2.1.

Definition of Effective Fluorescence Collection Volume

The effective fluorescence detection volume for an optical fiber that is used to supply excitation light to, and collect fluorescent emission from, a region adjacent to its tip is determined by both its illumination and collection volumes. The illumination intensity distribution adjacent to the tip of a flat-cleaved fiber is assumed to be axisymmetric with respect to a cylindrical polar coordinate system $(r,z)$ , in which $z$ is coincident with the axis of the fiber and the radial coordinate $r$ is transverse to it. Following the analysis proposed by Zhu and Yappert,¹⁷ the illumination efficiency is therefore represented as $I_e(r,z)$ , where $0<I_e<1$ , and the fluorescence collection efficiency is similarly represented as $I_c(r,z)$ , where $0<I_c<1$ . Provided that illumination intensities are sufficiently low to avoid saturation, the fluorescence signal (dF) collected from a small volume (dV), at location $(r,z)$ , can be expressed as:

Hence, the signal depends on the product of illumination and collection fields. The product

The relative fluorescence signal

2.2.

Measurement of Fluorescence Collection Fields

Characterization of fiber response fields was performed by placing a fiber tip into the sample chamber (containing tissue or test solution) of a combined 2P/confocal laser scanning microscope (Zeiss LSM 410, Jena, Germany).²⁰ A mode-locked Titanium:Sapphire $(\mathrm{Ti}:\mathrm{S})$ laser (Mira 700, Coherent, Santa Clara, California) tuned to $750\mathrm{nm}$ provided excitation for 2P excitation of fluorophores or three-dimensionally resolved photorelease of caged fluoroscein. In the former cases, fluorescence emission was collected by the optical fiber under investigation, and measured with a photomultiplier tube (PMT) after passing through a bandpass emission filter (centered at $535\mathrm{nm}$ and 50-nm wide, Chroma, Brattleboro, Vermont), see Fig. 1 . The microscope was used in 2P mode to create points of fluorescence emission by focusing the light of the $\mathrm{Ti}:\mathrm{S}$ laser through a $10\times$ objective into the sample. Since the fluorochrome concentration within the sample was uniform and the excitation profile resulting from 2P excitation was small compared to the length scale over which the collection field changed (i.e., it approximated a delta function), the fluorescence incident on the PMT was proportional to the value of $I_c(r,z)$ when the 2P excitation spot was placed at the coordinates $(r,z)$ [see Eq. 3]. In other words, the collection field was sampled by creating point sources of fluorescence of constant intensity at specified locations. To reconstruct the axisymmetric collection field of a flat-cleaved fiber, it was sufficient to scan only in the plane containing the fiber axis. On the other hand, to characterize the asymmetric collection field of an angle-cleaved optrode,⁹ it was necessary to collect a stack of such images.

Fig. 1

Schematic diagram of the imaging system. The system consists of a 2P confocal microscope, a PMT with a selectable emission filter F and an external laser source. For two-photon excitation studies, the 2P laser in the confocal microscope was used to stimulate fluorophores in the focal volume adjacent to an optical fiber (see inset A), and fluorescent emission collected by the fiber was detected by the PMT. Laser scanning was used to reconstruct fluorescence collection fields. For flash photolysis measurements, the 2P laser in the confocal microscope was used to release caged fluorophores in the same focal volume. For studies of fiber illumination profiles, continuous illumination was routed through the fiber from the external laser source, and the time-course of fluorescent emission by the uncaged fluorophore was monitored by the confocal microscope. A similar flash photolysis approach was used to provide additional characterization of fiber collection profiles (see Sec. 2 for details).

The scan time for a single $512\times 512$ (sampling resolution $2.5\mu \mathrm{m}$ ) 2P image was typically $16\mathrm{s}$ , and the image signal-to-noise ratio (SNR) was improved by averaging over 16 images. For quantitative analysis, residual background was subtracted using control images recorded in the absence of 2P illumination.

2.2.1.

Measurement of collection fields with minimal fluorescence reabsorption

Although the preceeding technique yielded data with good spatial resolution, it is possible that fluorescent emission from fluorophores experiences reabsorption by surrounding fluorophore. Therefore, a variation of the technique described earlier was used to determine $I_C$ while minimizing attenuation of the fluorescence signal due to reabsorption by dye in the sample solution. A point source of fluorophore was provided by 2P excited flash photolysis of caged fluoroscein.²¹ Short bursts (200-ms duration) of light from the $\mathrm{Ti}:\mathrm{S}$ laser were focused into a sample containing $1.67\mathrm{mM}$ of caged fluoroscein using a $40\times 1.2$ numerical aperture (NA) water immersion objective. The full-width half-maximum (FWHM) of the point spread function (PSF) of photorelease was $\sim 0.5\mu \mathrm{m}$ in the $x\text{‐}y$ plane and $1.4\mu \mathrm{m}$ in the $z$ -axis direction (coinciding with the axis of the objective).²⁰ The released fluorophore was excited through the same objective using 488-nm light from the argon ion laser in the confocal microscope, which was aligned confocal with the stationary photolysis spot. A more detailed account of these methods is given elsewhere.²⁰ During and after photolysis, the fluorescence received by the fiber was recorded while a line through the photolysis spot was repeatedly scanned with 488-nm light. For quantitative analysis, the line scan image was averaged temporally ( $50\mathrm{ms}$ , after termination of the 2P release flash) and spatially (over a $10\text{‐}\mu \mathrm{m}$ -wide region—see also Fig. 2 ) to obtain the collection efficiency at the photolysis spot. Due to diffusion, the full width of the fluorescence distribution at half maximal intensity was $\sim 10\mu \mathrm{m}$ at the time of signal analysis (Fig. 2). While it was impractical to sample collection fields fully using 2P flash photolysis, this technique provides a means of characterizing their principal spatial features (see Sec. 3). The collection field adjacent to the fiber tip was characterized by performing 2P flash photolysis measurements at points 10 to $50\mu \mathrm{m}$ apart in the axial direction and at selected transverse distances from the fiber axis. Since the fluorochrome concentration was insignificant at sites other than the photolysis spot, fluorescence reabsorption within the sample was negligible.

