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Michael AC, Borland LM, editors. Electrochemical Methods for Neuroscience. Boca Raton (FL): CRC Press/Taylor & Francis; 2007.

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# Chapter 10Biophysical Properties of Brain Extracellular Space Explored with Ion-Selective Microelectrodes, Integrative Optical Imaging and Related Techniques

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## Introduction: Extracellular Space

The extracellular space (ECS) may be imagined as a system of interconnected channels demarcated by cellular membranes (Figure 10.1) and filled with an ionic solution, primarily NaCl, along with macromolecules of the extracellular matrix, negatively-charged proteoglycans and glycosaminoglycans (Margolis and Margolis 1993; Ruoslahti 1996; Novak and Kaye 2000). The ECS mediates intercellular communication (Nicholson 1979) and the transport of nutrients and metabolites (Syková et al. 2000), and it forms a reservoir of ions that establishes the resting potentials of cells and mediate fluxes across the membranes. It also may serve to deliver therapeutic substances to cells (Ulbrich et al. 1997; Saltzman 2001). All these processes are primarily mediated by diffusion. The ECS diffusive properties are thus critical for neurotransmission over short and long distances (the latter being called volume transmission, Fuxe and Agnati 1991; Agnati et al. 2000), neurotrophic effects, general electrical activity, basic cellular homeostasis, and drug delivery. This chapter will introduce the brain ECS, describe techniques and preparations most commonly used to measure diffusion properties together with relevant theory and software for data analysis, review selected findings, demonstrate the utility of mathematical modeling and computer simulations, and outline some future directions for research.

#### FIGURE 10.1

Electron micrograph of a small region of the cerebral cortex of a rat with a prominent synapse. The black areas between cells indicate the ECS, which may have been reduced in size as a consequence of the histological processing. The asterisk indicates (more...)

### Brief History

The middle years of the 20th century witnessed great controversy about the ECS. Experiments that measured the distribution of sodium and chloride in the brain, both regarded as having a predominantly extracellular distribution, indicated that the ECS might occupy as much as 40% of brain volume while emerging electron microscopy of the CNS indicated that the spaces between cells were so small that the ECS might only occupy 5% of the brain volume or even be absent altogether (Horstmann and Meves 1959; Gonzalez-Anguilar 1969; Johnston and Roots 1972).

After it was recognized that sodium and chloride had an appreciable intracellular distribution, investigators turned to radiolabeled sucrose, inulin and other membrane impermeant markers to measure the ECS volume fraction, resulting in values of 15%–20% (Fenstermacher and Kaye 1988). At the same time it became evident that the ischemia that occurred during fixation and also the subsequent chemical processing associated with conventional electron microscopy cause a drastic reduction of ECS volume (Van Harreveld 1972). Later studies (ibid.) employed freeze substitution, which to some extent remedies the shortcomings of conventional fixation, and also arrived at an ECS volume fraction of 15%–20%. These estimates are close to the values typically obtained with the modern diffusion methods that will be described in detail in this chapter. The existence of a substantial ECS then became an accepted concept and, the structure of the ECS was likened to the water phase of a foam (Kuffler and Potter 1964).

Even while some questioned the existence of the ECS, others debated its content. It was recognized early it its history that the ECS could not merely be filled with just a facsimile of cerebrospinal fluid (CSF) but also likely contained a “ground substance” composed of “mucopolysacharrides” and related molecules (Schmitt and Samson 1969; Johnston and Roots 1972). Today the reality of what is now called the extracellular matrix (ECM) is accepted but the composition, amount, and distribution are still debated.

### Volume Fraction and Width

The macroscopic properties of the ECS are summarized by two aggregate parameters: the volume fraction and the tortuosity (discussed in the next section). The volume fraction α represents the proportion of tissue volume (V) occupied by the ECS; α = VECS/Vtissue. In healthy isotropic brain, α = 0.2 (Nicholson and Syková 1998; Nicholson 2001), which, as noted in the previous section, means that typically the ECS occupies 20% of brain tissue. From a practical point of view, α determines the dilution of substances released to the ECS.

In addition to α, the width and shape of a two-dimensional (2D) cross section of individual ECS channels can be estimated from electron micrographs. The gaps between the cells were estimated to be 10–20 nm wide (Brightman 1965; Johnston and Roots 1972, Van Harreveld 1972) when brain tissue was processed using freeze substitution or other techniques that were thought to preserve the dimensions. However, it is becoming evident that electron micrographs underestimate the width of ECS channels. Recent diffusion studies employing large ECS probes, such as the polysaccharide Ficoll (hydrodynamic diameter 41 nm, Hrabětová and Nicholson 2002) in rat neocortex in vitro or quantum dots (hydrodynamic diameter 34 nm, Thorne and Nicholson 2006) in rat neocortex in vivo suggest that at least some interstitial spaces are sufficiently wide and interconnected to permit these large entities to diffuse in the ECS. Regarding the shape of ECS channels, electron micrographs uncovered many aspects such as widely distended regions, “lakes” (Van Harreveld et al. 1965) and intercellular gaps of uneven width (Brightman 1965; Bondareff and Pysh 1968; Cragg 1979), although it cannot be excluded that some might be artifacts caused by ice crystal formation during the freeze substitution phase of the fixation procedure. Electron microscopy also revealed regions of the ECS surrounded by glial processes (Špaèek 1985; Kosaka and Hama 1986; Grosche et al. 1999). The functional significance of these morphological variations of the ECS channels remains unknown.

### Hindrance and Tortuosity

Molecular transport in brain ECS is hindered by the structure of the tissue compared to a free medium. Diffusion analysis of small extracellular marker molecules quantifies this hindrance, expressed as tortuosity λ = (D/D*)1/2, where D is the free diffusion coefficient and D* is the effective diffusion coefficient in tissue (Nicholson and Phillips 1981; Nicholson 2001). In isotropic healthy brain, λ is = 1.6 (Nicholson and Syková 1998; Nicholson 2001). From a practical point of view, λ determines the ease of spread of molecules in the ECS.

Unlike α, the nature of λ is incompletely understood; it has proven difficult to ascertain which structural elements of the brain determine λ. Cells and their processes create obstacles that diffusing molecules have to navigate around; thus geometry of the ECS, as determined by the shape of cells, undoubtedly contributes to the tortuosity of brain tissue. The open question is whether ECS geometry alone can account for the value of this parameter or whether other factors, such as viscosity of the extracellular fluid arising from the presence of the ECM, come into play. We shall return to this issue using computer simulation techniques in the Section: “Monte Carlo Simulation of Diffusion in 3D Media.”

### Bulk Flow

It has been postulated from time to time that there is a bulk flow of interstitial fluid within the ECS, but conclusive evidence is still lacking. The most probable origin of bulk flow would be an osmotically-balanced fluid that is secreted across the blood–brain barrier (BBB), although some flow might be generated by cells when water is produced through metabolism (Abbott 2004). The fluid then flows into the CSF or enters the lymphatics (ibid.). The actual route for the flow was indicated by Cserr and colleagues (Ichimura et al. 1991) who concluded that bulk flow was restricted to the Virchow–Robin perivascular space surrounding the capillaries, rather than the whole ECS. The flow velocity has been estimated at only about 10 μm min−1 towards the ventricle in white matter and may not occur in grey matter at all under normal conditions (Rosenberg et al. 1980). These conclusions make it unlikely that bulk flow affects the diffusion measurements to be described in this chapter.

### Volume Transmission and Drug Delivery

Electrically active neurons cause voltage changes in the immediate ECS, and synchronized populations can lead to quite widespread potentials; indeed this is the basis of the familiar EEG diagnostic recording. The invention of ion-selective microelectrodes (ISMs) established that the electrical fluctuations in the ECS were often accompanied by ionic variation, especially in extracellular potassium, which is typically at a low concentration of about 3 mM. Thus the idea arose that the electrical or ionic changes might transmit signals between brain cells along pathways, which were independent of the conventional synaptic connections, so the ECS reasonably might be regarded as a communication channel (Nicholson 1979). Little evidence has been found to support a role for these signal mediators in normal brain function, perhaps because they are non-specific, but the idea has been generalized into the concept of volume transmission (Fuxe and Agnati 1991) involving more complex molecules and receptors. The signaling domain can range from the local spillover of transmitter at a synapse, enabling the molecules to reach extrasynaptic receptors on the same or adjacent cell, to long distance communication using the cerebral ventricles as conduits for hormones (Agnati et al. 2000).

Volume transmission involves endogenous substances, either released by cells or entering from the ventricles. Obviously the ECS is also capable of distributing exogenous substances—drugs—so long as they can be introduced into the brain without undue damage (Saltzman 2001). Like volume transmission, drug delivery will rely on diffusion to transport substances but this is not effective over long distances. For this reason convection enhanced delivery (CED) is gaining popularity (Morrison et al. 1994). This uses pressure injection from a cannula to force a drug through the brain; after the pressure injection ends, the drug distribution is again dominated by diffusion (ibid.).

## Tools and Methods to Study Extracellular Space in Real Time

As mentioned above, two independent approaches have been adopted to study the ECS in the past: morphometric and diffusion methods. The morphometric method employs electron micrographs that directly visualize the interstitial channels in a small 2D region of fixed tissue, typically containing a few cells. The diffusion method uses the distribution of extracellular probe molecules or markers to quantify the aggregate parameters of the ECS from a larger 3D region, typically containing hundreds of cells. Individual interstitial channels are below the resolution of current diffusion methods. Diffusion methods offer two important advantages over the morphometric approach. First, diffusion methods can yield both α and λ, while the morphometric method quantifies only α (and, as noted, can be subject to systematic error caused by tissue fixation techniques). Second, and more importantly, contemporary real-time diffusion methods can be exploited to study dynamic changes of the ECS during normal and pathological conditions in living tissue, while morphometric methods are limited to fixed tissue.

