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1 November 2008 Diffusion measured by fluorescence recovery after photobleaching based on multiphoton excitation laser scanning microscopy
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Fluorescence recovery after photobleaching (FRAP) is a widely used method to measure diffusion. The technique is normally based on one-photon excitation, which limits diffusion to two dimensions due to extended photobleaching in the axial direction. Multiphoton excitation, on the other hand, creates a well-defined focal volume. In the present work, FRAP based on a scanning laser beam and two-photon excitation is used to measure diffusion of macromolecules in solution and gels, as well as in the extracellular matrix in multicellular spheroids and tumor tissue in dorsal chambers. The bleaching profile is determined experimentally in immobilized gels, and for small scanning areas (approximately twice the lateral radius of the laser beam) a Gaussian bleaching distribution is found. In addition, the bleaching profile is determined theoretically based on the convolution of the Gaussian point spread function and a circular scanning area. The diffusion coefficient is determined by fitting a mathematical model based on a Gaussian laser beam profile to the experimental recovery curve. The diffusion coefficient decreases with increasing complexity of the sample matrix and increasing the amount of collagen in the gels. The potential of using two-photon laser scanning microscopes for noninvasive diffusion measurements in tissue is demonstrated.



Diffusion is a fundamental physical transport process in tissue. Transport of molecules in cytosol, in membranes as well as in the extracellular matrix, is governed by diffusion. Intracellular diffusion of various molecules is crucial for cellular functions, and extracellular diffusion is the main transport mechanism for oxygen, nutrition, and small regulatory and therapeutic molecules. Thus, the development of techniques for noninvasive in-situ measurements of diffusion is of great importance.

Fluorescence recovery after photobleaching (FRAP) is a widely used method to measure translational diffusion.1 The technique is normally based on one-photon excitation and bleaching of fluorophore-labeled molecules using a high-intensity laser beam. The bleached area is monitored using a laser with attenuated intensity, and from the recovery of fluorescence into the bleached area the diffusion coefficient can be estimated. FRAP using one-photon excitation has been limited to diffusion in two dimensions (2-D), because the focused laser beam generates extended photobleaching in the axial direction above and below the focal plane. 3-D models using low numerical aperture objectives and one-photon stationary or scanning laser excitation have been developed.2, 3, 4 However, such models are based on extensive numerical calculations, and a major problem is that the bleaching distribution in the axial direction depends on the sample thickness, which usually is hard to define accurately. Multiphoton excitation, on the other hand, creates a well-defined focal volume, and bleaching occurs only in the focal plane.5 FRAP based on multiphoton excitation may therefore be used for accurate measurements of 3-D diffusion, and has been developed using a stationary laser beam.6, 7, 8, 9, 10

Confocal laser scanning microscopes (CLSM) equipped with multiphoton excitation lasers are becoming standard instrumentation in many laboratories and core facilities, and can be used for 3-D diffusion measurements. However, some commercial CLSMs have too slow image acquisition rates for accurate diffusion measurements of small molecules in solution. This becomes a major problem when parking the laser beam, and due to the fast diffusion, the recovery may not be recorded. Increasing the effective recovery time by increasing the size of the bleached area may overcome this problem,11 although caution should be taken if diffusion occurs during the bleaching process. Scanning a larger region of interest (ROI) also improves the signal-to-noise ratio. Another advantage of using a scanning laser beam is that any arbitrary shaped region can be bleached. The purpose of the present work was thus to extend FRAP based on multiphoton excitation and a stationary laser beam to a scanning laser beam.

By using a stationary laser beam, the bleached volume is well defined by the Gaussian laser beam, and the diffusion coefficient can be calculated based on the mathematical model developed by Brown 6 When applying a scanning laser beam, a theoretical estimation of the bleached volume is based on the convolution of the point spread function (PSF) of the laser beam and the scan area. The bleaching profile will, however, depend on several parameters such as the optical components, scanning process, refractive indices, and the intensity and wavelength of the laser. In the present work the bleached volume was determined using a semiempirical approach, and the axial and radial bleaching radii were determined experimentally for various diameters of the bleached ROI. In addition, the bleaching profile was calculated theoretically based on the convolution product between the Gaussian PSF and a circular scanning area of various radii. We wanted to apply a simple mathematical model describing the bleaching and recovery processes, as such a model could be adapted by more users. The model developed by Brown 6 for a stationary laser was therefore used, and the appropriate scanning criteria for using this model were determined. The diffusion coefficients were determined for high molecular weight molecules in solution and gels as well as in the extracellular matrix in multicellular spheroids and in tumor tissue in dorsal chambers. The potential of using a commercially or laboratory built CLSM equipped with multiphoton laser excitation for noninvasive measurements of diffusion in tissue in vivo is demonstrated. This is to our knowledge the first time FRAP based on scanning multiphoton excitation has been used to measure 3-D diffusion in tumor tissue in vivo.