Fig. 2

Schematic diagram illustrating the flash photolysis technique. The 2P laser was used to release caged fluorophores in the focal volume adjacent to the fiber, which were then stimulated (see Fig. 1). Continuous line scans were acquired transverse to the fiber axis at this location. The 2P flash duration was $200\mathrm{ms}$ , and the time-course of fluorescent emission for a typical such recording is shown at the center of the figure. Fluorescent emission at the focal point is shown in (a) as function of time, while (b) is a line scan acquired at the termination of the 2P flash. The relative fluorescence intensity at the point addressed by the 2P flash is estimated by integrating across the $10\mu \mathrm{m}\times 50\mathrm{ms}$ space-time window indicated by the white rectangle (see Sec. 2 for further details). Note that the traces shown in (a) and (b) have been averaged to improve signal quality.

2.3.

Determination of Illumination Fields

Fiber illumination fields were also determined by utilizing 2P excited flash photolysis of caged fluoroscein. The illumination field for excitation of the released fluorophore was provided for the fiber under investigation by coupling the light from the 488-nm line of an argon ion laser (N57-796, Edmund Optics Singapore PTE Ltd.) into the fiber entrance (see Fig. 1). Fluorescence emission from fluoroscein released at the stationary 2P photolysis spot was measured through the microscope objective using one of the PMTs in the confocal microscope. The intensity of the fluorescence signal measured directly after release of the caged fluorophore (see Fig. 2) was thus proportional to the illumination efficiency of the fiber at the location of the photolysis spot. This follows from Eq. 1, since $C(r,z)$ was considered to be a delta function and the collection efficiency of the recording objective was constant. The illumination profile of the fiber was reconstructed from sequential intensity measurements at different sites along the plane containing the fiber axis.

2.4.

Optical Fibers

The detection volumes of several different optical fiber-types were characterized. The flat-cleaved fibers used were: a single-mode (SM) fiber with core diameter $4.0\mu \mathrm{m}$ and NA of 0.12 (FS-SN-3224, 3M), multimode (MM) graded index fibers with core diameters of 50 and $105\mu \mathrm{m}$ and NA of 0.22 (AFS50/125Y, AFS105/125Y, Thorlabs, Inc.), and an MM step-index fiber with core diameter $100\mu \mathrm{m}$ and NA of 0.22 (F-MCB-T, Newport Corp.). A more complex optical probe designed for multisite intratissue fluorescence mapping⁹ was also characterized. This probe consists of angle-cleaved fibers (F-MCB-T, Newport Corp.) enclosed in a glass micropipette.

2.5.

Solution and Tissue Preparation

The probes were characterized in both nonscattering and scattering solutions, as well as in stained tissue. Nonscattering solution containing $500\mu \mathrm{M}∕\mathrm{L}$ rhodamine was used for most two-dimensional (2-D) confocal scanning studies, because the SNR was optimized at this fluorophore concentration. For flash photolysis studies, $1.67\mathrm{mM}∕\mathrm{L}$ of 10,000 MW 4,5-dimethoxy-2-nitrobenzyl (DMNB)-caged fluoroscein in dextran (Molecular Probes Corp.) were used. Scattering solutions were prepared by adding nonfluorescent $1\text{‐}\mu \mathrm{m}\text{‐}\mathrm{diam}$ microspheres at a concentration of $5.06\times {10}^9\mathrm{beads}∕\mathrm{mL}$ to these aqueous solutions, resulting in scattering properties that mimic brain tissue.²² In addition, 2-D confocal scanning studies were carried out for a flat-cleaved AFS105/125Y fiber in nonscattering solution at rhodamine concentrations ranging from 50 to $5000\mu \mathrm{M}∕\mathrm{L}$ to determine the effects of fluorophore concentration on fluorescence collection fields.

The tissue used in this study was taken from the hearts of 20-month-old Wistar Kyoto (WKY) rats. The rats were anaesthetized with halothane, decapitated, and the hearts were rapidly removed. Tissue slices ( $\sim 1\text{‐}\mathrm{mm}$ thick) were cut from the free wall of the left ventricle (LV), oblique to the epicardial surface, and stained with rhodamine $(1\mathrm{mM}∕\mathrm{L})$ for $24\mathrm{h}$ at $4°\mathrm{C}$ . Subsequently, fiber optic probes were inserted manually into the section thickness and 2P excitation studies were carried out. All procedures were approved by the University of Auckland Animal Ethics Committee.

2.6.