### Diffusion Measurements Reveal Extracellular Space Properties

The diffusion methods were pioneered by a group around Fenstermacher and Patlak (Rall et al. 1962, Levin et al. 1970) using radiotracers, while the contemporary, real-time techniques, which do not employ radiotracers, were developed by Nicholson and coworkers (Nicholson and Philips 1981; Nicholson 1993; Rice and Nicholson 1995). In principle, in all diffusion methods, a probe molecule is delivered to the ECS, and its subsequent distribution is measured and analyzed using the appropriate solution of the diffusion equation. The marker can be any molecule that stays predominantly in the ECS and can be quantitatively detected. In the most reliable early studies, radiolabeled sucrose or inulin were delivered into the ventricle and their distributions were obtained post mortem from sections of the caudate nucleus, the region adjacent to ventricles. The parameters measured in several species were in the range 15%–20% for α and 1.52–1.64 for λ (Fenstermacher and Kaye 1988). Although the diffusion of a radiotracer occurred in living tissue, the quantification was performed in post mortem tissue yielding a single time point. This drawback was overcome by Nicholson who introduced new methods based on release of a probe ion or molecule from a point source (Figure 10.2) and detection with ion-selective microelectrodes, voltammetry or fluorescence microscopy. In these techniques, the diffusion of probe molecules in the ECS of living tissue is detected in a real time and this facilitates experiments involving tissue manipulation. This chapter will largely focus on these contemporary real-time diffusion techniques.

#### FIGURE 10.2

Point source paradigm for diffusion studies. Glass micropipette (source micropipette) is filled with extracellular probe molecules that are released to the ECS by passing current (iontophoretic delivery of small charged molecules) or by pressure injection (more...)

### The Diffusion Equation and ITS Meaning

Typically, diffusion measurements are made over distances of the order of 100 μm and this ensures that the local anatomical properties are averaged, enabling a modification of the classical diffusion equation of Adolf Fick to be used (Nicholson and Phillips 1981; Nicholson 2001):

$∂C∂t=Dλ2∇2C+Qα-v·∇C-f(C)α.$
10.1

The concentration, C, is that which exists in the narrow interstitial space; it is a function both of time t and position, represented by x, y, z in a rectangular coordinate system, or the single coordinate r in spherically symmetric coordinate system centered on the point source, as is frequently assumed here. The symbols ∇ and ∇2 symbolize the first and second spatial derivatives in the appropriate coordinate system. The averaged tissue structure comes into Equation 10.1 through the two nondimensional parameters mentioned above, the volume fraction (α) and the tortuosity (λ).

The various terms of Equation 10.1 may be interpreted as follows. The term on the left of the equality sign represents the way the interstitial concentration changes with time at a given location. The first term on the right of the equality sign is the contribution of diffusion itself. The free diffusion coefficient is D and the effective diffusion coefficient in brain, D* = D/λ2, indicating the role of tortuosity in reducing the free diffusion coefficient. The next term is the source-term, Q, that may be used to describe iontophoresis from a micropipette (Nicholson and Phillips 1981; Nicholson 1992), or release of molecules by pressure ejection from a micropipette (Nicholson 1985, 1992). The third term represents the possible contribution of bulk flow. Flow is characterized by a velocity vector v that forms a scalar product with the concentration gradient ∇C. The final term, f(C), represents irreversible loss or clearance of material from the ECS. This may occur through uptake into cells, loss across the BBB, or simply by the destruction of the molecule. Often clearance is proportional to local concentration and we can write f(C) = −k′αC. On the other hand, if f(C) represents uptake that is governed by, say, Michaelis–Menten kinetics, Equation 10.1 becomes nonlinear and the analysis is more complex (Nicholson 1995). Equation 10.1 is quite general and must be solved for specific experimental situations. Some of the elementary solutions will be detailed in later sections.

### Choice of Molecular Probes for Extracellular Space

The aim of the diffusion study determines the type of molecular probe employed. In the first type of study, we can quantify two macroscopic parameters, λ and α, that encapsulate the properties of the ECS. For this purpose, the ideal probe should fulfill the following criteria: water solubility; small size; no interaction with the extracellular environment; no membrane permeability; and, for in vivo studies, no transport across the BBB. This probe would visit all channels of the ECS and report true, probe-independent, ECS parameters. In the second type of study, a probe significantly violating one or more criteria outlined above is used. The use of such non-ideal probes allows us to ask questions about: (1) the width of the ECS channels; (2) interactions (non-specific and specific binding); and (3) loss from the ECS (transport into cells or across the BBB). Such probes might not visit all channels of the ECS or might be detained in, or lost from, the ECS compartment. The experimental outcome combines the basic properties of the ECS with one or more probe-specific elements. In the following paragraphs, some examples will be given for the two categories of molecular probes; a more comprehensive overview has been given by Nicholson (2001).

The truth is that an ideal probe molecule that would belong to the first group does not exist and molecules used in this context do not fully comply with criteria for an ideal probe. Molecular probes most commonly used in real-time diffusion techniques (Table 10.1) are small but of finite size; may carry a charge, thus introducing the possibility of interactions with negatively charged molecules of the ECM; and are lost in small amounts from the ECS. Based on studies reported to date, however, they appear to be a good approximation of an ideal probe. For example, α-naphthalenesulfonate, the largest ion listed in Table 10.1, has a hydrodynamic diameter (calculation based on a free diffusion coefficient) of only ≈ 1.0 nm, about 10 times less than the lower estimate of the width of an ECS channel from electron micrographs (10–20 nm, Brightman 1965; Johnston and Roots 1972; Van Harreveld 1972). Because both monovalent cations and anions measure similar λ (Nicholson and Phillips 1981), no charge-based interactions are manifested. Finally, non-specific clearance (also called a non-specific uptake), which quantifies the loss of a probe from the ECS (Nicholson 1992, 2001), is small in both in vitro and in vivo preparations (k′ ≈ 10−3 s−1), or is even absent when brain slices are held in an interface rather than a submersion chamber (Nicholson et al. 2000). In summary, small ions can be used as extracellular probes in the diffusion analysis of ECS properties; in isotropic healthy brain tissue, α ≈ 0.2 and λ ≈ 1.6 (Figure 10.3, Nicholson 2001; Nicholson and Syková 1998). These estimates of α and λ agree well with radiotracer studies employing sucrose, a small (MW 342), neutral, membrane impermeable molecule (Table 10.1; Fenstermacher and Kaye 1988).

#### TABLE 10.1

Selected Measurements of Tortuosity and Volume Fraction of Brain ECS Measured with Molecules Approximating Ideal

#### FIGURE 10.3

Diffusion of tetramethylammonium (TMA) detected with an ion-selective microelectrode (ISM) in rat neocortex in vitro. Tetramethylammonium (MW 74) was delivered by an iontophoretic pulse (+00 nA, 50 s) and the concentration was measured with a TMA+-ISM (more...)

The studies employing “imperfect” probes from the second category have utility even for investigators not primarily interested in the structure of the ECS. Using inert globular molecules of increasing molecular weight, such as dextrans, violates the criterion of small size. We observe that as the hydrodynamic diameter increases, λ increases as the hindrance to diffusion rises (Figure 10.4). Note that the largest molecule shown in Figure 10.4 has a diameter approaching the average electron microscopic estimate of the width of an ECS channel. When even larger molecules are used, such as Ficoll or quantum dots (Hrabětová and Nicholson 2002; Thorne and Nicholson 2006) with hydrodynamic diameter significantly larger than the upper estimate of width of the ECS (20 nm), one is surprised that they diffuse in brain at all. This finding indicates that the width of at least some channels of the ECS is larger than previously thought, and that these wide channels are sufficiently interconnected to permit macroscopic diffusion processes. In summary, using probes of large sizes reveals new information about the biophysical properties of the ECS and has practical implications for effective design of macromolecular drug carriers and viral vectors underpinning gene therapy.

#### FIGURE 10.4

Tortuosity as a function of hydrodynamic radius. Diffusion of TMA+ (MW 74, white circles) in a rat neocortex in vivo and in vitro was quantified using real-time iontophoresis (RTI). The diffusion of fluorophore-labeled dextran molecules (MW 3,000, black (more...)

Examples of an interaction between the probe molecule, specific receptors and the extracellular environment include measurement of the diffusion of neurotransmitters (Rice et al. 1985; Cragg et al. 2001; Cragg and Rice 2004—see Section: “Diffusion Measurements Using Dual Probe Microdialysis Probes”); second messengers such as calcium (Morris and Krnjevic 1981; Nicholson and Rice 1987; Hrabětová and Nicholson 2004); and trophic factors (Krewson et al. 1995; Stroh et al. 2003; Thorne et al. 2004).

Finally, the older studies with radiotracers showed that non-specific uptake of a molecular probe into cells can be measured when mannitol or creatinin are used, and loss across the BBB can be determined when ethylene glycol or urea are employed (reviewed in Nicholson 2001). In midbrain, specific uptake of neurotransmitter dopamine (DA) can be quantified when the diffusion is measured in the absence and presence of dopamine uptake blockers (Figure 10.5; Cragg et al. 2001).

#### FIGURE 10.5

Diffusion analysis determines specific uptake of neurotransmitter dopamine (DA) in midbrain. Dopamine was released by pressure injection and detected with a carbon fiber microelectrode (CFM) using fast-scan cyclic voltammetry (FCV). To quantify the specific (more...)

By combining studies, the biophysical properties of the ECS can be explored using diffusion analysis. True, probe non-specific, diffusion parameters can be measured with small exogenous ions. In contrast, probe molecules of large size, strong charge, containing binding motifs or those that can penetrate cellular membranes or move across the BBB, explore molecular interactions with extracellular environment. Next, we will discuss the methods of delivery and the detection; a schematic of the recording setup used for diffusion measurements is shown in Figure 10.6.

#### FIGURE 10.6

Recording setup for diffusion measurements using real-time iontophoretic (RTI) and integrative optical imaging (IOI) methods. The specimen (dilute agarose gel, brain slice, or living animal) is placed on the stage of a compound microscope. For RTI, an (more...)

### Delivery of Molecular Probes: Iontophoresis and Pressure Ejection

Two methods of delivery are employed in point source real-time diffusion techniques: iontophoresis and pressure ejection, both from a glass micropipette. In iontophoretic delivery, charged molecules are released by passing a current (on the order of tens to hundreds of nA) through the micropipette. The advantage is that ions (not the solution) can be released in a precise amount by controlling the iontophoretic current and this amount can be quantified with an appropriate detector. The pressure ejection technique has broader applicability because solutions of both charged and neutral molecules can be delivered; the disadvantage, however, is the injected volume is hard to control and it varies with the medium into which the ejection is made.