Materials and Methods


Two-Photon Laser Scanning Microscopy and Fluorescence Recovery after Photobleaching

FRAP experiments and imaging were performed using an LSM 510 confocal scanning microscope (Carl Zeiss, Jena, Germany) with a C-Apochromat 40×1.2 water immersion (W) objective and a C-Achroplan 40×0.8W objective. The laser source was a Mira Model 900-F mode-locked Ti:sapphire laser (Coherent, Incorporated, Santa Clara, California) pumped by a solid-state Verdi V-5 laser at 532nm (Coherent, Incorporated). The Ti:sapphire laser produces 200-fs pulses with a repetition rate of 76MHz , and at 780-nm excitation the output power was 800mW at the laser cavity and 40 or 20mW at the objective for the C-Apochromat 40×1.2W and C-Achroplan 40×0.8W , respectively. Fluorescein isothiocyanate (FITC) was excited at 780nm , and the fluorescence was detected using a bandpass 500-to550-nm filter. The laser power was attenuated to 5% of the 100% bleach intensity by an acousto-optical modulator, to minimize bleaching and avoid saturation of the detector during imaging.

Each FRAP experiment started with ten image scans of the user-defined ROI, followed by a bleach pulse. The bleached ROI in the xy plane was circular with a radius of 8, 12, or 16pixels corresponding to 1.8, 2.7, or 3.6μm , respectively. Each ROI was bleached using maximum laser power, maximum scanning speed, and ten scanning iterations in a raster scan across the ROI. This corresponded to a total bleach scanning time for the three ROIs of 75, 115, and 150ms , respectively. The total bleach scanning time is the time from bleach start to end, which includes the actual bleaching time per pixel ( 1.6μspixel×10 iterations) and the time needed for repositioning of the laser to the next line of pixels. The bleach scanning time used was the shortest period that achieved maximum bleaching (difference between prebleached and postbleached intensity divided by prebleached intensity) of FITC-dextrans (Sigma-Aldrich, Saint Louis, Missouri) in spheroids.

To determine the recovery curve, series of up to 600 images of the bleached ROI were collected at the highest possible rate. Only the bleached ROI was imaged. For the three ROI sizes used, this corresponded respectively to 11.6, 17.4, and 25.1ms per image (duration of the acquisition of the image of one ROI plus the shortest possible time between each acquisition, which was 0.1ms ). The fluorescence intensity of the bleached ROI was determined as the average pixel value, and normalized to the average fluorescence intensity after reaching a stable level. The normalized recovery curves were fitted to estimate the diffusion coefficient (D) , the mobile fraction (Rmob) , and the bleaching parameter (β) , as described next.

A high numerical objective 40×1.2W was used in all experiments except for measurements in dorsal window chambers. These chambers required an objective with a long working distance, and here the C-Achroplan 40×0.8W objective was used. When performing FRAP experiments with this objective combined with tumors growing in dorsal window chambers, the image had to be magnified to decrease the pixel size to obtain sufficient laser intensity per pixel. The images were zoomed to pixel size 0.11μm . In all other experiments the pixel size was 0.45μm and the image size 512×512pixels . To ensure that the zooming did not affect the measured diffusion coefficient, the diffusion coefficient of 150-kDa FITC-dextrans in solution was determined for pixel sizes 0.45 and 0.11μm , and found to be the same (data not shown).

To ensure that no excitation saturation took place during bleaching, the bleach depth parameter (β) for 2-MDa FITC-dextran molecules in solution was measured as a function of laser power using the 40×1.2 objective. The bleaching and scanning of an ROI with radius 2.7μm was performed as described before. To further ensure that a two-photon bleaching process took place, the fluorescence intensity of 2-MDa dextran molecules (1mgml) immobilized in a 15% polyacrylamide gel was measured as a function of laser exposure time and bleaching laser power. The fluorescence intensity was measured as the average pixel value in an ROI with radius 1.8μm . The gels were made from a mixture of acrylamide (Sigma-Aldrich) and bisacrylamide (Bio-Rad Laboratories, Richmond, California) in the ratio 37.5:1 with 0.5% ammonium persulphate (Sigma-Aldrich) and 5% aqueous N, N, N , N -tetramethylethylenediamine (Sigma-Aldrich) as polymerization catalysts. The biexponential decay function [I(t)=aebt+cedt] was fitted to the measured fluorescence intensity to determine the bleaching rate. The bleaching rate b was much larger than d , thus b was used as the bleaching rate.


Experimental Bleaching Profiles

The bleaching profiles were determined experimentally using FITC-labeled dextrans with average molecular weights of 500kDa or 2MDa immobilized in a 15% polyacrylamide gel to a final concentration of 1mgml . The gels were prepared as described before. Bleaching was performed using the scanning conditions described before.

The radial bleaching profile when using a stationary laser beam was determined by parking the laser beam and generating the bleaching profile in the xy plane. The axial bleaching profile could not be obtained. The bleaching profiles when scanning the laser beam were generated for the three ROIs in both the radial and axial directions. To determine the axial bleaching profile, a z stack of the bleached area was acquired, and the number of consecutive confocal images and distance between each image necessary to cover the bleached axial area were used, which corresponded to 20 consecutive images with a distance of 0.74μm . Ten independent xy images and z stacks were acquired for each ROI to determine the average bleaching radial (ωr) and axial (ωz) radius, respectively, and the bleaching radii were determined as described next (theory: the FRAP analysis model). The bleaching radii were determined for both objectives used.