Flux Distribution Model for Illumination Field

One of the objectives of this study is to develop a model that can be used to estimate the illumination and effective fluorescence collection volumes in tissue for MM optical fibers of varying diameter and NA. In order to address this problem, we have set up a semiempirical model similar to that presented by Zhu and Yappert.¹⁷

The illumination intensity distribution at a given wavelength is represented as:

The illumination source function $E(r,z)$ is the convolution of a simple Gaussian distribution $S_1(r,z)$ , representing a single-mode illumination field, with a circular function $S_2(r,z)$ and is expressed as:

Fig. 3

Cross-sectional representations of the simple Gaussian function $S_1(r,z)$ and the spatial source term $S_2(r,z)$ that are convolved to form the illumination source function $E(r,z)$ . Note that functions are normalized to ensure flux conservation.

The function $S_2(r,z)$ describes the circular area over which the Gaussian function $S_1(r,z)$ is convolved (in the radial coordinate) to generate the radial distribution of illumination at each axial position, and it incorporates a scale factor that ensures that energy is conserved as specified in Eq. 7.

When ${\theta }_b=0$ , $S_2$ is independent of $z$ and $E$ corresponds to a distribution of essentially independent single modes originating at the face of the MM fiber. Under these circumstances, the predicted axial profile of the illumination intensity $E(r,z)$ is constant in the near-field, when the effective absorption coefficient ${\mu }_t=0$ . This does not match the fall in axial intensity in the near-field observed either in air or in nonscattering fluids (see Sec. 3), due, presumably, to redistribution of light in the radial direction that is not encompassed by the spread of a Gaussian function alone. To more closely match the fall of axial illumination in the near-field, we have allowed, as a first-order approximation, the radius of $S_2$ to vary linearly with $z$ . Note that, in our simulations, ${\theta }_b$ is relatively small in comparison with $\theta$ .

A similar model should also describe the fluorescence collection efficiency at a given emission wavelength. However, because fluorescent emission is distributed across a range of wavelengths, the fluorescent light collected from a point $(r,z)$ will be affected by wavelength variation in both the intensity of emission and the decay component $D\left(z\right)$ . To account for this, the collection efficiency $I_c$ should be evaluated as follows:

This model has been used to interpret and analyze experimental data and to investigate systematically the effects of fiber diameter and NA on both illumination and effective fluorescence collection volumes. Effective fluorescence collection volumes are estimated as the product of illumination and fluorescence collection fields.

The model has also been used to study the extent to which a wave of altered fluorescence emission propagating through the volume addressed by a fiber will be blurred by varying contributions of regions within the collection field. This process is simulated by allowing a plane-wave impulse to propagate at various angles and propagation speeds through the effective collection field defined by the model. At any time, the fluorescence signal, $dF(r,z)$ , is equal to the effective collection field [see Eq. 4] along the impulse wavefront and is zero elsewhere. The total fluorescence signal was estimated by integrating across the entire volume as indicated in Eq. 3. This process was repeated sequentially to reconstruct the time-course of the optical signal collected as the plane-wave impulse moves across the volume addressed by the fiber.

3.1.

Comparison of Illumination and Fluorescence Collection Profiles

Flash photolysis was used to characterize illumination and collection fields under identical experimental conditions for a flat-cleaved fiber (F-MCB-T) in nonscattering and scattering media. Illumination and collection profiles mapped out along the fiber axis using 2P flash photolysis are superimposed in Fig. 4a on the axial collection profile measured in nonscattering rhodamine solution $(100\mathrm{\mu }\mathrm{M}∕\mathrm{L})$ using 2P confocal microscopy. Note that these data were obtained with a 488-nm excitation source and that the fluorescence emission was collected at $535\mathrm{nm}$ . It is evident from Fig. 4a that there is no difference between axial illumination and collection profiles obtained with flash photolysis and that the axial collection data obtained with 2P confocal scanning in comparable conditions are consistent with these results.

Fig. 4

Comparison of illumination and fluorescence collection profiles for an F-MCB-T optical fiber in nonscattering fluid. (a) Axial illumination and fluorescence collection measurements: illumination $(◻)$ and collection $(◇)$ profiles obtained with 2P flash photolysis are compared with collection profile obtained with 2P confocal scanning (—). (b) Radial illumination and fluorescence collection measurements: illumination $(◻)$ profile obtained with 2P flash photolysis is compared with collection profile obtained with 2P confocal scanning (—) under comparable conditions.

The radial illumination profile in Fig. 4b, measured with 2P flash photolysis, exhibits a flat top over the central region and decays toward the periphery, almost exactly matching the corresponding collection profile obtained with 2P confocal scanning with the same fiber in nonscattering fluid. Results observed in scattering fluids were also similar to those presented here. SNR was relatively low in these experiments. This was due to decreased signal levels, but traces of incompletely caged fluorophore also increased the background noise.

3.2.