The release by iontophoresis or pressure injection can be triggered manually, but is typically controlled by the iontophoretic current generator or pressure ejection system from a MATLABbased data acquisition program (Wanda, see Section: “Specialized Software for Point-Source Diffusion Analysis”) running on a PC equipped with an A/D and D/A converter (model PCI-MIO- 16E-4; National Instruments Corp., Austin, TX), as the timing of delivery is critical for data analysis.

#### Iontophoresis

For iontophoresis, a micropipette is pulled from a borosilicate glass capillary (single barrel capillaries from A-M Systems, Inc., Carlsborg, WA, catalog # 617000 or double barrel capillaries from Harvard Apparatus Ltd, Holliston, MA, catalog # 30-0117) and the tip is broken (by bumping against the edge of a glass slide under a microscope with 100× total magnification) to have an outer diameter (o.d.) of 3–6 μm. We prefer tips of this size to a larger diameter because they create less damage to the tissue upon insertion, and because with a bigger tip, the iontophoretic current carries large amounts of ions creating uncontrollable flow. The micropipette is back-filled with a concentrated solution of the ion (100 mM—1 M made in distilled water), a chloridized silver wire is inserted into the barrel and immersed in the filling solution. The barrel is then sealed with a hard, fast-setting wax (Sticky Wax, KerrLab Corporation, Orange, CA, catalog # 00625) to prevent evaporation and mechanical artifacts from wire movement. The iontophoretic pipette is attached via the silver wire to an accurate source of constant current (e.g., an iontophoretic current generator ION-100 made by Dagan Corporation, Minneapolis, MN). Positively and negatively charged ions are ejected by passing positive and negative current, respectively. When a sensor selective for a particular ion (see below) is used, transport number (nt) of the iontophoretic electrode can be quantified in a free solution (where α = 1). For this purpose, dilute agarose gel (0.3% in 150 mM NaCl; NuSieve GTG, FMC BioProducts, Rockland, ME) rather than water or isoosmotic pH-balanced solution is used to avoid the influence of thermal convection on the measurement. In theory, nt should be about 0.5 if the dissociated cation and anion have similar mobility, because half of the current passing through the micropipette should be carried by ions of the same polarity leaving the micropipette through the tip, while the other half is carried by ions of opposite charge entering the tip. In practice, however, various factors lower nt, such as geometry of the tip and interaction with the glass.

Following are several practical tips based on many years of experience. First, even when a current generator indicates that the current is passing, sometimes only ions of the opposite charge are entering the tip of the micropipette but no ions are moving out. In general, without detecting released molecules with a selective sensor, no assumption can be made about whether there is release or how much (unfortunately this is equally true when microiontophoresis is used in contexts to release transmitter substances or their analogues). Second, it is a common practice in other work (e.g., iontophoresis of transmitters) to use a retaining current of the opposite polarity at the end of the delivery pulse to hold the charged substance in the barrel. While this strategy fulfills its purpose, this can allow the solution around the micropipette to enter and dilute the tip. Subsequently, the transport number of the micropipette changes, resulting in a different amount of ions being released by the next pulse. For diffusion measurements, the dilution is prevented by constantly passing a small forward bias current of the same polarity as the main ejection current (Nicholson and Phillps 1981). Third, iontophoresis can be used to deliver large charged molecules (proteins, dextrans); however, it is less reliable. Typically, a large current has to be applied and these molecules are most probably carried out by a water flow (electroosmosis) rather than true iontophoresis.

The solution to Equation 10.1 for iontophoresis is (Nicholson and Philips 1981; Nicholson 2001):

$C(t)=Q8πD*αr[erfc(r2D*t+k′t)exp(rk′D*)+erfc(r2D*t-k′t)exp(-rk′D*)].$
10.2

Here, erfc is the complementary error function. Equation 10.2 represents the behavior of the concentration C at distance r from the source, after the iontophoretic pulse is switched on and maintained, i.e., the current is a step function. To account for what happens when the iontophoretic current is switched off, e.g., to make the step function into a pulse of duration tp, a delayed version of Equation 10.2 is subtracted from the original function:

$C(t>tp)=C(t)-C(t-tp)$
10.3

It is important to note that Equation 10.2 can be extended to take account of anisotropy in the tissue; the details may be found in (Rice et al. 1993; Nicholson 2001).

For iontophoresis, the magnitude of the source Q is given by

$Q=Int/zF$
10.4

where I is the magnitude of the iontophoresis current, nt the transport number of the iontophoresis electrode, z the valency of the ion and F is Faraday’s electrochemical equivalent.

#### Pressure Ejection

For pressure ejection, a micropipette is pulled from a borosilicate glass capillary (single barrel capillaries, A–M Systems, Inc., catalog # 617000) and the tip is again broken to have an o.d. of 3–6 μm. Because sub-nanoliter volumes are typically injected (Cragg et al. 2001), only the very tip of the micropipette needs to be filled with solution. Extreme care must be exercised to remove all air bubbles from the solution by gently tapping the micropipette because, when compressed during the pressure pulse, air bubbles prevent the ejection of the solution from the micropipette. A Teflon® tube of suitable diameter (Small Parts Inc, Miami Lakes, FL, STT-30–50 PTFE Tube) is inserted in the open end of the micropipette that is then sealed with melted wax (as used for iontophoresis electrodes; Teflon® tubing is used so that the hot wax does not melt it) and connected to a pressure ejection system (e.g., Picospritzer III, Parker Hannifin Corp., Pneutronics Division, General Valve Operation., Fairfield, NJ). Compressed nitrogen is used in our laboratory but any non-toxic compressed gas would do.

The appropriate solution to Equation 10.1 for pressure ejection is for the concentration C at distance r from the source (Nicholson 1993, 2001):

$C(t)=Qα1(4D*tπ)3/2exp(-r24D*t-k′t)$
10.5

This assumes that the pressure pulse is very brief in duration and confined to a point but still releases a finite amount of substance, i.e., it is an impulse or mathematical delta-function in both time and space. While this latter assumption is obviously violated, it is generally an adequate approximation because measurements are made at some distance and time from the source. When the point-source assumption breaks down, a more realistic solution for a finite volume is available (Nicholson 1985, 1993, 2001).

For pressure ejection described by Equation 10.5, the source is defined as

$Q=UCf$
10.6

where U is the volume ejected and Cf is the concentration of the ejected solution. Unfortunately, U is generally not known accurately, and because α is also unknown it is evident from Equation 10.5 that only the ratio U/α can be estimated. Indeed, Cf may not be well defined either because of dilution of the solution in the tip of the ejecting electrode (a problem that is overcome with a bias current in iontophoresis).

### Detection of Molecular Probes: Ion-Selective Microelectrodes, Integrative Optical Imaging and Carbon-Based Microelectrodes

Three detection systems have been used: ion-selective microelectrodes (ISMs), integrative optical imaging (IOI) employing fluorescence microscopy and a (charge-couple device CCD) camera and carbon-based sensors combined with chronoamperometry or voltammetry.

The ISM is limited to detecting charged molecule for which a selective ion exchanger exists in a formulation that works in a micropipette (for overview see, Ammann 1986; Nicholson 1993). These probe molecules may be delivered by pressure ejection or iontophoresis; when an ISM (or occasionally a carbon-based sensor) is used in a tandem with iontophoresis, the technique is called the realtime iontophoretic (RTI) method. The IOI method quantifies the diffusion of neutral or charged fluorophore-labeled substances, usually in the form of macromolecules; these probe molecules are almost always delivered by pressure ejection. Carbon-sensor methods quantify the diffusion of electroactive substances. In following three subsections, ISMs, IOI and carbon-based sensors are described briefly; detailed descriptions of the techniques and theory can be found elsewhere (Nicholson and Phillips 1981; Nicholson and Rice 1988; Nicholson 1992, 1993, 2001; Nicholson and Tao 1993; Tao and Nicholson 1995).

#### Ion-Selective Microelectrodes

To fabricate an ISM, a micropipette is pulled from double-barreled capillaries (Harvard Apparatus Ltd, catalog # 30-0117). The capillaries are cleaned internally by flushing with acetone and dried by forcing compressed gas, pulled and the tip broken to have an o.d. of 3–6 μm. The reference barrel is filled with 150 mM NaCl and a chloridized silver wired sealed in with wax (see “Iontophoresis”). The barrel to be used for ion sensing is filled with a 100–150 mM solution of the ion to be sensed and a Teflon® tube (see “Pressure Ejection”) is temporarily sealed with wax into the back of the barrel. The inner surface of the tip of the ion-detecting barrel is then coated with a solution of chlorotrimethylsilane (Sigma-Aldrich, Inc., St Louis, MO, catalog # C72854; 4% v/v in xylene) by applying pressure and suction (a disposable syringe may be used) through the Teflon® tube to make the surface hydrophobic and organophilic and prepare it to retain the ion-exchanger. After silanization, the ion exchanger, a liquid-membrane containing carrier molecules selective for the ion (Ammann 1986; Nicholson 1993) is drawn in and finally the Teflon® tube is removed and a chloridized silver wire inserted in this barrel. The length of the exchanger column in the tip determines the resistance of the barrel and this resistance can be made small by inserting another capillary; this will shorten the response time of the ISM (Ujec et al. 1979). For example, the response time of an ion exchanger column ≈ 200 μm long might be on the order of hundreds of milliseconds (Nicholson 1993). However, when the length of the exchanger column is reduced to 10–20 μm, the response time is shortened below 10 ms (Ujec et al. 1979). The ion-selective barrel detects a potential corresponding to the concentration of the ion of interest together with the local potential in the tissue at the tip of the electrode while the reference barrel detects the local potential only. Consequently, a potential that exclusively reflects the ion concentration is obtained by continuously subtracting the potential measured by the reference barrel from the signal measured on the ion-selective barrel using a dual channel microelectrode preamplifier (model IX2-700; Dagan Corp., Minneapolis, MN). Each ISM is calibrated in a set of standard solutions in which the concentration of the ion of interest is increased by doubling its amount against background of 150 mM NaCl (the Fixed Interference Method, Nicholson 1993). Calibration parameters are used to obtain the slope and the interference of the electrode by fitting the data to the Nikolsky Equation (Ammann 1986; Nicholson 1993). The slope and interference are used for potential-to-concentration conversion. A detailed description of ISM fabrication can be found in (Nicholson and Rice 1988; Nicholson 1993).

