The Concentration of Soluble Extracellular Amyloid-β Protein in Acute Brain Slices from CRND8 Mice

Background Many recent studies of the effects of amyloid-β protein (Aβ) on brain tissue from amyloid precursor protein (APP) overexpressing mice have concluded that Aβ oligomers in the extracellular space can profoundly affect synaptic structure and function. As soluble proteins, oliomers of Aβ can diffuse through brain tissue and can presumably exit acute slices, but the rate of loss of Aβ species by diffusion from brain slices and the resulting reduced concentrations of Aβ species in brain slices are unknown. Methodology/Principal Findings Here I combine measurements of Aβ1–42 diffusion and release from acute slices and simple numerical models to measure the concentration of Aβ1–42 in intact mice (in vivo) and in acute slices from CRND8 mice. The in vivo concentration of diffusible Aβ1–42 in CRND8 mice was 250 pM at 6 months of age and 425 pM at 12 months of age. The concentration of Aβ1–42 declined rapidly after slice preparation, reaching a steady-state concentration within one hour. 50 µm from the surface of an acute slice the steady-state concentration of Aβ was 15–30% of the concentration in intact mice. In more superficial regions of the slice, where synaptic physiology is generally studied, the remaining Aβ is less than 15%. Hence the concentration of Aβ1–42 in acute slices from CRND8 mice is less than 150 pM. Conclusions/Significance Aβ affects synaptic plasticity in the picomolar concentration range. Some of the effects of Aβ may therefore be lost or altered after slice preparation, as the extracellular Aβ concentration declines from the high picomolar to the low picomolar range. Hence loss of Aβ by diffusion may complicate interpretation of the effects of Aβ in experiments on acute slices from APP overexpressing mice.


Introduction
Amyloid-b protein (Ab) molecules aggregate in solution, forming soluble oligomers and insoluble aggregates [1][2][3]. The latter are a major component of senile plaques, a hallmark of Alzheimer's disease (AD). The discovery that neurodegeneration and cognitive impairment correlate only weakly with plaque count and more strongly with soluble or total Ab load has led to renewed appreciation of the deleterious effects of soluble forms of Ab [4][5][6][7][8][9][10][11].
The effects of soluble Ab include impaired synaptic transmission, typically observed in mouse models in which point mutations in the amyloid precursor protein (APP) or other AD-linked genes lead to APP overexpression and Ab accumulation [12][13][14][15][16][17][18][19][20][21]. In these mouse models, neurons and synapses are exposed to elevated concentrations of Ab for months, but soluble Ab can also act rapidly. For example, exposure to soluble Ab for a few minutes impairs hippocampal synaptic plasticity [18,19,[22][23][24]. Hence both prolonged and rapid elevation of the extracellular concentration of soluble Ab species can alter neuronal function.
Much of our understanding of the effects of Ab on cellular and synaptic function originates from experiments on acute brain slices, prepared from APP overexpressing mice. Acute slices are often used to study synaptic physiology as cells and synapses are accessible and many short-range synaptic connections remain intact. One potential problem is the loss of Ab after slice preparation. Soluble Ab species are mobile in solution and extracellular soluble Ab may exit slices by diffusion. Hence soluble Ab is presumably lost from slices and the steady-state concentration of extracellular Ab in an acute slice is likely to be lower than that in the brains of intact mice (in vivo). However, neither the steady-state concentrations of Ab monomers and oligomers nor the kinetics of the decline in extracellular Ab concentration have been determined in acute slices from APP overexpressing mice.
Here I use simple numerical models, fluorescence microscopy and enzyme-linked immunosorbent assays (ELISAs) to determine the extracellular concentration of soluble Ab 1-42 in acute slices from CRND8 mice. I find that the extracellular Ab  concentration is elevated after slice preparation, but rapidly declines below the initial concentration, reaching steady-state within ,30-60 minutes. Hence in slice physiology experiments, in which measurements are typically made 30 minutes to several hours after slice preparation, the extracellular concentration of soluble Ab 1-42 is a small fraction of the in vivo concentration, raising the possibility that some of the effects of extracellular soluble Ab species are absent or attenuated.

Rate of diffusion of Ab in brain tissue
The rate of diffusion of a particle in solution is determined by its diffusion coefficient (D). The diffusion coefficient for amyloid-b (Ab) monomer in aqueous solution at 25uC has been estimated from NMR measurements as between 1.4610 26 and 2.1610 26 cm 2 /s [25,26]. The diffusion coefficient is linearly dependent on absolute temperature and is therefore approximately 4% greater at 37uC than at 25uC. Hence the literature indicates that the diffusion coefficient for Ab monomer in aqueous solution at 37uC is approximately 1.8610 26 cm 2 /s and this is the value used here.
Diffusion in brain tissue is impeded by a variety of barriers, including cells and extracellular matrix. The effective diffusion coefficient (D eff ) in brain tissue is related to the diffusion coefficient for free diffusion by: where l is the tortuosity factor. For brain tissue, l is ,1.7 [27,28]. Hence D eff = 0.623610 26 cm 2 /s for Ab monomer in brain tissue at 37uC.