To determine if there was any diffusion during bleaching, an image was taken of the bleached ROI immediately (0.1ms) after the bleach scan using a solution of 150-kDa , 500-kDa , and 2-MDa FITC-dextran. Radial profiles of the bleached area were made and compared to the profiles from the experiments using immobilized gels. Assuming that no diffusion occurred during bleaching, the bleaching profiles for the dextrans in solution and in immobilized gel should be the same.


Sample Preparation for Diffusion Measurements

Diffusion was measured for FITC-labeled dextran molecules of size 40kDa , 150kDa , 500kDa , 2MDa , and IgG (Sigma-Aldrich), either in solution [phosphate buffered saline (PBS)], in gelatin or collagen gel, or in tumor tissue growing as multicellular spheroids or in dorsal window chambers.

Gelatin gels were prepared from bovine gelatin powder (Kebo, Oslo, Norway) dissolved in PBS to 5% and incubated at 45°C for at least 20min . FITC-dextrans were added to the aqueous hydrogel to obtain a homogenous distribution before cooling to room temperature.

Collagen gels were prepared using Vitrogen 100 collagen type-1 solution (Cohesion Technologies, Palo Alto, California) as described previously.12 In brief, the pH and ionic strength were adjusted by addition of NaOH to pH 7.4 and 10× PBS. To concentrate the solution, the collagen was ultracentrifuged (Optima LE-80K Ultrasentrifuge, Beckman Coulter, Fullerton, California) at 5°C for 24to60h to obtain 1 to 2% gels. The collagen concentration in the pellet was determined by UV spectrophotometry by determining the difference in collagen content of the solution before ultracentrifugation and in the supernatant afterward.

Multicellular spheroids were made from the human osteosarcoma cell line OHS.13 Briefly, 2106 cells were seeded in 80-cm2 tissue culture flasks (Nunc, Tamro AS, Skårer, Norway) precoated with 1% agar to prevent cell attachment, and containing 25-ml growth medium [RPMI-1640 supplemented with 10% fetal bovine serum, 100-unitsml penicillin/streptomycin, and 1-mM L-glutamine (all from Sigma-Aldrich)], and kept at 37°C and 5% CO2 . Half of the medium was changed twice a week. Spheroids were harvested for experiments after 5to6days , when their diameter had reached 150to250μm .

Transparent window skinfold chambers were implanted on the back of athymic BALB/c nu/nu mice ( 10to16weeks old, 24to27g , Taconic, M&B, Ry, Denmark) as described by Endrich 14 Briefly, an extended double layer of skin was sandwiched between two symmetrical titanium frames, and a circular area of 15mm in diameter was removed from one layer of skin. The remaining layers of the other skin fold (thin striated skin muscle, subcutaneous tissue, dermis, and epidermis) were covered with a glass coverslip incorporated into one of the titanium frames. 24h after the implantation of the chamber, the coverslip was removed and 1.5106 OHS cells were placed into the center of the chamber. A new glass coverslip was replaced and the chamber closed. All surgical procedures were performed under sterile conditions, and the mice were anesthetized by i.p. injection of Fentanyl/Midazolam/Halsol/sterile water (3:3:2:4) in the amount of 12-mlkg body weight (Hameln Pharmaceuticals, Germany, Alpharma AS, Norway, and Janssen-Cilag AS, Norway). The animals were kept under pathogen-free conditions at constant temperature (24to26°C) and humidity (30 to 50%), and were allowed food and water ad libitum. All animal experiments were carried out with ethical committee approval.

FITC-dextran molecules were mixed with PBS or the gel to a concentration of 1mgml . Multicellular spheroids were incubated with 1-mgml FITC-dextrans for 6h on a roller at 37°C before measuring diffusion. Athymic mice received 200-μl i.v. injections in the tail vein of 20-mgml 150-kDa FITC-dextrans 24 before measuring diffusion. Tetramethyl rhodamine-dextran 580kDa (Invitrogen, Eugene, Origon) was injected i.v . ( 200μl of 10mgml ) to visualize the blood vessels.


Statistical Analyses and Fitting Routine

Two-sample two-tailed student’s t tests assuming nonequal variances were applied to the Gaussian distributed data to compare population means (Minitab, Minitab Incorporated, State College, Pennsylvania). All statistical analyses were performed using the significance criterion of p0.05 .

Fitting was performed using the Matlab 7.0 lsqcurvefit function (The Math Works, Natick, Massachusetts), and the results plotted using the SigmaPlot 9.0 software (Systat Software, Incorporated, Richmond, California).


Theory: Fluorescence Recovery after Photobleaching Analysis Model


Theoretical Bleaching Profile

The theoretical bleaching profile can be described by the bleaching light distribution B(r,θ) , which is given as a convolution of the laser beam PSF and the 2-D circular scanning area. The laser beam profile is assumed to be Gaussian in both the radial and axial directions:

Eq. 1

where b=2 for two-photon excitation, and ωr and ωz are the 1e2 radial and axial dimensions, respectively.