Fluorescence Collection Fields for Flat-Cleaved Optical Fibers

Fluorescence collection fields of optical probes were measured with 2P confocal microscopy as described earlier. A typical 2-D collection field along the axis of an MM flat-cleaved fiber in nonscattering fluid is shown in Fig. 5a . The axial intensity profile is shown in Fig. 5b, while radial intensity profiles at $z=50$ , 200, and $350\mu \mathrm{m}$ are superimposed in Fig. 5c. The axial intensity falls progressively with distance from the fiber tip. However, the rate of axial decay appears to be less in the near-field than the far-field. The radial intensity distribution has a flat top adjacent to the tip of the fiber. This plateau is reduced as light spreads radially with increasing axial distance from the fiber tip. In the far-field, the radial intensity profile resembles a Gaussian distribution. The transition between these forms takes place around $250\mu \mathrm{m}$ from the fiber tip in this case, and in general, the axial distance at which this occurs is directly related to the fiber radius.

Fig. 5

Fluorescence collection field for an F-MCB-T optical fiber characterized using 2P excitation. (a) Measured 2-D intensity map and the corresponding model fitting. (b) Axial intensity profile. (c) Radial intensity profiles at axial locations indicated in (b) 50, 200, and $350\mu \mathrm{m}$ from the fiber tip. Note that results were obtained with nonscattering $500\text{‐}\mu \mathrm{M}∕\mathrm{L}$ rhodamine solution. The solid lines in (b) and (c) were fitted using the model.

Typical collection fields for a MM fiber in scattering fluid and tissue are shown in Fig. 6a . Corresponding axial intensity profiles in nonscattering and scattering fluids and in stained tissue are overlaid in Fig. 6b. Note that the data here are smoothed using a moving average filter. There is significant attenuation of fluorescence collection intensity in scattering fluid and in tissue. However, the flat top of the radial intensity distribution adjacent to the fiber tip was preserved in all cases. It should be noted that the axial intensity profile in scattering medium exhibits considerably more decay than the corresponding profile in the nonscattering fluid.

Fig. 6

Fluorescence collection field for an F-MCB-T optical fiber characterized using 2P excitation in different media. (a) 2-D intensity maps for scattering fluid and stained tissue and the corresponding model fittings. (b) Axial intensity profiles for tissue, nonscattering fluids, and scattering fluids. Note that results obtained with nonscattering and scattering fluids contained $500\mu \mathrm{M}∕\mathrm{L}$ rhodamine, while the tissue was stained with $1000\mu \mathrm{M}∕\mathrm{L}$ rhodamine. The solid lines in (b) are the corresponding fitting results using the model.

3.3.

Model Validation

The semiempirical model described in the Sec. 2 was validated by comparing predicted collection profiles with data obtained experimentally. Because fluorescent emission was collected across a relatively narrow wavelength window $(535\pm 25\mathrm{nm})$ , it was assumed that the weighted axial component $D^{*}\left(z\right)$ in Eq. 15 could be represented by a simple exponential expression as in Eq. 8. For a given MM fiber, the parameter ${\theta }_b$ that controls the spread of the function $S_2(r,z)$ [see Eq. 13] was set as follows. Near-field axial intensity profiles observed in nonscattering fluid were matched to the model predictions by adjusting ${\theta }_b$ , having first estimated the effective absorption coefficient ${\mu }_t$ in this medium from measured photon loss. The effective absorption coefficient ${\mu }_t$ that best fit the model to experimental data was then determined for scattering fluid and tissue. For the AFS105/125Y fibers, this process was also carried out for nonscattering fluids at a series of rhodamine concentrations ranging from 50 to $1000\mu \mathrm{M}∕\mathrm{L}$ . Best-fit values for the effective absorption coefficient ${\mu }_t$ are given in Table 1 .

Table 1

Values of effective absorption coefficient that best fit model to experimental data. The average RMS error for the fit between model and data was 0.058±0.025 and 0.072±0.023 , for axial and radial profiles, respectively. The parameter θb was set to 2deg in all cases. The value indicated by * is the concentration of dye used to stain the tissue, not the actual dye concentration in the tissue.

Similar results were observed in the other flat-cleaved MM fibers tested. The average root-mean-square (RMS) error for the fit between the model and experimental data was $0.058\pm 0.025$ and $0.072\pm 0.023$ for axial and radial profiles, respectively. Background due to dark current was evident at low dye concentrations, but this was not subtracted from the experimental data.

3.4.

Quantification of Effective Collection Volumes

The effective fluorescence collection efficiency $I_{eff}$ is defined in Eq. 4 as the product of fluorescence collection and illumination efficiencies. Because illumination and fluorescence collection fields are very similar, $I_{eff}$ can be approximated as the square of the measured collection efficiency distribution $I_c$ and used to construct the relative fluorescence signal $R\left(z\right)$ —see Eq. 5.

Figure 7 shows relative fluorescence signals for a flat-cleaved F-MCB-T fiber in nonscattering and scattering fluids and in tissue. The parameters $Z_{50}$ and $Z_{80}$ , which are defined as the axial depths at which $R=50$ and 80%, respectively, are used to quantify the effective collection volume of the fiber. Values for $Z_{50}$ in stained tissue, scattering fluid, and nonscattering fluid are 75, 95, and $220\mu \mathrm{m}$ , respectively. Corresponding values of $Z_{80}$ are 170, 195, and $465\mu \mathrm{m}$ .

Fig. 7

Relative fluorescence signals for an F-MCB-T optical fiber in different media. $Z_{50}$ in stained tissue, scattering fluids, and nonscattering fluids are 75, 90, and $220\mu \mathrm{m}$ , respectively. Corresponding values of $Z_{80}$ are 175, 195, 90, and $465\mu \mathrm{m}$ . The fluorescence emission was measured at $535\pm 25\mathrm{nm}$ .