For diffusion measurements in agarose gel or brain, the iontophoretic microelectrode and the ISM are glued together to make a fixed array with inter-tip spacing 100–200 μm (Nicholson 1993). Alternatively, when robotic micromanipulators (such as MP 285; Sutter Instrument Co., Novato, CA) are available, the two microelectrodes can be held and moved independently (Hrabětová and Nicholson 2000, 2004). In the second scenario, the recording chamber has to be in a setup equipped with optics that allows the spacing between the tips of the electrodes to be set under visual control. In our laboratory we use a compound microscope (model BX61WI, Olympus America, Melville, NY) equipped with a water-immersion objective (Olympus, UMPlanFl, 10×, NA 0.3) and infrared (IR) differential interference contrast (DIC) optics. Independent microelectrodes offer great advantages when records are taken along x-, y-, and z-axes to test for anisotropy (i.e., when the diffusion is different along three orthogonal axes) of the region because the microelectrodes can be repositioned easily.

During diffusion measurements, a continuous bias current of 20 nA, with polarity appropriate to the ion, is applied from a constant-current, high-impedance source (see above). This iontophoretic current is stepped up to 60–100 nA for 50–100 s to obtain diffusion curves in dilute agarose gel and in brain. As mentioned above, the ion signal is extracted by continuous subtraction of the potential on the reference barrel from the potential on the ion-detecting barrel using a dual-channel microelectrode amplifier (model IX2-700; Dagan Corp., Minneapolis, MN). The headstage buffer amplifier connected to the ISM must have adequate impedance and a sufficiently low leakage current to accommodate the impedance of the ISM, which equals or exceeds 1000 MOhms. The ion and DC signals are amplified, low-pass filtered (6 Hz, 8 pole Bessel filter) using a CyberAmp 320 (Axon Instruments, Inc.), and continuously monitored on a chart recorder. The ion diffusion curves are digitized using an A/D converter installed in a personal computer and saved using a MATLAB-based (MathWorks, Natick, MA) program Wanda and analyzed with the MATLABbased program Walter, both described in “Specialized Software for Point-Source Diffusion Analysis” in this chapter. Typical examples of records obtained in dilute agarose and brain are shown (Figure 10.3).

#### Integrative Optical Imaging

In the Integrative Optical Imaging (IOI) method (Nicholson and Tao 1993), a molecule of interest with an attached fluorescent marker is pressure ejected from a glass micropipette into agar or brain and its distribution is detected optically over a sequence of intervals using an epifluorescent microscope equipped with a CCD camera. The distributions are fitted to the diffusion equation to obtain an estimate of the effective diffusion coefficient (D*). Various fluorophore-labeled molecules have been used: dextrans (MW 3 000–70 000; Nicholson and Tao 1993; Tao and Nicholson 1996; Tao 1999; Hrabětová et al. 2003; Thorne et al. 2004; Hrabìtová 2005; Thorne and Nicholson 2006), proteins (albumins and EGF; Tao and Nicholson 1996; Prokopová-Kubinová et al. 2001; Thorne et al. 2004), and synthetic polymers (PHPMA (Prokopová-Kubinová et al. 2001); polyethylene glycol (PEG) (Thorne et al. 2005)). These macromolecules, labeled with Texas Red®, tetramethylrhodamine, fluorescein, or Oregon Green® 514 fluorophores, were obtained from Molecular Probes, Sigma, Nektar Therapeutics (PEG) or custom made (PHPMA) at the Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic. The working concentration was 0.1–1 mM in 150 mM NaCl (dextrans and PHPMAs) or in PBS (proteins and PEG) with the exceptions of EGF, which was used at the concentration of 0.6×10−3 mM, and PEG, which was used at concentrations of 5–50 mM. In our experience, the small dextrans and PHPMAs are the easiest to work with. Tao and Nicholson (1996) reported that albumins tend to block the pipette, and Thorne et al. (2004) noted that solutions of proteins have to be carefully checked for aggregation.

The fluorophore-labeled probes are injected into dilute agarose gel or brain by a brief pulse (typically 10–200 ms, 10–20 psi) of compressed nitrogen applied to the micropipette containing the probe solution. The injected volume is less than 1.0 nL (Nicholson and Tao 1993) and typically 25– 50 pL (Cragg et al. 2001). For fluorescence imaging, the excitation light from a 75 W xenon or 100 W halogen source is directed to the specimen using a dichroic mirror with a filter set corresponding to the fluorophore used. A shutter (Uniblitz®, Vincent Associates, Rochester, NY) is located in the light path to prevent any photobleaching of the fluorophore that can occur when a set of images is acquired over a long period and the fluorophore is prone to photobleaching, e.g., fluorescein. The light emitted by the diffusing molecules is imaged by a CCD camera (CH350 or Cool-Snap HQ Monochrome, Photometrics, Tuscon, AZ are used in our laboratory). A set of 11 or more images is obtained at preset intervals for each injection. The first image, taken before the injection, serves as a background that is subtracted from subsequent images. The image acquisition is controlled by the program Vida, written in the V++ language (Digital Optics Ltd, Auckland, New Zealand) and the overall control of the experiment and analysis of the data is implemented in the MATLAB-based program Ida (see “Software for Integrative Optical Imaging-Vida and Ida”).

Given that the pressure ejection paradigm underlies the IOI method, Equation 10.5 with k′ = 0 forms the basis for interpreting the results. Ideally, the 3D distribution of molecules described by this equation forms a spherical cloud that is projected onto a plane in the camera attached to the fluorescent microscope with the intensity distribution modified by the defocussed point spread function of the microscope objective (Nicholson and Tao 1993; Tao and Nicholson 1995). Although the details are complicated, the intensity distribution (Ii) can be represented by the simple expression for the ith image at time ti after the initial injection of the fluorescent molecules:

$Ii(r,γi)=Eiexp(-(r/γi)2)$
10.8

where r is the distance from the point injection site and

$γi2=4D*(ti+t0).$
10.9

The offset t0 represent a virtual time origin prior to the actual time of injection to approximate the fact that the initial injection will have a finite volume (Prokopová-Kubinová et al. 2001).

The analysis of a set of IOI images is shown in Figure 10.7. The images are 2D projections of the diffusing molecules and the program fits an appropriate solution of the diffusion equation to intensity curves obtained along horizontal, vertical, and two diagonal lines running through the center of each image (Hrabětová et al. 2003). Diffusion coefficients (D for agarose gel and D* for brain tissue) were obtained by (a) fitting Equation 10.8 to each image to obtain a sequence of γi for each ti and then (b) performing a linear regression based on Equation 10.8 to obtain D or D* and t0 (Nicholson and Tao 1993; Prokopová-Kubinová et al. 2001). Unlike the RTI method, the IOI method does not determine α and k′. Potentially, a finite k′ would affect the accuracy of λ but it is anticipated that most macromolecules cannot easily leave the ECS so k′≈0. Eventually some endocytocis may be expected.

#### FIGURE 10.7

Diffusion of fluorophore-labeled dextran measured with IOI in dilute agarose gel and in rat neocortex in vitro. At time zero (0 s), a small volume of dextran (MW 3000) labeled with the fluorophore Texas Red® was pressure ejected from a glass micropipette (more...)

#### Carbon-Based Microelectrodes

Because there have been few diffusion measurements using electroactive compounds, this section will discuss both methods and results. Carbon epoxy electrodes or carbon fiber microelectrodes (CFMs) can sense electroactive substances and function much like an ISM. Such electrodes can therefore be used as sensors in the RTI or pressure ejection method, however, the theory and technology are distinctly different from those associated with ISMs; they are described in detail in Chapter 11 by Rice and colleagues.

The earliest diffusion measurements using electroactive probes combined with a point-source paradigm were described by Dayton et al. (1983) who employed CFMs, 7–10 μm diameter, and pressure ejected ascorbic acid (AA), dihydroxyphenylacetic acid (DOPAC) and α-methyldopamine (α-MeDA) from a 20 gauge needle (0.7 mm diameter) some 0.5–1.0 mm distant from the CFM in the rat cortex. The resulting signals were analyzed by chronoamperometry and Equation 10.5 with k′ = 0 was utilized to interpret the data. Generally, the investigators relied on tmax, the time of the peak of the diffusion curve, to estimate D* rather than attempting detailed curve fitting. The relationship for this is (Nicholson 2001):

$Dk′=0*=r2/6tmax.$
10.7

For the three substances studied, the values of D* are consistent with tortuosities of ~ 1.8. Rice et al. (1985) used a more extensive range of electroactive molecules in a similar approach but with graphite epoxy sensing electrodes some 100–250 μm in diameter, a pressure ejection micropipette of 5 μm diameter, and injection and sensing locations spaced 0.4–0.8 mm apart. Measurements were made in the rat caudate nucleus (striatum) and, when diffusion of DA or α-MeDA was measured, cellular uptake was blocked pharmacologically. Chronoamperometry was again used, and the interpretation of the data based on Equation 10.5 and Equation 10.7. Rice et al. (1985) found that, for anionic electroactive compounds, including AA and DOPAC, they obtained results similar to those of Dayton et al. (1983) but for cationic compounds, including DA and α-MeDA, D* was about three times less than for the anions. The authors attributed these results to differential interaction with the negatively charged ECM but the results have remained controversial and other factors, such as incomplete blockage of cellular uptake, lack of detailed curve fitting and large sensing electrode may have distorted the results, although the systematic differences between anions and cations remain striking.

These point-source paradigms were refined by Rice and Nicholson (1989) who used CFMs of 10–12 μm diameter and an iontophoretic micropipette some 150–200 μm away, combined with improved CFM fabrication techniques, signal processing and fast-scan cyclic voltammetry (FCV) to make accurate diffusion measurements of DA at concentrations as low as 35 nM in agarose gels. The analysis was based on fitting the experimental curves to Equation 10.2 through Equation 10.4 with k′ = 0. The techniques were also reviewed by Rice and Nicholson (1995).