The two-dimensional radial convolution product in the scanning plane is obtained by integrating over the circular scanning area of radius R and all angles, and gives the bleaching light distribution:

Eq. 2


Eq. 3

where r is the position of the bleached point, r is the position of the laser beam, and θ is the angle between the two vectors r and r . Jo is the Bessel function. Equation 3 shows that for small R , the bleaching light distribution is dominated by the Gaussian function. Equation 2 was plotted for various scanning radii R and ωr=1μm using Maple (MapleSoft, Waterloo, Canada), and is shown in Fig. 1 . The radius of the laser beam was chosen to ωr=1μm because the bleaching profile using a stationary laser beam had a radius ωr=1μm at 1e2 intensity (Fig. 1), i.e., the Gaussian laser beam profile was assumed to be equal to the bleaching profile of a stationary laser beam.

Fig. 1

Simulation of bleaching light distribution from Eq. 2. The Gaussian laser beam profile is integrated over a circular ROI with radius R=1.0μm (—), R=1.5μm (–––), R=1.8μm (⋯), R=2.7μm (–⋅–), and R=3.6μm (---). The bold solid line represents the bleaching profile when parking the laser beam.



Determination of Radial and Axial Bleaching Radii

The actual bleaching profile obtained when using a laser scanning microscope will not be exactly described by the bleach light distribution.2 The bleaching profile was thus determined experimentally using immobilized FITC-labeled dextran molecules in a polyacrylamide gel, as described earlier. The axial and radial bleaching radii were determined by fitting the expression for the concentration of unbleached molecules in the circular area of interest [Eq. 5] to the experimental bleaching profile, according to Braga, Desterro, and Carmo-Fonseca.4 The photo-bleaching process was assumed to follow a first-order differential equation:

Eq. 4

where b is the order of bleaching, δb is the absorption cross section of the fluorophore being bleached, and qb is the quantum efficiency for two-photon bleaching. This requires that there is no saturation in the excitation process. Ib(r,z) represents the 3-D Gaussian distribution of the laser intensity given by Eq. 1.

Assuming two-photon excitation (m=2) and bleaching (b=2) , the solution to the differential in Eq. 4 is expressed as:

Eq. 5

where C0 is the initial concentration of fluorophores, ω1 and ω2 are characteristic parameters related to the width of the bleaching profile in radial and axial directions, respectively, and r0 and z0 are fitting parameters to define the zero-position corresponding to the center point of our experimental data. β is the bleach depth parameter given by:

Eq. 6

where Δt is the duration of the bleach pulse. The radial and axial bleaching radii are considered to be independent, and in the fitting, Eq. 5 was separated into the axial and radial part. The parameter C0 was set to unity, since the bleached region was normalized to the fluorescence signal prior to the analysis. The axial part was first fitted using data from the z -stack images using βz , ω2 , and z0 as free parameters. βr , ω1 , and r0 were subsequently fitted by using the dataset corresponding to z=0 . βr and βz were found to be almost the same. Assuming a Gaussian laser bleaching profile, the 1e2 radial and axial radii were determined by solving Eq. 5 for r=ωr and z=ωz . For larger ROIs (12pixels) , the radial part was fitted using a product of two independent Gaussian bleaching profiles separated a distance 2ωs . Together they defined a total 1e2 radial width of ωr+ωs .


Determination of Diffusion Coefficient and Bleaching Parameter

The equation describing the recovery of fluorescence into the area bleached by a stationary one- and two-photon laser has been developed by Axelrod 1 and Brown, 6 respectively. We found that two-photon laser scanning over a small circular area generated a bleaching profile well fitted by an exponential of a Gaussian [Eq. 5], i.e., the bleaching profile generated by an idealized stationary Gaussian laser beam (see Sec. 4). Assuming a Gaussian bleaching profile, the recovery curve can be fitted to the following equation from Brown, 6 including the mobile fraction Rmob=[FF(0)][F0F(0)] :

Eq. 7

This equation is valid for a single freely diffusing species in three dimensions and is generalized for the m ’th photon excitation and the b ’th photon bleaching. Here, F0 is the prebleach intensity, F(0) the intensity at time t=0 after bleaching, and F the intensity of the bleached spot at infinite time after bleaching. β represents the bleaching parameter. For two-photon excitation and two-photon bleaching, and with the characteristic radial diffusion time τD=ωr28D , Eq. 7 yields:

Eq. 8

D is the diffusion coefficient of the fluorophore, and ωr and ωz are the radial and axial 1e2 radii of the bleached area, respectively. D and β were determined by fitting Eq. 8 to the experimental recovery curve. The series expansion typically was summed up to n=7 ; extending the summation did not change the fit. Equation 8 is based on a Gaussian bleaching profile. However, we also investigated if reliable diffusion coefficients could be obtained by fitting Eq. 8 to recovery curves based on non-Gaussian bleaching profiles, which were generated by scanning the laser across larger ROIs. The experimentally determined ωr and ωz for such bleaching profiles were inserted in Eq. 8.




Excitation and Bleaching Processes

The FRAP analysis is based on the assumption that there is no excitation saturation during bleaching. The linear relation in Fig. 2 demonstrates that this was the case. From the bleaching rate one may estimate the “multiplicity” of the excitation process, i.e., two-photon, three-photon, mixed variants, etc. The bleaching decay and corresponding bleach rate as a function of laser power is shown in Fig. 3 . The slope of this log-log plot was 2.1, demonstrating a two-photon bleaching process as described by Patterson and Piston.15

Fig. 2

The bleach depth parameter β as a function of laser power for 2-MDa FITC-dextrans in solution. The linear relationship demonstrates that there is no excitation saturation during bleaching.