The distribution of Ab after release from a point source may be calculated using the following equation [27,29]: where C is the concentration of the diffusing species S is the amount of the diffusing species D eff is the effective diffusion coefficient t is time r is the distance (radius) of diffusion in three dimensions

Random walk model of Ab diffusion
The distribution of Ab monomer (calculated using equation 2) as a function of distance from a point source (distance = 0) after 1ms of diffusion is shown in figure 1A. Integration of this curve from the point of release to infinity gives the cumulative probability of finding a single molecule of Ab with increasing distance from the point of release (figure 2B). (Note that the cumulative probability approaches 0.5 as distance tends towards infinity. This is because half of the particles diffuse towards infinity and half in the opposite direction, towards minus infinity.) From the cumulative probability plot shown in figure 2B I derived the median radius of diffusion (r). The median radius is the distance traveled by half the molecules, i.e. cumulative probability = 0.25. For Ab monomer in brain tissue, the median radius of diffusion in 1 ms is 238nm. Hence half of the diffusing Ab molecules are found between +238 and 2238 nm of the point source 1ms after their release.
In the model Ab diffuses from brain tissue into ACSF or a layer of agarose. Tissue, ACSF and agarose are homogenous media separated by planar boundaries that are parallel to each other and the concentrations of Ab are uniform across each plane parallel to these boundaries (and perpendicular to the axis of diffusion). This greatly simplifies the model since Ab that moves laterally away Cumulative distribution of Ab monomer 1ms after release from a point source (at distance = 0), derived by integrating the curve in panel A from zero to infinity. Half the diffusing molecules (cumulative probability 0.25) are within 238 nm of the point source. (C) Calculation of mean diffusion distance in one dimension from that in three dimensions. A particle is released from a point source. In time = t, it diffuses a mean distance of r (black arrow) in a random angle (h) to the axis of diffusion (dashed line). The mean distance of diffusion in one dimension (r9; black arrow) occurs when h = 45u. (D) Diffusion of four molecules through two iterative steps of the random walk model. At each step half of the molecules in each location move r9 towards infinity and half towards negative infinity. doi:10.1371/journal.pone.0015709.g001   was ejected from the pipette with a single 20 ms pulse of pressure at the arrow head. The fluorescence amplitude peaked several milliseconds later (dashed line). The widths of the Gaussian fits were variable before pressure ejection because the fits to the low background fluorescence (probably autofluorescence of the slice and leakage of fluorescence from the pipette tip) were variable. During and after pressure ejection the width became more consistent, indicating reliable Gaussian fits, as seen in panel E. The width of the Gaussian fits increased over time. For comparison, the expected rates of expansion of the widths of the Gaussian fits, starting from the time at which the fluorescence intensity peaked, were calculated for monomer and 24mer and are overlaid in red. doi:10.1371/journal.pone.0015709.g002 from the axis of diffusion will, on average, be replaced by an equal amount of Ab moving towards the axis of diffusion. Hence diffusion can be described using a relatively simple onedimensional model with the axis of diffusion perpendicular to the boundaries between tissue, ACSF and agarose. However, the radius of diffusion calculated from equation 2 is for diffusion in three dimensions. I therefore converted the radius of diffusion (in 3 dimensions) to an equivalent distance of diffusion in one dimension, along the axis (r9). This is illustrated in figure 2C. In time = t, a particle diffuses distance r (grey arrow) in a random direction, i.e. at a random angle h with respect to the axis of diffusion in one dimension (dashed line in figure 2C). Hence after time = t, the particle is at a random location on the surface of a sphere of radius r (drawn as a circle in figure 2C). On average the particle travels at 45 degrees to the axis of diffusion in one dimension (there being an equal number of angles between h and zero and h and 90 degrees and, therefore and equal probability of h being less and greater than 45 degrees). The mean distance traveled along the axis of diffusion in one dimension (r9) will therefore be Hence the mean distance of one-dimensional diffusion of Ab in 1 ms (for which r = 238 nm) is 168 nm. One-dimensional random-walk models were constructed to quantify diffusion. Diffusion was simulated as a series of stepwise displacements of Ab, where in each time-step half of the material moved towards infinity and the other half towards negative infinity. This is illustrated in figure 2D in which the position of 4 molecules are shown at three subsequent time points. Calculations were performed with 5-10 nm spatial resolution and 1 ms to 1 s time steps.
Ab multimers were assumed to diffuse as globular proteins, for which where n is the number of monomers that aggregate to form the multimer.
Hence for an oligomer composed of 24 monomers (24mer), D = 0.624610 26 cm 2 /s and D eff = 0.216610 26 cm 2 /s. Constant turn-over of Ab was included in some model calculations by adjusting the concentrations of Ab at every location within the tissue after each iterative step of the model. Monomer and oligomers were all assumed to turn-over at a constant rate of 0.015% per second (unless otherwise stated) which corresponds to a half-life of 1.28 hours. Published measurements for the half-life of Ab are 0.7 to 3.8 hours in mice [30,31] and ,8-9 hours in human CNS [32].
Several similar models were employed to calculate Ab concentrations under slightly different conditions. Model #1 simulated a brain slice bathed in a practically infinite volume of circulating ACSF, such as in a large-volume holding chamber. After each iterative step of this model, Ab that had exited the slice into the ACSF was eliminated from the model. Diffusion of Ab from the brain in an intact mouse was simulated in model #2, in which the brain surface was covered with a layer of agarose. Ab exiting the brain entered the agarose. Diffusion through the agarose was modeled as diffusion in free solution, using a diffusion coefficient of 1.8610 26 cm 2 /s for Ab monomer and 0.624610 26 cm 2 /s for 24mer. A half-life for Ab of 1.28 hours was employed in model #2. Model #3 simulated a variable 'recovery' period and subsequent 'measurement' period. The tissue concentration of Ab was first calculated (using model #1) during a recovery period in which the slice was in a practically infinite volume of circulating ACSF. The slice was then transferred to a smaller-volume chamber containing 250 ml ACSF, in which Ab which exits the slice accumulates in the ACSF. The ACSF was assumed to be well-mixed and after each iterative step Ab that had exited the slice into the ACSF was therefore redistributed evenly throughout the ACSF. Model #4 simulated a slice placed in a small-volume chamber containing 250 ml ACSF immediately after preparation. The ACSF was assumed to be well-mixed and after each iterative step Ab that had exited the slice into the ACSF was therefore redistributed evenly throughout the ACSF. Model #5 simulated exogenous application of Ab to a slice. The initial concentration of Ab in the slice was zero. The slice was bathed in an infinite volume of Ab-containing ACSF and after each iterative step, the Ab concentration in the ACSF was restored to 1. Turn-over of Ab was excluded from this model.