Table 2 presents $Z_{50}$ and $Z_{80}$ for MM fibers of various diameters in nonscattering fluid. The radial FWHM of the effective collection profile at $Z_{50}$ is also given. The results indicate that effective fluorescence collection volume of an MM fiber is directly related to its core diameter. While the data in Table 2 were acquired in nonscattering fluids at an emission wavelength of $535\pm 25\mathrm{nm}$ , comparable results were also obtained in scattering fluid and at different emission frequencies. $Z_{50}$ increased by less than 10% at $600\mathrm{nm}$ and was approximately halved in scattering fluid.

Table 2

Effective collection dimensions for MM optical fibers of different diameter in nonscattering fluid at a rhodamine concentration of 500μM∕L . Note that fluorescence emission was measured at 535±25nm .

3.5.

Effective Fluorescence Collection Volume of Purpose-Built Optical Probe (Optrode)

The fluorescence collection profile of a purpose-built optical probe (optrode), in which hexagonally packed, angle-cleaved F-MCB-T fibers are enclosed in a glass micropipette, was also measured.

The fluorescence collection field in nonscattering fluid for a single optrode channel is presented in Fig. 8a . The collection field is oriented radial to the probe axis and is more complex (and less symmetric) than for a flat-cleaved MM fiber. In Fig. 8b, axial profiles for a single optrode channel in nonscattering fluid and tissue are compared with the profile in nonscattering fluid for a flat-cleaved F-MCB-T fiber. Effective collection dimensions for a single optrode channel were estimated from three-dimensional (3-D) two-photon excitation data. These results are given in Table 3 .

Fig. 8

Fluorescence collection field for a purpose-built fiber optic probe (optrode) characterized using 2P excitation. (a) 2-D intensity map in nonscattering fluid. (b) Axial intensity profiles for optrode in tissue and nonscattering fluid compared with profile for a flat-cleaved F-MCB-T optical fiber in nonscattering fluid. Note that the optrode is constructed from angle-cleaved F-MCB-T fibers (for further details see Sec. 2).

3.6.

Effective Collection Volume and Propagated Impulse Response

The volume from which fluorescence is collected determines the spatial resolution of functional fluorescence measurements. However, when propagating events such as calcium waves or electrical depolarization are being mapped, the time-course of the fluorescence signal may be blurred by the spread of the wave across the collection volume. To investigate this, we have simulated the temporal response of an impulse of fluorescence emission propagating as a plane-wave across optical fiber collection volumes of varying dimensions (see Sec. 2). The results of this study are presented in Fig. 9 .

Fig. 9

Simulated fluorescent response of impulse propagating as a plane-wave across a fiber collection volume. (a) Impulse response for propagation at $0\mathrm{deg}$ , $45\mathrm{deg}$ , and $90\mathrm{deg}$ with respect to the fiber axis ( $Z_{50}=80\mu \mathrm{m}$ , fiber $\mathrm{diam}=100\mu \mathrm{m}$ , and propagation $\mathrm{velocity}=0.4\mathrm{m}∕\mathrm{s}$ ). (b) FWHM of the impulse response as a function of fiber diameter ( $Z_{50}=80\mu \mathrm{m}$ and propagation $\mathrm{velocity}=0.4\mathrm{m}∕\mathrm{s}$ ). (c) FWHM of the impulse response as a function of $Z_{50}$ (fiber $\mathrm{diam}=100\mu \mathrm{m}$ and propagations $\mathrm{velocity}=0.4\mathrm{m}∕\mathrm{s}$ ). Results for the propagation directions $0\mathrm{deg}$ and $90\mathrm{deg}$ are superimposed in both (b) and (c).

Table 3

Effective collection dimensions for a single optrode channel in nonscattering fluid and tissue.

Figure 9a shows a simulated fluorescent signal propagating across a fiber collection volume at $0\mathrm{deg}$ , $45\mathrm{deg}$ , and $90\mathrm{deg}$ with respect to the fiber axis. Propagation across the collection volume broadens the fluorescent signal, and this blurring varies with impulse direction. The broadening is quantified by estimating the FWHM of the impulse response. FWHM is presented as a function of fiber diameter in Fig. 9b ( $Z_{50}=80\mu \mathrm{m}$ and propagation $\mathrm{velocity}=0.4\mathrm{m}∕\mathrm{s}$ ) and as a function of $Z_{50}$ in Fig. 9c (fiber $\mathrm{diameter}=100\mu \mathrm{m}$ and propagation $\mathrm{velocity}=0.4\mathrm{m}∕\mathrm{s}$ ). Results for the propagation directions $0\mathrm{deg}$ and $90\mathrm{deg}$ are superimposed in both cases.

As expected, these results show that the blurring introduced by propagated fluorescent activity increases as the collection volume is enlarged. However, for a propagation velocity of $0.4\mathrm{m}∕\mathrm{s}$ (a velocity typical for spread of electrical excitation through cardiac tissue), the FWHM of the impulse response was less than $0.5\mathrm{ms}$ across the range of fiber diameters and $Z_{50}$ values tested. Last, sensitivity to the direction of propagation depends on the effective collection dimensions. With a fiber diameter of $100\mu \mathrm{m}$ and $Z_{50}$ equal to $80\mu \mathrm{m}$ , the effective collection volume is approximately isotropic, and the fact that the FWHM of the impulse response shown in Fig. 9a is not affected significantly by the propagation direction under these circumstances reflects this.