Fig. 3

(a) The fluorescence decay of 2-MDa FITC-dextrans immobilized in acrylamide gel and illuminated with a laser power of 13.8mW (●), 15.0mW (○), 18.0mW (◼), 20.0mW (◻), and 28.0mW (▼). The biexponential function I(t)=aebt+cedt was fitted to the experimental data to determine the bleaching rate. b is much greater than d and was used as the bleaching rate. (b) The bleaching rates obtained from (a) as a function of laser power.



Theoretical Bleaching Profiles

The bleaching profile when parking the two-photon laser beam was determined experimentally using FITC-dextran molecules in an immobilized gel. β was found to be β=0.68±0.04 , and the 1e2 radius ωr=0.98±0.23μm (bold line in Fig. 1). This radius is approximately three times wider than the theoretical laser beam radius of 0.32μm given by Williams, Piston, and Webb.16 In the simulations, the radius of the laser was chosen to be ωr=1μm .

The theoretical bleaching profile based on the convolution product of the PSF and a circular scanning area of various radii are shown in Fig. 1. A Gaussian bleaching profile was obtained when the scanning radius R was equal to the 1e2 radius of the laser beam. The bleaching profile became broader and the discrepancy from the Gaussian profile increased with increasing bleaching radius. For large R , the bleaching profile can be characterized by the uniform disk model described by Braeckmans, 3 where the intensity profile was approximated by a discontinuous step function, which was one inside the bleaching disk and zero outside.


Experimental Bleaching Profiles

Experimentally obtained bleaching profiles will not be exactly described by the convolution product since the laser beam has to pass various optical components and was therefore measured for three different scanning radii (Fig. 4 ). For a small ROI (radius=1.8μm) , the bleaching profile was well fitted to a distribution corresponding to a Gaussian laser beam [Fig. 4a]. Comparing the theoretical and experimental bleaching profile for R=1.8μm (Fig. 5 ), some discrepancy was found. The theoretical bleaching profile was steeper than the experimental profile, although the profiles showed a great deal of overlap. Based on the good fit of Eq. 5 to the experimental data [Fig. 4a] and overlap between the experimental and theoretical profiles, the bleaching profile was approximated to a Gaussian distribution, and the 1e2 radial and axial bleaching radii estimated. When the ROI increased, the bleaching profile became broader in agreement with the simulations. To determine the bleaching radii, the experimental bleaching profile was, as a first approximation, fitted by two functions given by Eq. 5 [Figs. 4c and 4e]. The edges of the bleaching profiles followed a Gaussian slope, whereas the bottom part was flat and could not be fitted to a Gaussian. Based on two Gaussian functions separated by a distance 2ωs , the radial and axial bleaching radii were estimated. The axial fit was well described by Eq. 5 for all three ROI sizes.

Fig. 4

Radial and axial fluorescence intensity profiles of different sized bleached ROIs using 2-MDa FITC-dextrans immobilized in acrylamide gel. The profiles are the average of ten independent profiles. (a) and (b) ROI radius 1.8μm , (c) and (d) ROI radius 2.7μm , and (e) and (f) ROI radius 3.6μm . Dots represent the experimental data, and solid lines represent the fit from Eq. 5. The fluorescence intensity is normalized to the intensity of the unbleached pixels. (e) shows the two Gaussian profiles separated a distance 2ωs , each with 1e2 radius ωr .


Fig. 5

Simulated bleach light distribution from Eq. 2 with R=1.8μm and ωr=1μm (solid line), and experimental bleach profile R=1.8μm (broken line).


The radial and axial bleaching radii ωr and ωz are listed in Table 1 . The bleaching radii were essentially identical for the two dextran molecules used, and the data presented in Table 1 are therefore an average of the measurements using the two molecular sizes. The radial radii ωr increased significantly with increasing ROI, whereas the axial radii ωz remained equal for all the ROI sizes within experimental error. No significant differences in the bleaching radii were found for the two objectives used.

Table 1

Experimental values of ωr and ωz determined as the 1∕e2 radii of the bleaching profile of immobilized FITC-dextran molecules.

ROI rad (μm) C-Apochromat 40×∕1.2W C-Achroplan 40×∕0.18W
ωr (μm) ωz (μm) ωr (μm) ωz (μm)
1.8 2.66±0.17 3.87±0.31
2.7 3.18±0.08 4.03±0.50
3.6 4.41±0.26 3.76±0.27 3.82±0.51 4.23±0.55


Diffusion during Bleaching

The characteristic diffusion time τD=ωr28D was calculated based on the measured diffusion coefficient for 150-kDa , 500-kDa , and 2-MDa FITC-dextran molecules and the radial bleaching radii. For 150- and 500-kDa dextran molecules in solution, the bleaching time was larger than the characteristic diffusion time, whereas for 2MDa , the bleaching time and characteristic diffusion time were approximately the same. In gels, the bleaching time was shorter than the characteristic diffusion time, except for 150-kDa dextrans in the lowest concentration of collagen gels (0.24%). According to Meyvis, 17 the characteristic diffusion time should be 15 times longer than the bleaching time, and that was not the case in our experimental setup.