Simplifying assumptions were used to calculate the rates of diffusion across boundaries between media in which the diffusion coefficients for Ab were different. In slice experiments the ACSF was constantly stirred. In the corresponding models (#3 and #4) I therefore assumed that the ACSF contained a homogenous concentration of Ab. Hence at the end of each iteration of the model, during which Ab would have been exchanged between tissue and ACSF using the effective diffusion coefficient for the tissue, the concentration of Ab at all locations in the ACSF was set to the mean ACSF Ab concentration. In the in vivo model (#2), it was necessary to simulate diffusion through across a tissue-agarose boundary and through the agarose. After each iteration, the concentration across all locations within one diffusion coefficient of the boundary was averaged. To prevent any Ab crossing the agarose-coverslip boundary, particles less than one diffusion coefficient from the boundary were permitted to move only away from the coverslip, 50% of the particles remaining stationary in each step.
All simulations were implemented in Igor Pro 6.0 (Wavemetrics, Eugene, OR) using custom-written routines. The code for model #1 is available as supporting information (Code S1).

Diffusion of fluorescent Ab 1-42 in acute slices
Diffusion of fluorescently labeled Ab 1-42 (HiLyte Fluor 488 labeled Ab(1-42), Anaspec, Fremont, CA) was modeled using the diffusion coefficients for unlabeled Ab 1-42 . This is an approximation since the fluorescent tag increases the relative molecular mass of the molecule from 4514.1 to 4870.5, which presumably results in a small reduction in the diffusion coefficient. When modeling diffusion of fluorescent Ab 1-42 I also assumed that there was no turn-over of this molecule in brain tissue.
I measured diffusion of fluorescent Ab 1-42 in acute slices. An adult wild-type mouse was deeply anesthetized with isoflurane and decapitated and the brain was rapidly removed into cold artificial cerebrospinal fluid (ACSF). ACSF composition was (in mM): 125 NaCl, 2.5 KCl, 1.25 NaH 2 PO 4 , 20 NaHCO 3 , 5 HEPES, 25 Glucose, 2 CaCl 2 , 1MgCl 2 , pH 7.3, oxygenated with 95% O 2 /5% CO 2 . Parasagittal slices 200 mm thick were prepared using a vibrating slicer (Vibratome, St. Louis, MO) and maintained in the above solution at room temperature for 2 hours before transfer to the microscope stage. On the microscope stage the slice was constantly perfused with the above solution at 35-37uC.
Fluorescence was measured by 2-photon fluorescence microscopy using a custom built microscope based on an Olympus BX51 frame. The specimen was illuminated with 840 nm light from a Ti:sapphire laser (Chameleon Ultra, Coherent, 80 MHz repetition rate; 100-150 fs pulse width). Excitation light was focused onto the specimen using a 640, NA 0.8 or 620, NA 1.0 water-immersion objective (Olympus, Center Valley, PA). Emitted light was collected in the epifluorescence configuration through a 680 nm dichroic reflector and an infrared-blocking emission filter (ET700sp-2p, Chroma Technology, Bellows Falls, VT). Fluorescence was detected via a 490-560 bandpass filter (Chroma Technology) using a photomultiplier tube (R6357, Hamamatsu). Scanning and image acquisition were controlled using custom software written in Labview (National Instruments, Austin, TX).
The rate of diffusion of fluorescent Ab 1-42 was measured in acute slices by 2-photon fluorescence microscopy. Fluorescent Ab 1-42 was applied locally by pressure ejection from a pipette. HiLyte Fluor 488 labeled Ab  was dissolved at 100 mM in modified ACSF (in mM): 135 NaCl, 5.4 KCl, 1 MgCl2, 1.8 CaCl2, 5 HEPES, pH 7.3. A pipette with a tip diameter of ,1 mm was filled with 100 mM fluorescent Ab 1-42 and the tip was positioned in stratum radiatum of the hippocampus, ,100 mm below the surface of the slice. Fluorescent Ab 1-42 was ejected from the pipette with brief a pulse of positive pressure (10-50 ms, 20-30 psi) controlled by a pressure ejector (Toohey Spritzer, Fairfield, NJ). Fluorescence was measured a few mm from the pipette tip using line scans, in which the tissue near the tip of the pipette was scanned repeatedly at 2ms intervals by the excitation laser.
I also measured diffusive loss of fluorescent Ab 1-42 from acute slices. Slices were prepared as described above and immediately placed into 50 mM fluorescently labeled Ab 1-42 in modified ACSF(in mM): 135 NaCl, 5.4 KCl, 1 MgCl2, 1.8 CaCl2, 5 HEPES, pH 7.3, oxygenated with 95% O 2 /5% CO 2 . Solution was constantly stirred by placing the plate on an orbital shaker. After 2 hours (to allow the concentration of fluorescent Ab 1-42 throughout the slice to approach 50 mM), the slice was transferred to the stage of the 2-photon microscope. The fluorescence within the slice was imaged at regular intervals during perfusion with ACSF (no Ab or fluorophore) warmed to 35-37uC.