4.1.

Characterization of Illumination and Fluorescence Collection Fields

The experimental techniques employed in this study to determine the illumination and fluorescence collection profiles incorporate a number of novel aspects that add to the precision of this study. First, 2P excitation illuminates an ellipsoidal volume that is small $(<1\%)$ in comparison to the diameter of the optical fiber and can therefore be considered to be a point source. Because the 3-D location of the 2P focal volume can be precisely controlled, it is possible to reconstruct the fluorescence collection fields of optical fibers and fiber optical probes with high spatial resolution. Second, the 2P excitation wavelength $\left(750\mathrm{nm}\right)$ is well separated from the spectral range of fluorescent emission for both rhodamine and fluoroscein. Last, with 2P release of caged fluorophores, it is possible to sample fiber illumination profiles in a fashion similar to that used to characterize fluorescence collection fields. Moreover, the use of flash photolysis also means that confounding effects of light absorption by the fluorophore are minimized, since caged fluorophores outside the region addressed by the 2P focal volume do not respond to excitation. In any case, this study demonstrates that the three-dimensionally resolved nature of 2P excitation allows accurate characterization of illumination and collection fields of arbitrary probes or other optical elements as long as the fields are moderately larger than achievable 2P point spread functions $(<1\mu \mathrm{m})$ .

The ellipsoidal shape of the 2P focal spot may affect the resolution with which illumination and fluorescence collection volumes can be characterized experimentally. With a $40\times$ 0.9 NA objective, the FWHM of the 2P focal volume was $0.5\mu \mathrm{m}$ in the $x\text{‐}y$ plane but $1.4\mu \mathrm{m}$ in the $z$ -axis direction, and for lower magnification objectives, where NA is reduced, elongation in the $z$ direction was proportionately worse. This has little impact on the axially symmetric illumination and fluorescence collection fields associated with the individual optical fibers studied here. In this case, it can be shown that the $x\text{‐}y$ resolution of the objective applies, except in the near-field. No significant differences were observed between images obtained with the $10\times$ and a $40\times$ objective, and we therefore used a $10\times$ objective since it covered a significantly larger field of view. However, the spread of the 2P focal volume in the $z$ direction does give rise to signal blurring when characterizing illumination and collection fields that are not axially symmetric. The resultant degradation of resolution can always be overcome by using high NA objectives and employing image tiling to cover the required field of view.

4.2.

Effective Collection Volume

The fact that measured illumination and fluorescence collection fields for flat-cleaved MM fibers were very similar in both nonscattering and scattering fluids is noteworthy. Absorption and scattering are wavelength-dependent, and the wavelengths of excitation light and fluorescent emission are different. In this study, however, fluorescence was collected over a relatively narrow wavelength window close to the excitation wavelength. Moreover, we have shown that collection volumes are strongly determined by scattering, which is more weakly dependent on wavelength than absorption (see the following discussion). Within this context, it is not surprising that the techniques employed could not resolve differences between illumination and collection volumes. We have also independently reconstructed the 3-D illumination field of an optical fiber in air by imaging light intensity scattered from a thin film transverse to it at a series of different axial locations. A confocal microscope in wide-field mode was used for this purpose. The results (not shown) match the detailed fluorescence collection fields obtained with 2P excitation when absorption and scattering are minimized. These results provide objective justification for the reconstruction of effective fluorescence collection volumes from measured fluorescence collection fields using the relationship [see Eq. 1]:

4.3.

Comparison with Previous Work

A number of studies of effective fluorescence collection dimensions have been published. ^{7, 9, 10, 13, 15, 23, 24, 25, 26} The effect of the propagation of cardiac electrical activation across the effective collection volume on measured optical potentials has also been considered, ^{13, 15, 25, 26} with reported collection depths in tissue ranging from $\sim 100\mu \mathrm{m}$ (Ref. 9) to $>1\mathrm{mm}$ (Ref. 25). In part, this variance reflects the different means of fluorescence excitation and collection used. Focusing on previous results that are most comparable with ours, Krauthamer ⁷ measured illumination intensity profiles for SM and MM optical fibers in air but report significantly greater axial attenuation than we do and observed no significant effect of fiber diameter on collection depth. Like us, Byars ¹⁰ characterized the effective collection volumes of MM optical fibers in cardiac tissue. Their results for a flat-cleaved fiber with core diameter $240\mu \mathrm{m}$ and NA 0.51 are consistent with a value of $Z_{50}$ around $800\mu \mathrm{m}$ , while we would expect $Z_{50}$ to be less than $300\mu \mathrm{m}$ for an optical fiber with these specifications in tissue. Byars characterized axial attenuation by passing a fiber through an excised LV wall segment from the endocardial surface until it touched a layer of absorbent paper soaked with fluorophore attached to the epicardial surface. Measurements were then made at a number of points through the LV as the fiber was withdrawn. We note that their axial steps were very coarse $\left(500\mu \mathrm{m}\right)$ , and it is unknown to what extent the optical properties of tissue might have been affected by the track previously made by the fiber.