Diffusion during bleaching was therefore determined by imaging the fluorescence intensity profile immediately (0.1ms) after bleaching of 150-kDa , 500-kDa , and 2-MDa FITC-dextrans in PBS and 0.24% collagen gel. ROI sizes of 1.8 and 3.6μm were used. Equation 5 was fitted to the intensity profiles to obtain ωr and β . This was done for ten independent profiles to estimate the average value of ωr and β . These average values obtained for dextran molecules in solutions and collagen gels were compared with the corresponding average values for dextran molecules immobilized in gels. The bleaching profiles obtained in solution using ROI 1.8μm are shown in Fig. 6 . In solution and immobilized gels, ωr was statistically equal for all molecules tested when using ROI 1.8μm . However, β decreased with decreasing molecule size, indicating that some diffusion occured during bleaching. For immobilized gels, β=0.86±0.16 , while for dextrans in solution, β was reduced to 0.51±0.12 , 0.44±0.11 , and 0.27±0.07 for dextran sizes 2MDa , 500kDa , and 150kDa , respectively. In collagen gels, β was only reduced for 150kDato0.39±0.10 . When increasing the ROI to 3.6μm , approximately the same bleaching profiles were obtained in solution and in immobilized gels, both with respect to ωr and β (in solution ωr=4.40±0.77μm and β=0.55±0.16 ). This suggests that although the bleaching time increased with increasing ROI, the recovery time will also increase, and no significant diffusion took place during the bleaching process.

Fig. 6

Fluorescence intensity profiles immediately (0.1ms) after bleaching of 150kDa (◇,–––), 500kDa (●,—), and 2MDa (◻,⋯) dextrans in solution, using a ROI of 1.8μm . The profiles are the average of ten independent profiles. Equation 5 has been fitted to the experimental intensity profiles. The fluorescence intensity is normalized to the intensity of the unbleached pixels.



Measurements of Diffusion in Solution

The diffusion coefficients for IgG and dextran molecules of various sizes in solution were determined for the smallest ROI used (1.8μm) . The experimentally estimated radial and axial bleaching radii were substituted in Eq. 8, which was fitted to the experimental recovery curve to obtain the diffusion coefficient. Our experimental diffusion coefficients were compared to the literature and theoretical values. The theoretical diffusion coefficient was determined based on the Stokes-Einstein equation:

Eq. 9

where Rh is the hydrodynamic radius, η the viscosity, k the Bolzmann constant, and T the absolute temperature. The hydrodynamic radii were found from previously published data.18, 19

A typical recovery curve showing the experimental data and the fit based on Eq. 8 are presented in Fig. 7 . The measured diffusion coefficients as well as the literature and theoretical values are presented in Table 2 . The diffusion coefficients previously reported are based on 2-D FRAP measurements using a conventional fluorescence microscope with a stationary laser or CLSM with a scanning laser. Within experimental error and the fact that some diffusion occurs during bleaching, the values reported here using two-photon excitation laser scanning microscopy are in accordance with 2-D FRAP measurements. The theoretical values for the larger dextran molecules are somewhat smaller than the experimental values, probably due to the hydrodynamic radii used. In the literature these radii vary, and we chose to refer to values from only two different papers.18, 19 The diffusion coefficient decreased nonlinearly with increasing molecular weight. The experimental data can be described by a power law expression.20

Eq. 10

This analysis was carried out for the linear dextran molecules, and Eq. 10 was fitted to the experimental data presented in Fig. 8 . This resulted in a=1.3105 and b=0.32 .

Fig. 7

Typical fluorescence intensity, bleaching, and recovery curve for 2-MDa FITC-dextran in solution. The solid line represents the fit [Eq. 8] and the dots represent the experimental data. The images show the fluorescence intensity in the bleached ROI at different times: (a) before bleaching, (b) 0.1ms after bleaching, (c), (d), and (e) during recovery, and (f) full recovery.


Fig. 8

Diffusion coefficients of 40-kDa (●), 150-kDa (◆), 500-kDa (▼), and 2-MDa (▲) dextrans. The vertical bars indicate the standard deviation, and the solid line indicates the fit D=1.3105 MW0.32 . The correlation index R2=0.93 .


Table 2

Diffusion coefficients at 20°C for various dextran molecules and IgG in PBS. Experimental data are compared to literature values (with standard deviations when reported) and theoretical values calculated from the hydrodynamic radius Rh of the molecule and the Stokes-Einstein equation. n is the number of measurements for each case.

MW [kDa] Rh [Å] Dexp [10−8cm2∕s] Dlit [10−8cm2∕s] Reference Dtheo [10−8cm2∕s]
4047.8 45.0±1.6 46.3±4.6 1944.8
(n=10) 51.5±2.3 4
150 (IgG)52.9 36.7±11.6 40.0±5.0 3040.6
(n=59) 45.031
15090.7 23.2.0±5.1 18.8±0.17 323.7
(n=20) 35.032
500159.0 22.4±2.6 23.2±1.1 413.5
(n=30) 22.032
2000268.9 12.9±3.6 6.4±0.09 38.0
(n=20) 15.032
9.6±1.0 23

Although the larger ROIs introduce non-Gaussian bleaching profiles, Eq. 8 was used to determine the diffusion coefficient to estimate the deviation from the diffusion coefficient obtained for the smallest ROI. The appropriate radial and axial bleaching radii were inserted into Eq. 8. No significant difference in the diffusion coefficients was found for the different ROIs, except for 40-kDa dextran (Table 3 ). This indicates that Eq. 8 may be used to determine the diffusion coefficient for larger molecules, even for the non-Gaussian bleaching profiles obtained using a scanning radius up to 3.6μm .

