.
Under measurement conditions similar to ours(neutral pH, 1.5% agar gel), Lead et al.18 measuredthe diffusion coefficient of an 860-Da humic acid(SRFA) by use of f luorescence correlation spectroscopyand by use of classical diffusion chambers. Theirresults were 2.5 6 0.1 and 2.2 6 0.2 in units of10210 m2s for the two methods, respectively, which isin good agreement with our results for a similar-sizedmolecule.
We tested the robustness of the method for estimat-ing D by investigating how the standard deviation inD, sD , varied with the number of scan series usedfor model f itting and the number of A-scans used forspeckle averaging. Reduced data sets will facilitatemeasurement of more-rapid diffusion processes. With
only one series the smallest value of sD was observedfor the series acquired 5–7 min after dye deposition,indicating that there is an optimal time window fordata acquisition, probably related to the diffusion rate,signal-to-noise ratio, and length of the depth scans.Reducing the number of A-scans in each series to 4resulted in a 30% increase in sD caused by increasedspeckle noise, whereas no change was observed for16 scans.
In conclusion, we have applied LCI to measure thediffusion coefficient of a phthalocyanine dye in agargel as a f irst step toward in situ concentration moni-toring. We plan to extend the study to more-realisticmodels of tissue, including depth-dependent scatteringand ref lectivity. We will then need simultaneous mea-surements at the two wavelengths.
This work was supported by the Research Coun-cil of Norway. Trude Støren’s e-mail address [email protected].
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