Measuring loss of fluorescent Ab 1-42 required repeated imaging in the single location for up to ,1 hour. Hence photobleaching and drift in x-, y-and z-axes could potentially confound the results. To minimize photobleaching I used a relatively high concentration of Ab 1-42 , which was bright enough to enable the use of low intensity laser excitation. In addition after imaging the decline in fluorescence through time I verified that the intensity in the imaged region was similar to that in neighbouring regions of the slice. The presence of identifiable structures in the image throughout the imaging period suggested that drift was negligible, particularly parallel to the optical axis of the microscope (x-and yaxes). To further exclude the possibility of drift perpendicular to the optical axis (z-axis) I checked that the slice surface was still 100 mm from the imaging site after measurements of fluorescence through time were complete. Through these controls I ensured that neither photobleaching nor drift in the relative position of the slice and focal position effected these measurements significantly.

Measurement of Ab 1-42 release from acute slices
Coronal slices 300 mm thick were prepared from CRND8 mice as described above. To measure release of Ab species, each slice was transferred to one well of a 24-well plate immediately after preparation. The well contained 250 ml of modified ACSF (in mM): 135 NaCl, 5.4 KCl, 1 MgCl2, 1.8 CaCl2, 5 HEPES, pH 7.3, oxygenated with 95% O 2 /5% CO 2 . Solution was constantly stirred by placing the plate on an orbital shaker. Duplicate measurements of Ab 1-42 concentration in the ACSF were made using a sandwich enzyme-linked immunosorbent assay (ELISA) kit (high sensitive human beta amyloid(1-42) ELISA kit, Wako Chemicals, Richmond VA), which measures Ab  concentrations in the 1-20 pM range. To correct for differences in the volumes of individual slices, we measured the wet weight of each slice at the end of the experiment and normalized Ab  release to that of a 25 mg slice.
Hence Ab 1-42 concentration was measured after release into a relatively small volume of ACSF. 250 ml was selected as the volume of ACSF as this was the smallest volume compatible with duplicate measurements using the ELISA kit, which requires 100 ml per measurement. Hence the volume of a 25 mg slice was 10% of the volume of ACSF. Minimizing the volume of ACSF was necessary as the total amount of Ab 1-42 released from a brain slice is likely to be small.
In some experiments slices were pre-washed in ACSF before the measurement of Ab 1-42 release into ACSF. The pre-wash consisted of incubating the slices in a large volume (,250 ml) of constantly-circulating ACSF at 35uC.

Measurement of Ab release in vivo
To measure the extracellular diffusible Ab 1-42 concentration in vivo (i.e. in intact animals) I implanted a small chamber over frontal cortex, placed a small volume of agarose in the chamber and in contact with the brain for 48 hours and later assayed the concentration of Ab 1-42 in the agarose.
A mouse was deeply anesthetized with isoflurane (2-2.5%, inhaled) such that pinch-withdrawl and palpebral reflexes were eliminated. The skin was reflected and a small aluminium plate was attached to the skull with dental acrylic. A large craniotomy (diameter ,5-6 mm) was opened over frontal cortex, centered ,1 mm posterior to bregma. The brain was covered with 40-60 mg of 0.75-1% agarose (w/v) in ACSF. The craniotomy and agarose were covered with an 8 mm-diameter coverslip and sealed using dental acrylic. The mouse was allowed to recover and was returned to its home cage.
The model predicts that the concentrations of diffusible extracellular Ab species in the agarose are close to the concentrations in brain tissue after ,30 hours and the mouse was therefore left in its home cage for ,60 hours. It was then reanesthetized. The agarose was collected and placed in 350 ml of ACSF for $4 hours, which is sufficient for the Ab 1-42 to diffuse into the ACSF and reach an equilibrium concentration. The ACSF was then collected, diluted (to bring the final ACSF Ab 1-42 concentration into the appropriate range for the ELISA kit) and its Ab 1-42 content was determined with the ELISA kit. The agarose was subsequently weighed to determine the dilution factor into ACSF and thereby calculate the concentration of Ab 1-42 in brain tissue.
A total of 9 mice were used for in vivo experiments, with ages from P100 to P370.

Rate of diffusion of Ab 1-42 in brain tissue
To model the diffusion of Ab out of a brain slice, we need to know the effective diffusion coefficient of Ab in brain tissue. The diffusion coefficient for Ab 1-42 monomer in aqueous solution is available from the published literature and from this value the effective diffusion coefficient of Ab 1-42 monomer in brain tissue was estimated at 0.623610 26 cm 2 /s (see Methods section). Soluble forms of Ab include monomers and oligomers of up to 24 monomers [1] (24mers). Hence I also considered diffusion of 24mers of Ab 1-42 with an effective diffusion coefficient in brain tissue of 0.216610 26 cm 2 /s (see Methods section). To verify that these estimates of effective diffusion coefficients accurately represent diffusion of Ab 1-42 in the brain, I measured the diffusion of fluorescently-tagged Ab 1-42 in hippocampal tissue and compared the spread of fluorescent Ab 1-42 with the calculated distribution expected from theory.