Table 3

Diffusion coefficient [10−8cm2∕s] at 20°C in PBS for various dextran molecules using three different ROIs.

Dextran sizeROI 1.8μm ROI 2.7μm ROI 3.6μm
40kDa 45±1.6 37.8±1.2 35.1±0.6
150kDa 23.2±5.1 20.8±3.9 24.8±2.4
500kDa 22.4±2.6 20.4±4.9 21.5±5.3
2MDa 12.9±3.6 12.9±2.9 14.6±4.1


Comparing Diffusion in Solution, Gels, and Tissue

Diffusion of 150-kDa and 2-MDa dextran molecules in solution were compared with biologically more relevant systems such as collagen and gelatin gels, multicellular spheroids, and tumors growing in dorsal window chambers [Fig. 9a ]. The FRAP measurements and estimation of diffusion coefficients were carried out as for solution. In spheroids and tumors, the ROIs were placed in the extracellular matrix to determine interstitial diffusion. The diffusion coefficient decreased as the complexity of the system and collagen concentration increased. The diffusion coefficient was reduced for all the systems compared to values in solution. Diffusion of 150-kDa dextrans was reduced 20 to 60% in spheroids and gels, and more than 70% in tumor tissue. The diffusion of 2-MDa dextrans was slower than the case of 150-kDa dextrans, and the diffusion coefficient was reduced 40 to 90% in spheroids and gels compared with solution. This large molecule was not injected into mice, because only a minor fraction of the molecule would be able to cross the capillary wall. The mobility fraction was high (86 to 99%) in all cases except in tumors growing in window chambers, where the mobile fraction was approximately 50% [Fig. 9b].

Fig. 9

Diffusion coefficients (a) and mobile fractions (b) of 150-kDa (black columns) and 2-MDa (gray columns) dextrans in solution, collagen gels, gelatin gels, spheroids, and tumors. Each value is the mean of 20 to 90 measurements. Standard deviations are indicated as error bars.




The results presented demonstrate that 3-D diffusion readily can be measured by FRAP using a two-photon laser scanning microscope. Our semiempirical approach showed that when a high numerical aperture objective is used, the laser may be scanned over an area with a radius less than twice the 1e2 laser radius, and can still maintain an approximately Gaussian bleaching profile. Increasing the scanning area, the bleaching profile could be approximated by a discontinuous step function described as the uniform disk model.3 The experimental bleaching profiles for small scanning radii were confirmed by the theoretical bleaching profiles obtained by the convolution of the Gaussian laser beam and the circular scanning area, although some discrepancy was observed. The discrepancy is due to the optical components along the light path, the refractive indices of the various media, the scanning procedure, the photobleaching process, the intensity and wavelength of the laser, and diffusion during bleaching. These factors were also shown to broaden the stationary laser beam profile by approximately a factor of 3. The scanning process and laser power used are important for the bleaching profile. In the present work using a commercial system, the scanning speed and repetition rate were set to maximum values, and 100% laser power was used, as this power did not induce excitation saturation. Various bleaching iterations were tested, and ten iterations gave the shortest bleaching time and maximum bleaching. Thus scanning and bleaching conditions were chosen to obtain as fast and effective bleaching as possible.

The estimated bleaching radii are of critical importance for the determination of the diffusion coefficient, as these radii are inserted in Eq. 8 and used to fit to the experimental recovery curve. The radial bleaching radius (ωr) is approximately equal to the the sum of the ROI radius and the laser beam diameter (1μm) . ωr increased with increasing ROI, whereas ωz was independent of ROI within experimental error. The relatively large ωz may be due to the fact that as long as the scan area is small with relatively close proximity between the partially overlapping rasters/pixels, also a considerable broadening along the axial dimension is expected, depending on residence time in each pixel, laser intensity, etc., and in addition, ωz depends on any variations in axial positioning of the laser beam during scanning. It should be emphasized that the bleaching radii have to be determined for each objective used, as the size of the bleaching spot increases with decreasing numerical aperture.16 However, when bleaching a large ROI, the size of the ROI will dominate the effect of the numerical aperture.

Although the largest ROI generated a truncated top-hat bleach profile, we attempted as a first approximation to use Eq. 8 (based on a true Gaussian beam profile) to compare the macromolecular diffusion coefficient in all cases. Surprisingly, all three ROIs tested could be used in determining the diffusion coefficient of larger molecules (Mw150kDa) , as Eq. 8 fitted well to the experimental recovery curve and gave the same diffusion coefficient independent of ROI. However, for smaller molecules ( 40-kDa dextrans), the diffusion coefficient decreased with increasing ROI. The consistence between the diffusion coefficients for the various ROIs indicates that for large molecules, Eq. 8 is not that sensitive to ωr and may also be used for the truncated bleaching profile. However, a mathematically more correct model would be the uniform disk model described by Braeckmans 3 In the present work we wanted to establish a user-friendly approach to determine the diffusion coefficient, and the model developed by Brown 6 is mathematically simpler. As a rule of thumb we found that this model can be applied when scanning a ROI with radius up to 1.8 times the laser radius.