Ding ²⁴ used Monte Carlo simulations to investigate the effects of illumination and collection diameters on effective collection depth. Their predicted $Z_{50}$ for illumination and collection diameters of $100\mu \mathrm{m}$ was around $200\mu \mathrm{m}$ , which closely matches our results with optical fibers of comparable dimension. Hooks^{9, 27} measured effective collection volumes for flat-cleaved MM optical fibers in nonscattering fluid by recording the fluorescence collected as the fiber was moved toward the edge of a cuvette containing dye solution. The results— $Z_{50}$ was $140\mu \mathrm{m}$ and $224\mu \mathrm{m}$ , respectively, for MM optical fibers with core diameters of $65\mu \mathrm{m}$ and $100\mu \mathrm{m}$ —are very similar to those given here. Knisley ²³ employed a similar approach to estimate effective fluorescence collection depths in cardiac tissue. They used a $100\text{‐}\mathrm{\mu }\mathrm{m}$ laser beam to excite regions of different tissue thickness along a wedge of LV myocardium stained with di-4-ANEPPS and stated that the thickness of tissue contributing to fluorescence was around $300\mu \mathrm{m}$ . In comparison, the $Z_{80}$ reported here for a $100\text{‐}\mu \mathrm{m}$ MM fiber is $170\mu \mathrm{m}$ . This difference is due to the fact that Knisley collected fluorescence emission with a PMT, which has a much larger detection area than the optical fiber used in our case. When the effects of collection dimension are accounted for using the theoretical analysis outlined earlier,²⁴ their results are entirely consistent with ours.

4.4.

Validation of the Model

The model outlined in this paper provides a semiempirical description of the illumination field adjacent to the tip of a flat-cleaved MM fiber. The approach employed has a number of similarities to a model previously presented by Zhu and Yappert.¹⁷ These investigators represented the distribution of light radial to the fiber axis as a Gaussian function with a standard deviation that was a linear function of axial position. As in our model, losses due to absorption and scattering were lumped as a single axial exponential decay. Such a model could not account for the complexity of the illumination field observed near the tip of an MM fiber, and Zhu and Yappert¹⁷ acknowledged that “for multi-mode fibers a multi-Gaussian distribution should be used.” In our model, we have constructed such a distribution by convolving a Gaussian function (representing single-mode propagation) with a circular source area transverse to the fiber axis. The diameter of the radial distribution source is identical to that of the fiber core at its tip but is also a linear function of axial position, which enabled us to accommodate redistribution of light in the radial direction that is not encompassed by the spread of a Gaussian function alone.

The model provided a good fit to the fluorescence collection fields measured using 2P excitation and fit the 2P flash photolysis data equally well. The radial distribution function accurately represented experimental intensity profiles transverse to the fiber axis in near- and far-fields. In particular, it reproduced the flat-topped radial intensity distribution adjacent to the fiber tip and the Gaussian distribution distant from it. It also predicted the axial distance at which the transition between these two profiles was observed. However, when ${\mu }_t$ was low, there were small but systematic differences between observed and predicted axial intensity profiles. Under these circumstances, the rate of axial attenuation was slightly less in the near-field than in the far-field, but this is not completely matched by the model [see Fig. 5b]. Residual asymmetries that are probably the result of modal fluctuation,^{28, 29, 30} and are most evident in radial intensity profiles, also cannot be reproduced by the model [see Fig. 5c]. For individual model fits, however, the RMS error was on the order of the regional variation introduced by modal fluctuation and optical scattering. Taken together, these observations confirm the robustness of the model.

4.5.

Scattering and Absorption

Tissue exhibits marked scattering of light and weaker frequency-dependent light absorption. Optical scattering is a complex process, and this is reflected by the fluorescence collection profiles presented in Fig. 6 for a flat-cleaved fiber (F-MCB-T) in scattering fluid and stained tissue. There is much greater regional variability in the fluorescence collection field for the scattering fluid [see Fig. 6a] than in equivalent results for a nonscattering fluid [see Fig. 5a]. It is also evident from these figures that fluorescent light originating from outside the NA of the fiber has a higher probability of being collected in the scattering fluid than in the nonscattering fluid. The scattering fluids prepared for this study provide an isotropic scattering environment,²² whereas scattering in tissue is generally highly anisotropic. For the data shown in Fig. 6a, at least, there appears to be greater regional variability for the stained tissue than for scattering fluid.

Despite the complexity of these processes, it has been suggested that optical scattering can be treated as an isotropic “absorption-like” event, provided that the dimensions of the volume of interest are greater than the scattering mean-free-path length.^{17, 19} According to Patterson and Pogue,¹⁹ the mean-free-path-length in biological tissues is $71\mu \mathrm{m}$ at $500\mathrm{nm}$ . On this basis, scattering was lumped with absorption as a single axial decay term in our model. While scattering markedly affects fluorescence collection characteristics and gives rise to substantial regional variability (Fig. 6), the model nonetheless provides a good estimate of the average intensity distribution. This validates the assumption that optical scattering in large volumes can be approximated as isotropic absorption.