I first calculated the expected distribution of Ab 1-42 in brain tissue after release from a point source. This distribution is Gaussian, with the concentration declining from a peak at the site of release. As molecules diffuse away from the source of release over time, the shape of the distribution remains Gaussian, with the peak amplitude declining and the width increasing with time ( figure 2A and 2B). The amplitude and width of the Gaussian distribution at any moment in time will depend on the diffusion coefficient of the diffusing species. I therefore calculated the expected distributions for both monomer and 24mer using their respective diffusion coefficients. I then measured the rate of diffusion of fluorescent Ab 1-42 in the acute hippocampal slice. A pipette with a tip diameter of ,1 mM was filled with 100 mM fluorescent Ab 1-42 (in ACSF) and was inserted into stratum radiatum, such that the tip was ,100 mm below the surface of the slice. A small volume of fluorescentlytagged Ab 1-42 was ejected into the slice by the brief application of pressure (10-50 ms, 20-30 psi). The diffusion of Ab 1-42 through the tissue near the tip of the pipette was measured by 2-photon fluorescence microscopy ( figure 2C and 2D).
During pressure ejection the fluorescence intensity near the tip of the pipette increased rapidly, reaching a peak a few milliseconds after ejection ended (figure 2D-F). As expected, the fluorescence intensity was well-described by a Gaussian distribution with the peak of the fluorescence close to the tip of the pipette. Furthermore the peak amplitude declined and the width increased with time (figure 2D-F). Hence pressure ejection of Ab 1-42 approximates release of Ab 1-42 from a point source. Importantly, the rate at which the width of the Gaussian fit increased was similar to that expected from theoretical calculations (figure 2F). I measured diffusion in 6 locations, following Ab 1-42 diffusion for 7056 145 ms, during which time the width of the Gaussian fit increased from 6.561.0 mm to 13.762.46 mm. By comparison, expansion of the Gaussian distribution from 6.5 to 13.7 mm was expected, from theoretical calculations, to take 580 ms for monomer and 1.68 seconds for 24 mer. Hence these measurements confirm that Ab 1-42 diffuses rapidly through the extracellular space in brain tissue and that this diffusion can be accurately modeled using effective diffusion coefficients of 0.623610 26 cm 2 /s for monomers and 0.216610 26 cm 2 /s for 24mers of Ab 1-42 .
Model predicts substantial loss of extracellular diffusible Ab species from acute brain slices within 1 hour Using these diffusion coefficients, I calculated the expected rate of loss of diffusible Ab species from the extracellular space of an acute slice, using a simple random walk model of diffusion. The aim was to model loss of Ab in slice physiology experiments, such as experiments in which many investigators have studied the effects of Ab species on synaptic transmission. In such experiments, the brain is typically cut into slices that are 200-to 400 mm-thick and these slices are then incubated in a large volume of ACSF for 30 minutes to several hours before electrophysiological recordings are obtained. I therefore modeled diffusion from a 300 mm-thick section of tissue, bathed on both sides by ACSF ( figure 3A). Ab which exited the tissue into the ACSF was instantly removed. The initial concentrations of Ab were one in tissue and zero in ACSF (figure 3B), with the result that all concentrations in the model are normalized to the initial concentration of Ab in the extracellular space. Using this approach I calculated the concentrations of monomer and 24mer throughout the slice ( figure 3C-H).
Ab concentrations were first calculated for the simplest scenario, in which Ab is not gained or lost (other than by diffusion) from the tissue (no turn-over; figure 3C, D). However, both Ab 1-42 and Ab 1-40 have relatively short half-lives in the CNS (0.7 to 4 hours in APP overexpressing mice [30,31]). Hence Ab concentrations were also calculated with constant turn-over of Ab, assuming a half-life of Ab of 1.28 hours ( figure 3E,F). In both scenarios, the concentrations of monomer and 24mer were greatly reduced at the edges of the slice within a minute of slice preparation ( figure 3C-F). Even deep in the slice, the concentrations of monomer and 24mer were dramatically reduced. Monomer reached a steady-state concentration throughout the slice in 30-60 minutes and 24mer after 1-2 hours ( figure 3G,H). Hence the model predicts that the diffusive loss of Ab from the extracellular space in an acute slice is rapid and profound.

Loss of fluorescently-tagged Ab 1-42 from acute brain slices
To confirm that soluble Ab is indeed rapidly lost from the extracellular space in acute slices I loaded acute slices with fluorescent Ab 1-42 and monitored the decline in intensity as fluorescent Ab 1-42 diffused out of the slice. Immediately after preparation acute slices were placed in ACSF containing 50 mM fluorescent Ab 1-42 . After 2-3.5 hours (long enough for the concentration of fluorescent Ab 1-42 in the extracellular space to approach 50 mM; see 'Kinetics of exogenous Ab accumulation in slices' section, below) the slice was placed on the stage of the 2photon microscope and the fluorescence intensity 100 mM below the upper surface of the slice was monitored for up to ,1 hour as the slice was perfused with fresh ACSF, containing no Ab or fluorophore. As expected the fluorescence intensity declined with time (n = 3 slices; figure 4). Furthermore, the decline in intensity with time matched the rate of loss of Ab expected from the model (figure 4B), occurring with a time constant of 18.067.0 minutes (n = 3 slices). I conclude that extracellular Ab is lost rapidly from acute slices with the time course predicted by the diffusion model.

The concentration of diffusible extracellular Ab 1-42 in CRND8 mice in vivo
Hence it seems likely that endogenous soluble Ab species are rapidly lost from the extracellular space in acute slices with the time course predicted by the model, but the diffusion of fluorescent Ab 1-42 might differ from that of endogenous Ab 1-42 . For example, the fluorescence tag might affect diffusion or the high concentration of fluorescent Ab 1-42 that was, necessarily, added to the slice might stimulate or saturate pathways of Ab metabolism or clearance. I therefore sought to determine the rate of loss of Ab 1-42 from acute slices from CRND8 mice, which overexpress APP and accumulate Ab 1-42 in the hippocampus and neocortex. The concentration of endogenous Ab 1-42 in the extracellular space of an acute slice is difficult or perhaps impossible to measure directly. My aim was therefore to combine the model with measurements of Ab 1-42 release from acute slices to determine what proportion of diffusible Ab remains in the extracellular space as a function of time after slice preparation.