The FRAP method is based on the assumption that no diffusion takes place during bleaching. For smaller molecules 150kDa in solution and low density collagen gels, diffusion was found to occur during bleaching, and this is probably also the case in most FRAP experiments reported in solution.4, 21 Braga 4 developed a method to correct for this diffusion using the larger width of the bleaching profile. In our work, no increase in the width of the bleaching profile was observed, only the bleaching parameter was reduced. However, when measuring diffusion of macromolecules in tissue, diffusion is slow and diffusion during bleaching is not likely to cause a problem.

The 3-D diffusion coefficients obtained in solution in the present work were consistent with 2-D diffusion measured by FRAP based on a conventional fluorescence microscope with a stationary laser or a CLSM with a scanning laser, and were also in the same range as the theoretical diffusion coefficient. The large standard errors found in our results are partly due to day-to-day variations in the measurements and partly due to large fluctuations in the fluorescence intensity in the ROIs used to generate the recovery curve, although the fit was good. The diffusion coefficient decreased with increasing molecular weight as reported by others, and the obtained values for the parameters a and b in Eq. 10, which depend on the medium the molecules are diffusing in, are in accordance with earlier investigations in solution.22, 23 Previously published diffusion coefficients show considerable variations, and our results are slightly in the lower range. Thus, 3-D diffusion measurements do not imply a higher diffusion coefficient than 2-D diffusion.

The potential of using two-photon scanning laser excitation and FRAP to measure diffusion in biological tissue was demonstrated in gels, multicellular spheroids and tumors growing in transparent window chambers. Collagen and gelatin gels are well-characterized models for the structural protein network in the extracellular matrix.12 The transport in the avascular multicellular spheroids is only governed by diffusion, whereas in the vascularized tumor tissue growing in transparent window chambers, both diffusion and convection driven by the pressure gradient take place.24, 25 The values of the diffusion coefficients of the dextran molecules were smaller in all the biological systems tested compared to solution, demonstrating that the extracellular matrix represents a barrier for diffusion. For the larger 2-MDa dextran molecule, the reduction in diffusion coefficient increased with an increasing amount of collagen in the gels. This is in agreement with previously published results of diffusion in tumor tissue, using a conventional microscope and FRAP.26 The multicellular spheroids and tumor tissue growing in dorsal chambers represent more complex systems than gels, consisting of a structural collagen network embedded in a hydrophilic gel of glycosaminoglycans, as well as cells. The collagen concentration in the spheroids has been measured to 0.05% (unpublished results), and the more retarded diffusion in spheroids compared to 0.24% collagen gels is thus due to the more complex extracellular matrix of the spheroids. The diffusion of the smaller dextran molecule (150kDa) was also retarded in gels and tissue compared to solution, but not to the same extent as the larger molecule, demonstrating that the extracellular matrix is a more severe barrier for larger molecules. The diffusion coefficient of the 150-kDa dextran molecules in tumors growing in dorsal chambers was approximately three times lower than in spheroids, probably due to the higher level of collagen (0.16%),27 and the more complex structure of the extracellular matrix.

The recovery curve obtained for tumors in dorsal chambers revealed a high extent of immobilized molecules. This is probably due to interactions between the dextran molecules and the extracellular matrix and tumor cells, as well as steric exclusion and tortuosity of pathways.26, 28 The lack of full recovery of fluorescence into the bleached area may also be due to a fraction of very slowly diffusing molecules. Based on two-photon fluorescence correlation spectroscopy, a two-phase nature of diffusion in tumors has been reported.29 The slow component had an approximately 2 orders of magnitude lower diffusion coefficient than the fastest component. This slow component is not detectable using FRAP, and would appear as incomplete recovery in the data.

Multiphoton microscopy has several advantages over CLSM regarding the study of thick samples. It is possible to image farther into the tissue due to enhanced light penetration, and detection of the emitted light is more efficient, as no pinhole is required in front of the detector. Combined with tissue growing in transparent window chambers, it provides a valuable tool for molecular imaging in vivo, and the present work demonstrates for the first time the potential of FRAP based on two-photon excitation using a scanning laser in studying dynamic transport processes such as diffusion.


We thank Asle Sudbø and Magnus Lilledahl for valuable discussions regarding the convolution product and the theoretical bleaching profile, Tore Lindmo for comments on the manuscript, and Martin S. Grønsleth for his programming expertise. The work has been supported by the Norwegian Cancer Society, and The Norwegian Research Council (NanoMat program contract 163529).



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©(2008) Society of Photo-Optical Instrumentation Engineers (SPIE)
Edrun A. Schnell, Live Eikenes, Ingunn Tufto, Arne Erikson, Aphirak Juthajan, Mikael Lindgren, and Catharina de Lange Davies "Diffusion measured by fluorescence recovery after photobleaching based on multiphoton excitation laser scanning microscopy," Journal of Biomedical Optics 13(6), 064037 (1 November 2008).
Published: 1 November 2008

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