These developments led to a more sophisticated measurement of DA diffusion in brain tissue with the point-source paradigm using CFMs and a micropipette pressure source by Cragg et al. (2001). Here, the full version of Equation 10.5 was used to determine both λ and k′. A value for α was obtained from iontophoretic measurements with TMA+, along with λ for that ion. Moreover, 3000 MW fluorescent dextran was included with the DA to enable the ejected volume to be estimated (using the TMA+ data) and controlled; the ejected volume was maintained in the range 25–50 pL. These combined measurements led to a detailed analysis of the diffusion properties of the guinea pig midbrain. The TMA+ measurements revealed that the substantia nigra compacta (SNc), substantia nigra reticulata (SNr) and ventral tegmental tract (VTA) had tortuosities in the range 1.58–1.69, quite comparable to the value in the guinea pig cortex (1.59) but the volume fraction in all three midbrain regions was a surprising 0.3, almost 50% greater than that seen in the cortex (0.22). The values of λ measured with DA (range 1.68–1.80) were slightly higher than those measured with TMA+ and the reason is not clear, however, they were nowhere near as high as the value obtained in rat caudate in the earlier work of Rice et al. (1985).

The studies of Cragg et al. (2001) also revealed distinct DA clearance mechanisms (Figure 10.5). In contrast to the striatum (caudate) where pressure-injected DA is rapidly removed (Van Horne et al. 1992) and Michaelis–Menten kinetics dominate (Nicholson 1995), in the midbrain simple clearance kinetics proportional to extracellular DA concentration were adequate to describe the data but the value of k′ was 0.085 s−1 in SNc and 0.093 s−1 in VTA, in marked contrast to the value of 0.006 s−1 in SNr. Indeed, the DA clearance value in SNr was comparable to the non-specific loss measured for TMA+ (k′ = 0.005–0.007 s−1).

One other study (Xin and Wightman 1997) employed modified CFMs coupled to an enzyme to attempt to measure the diffusion of choline and acetylcholine in rat brain slices, using a point source paradigm with pressure ejection. Although the study indicated some potential for the approach, the slow response of the sensors and a separation between sensor and source comparable to the thickness of the brain slice (see Section: “Effects of Boundary Conditions on Diffusion Measurements in Brain Tissue”) compromised the study. Finally we note that the diffusion of the electroactive neurotransmitter serotonin (5-hydroxytryptamine) can be measured in brain tissue with the RTI method, using the same ion-exchanger as is employed to measure TMA+ (Rice and Nicholson 1986).

### Diffusion Measurements Using Dual Microdialysis Probes

Because of the size of a microdialysis probe (typically 4 mm long, 0.24 mm o.d., Höistad et al. 2002) diffusion measurements with two such probes violates the point source/point sensor paradigm described above; however, the underlying concept is similar and this approach is briefly described here. Note that in order to make a correct quantitative assessment of microdialysis results in brain tissue in general, it is essential to take into account diffusion but these issues will not be addressed here; some discussion is given by Nicholson (2001). In this section we will only be concerned with the use of microdialysis to make actual diffusion measurements.

In principle it is possible to measure in vivo diffusion coefficients using two microdialysis probes (Höistad et al. 2000; Kehr et al. 2000) but there is only one study where an effective diffusion coefficient has been measured (Chen et al. 2002; Höistad et al. 2002). The potential advantages are that any substance that can be detected by microdialysis could be measured. The disadvantages are that the microdialysis probes occupy quite a large volume and, because they remove an appreciable amount of the substance being measured, they perturb the diffusion field. These two issues greatly complicate the theoretical analysis of the diffusion problem, however, they are not insurmountable given the availability of personal computers (Chen et al. 2002). A further disadvantage is that the time-scale of diffusion measurements with dual probes is long so they are unsuitable for measuring transient changes in diffusion conditions. Finally, the insertion of the probes leads to the formation of glia scar tissue around them over hours and days and this may alter the local environment in ways that have yet to be characterized fully.

In the study by Höistad et al. (2002), a radiolabeled tracer, 3H-mannitol, was continuously infused at different concentrations via a probe acutely implanted into the striatum of anesthetized male rats. Control measurements were made in a dilute agar gel. Samples were collected by a second probe 1 mm away from the first, and the recovered 3H-mannitol was measured by liquid scintillation counting. In the striatum, the delivery of 3H-mannitol was counteracted by its removal from the ECS by passive uptake into cells and clearance into the microcirculation, causing the diffusion profile to approach quasi steady-state levels within two hours. Diffusion data from brain and agar were analyzed using a mathematical model (Chen et al. 2002). The effective diffusion coefficient for 3H-mannitol was D* = 2.9×10−6 cm2 s−1, the effective volume fraction α* = 0.30 and the clearance rate constant k = 2.3×10−5 s−1. From these data, the tortuosity value was λ = 1.81, and penetration distance, Γ = 4.2 mm, where Γ = (D*/k)0.5 (Bungay et al. 1990). Mannitol has been widely used in various brain studies, including radiotracer diffusion measurements (Patlak and Fenstermacher 1975), but there is evidence that this substance enters cells to some degree and sucrose might be a better choice of probe substance for future diffusion measurements using dual probes (see, Höistad et al. 2002, for further discussion).

## Choice of Brain Tissue Preparation

Two classes of brain preparation have been used in past diffusion studies: tissue slices and slabs (in vitro preparations) and living animals (in vivo preparation). The major advantages and disadvantages for diffusion studies are outlined.

### Tissue Slices and Slabs

Brain slices, either 400 μm thick representing normal tissue or 1000 μm thick representing a model of ischemia, prepared from adult animals have been used extensively in diffusion studies. Detailed protocols are given by Rice and Nicholson (1991); Pérez-Pinzón et al. (1995). Slices are prepared according to a standard procedure and then kept at room temperature submerged in artificial cerebrospinal fluid (ACSF). Normal, 400 μm thick, slices are left to recover for at least an hour prior to recording. For the diffusion measurement, a single slice is transferred to the recording chamber where the temperature usually is maintained at 32°C–34°C. This temperature is below the typical physiological temperature of in vivo preparation (37°C) to prolong the survival of brain slices prepared from adult animals. Some studies, where the diffusion measurements were combined with patch clamp recordings, have been done at room temperature (Vargová et al. 2001).

Two types of tissue slab preparations have been used: isolated turtle cerebellum (Rice et al. 1993; Križaj et al. 1996) and isolated spinal cord from juvenile rats (P4-P21; Syková et al. 1999). Diffusion measurements were done at room temperature, which in the case of the turtle is the physiological temperature. Both these slab preparations exhibit a high resilience to ischemia so that a volume of healthy tissue containing intact 3D neuronal circuits can be obtained.

In general, in vitro preparations are employed when substantial manipulation of ACSF composition is required or when prompt and uniform drug delivery is needed. In the tissue chamber, one or both tissue surfaces (see below) are exposed to flowing ACSF to which substances can be added for rapid delivery to the tissue. Another advantage is region accessibility. In the intact brain, deep brain structures constitute a challenge for the RTI method. In typical use, the spacing between the tips of an array, comprising an ISM and a source micropipette, is stabilized by putting a drop of dental cement about 500 μm from the tips (Nicholson 1993). Such a strategy has to be abandoned when placing the array deep in the brain; consequently the spacing of the tips is likely to change as the microelectrodes are lowered into the tissue. Using a slice, however, almost any chosen brain region can be selected and it is possible to use two independent micromanipulators to further guarantee accuracy of placement for the micropipettes. Finally, brain slices are easier to work with than anesthetized animals. Once a successful dissection is done, slices are kept in the holding chamber and are used throughout the experimental day without need for further care.

It is important to keep in mind, however, that even the best prepared and maintained slices have limitations. Brain slices undergo an ischemic insult during preparation and, in addition, the tissue is traumatized by slicing. Consequently, the protoplasmic extensions of many cells are severed and the cut surface of the slice contains dying cells. It is pertinent to mention here that some of the recently introduced vibrating-blade microtomes are designed to have minimal movement perpendicular to the cutting plane, which reduces surface damage. Oxygen tension in the isolated preparation is usually un-physiological and the ACSF certainly lacks some chemical ingredients that are present in the real CSF (Artru 1999). Working with the slice preparation is always a race against time as the tissue slowly deteriorates. Finally, physiological questions that involve blood flow cannot be addressed in a slice.

### Effects of Boundary Conditions on Diffusion Measurements in Brain Tissue

Diffusion measurements made close to the interface between the tissue and the environment will be influenced by the conditions at the boundary. Some of these issues were addressed previously in the context of bath-application of pharmacological agents to slices (Nicholson and Hounsgaard 1983; Lipinski and Bingmann 1987). In this chapter, mathematical modeling has been used to investigate the effect of boundary conditions on diffusion measurements. The outcome is most relevant to brain slices but it is applicable also to brain slabs and in vivo preparations when the measurements are close to the tissue surface.

Diffusion measurements in slice preparations are performed in the center of the slice (i.e., for 400 μm thick slices at a depth of 200 μm). In the recording chamber, the brain slice is usually supported on a mesh with ACSF flowing underneath. The upper surface of the slice is either also exposed to flowing ACSF or it is exposed to a humidified and pre-warmed atmosphere (95% O2/5% CO2; Figure 10.8). The first type of chamber is called a submersion chamber, because the slice is totally submerged in flowing ACSF; the second type of chamber is called an interface chamber, because the moist upper surface of the slice forms an interface with the humidified gas. The two different scenarios at the upper surface result in distinct boundary conditions. In one case, the flowing solution interface functions as a sink that takes molecules away from the slice, while in the other case, the tissue–gas interface functions as an impermeable boundary where diffusing molecules within the slice get reflected back. These two scenarios were simulated to investigate their effect on diffusion measurements.

#### FIGURE 10.8

Submersion and interface tissue chambers. (a) In a submersion tissue chamber, the brain slice rests on a supporting mesh and the ACSF flows both underneath and over the slice. The slice is held down with a weight in the form of a “harp” (more...)

This analysis will be restricted to iontophoresis. Starting with Equation 10.2 and assuming the loss of probe ion is negligible (k′ = 0), Equation 10.2 reduces to:

$C(t)=Q4πD*αrerfc(r2D*t).$
10.10

Again, the falling phase of the concentration curve after the iontophoresis ceases after a pulse of duration tp, will be described by Equation 10.3.