First I measured the concentration of diffusible, extracellular Ab 1-42 in the intact brain. A craniotomy was cut over frontal cortex and a small reservoir of agarose was placed in contact with the brain surface. In this preparation Ab species diffuse from the brain and into the agarose and the concentrations of the mobile Ab species in the agarose increase until their concentrations equal those in the brain. The agarose was then removed and the concentration of Ab 1-42 in the agarose was measured with an ELISA kit. First the numerical model was employed to estimate the duration for which agarose needs to be in contact with the brain for the concentration of Ab in the agarose to reach steady-state. The model (#2) simulated diffusion from the brain into a 1 mmthick layer of agarose (figure 5A). As in the previous model, the tissue Ab concentration was initially set to 1 and the concentration in agarose to 0, thereby normalizing Ab concentrations to the initial tissue concentration ( figure 5B). In the model, the Ab concentration in the superficial millimeter of brain tissue declined rapidly as Ab was lost by diffusion into the agarose (figure 5C).
The model predicts that after 30 hours, the concentrations of Ab in the agarose would be 97% and 82% of the initial Ab concentration in the brain for monomer and 24mer, respectively (figure 5D). In experiments, agarose was therefore placed in contact with the brain surface for ,60 hours ensure that the Ab concentration in the agarose was an accurate reflection of the concentration in the brain. The concentration of Ab 1-42 in the agarose, and therefore of extracellular soluble Ab 1-42 in the brain, increased with age (figure 5E). Mean concentrations were derived from a regression line and were 160 pM at 3 months, 250 pM at 6 months and 425 pM at 12 months of age (figure 5E). These are the first measurements of the concentration of soluble Ab in the extracellular space in CRND8 mice. The results indicate that the concentration of soluble Ab 1-42 in the extracellular space increases Release of Ab 1-42 from slices is greater than expected from the model Next I measured the amount of Ab 1-42 released from acute brain slices from CRND8 mice. In physiology experiments, slices are typically cut from the brain and allowed to recover from the slicing process for at least 30 minutes, often several hours, before further use. To mimic a typical slice physiology experiment, after preparation, slices were placed in a large-volume holding chamber for 15-120 minutes (hereafter referred to as the recovery period). Slices were then transferred to a smaller chamber containing 250 ml ACSF and after 30 minutes the concentration of Ab 1-42 in the ACSF was measured using an ELISA kit.
The concentration of Ab 1-42 in the ACSF was between 4 and 8 pM for slices from 4.5 month-old (P132) mice ( figure 6A) and, as expected, the amount of Ab 1-42 released from slices from older mice was greater, at 11-15 pM for 5.5 month-old (P163) mice ( figure 6A). The duration of the recovery period had little effect on the amount of Ab 1-42 released during the measurement period ( figure 6A). This weak effect of the recovery period on the Ab 1-42 concentration in the ACSF is expected from the model, in which the Ab concentration in the slice declines rapidly towards a steadystate concentration, in around 30 minutes. The steady-state concentrations of Ab 1-42 in ACSF (mean 6 SEM of concentrations after 60, 90 and 120 minute recovery periods) were 7.060.4 pM for P132 mice and 13.061.1 pM for P163 mice. These concentrations are 3.5% and 5.6%, respectively, of the concentrations in vivo at these ages (calculated from the regression line in figure 5E), yielding a mean normalized concentration of Ab  in ACSF bathing slices of 0.045.
The measured concentrations of Ab  in the ACSF are greater than expected from the model. The model (#3) predicts that the normalized concentration of Ab  in the ACSF with a 60-120 minute recovery period will be 0.026 for the monomer (figure 6B) and 0.025 for the 24mer (not shown). Ab  in vivo at P132 and P163 were 203 and 234 pM. Hence the expected concentrations of Ab  in ACSF are 5.2 and 6.0 pM for P132 and P163 mice ( figure 6B) and more Ab 1-42 is released into the ACSF than expected.
Could the rate of diffusion of endogenous Ab 1-42 be greater than expected and might this result in a greater concentration of Ab   The steady-state Ab concentration in a slice is ,20% of the concentration in vivo  and 24mer, respectively ( figure 7C). If small oligomers predominate in CRND8 mice, the mean tissue concentration of Ab at steady state in an acute slice is likely to be towards the lower end of this range, around 20% of the in vivo concentration.
In slices, a concentration gradient of Ab will presumably exist, with relatively high concentrations of Ab species deep in the slice and lower concentrations towards the edges of the slice. From the model we calculated these concentration gradients for both monomer and 24mer, using a half-life of 0.64 hours ( figure 7D). The highest concentration of Ab will be at the center of the slice. The concentration of monomer at the center of the slice was 23% of the initial concentration. Hence physiology experiments will typically report the properties of cells and synapses in relatively low concentrations of extracellular soluble Ab. This problem may be particularly acute with whole-cell recordings, which are typically obtained from somata approximately 50 mm from the cut surface of the slice. 50 mm from the edge of the slice, the concentration of monomer was 15% of the initial concentration. For 24mer, the concentration at the center of the slice was 45% and 50 mm from the edge of the slice was 28% of the initial concentration. Hence the concentrations of soluble Ab throughout the slice are low and in the superficial regions of the slice, where most whole-cell electrophysiological recordings are obtained, concentrations are likely to be extremely low: probably 10-20% of the concentration of Ab in vivo.