While it is difficult to quantify actual fluorophore concentration in stained tissue, dye absorption will likely reduce effective collection volumes for such membrane potential sensitive dyes as di-4-ANNEPS, which is typically administered at relatively high concentrations.¹² Effective collection volume is therefore reduced at high fluorophore concentrations, but this can be ignored at concentrations $<\sim 100\mu \mathrm{M}∕\mathrm{L}$ . Ding have measured the optical properties of di-4-ANEPPS stained rabbit heart and report values for absorption ${\mu }_a$ and scattering ${\mu }_s$ of $0.52\pm 0.13{\mathrm{mm}}^{-1}$ and $23.0\pm 2.5{\mathrm{mm}}^{-1}$ , respectively, at $488\mathrm{nm}$ . Corresponding values at $669\mathrm{nm}$ were ${\mu }_a=0.10\pm 0.04{\mathrm{mm}}^{-1}$ and ${\mu }_s=21.8\pm 2.5{\mathrm{mm}}^{-1}$ (Ref. 24). Our estimates of total absorption for cardiac tissue were $3.5{\mathrm{mm}}^{-1}$ , which is much larger than the value for absorption ${\mu }_a$ quoted earlier, but significantly less than quoted scattering coefficients indicating that some scattered light is collected by the fiber optic probe. This suggests that scattering plays an important role in the axial decay of effective fluorescence collection efficiency for fiber optic probes in tissue. This view is reinforced by the observation that very similar estimates of total absorption were made for probes in scattering fluid at the same dye concentration.

4.6.

Determinants of Effective Fluorescence Collection Volumes in Fiber Optic Probes

The experiments and modeling studies reported here demonstrate that the dimensions of the effective fluorescence collection volume for a flat-cleaved optical fiber are mainly determined by its core diameter and by the optical properties of the medium in which it is used. We have shown that a change in fiber diameter from 50 to $100\mu \mathrm{m}$ gives rise to changes of similar magnitude in collection volume dimensions (see Table 1). However, we found that the influence of fiber diameter on effective collection volume was reduced when the axial decay constant was high, for instance, in cardiac tissue. These experimental observations are matched by the model, which also predicts similar trends when the fiber diameter is increased from 100 to $200\mu \mathrm{m}$ . While the effect of altering NA was not considered experimentally, the model predicts a relatively modest reduction in $Z_{50}$ when NA is doubled—less than 10% in a nonscattering fluid and less still in scattering fluids and tissue.

The fluorescence collection field of a purpose-built fiber optic probe (optrode) that consists of multiple angle-cleaved MM fibers enclosed in a glass micropipette⁹ was similar in shape but significantly smaller. We ascribe these differences to lensing effects as light passes through the curved surface of the fiber and also to scattering at interfaces in the probe, where there is a refractive index mismatch.

4.7.

Effects of Propagation on Fluorescence Impulse Response of Optical Probes

It has been noted previously that the upstroke of the cardiac action potential observed in optical recordings is generally much slower than rapid depolarization in corresponding intracellular measurements.^{12, 13} This has been attributed to the spread of electrical activation through the region addressed by the imaging system. ^{13, 15, 16, 31, 32} According to our analysis, propagation at $0.4\mathrm{m}∕\mathrm{s}$ produces blurring of the impulse response that is dependent on the direction of propagation and the dimensions of the effective collection volume. However, the FWHM for the impulse response in tissue was less than $0.5\mathrm{ms}$ for an optical fiber with a core diameter of $100\mu \mathrm{m}$ and $Z_{50}$ of $80\mu \mathrm{m}$ , regardless of propagation direction. It is noteworthy that directional sensitivity of fiber optic probes is minimized when the collection volume is relatively isotropic. However, it would be possible to design probes with specific directional sensitivity on the basis of the information provided in this paper. Electrical activation propagates through a normal ventricular myocardium at 0.2 to $1\mathrm{m}∕\mathrm{s}$ (Refs. 33, 34, 35), and rise-times of ventricular action potentials recorded using fiber optic probes with core diameters of $100\mu \mathrm{m}$ range from 5 to $10\mathrm{ms}$ (Refs. 7, 9, 10). Less than $1\mathrm{ms}$ of this upstroke time can be explained by the spread of activation across the relatively small collection volume of these probes. This discrepancy is perplexing and should be subjected to more-detailed investigation in the future.

4.8.

Limitations of the Study

There are some obvious limitations to the results presented in this study, and these will now be discussed. First, the optical fibers tested all had the same NA. While the model suggests that NA has a limited affect on $Z_{50}$ in tissue, this prediction has yet to be tested experimentally. Second, the experimental results presented here were obtained at a limited number of wavelengths. A variety of excitation wavelengths are used with functional fluorescence recordings, and emission spectra for most fluorophores occupy a relatively broad range of wavelengths. In addition, the absorption of light in tissue is strongly frequency dependent, whereas scattering is much less so.^{18, 19} However, we have demonstrated that effective collection volumes in tissue are dominated by scattering for fiber optic imaging systems (although this is less so for systems with larger collection dimensions). We have also shown that the variation of ${\mu }_t$ with wavelength is relatively small, and should therefore be possible to account for using a simple linear correction. A further difficulty is that estimates of ${\mu }_t$ at low dye concentrations are based on fluorescence recordings with low SNR. The reliability of the absolute ${\mu }_t$ values presented here may therefore be limited by confounding effects of dark-noise. Last, while the model used in this study was useful and relatively robust, it is nonetheless semiempirical. To represent illumination and fluorescence collection fields more completely, it will be necessary to use a representation that is based firmly on the fundamental principles of physical optics.