The derivation of Equation 10.10 assumes that the source and measuring electrode are both far from any boundaries, i.e., they are embedded in an infinite medium with effective diffusion coefficient D*. If the source and measuring electrodes are moved close to, say, the upper boundary then the influence of this boundary may be accounted for in two limiting cases: (a) all diffusing crossing the boundary are swept away by the superficial flow of bathing medium; this is the absorbing boundary; or (b) all molecules reaching the boundary are reflected back into the tissue so the boundary is non-permeable. Both these conditions can be modeled by placing a virtual point source above the location of the boundary as a mirror image of the actual source and then removing the boundary so there is an infinite medium with two sources. When the virtual source is −Q in magnitude, equal and opposite to the real source, then on the boundary the concentration will be zero, equivalent to the loss of all particles, i.e., case (a) the absorbing boundary. When the virtual source is equal to +Q, then the flux across the boundary will be zero, so there will be no loss of molecules equivalent to case (b), an impermeable boundary.

In the simulations we have taken Equation 10.10 with images, calculated the concentrations and then analyzed the results by fitting Equation 10.2, which embodies a loss term but returns to an infinite medium representation, to determine whether the effects of a bounded medium without loss are equivalent to an infinite medium with a “virtual” loss.

The equation with an image source has a contribution from the actual source at distance rdirect and the virtual, image, source at rimage (See figure with Table 10.2).

#### TABLE 10.2

Simulations of the Effect of Absorbing and Non-Permeable Boundaryon λ, α, K′. Determined by Curve-Fitting for Input Parameters: λinput = 1.6, αinput = 0.20, Kinput = 0 s−1.Molecule is TMA+.

$C(t)=Q4πD*αr[erfc(rdirect2D*t)±erfc(rimage2D*t)]$
10.11

where

$rdirect=(a-b)2+h2 and rimage=(a+b)2+h2$
10.12

and a is the vertical distance of the source from the brain/external medium interface, equal to the distance from the interface to the image source or sink on the opposite side of the interface, b is the vertical distance of the sensor from the brain/external medium interface and h is the horizontal distance between source and sensor in the plane of the surface (see figure with Table 10.2).

When the source and sensor electrodes are lowered to the same depth, a, (i.e., a = b) with fixed spacing h, Equation 10.12 reduces to

$rdirect=h and rimage=4a2+h2.$
10.13

Table 10.2 depicts the results of calculations for a variety of configurations of source and sensor with respect to the interface. In all cases it is assumed that the other, lower, interface of the slice is so distant that the medium can be regarded as being semi-infinite. The procedure was (a) use Equation 10.10 through Equation 10.12 and Equation 10.3 to compute a concentration verses time curve; and (b) use the program Walter (described in “Software for RTI and Pressure Ejection- VOLTORO, Wanda and Walter”) to try to fit Equation 10.2 and Equation 10.3 to this curve. This means that we tried to fit the solution to the diffusion for a source in an infinite medium with loss of ions to the curves derived for a source in a semi-infinite medium with two types of boundary condition (absorbing and non-permeable) without loss. We required a “good” fit but not a perfect one.

The outcome of modeling is summarized in Table 10.2 and examples of simulated diffusion curves shown in Figure 10.9. Before describing the results in detail, we would like to stress that the two types of boundary conditions modeled here mimic extreme conditions and at the same time over-simplify the typical experimental situation. In an experimental setting, a slice in the submersion chamber has a stagnant or unstirred layer of solution (Lipinski and Bingmann 1987) on the surface rather then laminar flow of the ACSF and this leads to incomplete absorption, while a slice in the interface chamber has a thin layer of the ACSF resulting in incomplete reflection of diffusing molecules. In addition, another complexity not taken into account in the present modeling is that the boundary conditions at the lower surface may affect the measurements in the center of the slice under some conditions. Therefore the models provide trends rather than a precise description.

#### FIGURE 10.9

Effect of boundary conditions at tissue-environment interface on the diffusion measurements. The diffusion curves (solid grey line) were generated (see text for details) with the following input parameters: nt = 0.3, r = 50, 100 or 150 μm, λ (more...)

In the simulation mimicking an absorbing boundary, the sink effect of the boundary conditions had minimal effect on the extracellular diffusion measured 200 μm underneath the boundary in a plane parallel with the surface of the slice (Table 10.2, Simulation number 1–3; Figure 10.9a). The tortuosity derived from curve fitting was slightly higher (<+4%) for spacing r = 50–150 μm while α was slightly lower (−7%) only for r = 150 μm. The sink effect of absorbing boundary was to be found in the apparent value of k′ (recall that the model had k′ = 0).

We also tested a special case, where the source microelectrode was kept in place but the detector was moved between the source and the boundary to measure the diffusion in a plane perpendicular to the surface of the tissue. This configuration is used in experiments testing for tissue anisotropy, where the diffusion is measured along three perpendicular axes (Rice et al. 1993). Our simulations show (Table 10.2, Simulations 4–6) that for r = 50 and 100 μm, the boundary conditions influence the diffusion measurements only slightly. The results are similar to those obtained in the parallel plane. However, when the spacing was increased to 150 μm (i.e., the detector is only 50 μm away from the tissue surface, Simulation 6), λ was significantly reduced and an increase in k′ was observed, attributable to an accentuated sink effect close to the boundary.

Next, we tested a case when both microelectrodes were placed just beneath the slice surface (Table 10.2, Simulation 7–9; Figure 10.9b). The sink effect of an absorbing boundary clearly dominated. The diffusion curves were greatly distorted both in shape and amplitude. A large drop in λ and an increase in k′ were observed while the values of α were meaningless (α > 1.0).

In the simulation mimicking a non-permeable boundary, a reasonable result for the plane parallel with the tissue surface (depth of 200 μm) was obtained only when r was small (Table 10.2, Simulation 10–12; Figure 10.9c). At r ≥ 100 μm, the tortuosity was increased by more than 8%.

Finally, when both source and measuring electrodes were placed close to the non-permeable boundary, a correct result for λ and k′ was obtained (Table 10.2, Simulation 13–15; Figure 10.9d). To get a correct value of α, the volume fraction extracted from such records has to be doubled; this is because of the reflective properties of the tissue–gas interface causing the sensor to detect twice as many molecules as would have been detected in an infinite medium.

It is obvious that great caution has to be taken when designing and carrying out diffusion experiments in brain slices to avoid possible effects from conditions at tissue–environment interfaces. When using a submersion tissue chamber, the iontophoretic microelectrode is typically positioned at a depth of 200 μm, i.e., the source of ions is equidistant from the upper and lower surface of the slice, for all measurements. The ISM is positioned typically 100 μm away along x-, y-, and z-axes. The recordings are thus obtained from the central portion of the slice where the influence of boundary conditions is minimal.

### In Vivo Preparations

Many diffusion studies have been done in in vivo preparations. In fact, the very first major study using the TMA+ RTI method investigated diffusion properties of the ECS in rat cerebellum in vivo (Nicholson and Phillips 1981). Subsequently, large numbers of in vivo studies have been done in Professor Syková’s laboratory in Prague where the research topics include the changes in the ECS during development, learning and memory, aging, as well as in animal models of various pathological conditions such as ischemia and terminal anoxia, stab wound, and transgenic models of Alzheimer’s disease (for a review of these topics see Syková 2004). Other important in vivo diffusion studies using point source diffusion methods include a study of Ca2+ diffusion using a pressure source (Nicholson and Rice 1987), a study of volume regulation (Cserr et al. 1991), a study of changes in diffusion parameters during development (Lehmenkühleret al. 1993), and studies of epileptic foci (Lehmenkühleret al. 1988; Schwindt et al. 1997). In vivo diffusion studies with CFMs have been reviewed in Section: “Carbon-Based Microelectrodes”.

For diffusion measurements, animals are anesthetized, placed in a stereotaxic apparatus and the cortex or other brain region is exposed, then the dura mater is carefully removed. Body temperature is maintained at 37°C with a heating pad and the exposed cortex is superfused with ACSF warmed to 37°C and bubbled with 5% CO2/95% O2 to maintain pH at 7.4. The precise details of the preparation will vary with the animal and laboratory and original papers should be consulted. Diffusion measurements are done using a glued array of an iontophoretic or pressure ejection electrode and ISM. The array is calibrated in agar or agarose gel first and then it is lowered into the brain to measure the diffusion properties of the ECS. More detail is provided by Nicholson and Phillips (1981); Cserr et al. (1991); Lehmenkühleret al. (1993), and Voøíšek and Syková (1997), among others. Recently the IOI technique has been adapted to the in vivo preparation (Thorne and Nicholson 2006).

The living animal does not have the same limitations as brain slices and the brain tissue remains in a framework of the whole organism. Its necessity is undisputed when questions involving BBB, bulk flow or diffusion over long distances are addressed. In addition, while in vitro models of various pathological conditions exist, they are usually an approximation of in vivo conditions. However, difficulties with drug application, inaccessibility of deep brain tissues, and finally a need for non-trivial surgery combined with the necessity to maintain careful control of the preparation throughout the experiment can make use of in vivo preparation truly challenging.

## Specialized Software for Point-Source Diffusion Analysis

### Software For Iontophoresis and Pressure Ejection—VOLTORO, Walter and Wanda

The earliest curve fitting was accomplished on hand-held electronic calculators (Nicholson and Phillips 1981) and did not allow for a non-zero clearance parameter (k′). When personal computers became available it was possible to adopt a more sophisticated approach using the non-linear simplex approach to fit Equation 10.2 and Equation 10.3 or Equation 10.5 to data that were acquired with a digital oscilloscope. A succession of personal computers were programmed by Dr C. Nicholson using the language Pascal, and this phase reached its final sophistication in the program VOLTORO that accommodated iontophoresis, pressure ejection, ISMs and CFMs combined with FCV. The first version of VOLTORO ran on a DEC PDP 11 computer and subsequently on an Apple II. In its final incarnation the program ran under the DOS operating system on the IBM PC. It is still possible to run the program under Windows XP but support for the programming language has long ceased and VOLTORO can no longer be extended to operate contemporary data acquisition boards or many features of modern PC hardware.