Kinetics of the changes in Ab concentration after slice preparation
Having determined that most of the extracellular soluble Ab is eventually lost from acute brain slices, I next investigated the rate at which the Ab concentration approaches steady-state after slice preparation. I first used the model to calculate the expected time course of Ab in the tissue. The tissue concentrations (averaged throughout the depth of the slice) of monomer and 24mer both declined rapidly towards steady-state concentrations (figure 8A). The time taken to reach a steady-state concentration of Ab in the tissue was expressed as the 90% completion time, defined as the time taken for the concentration of Ab to decline by 90% of the difference between the initial concentration and the steady-state concentration. The steadystate concentrations decreased and the 90% completion time increased with increasing half-life ( figure 8A and B). As half-life tended towards infinity (no turn-over of Ab), the 90% completion time tended towards asymptotes of 23.3 minutes for the monomer and 66 minutes for the 24mer. Hence the model predicts that the 90% completion times in slices (in which there is turn-over of Ab) are less than 23.3 minutes for monomer and 66 minutes for 24mer. For a half-life of 0.64 hours, the predicted completion times were 18.3 minutes for monomer and 37.3 minutes for 24mer.
To measure the kinetics of the changes in Ab concentration after slice preparation, each slice was placed in a small-volume chamber, containing 250 ml of ACSF, immediately after preparation. The concentration of Ab 1-42 in the ACSF was measured after 1 to 180 minutes in the well. The model (#4) predicts an initial rapid increase in the concentrations of monomer (figure 8C) and 24mer (not shown) in the ACSF. This initial phase lasts less than 30 minutes (figure 8C). Thereafter the ACSF concentration continues to rise approximately linearly, with the gradient increasing with decreasing half-life of Ab ( figure 8C).
In contrast to the predictions of the model, the measured Ab 1-42 concentration decreased during the first hour after slice preparation. ACSF Ab 1-42 concentration declined by approximately 40%, from a peak of 4.560.85 pM at 5 minutes to 2.460.31 pM at 1 hour (P,0.05, paired t-test; figure 8B). Thereafter the ACSF Ab 1-42 concentration stabilized and possibly increased slightly during the next hour (to 3.060.5 pM at 2 hours; not significantly different from concentration at 1 hour).
This decline in ACSF Ab 1-42 concentration with time was unexpected. I considered whether this might be a methodological artifact, such as binding of Ab to the plastic walls of the 24-well plate. To test for this, measurements were repeated with a recovery period in which each slice was placed in a large volume of ACSF for 15 minutes before being transferred to the 24-well plate. A methodological artifact, such as binding to plastic, would be unaffected by the recovery period and the decline would therefore persist. In contrast, much of the Ab 1-42 from an initial elevation in Ab 1-42 would exit the slice within ,30 minutes, reducing the Ab 1-42 concentration in the first few wells and suppressing the decline in Ab  .
Normalized ACSF concentrations of Ab 1-42 were initially (t#10 minutes) lower with than without the recovery step (figure 8E) and the decline in ACSF Ab 1-42 concentration with time was eliminated. For both conditions, the Ab 1-42 concentration increased gradually after 1 hour (figure 8E). Hence a 15 minute recovery period reduced the initial elevation and subsequent decline in Ab 1-42 concentration, indicating that the changes in Ab 1-42 concentration are not a methodological artifact. Hence Ab 1-42 was removed from the ACSF during the first hour after slice preparation. Immediately after slice preparation the extracellular diffusible Ab 1-42 concentration therefore follows a complex kinetic scheme: it initially increases, likely as a result of release of Ab 1-42 during slice preparation, and then declines over the first 30-60 minutes before reaching a steady-state condition.
Why does the concentration of Ab 1-42 in the ACSF decline with time? Presumably Ab 1-42 is taken-up by the tissue or catabolized via one of the many pathways for enzymatic degradation of Ab 1-42 [33]. Uptake and metabolism of Ab are incorporated into the model through the half-life calculations. In the model, as Ab concentration in the tissue declines, the balance between Ab release/synthesis and Ab uptake/catabolism tips towards net release/synthesis. This imbalance, along with diffusion of Ab, results in the increase in ACSF Ab concentration with time ( figure 8C). The measured decline in Ab 1-42 concentration (figure 8D) suggests that uptake/catabolism and release/synthesis are unbalanced in the other direction, resulting in net uptake/ catabolism. Such an unbalance might occur if the concentration of Ab is elevated. Slicing brain tissue inevitably results in the death of neurons, particularly near the cut surfaces of the slice, and dying cells may release Ab. In addition, slice preparation may elevate synaptic activity, which can release Ab from synaptic terminals [20,34]. Ab concentration may therefore be elevated immediately following slice preparation.
Elevating the initial Ab concentration in the model resulted in a decline in ACSF Ab concentration with time ( figure 9). The decline became faster and more pronounced as the initial Ab concentration was increased and the half-life for Ab was decreased (figure 9), but the decline predicted by the model was slow, presumably because the slice occupies only a small proportion (10%) of the volume of the chamber. Elevating the initial Ab concentration in the model, even by several orders of magnitude, failed to reproduce the observed rapid decline in ACSF Ab  concentration. Hence release of Ab during slice preparation is unlikely to entirely account for the decline in ACSF Ab concentration during the first hour and additional factors, that are not incorporated in the model, are likely responsible for the decline in Ab 1-42 concentration after slice preparation.