To take full advantage of the Windows operating system, the analysis part of VOLTORO was re-written in the MATALAB (MathWorks, Natick, MA) programming environment by Dr Nicholson and became the program Walter. Walter was considered a means of reading in substantial numbers of data files generated by VOLTORO and re-analyzing them, still using the simplex curve fitting, but removing any operator bias by randomizing starting parameters. The program also allowed better visualization of the data and storage of all data and fitting parameters in an Excel spreadsheet.

Initially, Walter lacked any means of acquiring data because MATLAB did not support this function. This was remedied by Dr C. Chen who used the LabView language (National Instruments) to write a data acquisition program called Wanda which generated files that could be read by Walter. Some years later the Data Acquisition Toolbox became available for MATLAB and Wanda was re-written in MATLAB by Dr L. Tao. Today both Wanda and Walter run under MATLAB and support a variety of data acquisition boards from National Instruments (and some other manufacturers). The programs are available from Dr C. Nicholson.

### Software for Integrative Optical Imaging—Vida and Ida

The IOI technique was introduced more than 10 years after the RTI method, when PCs were in general use, and the first version of the accompanying program was written in the C language by Drs L. Tao and C. Nicholson. This program both operated the CCD camera and fitted the resulting image data to Equation 10.8 and Equation 10.9.

After the experience with MATLAB and the Walter program it was also decided to move the IOI program to this environment. Because MATLAB lacked, and still lacks, any toolbox to control a Photometrics CCD camera, the programming language V++ (formerly V-Pascal) from Digital Optics, Auckland, New Zealand, was adopted for this part of the package. The main part of the program, including data analysis, was written in MATLAB by Dr C. Nicholson and called Ida; and the camera operation program, called Vida, was written in V++, also by Dr Nicholson. Vida runs under control of Ida using the DDE (Dynamic Data Exchange) facilities of the two programming environments. This arrangement has the disadvantage that two program environments are required (V++ and MATLAB) but it has the advantage that the camera can be run independently under the extensive turnkey V++ package for other image analysis tasks. Ida and Vida are available from Dr C. Nicholson.

## Diffusion Properties of Brain Extracellular Space

Here, selected examples of results of the diffusion studies in brain tissue in vivo and in vitro are given. More comprehensive reviews may be found elsewhere (Syková 1997; Nicholson and Syková 1998; Nicholson 2001; Syková 2004).

### Brain Under Physiological Conditions

The first major RTI study quantified the diffusion of small cations and anions in rat cerebellum in vivo (Nicholson and Phillips 1981). These investigators found that all ions tested gave similar results and the values were pooled to α = 0.21 and λ = 1.55. It may be noted that this early study looked for anisotropy in the molecular layer of the cerebellum but failed to find it, primarily because the data analysis techniques were not sensitive enough; later studies did reveal anisotropy both in the cerebellum and elsewhere (see below). Extensive subsequent studies in in vivo and in vitro preparations of rat, mice, guinea pig, and turtle brain revealed that most commonly, λ is about 1.6 and α is about 0.2 in isotropic tissue under physiological conditions (Nicholson and Syková 1998; Nicholson 2001). However, there are exceptions. For example, α is sometimes larger, about 0.3 in in vitro substantia nigra in guinea pigs (Cragg et al. 2001) and about 0.4 in in vivo neocortex of rat pups (age 2–3 days postnatal; Lehmenkühleret al. 1993); interestingly, λ remains close to 1.6 in these preparations (1.60 in SNr VTA, 1.69 in SNc, 1.59–1.68 in layers III–VI of the somatosensory neocortex). In contrast, a low tortuosity, λ = 1.46, is observed in in vitro preparations of the CA1 region of hippocampus (Hrabìtová 2005, but see Mazel et al. 1998), the region vulnerable to ischemia and epilepsy (Traynelis and Dingeldine 1988; Kawasaki et al. 1990; Schmidt-Kastner and Freund 1991). We may speculate that the tissue hindrance corresponding to λ ≈1.6 is optimal for intercellular communication and maintenance of extracellular environment, and that smaller values (as seen in the hippocampus) or higher values (as seen in many pathological conditions) have negative consequences for brain function. The maintenance of stable tissue hindrance for diffusion might be achieved by various, as yet unknown, regulatory mechanisms.

Tissue anisotropy, which signifies that diffusion is different in different axes, can be studied with RTI methods. This was first demonstrated by Rice et al. (1993) in the molecular layer of cerebellum of the turtle (where λx = 1.44, λy = 1.95, λz = 1.58), followed by studies in corpus callosum (Voøíšek and Syková 1997), hippocampus (Mazel et al. 1998 but see Hrabìtová 2005), and auditory cortex (Voøíšek et al. 2002). Diffusion anisotropy most often correlates with a directional arrangement of fibers (Beaulieu 2002).

The studies employing IOI explored how macromolecules are hindered in the ECS. Nicholson and Tao (1993) found that the tortuosity increases as the diameter of globular dextran increases (Figure 10.4). In contrast, long-chain flexible synthetic polymers (PHPMA) of molecular weight ranging from 7800 to 1 057 000 are hindered similarly to the small TMA cation (Prokopová-Kubinová et al. 2001), although their actual effective diffusion coefficient is small. The mechanism of low hindrance for long-chain polymers is unknown. Finally, studies employing proteins show that albumins with MW ranging from 14 500 to 66 000 are hindered more (λ = 2.24–2.50; Tao and Nicholson 1996) than the smaller epidermal growth factor, MW 6600 (λ = 1.8; Thorne et al. 2004).

### Brain During Reversible Osmotic Challenge

The ECS is not a static structure but one in which properties change during brain activity and under various challenges. One might conjecture that there is a relationship between λ and α; that λ will increase as α gets smaller and vice versa. However, while this is true under some circumstances it is not a universal fact and generally there is no simple relation between λ and α in brain (Figure 10.10). This was demonstrated using the TMA–RTI method by Kume-Kick et al. (2002) in neocortical slices where α can be reversibly reduced by hypotonic stress (150 mosmol kg−1) to as low as 0.12. As might be expected, λ increases in value to reach 1.86. But enlarging α from a control value of 0.24–0.42 has no effect on λ, which remains, even in the most severe hypertonic stress (500 mosmol kg−1), close to the control value of 1.68. Tao (1999) measured qualitatively the same behavior of λ with fluorophore-labeled MW 3000 dextran (see, Kume-Kick et al. 2002, for discussion).

#### FIGURE 10.10

Independence of extracellular tortuosity and volume fraction during osmotic stress in rat neocortex in vitro. Diffusion of TMA+ was measured using RTI; diffusion of fluorophore-labeled 3000 MW dextran was measured using IOI. Osmotic stress was induced (more...)

Similar trends in osmotic behavior were also seen in a prior study on the isolated turtle cerebellum using the TMA–RTI method (Križaj et al. 1996). Both this and the Kume-Kick et al. (2002) study combined TMA+ diffusion measurements with determinations of wet-weight and dryweight of tissue to arrive at a description of both the extra- and intracellular compartments. This approach had been pioneered in an earlier in vivo study of hypernatremia and brain volume regulation (Cserr et al. 1991).

### Brain in Pathological States

The dissociation of α and λ can become even more pronounced under pathological situations (Syková et al. 2000) suggesting that a combination of factors is in play. During terminal anoxia in rat cortex, α is reduced to as low as 0.05, while X-irradiation increases α to 0.50. Yet, the tortuosity of tissue under both these pathological conditions is elevated (to 2.0 in terminal anoxia and to 1.8 in X-irradiated tissue).

Diffusion is significantly hindered and the volume of the ECS is reduced in many neuropathological states associated with cellular edema (Nicholson and Syková 1998; Syková et al. 2000). Several studies, employing in vitro and in vivo models of ischemia, showed that the tortuosity increases and the volume fraction decreases (Lundbæk and Hansen 1992; Syková et al. 1994; Pérez-Pinzón et al. 1995; Hrabětová and Nicholson 2000). In thick-slices (1000 μm), which model aspects of ischemia in vitro (Newman et al. 1988, 1989, 1991), λ measured with TMA+ increases to 1.9 while α is reduced to 0.14 (Hrabětová and Nicholson 2000; Hrabětová et al. 2002). It has been proposed that the large hindrance observed during ischemia arises from formation of dead-space microdomains in the ECS, dead-end pores, formed by swelling cells, where diffusing molecules are delayed (Hrabětová and Nicholson 2000; Hrabětová et al. 2003). In in vivo models of anoxia and ischemia, λ measured with TMA+ increases to about 2.0 while α is reduced to 0.05 (Syková et al. 1994; Voøíšek and Syková 1997). Such dramatic changes in diffusion parameters occur even in rat pups (P4-6) but it takes longer for the changes to develop (Voøíšek and Syková 1997). Changes in α seen in in vivo ischemic models are more pronounced than in in vitro studies. This difference may reflect the fact that, in vivo, the spatial expansion of edematous ischemic tissue is restricted by the skull, but no such constraint is imposed on a slice (Hrabětová et al. 2002). Among the many consequences of the impaired transport in ischemia is the disruption of nutrient and metabolic trafficking; such disruption augments brain dysfunction and delays recovery during vascular reperfusion.

A large alteration in diffusion parameters is also observed during astrogliosis induced in rat spinal cord (Syková et al. 1999; Vargová et al. 2001), in anaplastic astrocytoma in humans (Vargová et al. 2003), and in experimental autoimmune encephalomyelitis that is used as a model for multiple sclerosis (Šimonová et al. 1996). Under all these, as well as many other conditions, the ECS environment changes, altering the diffusion of substances and the homeostasis of the extracellular environment.