Kinetics of exogenous Ab accumulation in slices
Finally I used the model to calculate the rate at which Ab enters a slice when the slice is bathed in ACSF containing exogenous Ab. In this model (#5) the initial concentration of Ab in the slice was set to zero and the concentration in ACSF to one (figure 10A). The concentration of Ab in the slice approached the concentration in the ACSF within 30 minutes (monomer) to 1 hour (24mer) throughout the depth of the slice ( figure 10B and 10C). Hence diffusion of soluble Ab species in brain tissue not only results in the rapid loss of endogenous extracellular Ab from slices, but also a rapid rise in Ab concentration within the slice during the application of exogenous Ab.

Discussion
I have measured extracellular Ab 1-42 concentrations in vivo and in acute brain slices from CRND8 mice. The extracellular concentrations of Ab species have been measured previously in the intact mouse brain by microdialysis and ELISA [30], but the size of microdialysis probes precludes their use in acute slices, which are only 200-400 mm thick. Instead I determined the extracellular concentration of Ab 1-42 in acute brain slices indirectly, measuring the concentration of Ab 1-42 released and using a numerical model to calculate, from these measurements, the extracellular concentration of diffusible Ab 1-42 in the tissue.
My principal conclusion is that the concentration of Ab 1-42 in acute slices is reduced by the loss, by diffusion, of most of the soluble extracellular Ab 1-42 . The steady-state concentration of Ab 1-42 in the slice was dependent on the parameters used in the model, but the calculated Ab 1-42 concentration was greatly reduced across a wide range of model parameters. Near the surfaces of the slice, the calculated steady-state Ab 1-42 concentration was less than 20% of the concentration in vivo and even deep within the slice the concentration was only ,20-30% of the in vivo concentration.

Heterogeneous distribution of Ab and plaques in CRND8 mice
CRND8 mice carry two APP mutations (KM670/671NL 'Swedish' and V717F 'Indiana') and elevated levels of Ab in hippocampus and through much of the neocortex from a young age [35]. CRND8s develop many of the histological and cognitive deficits of patients with AD, including amyloid plaques in hippocampus and neocortex [16,[35][36][37]. CRND8 mice do not carry mutations in other genes or exhibit tau hyperphosphorylation or neurofibrillary tangles and are therefore an excellent model in which to study the effects of Ab.
In my models I treated the slice as homogenous, but for slices from CRND8 mice, hippocampus and neocortex are likely to be the main sources of Ab released into the ACSF. Hippocampus and neocortex together account for ,50% of the coronal slices used in my experiments. Hence if little or no Ab is released from other regions, the amount of Ab released per unit volume of hippocampal and neocortical tissue might be higher than calculated in the results. The loss of Ab from slices may therefore occur more rapidly than suggested by my simplified models and my calculations likely provide a conservative estimate of the rate of Ab loss from acute slices.
Amyloid plaques first appear in the hippocampus and neocortex of CRND8 mice at 2-3 months of age and the plaque load increases steadily thereafter. Most of the mice used here were 2-6 months old, ages at which the plaque load is light. Whether plaques affect diffusion of soluble Ab species is unclear, but it is possible that the increasing plaque load with age would impede diffusion of Ab and retard loss of Ab from slices. However, given the relatively small percentage of the total volume of hippocampus and neocortex occupied by plaques, even in old mice, it seems unlikely that plaques would substantially alter diffusive loss of Ab from acute slices.  may be lost or moderated after slice preparation. For example, in acute slices from 2-6 month-old CRND8 mice the effect on LTP may be lost as the concentration of soluble extracellular Ab 1-42 declines from 300-500 pM in vivo to 100 pM or less in slices. As the mice age, and the soluble extracellular Ab 1-42 increases, the effects of loss of Ab may change. The rate at which loss of Ab alters synaptic physiology in slices is unclear. Loss of Ab 1-42 from acute slices is rapid, but Ab molecules bind to plasma membranes [38]. Hence Ab might remain bound to target molecules despite the falling extracellular concentrations of diffusible Ab species.
I also estimated the rate of entry of exogenous Ab, applied in the ACSF. The model predicts that exogenous Ab enters the slice quickly, reaching a steady-state concentration equivalent to the concentration in ACSF in 30 minutes to an hour. Unsurprisingly, this time course is similar to that for loss of endogenous Ab. Turnover of Ab was not considered in the model of exogenous Ab entry. Some authors have applied exogenous Ab at extremely high concentrations, far higher than endogenous concentrations of Ab (e.g. [43]). At these concentrations the relevant metabolic pathways may be unable to significantly affect Ab concentration and it is therefore unclear how to model turn-over in these cases. Fortunately in my models the effect of turn-over on the rate of change of concentration is relatively weak (see figure 8 A and B, for example). My results suggest that effects of exogenously applied Ab observed in less than 30-60 minutes are likely to result from lower concentrations of Ab than those in the perfusing ACSF and that this Ab must be acting extremely quickly. For example, effects of exogenous Ab on synaptic plasticity have been reported following application of only 50-200 pM exogenous Ab for 20 minutes (e.g. [22,24]). My results suggest that such effects must result from a site of action which is exclusively on the surface of the slice or from an extremely rapid action of lower concentrations of soluble Ab species than are present in the extracellular space in many APP mouse lines.
In summary, here I have shown that the concentration of soluble Ab 1-42 in the extracellular space declines during the first hour after preparation of acute brain slices from CRND8 mice. Electrophysiological recordings in acute slices are usually obtained at least 30 minutes and often several hours after slice preparation. Hence recordings in acute slices from APP overexpressing mice are obtained when extracellular Ab is at a steady-state concentration, which is far lower than the concentration of Ab in vivo. In acute slices prepared from CRND8 mice or other APP overexpressing mouse lines, diffusive loss of Ab from the tissue after slice preparation may eliminate or moderate some of the effects of Ab, greatly complicating interpretation of physiology experiments aimed at elucidating the effects of Ab.