## Monte Carlo Simulation of Diffusion in 3d Media

Over the years, diffusion measurements have generated lots of interesting but often counter intuitive data. Some striking examples include the findings that α and λ are independent parameters (Chen and Nicholson 2000; Kume-Kick et al. 2002, see above); and the discovery that, while the hindrance for globular molecules increases with the molecular weight (Nicholson and Tao 1993; Tao and Nicholson 1996), PHPMA linear polymers (MW 7800–1 057 000) have the same tortuosity as the small TMA cation (MW 74; Prokopová-Kubinová et al. 2001). As a final example it was established that the hindrance of ischemic tissue is actually reduced when large molecules are introduced into the ECS (Hrabětová and Nicholson 2000; Hrabětová et al. 2003). The interpretation of these results is challenging because the ECS parameters obtained with diffusion analysis are the result of macroscopic averaging and consequently they do not provide information about the features of individual ECS channels. We know very little about the local structure of these channels, whether they are homogeneous or heterogeneous and how they are organized in 3D. Microscopic heterogeneity would imply the existence of regions with distinct physiological characteristics. Although λ was originally thought of as a reflection of how molecules navigate around round cells, other biophysical features of the ECS may contribute to hindrance in brain tissue (Chen and Nicholson 2000; Hrabětová et al. 2003; Hrabì et al. 2004; Tao et al. 2005).

To test ideas and to facilitate data interpretation, modeling has come to our aid. Here we will focus on the parameters α and λ although the modeling approach has also been applied in other contexts to describe, e.g., the interaction between diffusion and Michaelis–Menten mediated uptake of dopamine from the ECS (Nicholson 1995; Cragg et al. 2001; Cragg and Rice 2004).

### MCell and DReAMM

The Monte Carlo Simulation of Cellular Neurophysiology (MCell) software package was developed to numerically simulate the events at the neuromuscular junction. These include: acetylcholine release, diffusion, and binding to receptors as well as channel kinetics (Stiles et al. 1996; Stiles et al. 1999; Stiles and Bartol 2001). This software (available at www.mcell.psc.edu) was designed for biologically oriented users and therefore it uses a simple programming language or model description language (MDL) to specify simulation components such as 3D surfaces and reaction mechanisms.

Tao and Nicholson (2004) adopted the MCell software package to simulate the diffusion of probe molecules in virtual models of brain tissue. Instead of using brain ECS reconstructed from electron micrographs, Tao and Nicholson (2004), and later others (Hrabì et al. 2004; Tao et al. 2005), built simple virtual models where the features of ECS structure can be fully controlled. During Monte Carlo simulation of diffusion, probe molecules are released into the artificial 3D media from a point source representing conditions of the biological experiment (pressure pulse release of molecules) and they perform random walks. During one time step, each diffusing molecule moves in a randomly chosen direction and travels a randomly chosen step. When a marker encounters the cell boundary, it is reflected at the appropriate angle, just like a billiard ball colliding with a smooth wall. The positions of individual molecules are recorded over time. The effective diffusion coefficient, D*, is calculated from the distribution of molecules as a function of time and distance, and λ derived (Tao and Nicholson 2004). The results are visualized using the Data Explorer-based program DReAMM (www.mcell.pcs.edu).

All simulations so far are based on simple 3D models that test the effect of certain geometrical features on the diffusion of a point particle. Interaction with the extracellular matrix, transport across the cellular wall or effects caused by finite size of the marker are neglected.

### Three-Dimensional Media Composed of Convex Cells

In the first study, the ECS was depicted as interconnected spaces of uniform size that surround convex elements (Figure 10.11; Tao and Nicholson 2004). The simplest 3D media was built from cubes (Figure 10.11a). Two additional geometries were generated (Tao and Nicholson 2004): one built from truncated octahedra and the other built from three elements of different size, rhombicuboctahedra, cubes and tetrahedra (Figure 10.11b). All three ensembles of geometrical objects can pack 3D space completely without gaps (in contrast to spheres) so α can be varied from 0 to 1. Ensembles of cubes have through-channels that might be supposed to decrease tortuosity but the two additional geometries did not have such channels and the last geometry also did not have just one type of “cell”. Surprisingly, all three geometries gave identical results, a simple relationship between the two ECS parameters, λg defined as the tortuosity arising only from the geometrical constraints, and α, the total volume fraction

#### FIGURE 10.11

Computer simulation of diffusion in virtual 3D media composed of convex cells. (a) ECS model constructed with uniformly spaced cubic cells (left). Projection of the 3D distribution of the diffusing particles onto the xy plane recorded at time (more...)

$λg(α)=(3-α)/2$
10.14

giving an upper limit of 1.225 for λg (Figure 10.11). For α = 0.2 (a typical value in brain), λg = 1.183 (Figure 10.11c). Interestingly, Maxwell (1954) derived the same formula from a study of electric conductivity of a medium containing a dilute suspension of spheres. Tao and Nicholson (2004) proposed that this result is generally applicable to 3D media composed of convex elements separated by uniform spaces. Indeed, simulations in a 3D model composed of random convex polyhedra confirmed this prediction (Figure 10.12; Hrabì et al. 2004). In brain, however, α and λ are independent parameters, and λ>λg. The 3D models built from convex cells thus did not capture all the critical structural features of brain ECS.

#### FIGURE 10.12

Virtual 3D media composed of random convex polyhedra. The gaps between the polyhedra are uniform. (Reproduced from Hrabì, J., et al. Biophys. J., 87, 1606–1617, 2004. With permission.)

### Dwell-Time Diffusion Theory

The surprising result of Tao and Nicholson (2004) together with the experimental finding that the introduction of background macromolecules to ischemic tissue decreases extracellular tortuosity (Hrabětová and Nicholson 2000; Hrabětová et al. 2003) led to the development of a simple semiquantitative model of tortuosity based on diffusion dwell-times (Hrabětová et al. 2003; Hrabì et al. 2004). It is assumed there that the ECS is composed both of well-connected spaces where molecules move with a hindrance given by simple geometric considerations and it also contains dead-spaces where diffusing molecules are delayed (Figure 10.13a). This model offers an explanation as to why the limiting tortuosity of 3D media composed of convex elements is 1.225 and shows how the addition of dead-spaces increases λ. Furthermore, the model predicts a relationship between λ and α in isotropic media containing dead spaces:

#### FIGURE 10.13

Computer simulation of diffusion in virtual 3D media containing concave slit-like dead spaces. (a) Two-dimensional sections through 3D model of the ECS. In the media composed of uniformly spaced convex elements, the ECS is composed of well-connected spaces (more...)

$λg=λ0α/α0,$
10.15

where λ0 and α0 are the values associated with the well-connected space under normal conditions. The volume fraction α here is the sum of volume fraction occupied by dead-spaces, αd, and well connected spaces, α0. Dead-space volume fraction is thus a proportion of the ECS, forming the space inside dead-end pores, leaving the remainder of the ECS to constitute the well-connected space (Figure 10.13a). The dwell-time diffusion model supports the interpretation of experimental data. When the background macromolecule, 70,000 MW dextran, was added to thick slice models of ischemia, α decreased from 0.14 to 0.10 and λ decreased from 1.9 to 1.54; the model predicts a decrease to 1.61 (Hrabětová et al. 2003). The model also allows us to make predictions about the distribution of volume fraction between well-connected spaces and dead-spaces based on λ measured in brain (Hrabětová et al. 2003; Hrabětová and Nicholson 2004). Note that this model assumes that molecules exchange rapidly between dead-space and well-connected space.

### Three-Dimensional Media Containing Concave Cells and Lakes

To confirm the dwell time diffusion model by Monte Carlo simulations, concave elements were incorporated into the virtual 3D media (Figure 10.13a; Hrabì et al. 2004). The ECS of such media is composed of well-connected spaces (α0) and dead-spaces within concavities (αd) that delay diffusing molecules by transiently trapping them. Pathways between the elements were kept narrow to comply with assumption of the dwell-time diffusion model that ECS width is significantly smaller than the side of an element. Dead-spaces were mimicked by introducing slits into the faces of cubes (Figure 10.13a). Monte Carlo numerical simulations showed that dead-spaces can increase the tortuosity to, and even well beyond, values measured in brain; for a given α, the increase in λ was dependent on the volume occupied by dead-spaces (Figure 10.13b). Monte Carlo numerical simulations thus fully support the dwell-time diffusion model.

A related study (Tao et al. 2005) employed a set of 3D models composed of cubes where deadspace microdomains were mimicked by open rectangular cavities inserted either into the face of a cube or at the vertex (Figure 10.14). At the vertex, the dead-spaces do not have the form of narrow slits but rather form a residual extracellular space volume or lake. Such models have physiological relevance; it has been proposed that such lakes are formed during hyperosmotic stress and function as transient traps for diffusing molecules, keeping values of λ constant while α increases (Chen and Nicholson 2000; Kume-Kick et al. 2002). In addition, dilated regions of the ECS have been observed in electron micrographs of brain tissue (Brightman 1965; Van Harreveld et al. 1965; Cragg 1979). Simulations (Tao et al. 2005) revealed that in such ECS models, λ can equal or exceed the typical experimental value of about 1.6. The equation derived from dwell-time diffusion theory no longer fits the data because the assumption of narrow spaces is violated here (Tao et al. 2005). To accommodate wide dead-spaces, λ0 was replaced with Maxwell’s expression and a reasonably good fit was obtained (Figure 10.13a). In addition, Tao et al. (2005) introduced an empirical factor β that correlated with the ease with which a molecule could leave a cavity and its vicinity (Figure 10.13b; Tao et al. 2005).

#### FIGURE 10.14

Computer simulation of diffusion in virtual 3D media containing concave lake-like dead spaces. (a) A model of the ECS was built from cubes containing cavities in the center of each face; lakelike dead-space is surrounded by two cubes. (b) A model of the (more...)

## Future Directions

Research done over past 25 years gives a solid picture of the macroscopic biophysical properties of the ECS (Nicholson and Syková 1998; Nicholson 2001). We have learned that the ECS occupies about 20% of tissue volume and that inert molecules are hindered so that the diffusion coefficient, D, is reduced by about 2.6 (i.e., λ2). We know now that these values change-sometimes independently-during brain activity and in various pathological states. In addition, when the size, shape, and charge of the probe molecule are altered, additional information reflecting the interaction of the probe with the extracellular environment, including the matrix, may be obtained. However, an unequivocal interpretation of all the results has proved elusive. It is clear that better understanding of ECS microstructure, in terms of geometry and molecular composition, would facilitate the interpretation of data. Diffusion studies in combination with modeling offer a unique exploratory tool. Future results will be highly relevant for understanding the transport of substances, including neurotransmitters, neuromodulators and exogenous drugs. The work to date does establish, however, that the ECS is an essential component of brain function. Supported by NIH NINDS grants NS47557 (S.H.) and NS28642 (C.N.).

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