#### Mathematical model of steady-state microdialysis with probe implantation trauma

The principal effects of probe implantation trauma are assumed to occur in a concentric layer of abnormal tissue interposed between the probe and surrounding normal tissue. The mathematical model to be developed is an extension of one previously proposed to describe microdialysis in a single tissue region of uniform properties (Bungay *et al*. 1990) on the spatial scale relevant to current microdialysis probes whose membrane diameters and lengths are greater than 0.2 mm and 1 mm, respectively. In these models the local variations in properties and concentrations associated with the individual cells and discrete sites of analyte supply and removal, such as blood vessels and synapses, are spatially averaged, since associated length scales separating these sites are on the order of 0.05 mm or smaller. A number of simplifications have been employed, such as the linearity in analyte concentration dependence to describe analyte clearance processes. These have been described in the earlier presentations (Bungay *et al*. 1990; Morrison *et al*. 1991). Few additional assumptions are invoked in adding the traumatized tissue layer. The types of trauma to be modeled are alterations in the rates of analyte supply and removal processes in this layer. Although it is possible to model these alterations with a continuous and diffuse spatial variation, for simplicity the trauma layer is assumed to be thin with uniform properties that are distinct from those of the surrounding tissue.

The model is formulated in cylindrical coordinates with *r* representing radial distance from the probe axis and *z* representing axial distance from the inlet end of the membrane. As indicated in Fig. 5, the inner and outer surfaces of the membrane are located at radial positions, *r*_{i} and *r*_{o}, respectively, and the thickness of the trauma layer is δ. The length of membrane accessible for diffusional exchange between perfusate and tissue is *l*_{m}. The diffusional permeability of the perfusate and membrane will be combined into a probe permeability, *P*_{p}, defined as the proportionality coefficient between the flux of analyte into the probe and the concentration difference driving the diffusion,

- ((A1))

The diffusive flux, *J*_{p}, is the mass flow rate of analyte into the probe per unit area of membrane outer surface. The driving force is given by the difference between the extracellular analyte concentration at the probe outer surface, , and the dialysate concentration, *C*_{d}. The flux and the concentrations vary with the axial location, *z*, but probe permeability is assumed to be uniform. The flux into the probe increases the dialysate concentration along a differential length, d*z*, according to the balance,

- ((A2))

in which *Q*_{d} is the perfusate flow rate. Diffusion in the axial direction is neglected.

The flux, *J*_{p}, is determined by a combination of the probe and tissue permeabilities. Solving mass balances for the two tissue layers will yield expressions for the tissue permeabilities. For simplicity, the spatially averaged rate of analyte supply from sources other than the probe will be assumed uniform, but different, in each layer and independent of the local analyte concentration. The non-diffusional local rate of analyte removal per unit volume of extracellular space (ECS) will be described by the product, *k*_{e}·*C*_{e}, in which *k*_{e} is the rate constant for clearance by all processes other than diffusion, and *C*_{e} is the local analyte concentration in the ECS. Thus, *C*_{e} is assumed to be sufficiently below the Michaelis–Menten *K*_{m} value for any saturable clearance mechanisms, such as cellular uptake or microvascular efflux transporters. The value of the clearance rate constant is assumed to be uniform, but different, within the two layers. As yet the model neglects mechanisms for regulating analyte supply and removal rates. Since the trauma layer is assumed to be thin, the curvature of the layer will be neglected. With these simplifications, the steady-state ECS mass balances for the analyte in the ECS are of the form,

- ((A3))

- ((A4))

with *D*_{e} denoting the analyte diffusion coefficient in the ECS and *S* denoting the supply rate per unit volume of ECS. The balances are to be solved subject to a number of constraints. Far from the probe the diffusional term in equation A4 vanishes as the concentration approaches a uniform level of determined by the balance of supply and removal rates,

- ((A5))

Analogously in the trauma layer, the supply and removal processes are associated with a potential steady-state concentration , defined by

- ((A6))

At the interface between the two tissue layers, *C*_{e} = in both layers and the analyte flux across the interface is

- ((A7))

in which δ_{–} and δ_{+} indicate evaluations in the trauma and surrounding tissue layers, respectively, at the interface, *r* = *r*_{o} + δ. At the membrane–tissue interface the flux leaving the tissue is the same as that entering the probe,

- ((A8))

In the perfusate at the inlet and outlet ends of the membrane, the analyte concentration is indicated by

- ((A9))

Microdialysis induces spatial variations in extracellular concentration. The length scales over which the concentration varies are characterized by ‘penetration depths’,

- ((A10))

The ease of permeation of each layer is given by a permeability,

- ((A11))

and

- ((A12))

The modified Bessel functions of the second kind, *K*_{0} and *K*_{1}, each with dimensionless argument, *r*_{o}/Γ_{n}, appear in equation A12 as a consequence of the cylindrical geometry. The ratio, *K*_{1}/*K*_{0}, approaches unity for large values of the argument.

The overall permeability for diffusion through the probe and composite tissue in series is given by the inverse of the sum of reciprocals of the probe and tissue permeabilities,

- ((A14))

In the above, the tissue permeability, *P*_{t}, is a composite of the permeability properties of the two tissue layers,

- ((A15))

Equation A15 was obtained by solving the tissue balances (equation A3 and equation A4) together with their boundary conditions. For conciseness the hyperbolic functions have been abbreviated by

- ((A16))

The apparent extracellular concentration of the analyte is given by

- ((A18))

In equation A18, *w* is a dimensionless weighting factor,

- ((A19))

Substituting equation A13 into the perfusate balance A2 and integrating from *z* = 0 to *z* = *l*_{m} yields equation 1 of the Methods section,

- ((A20))

in which *E*_{vivo} is the extraction fraction *in vivo* that depends exponentially on , *Q*_{d} and the membrane outer surface area, *A*_{o}, according to

- ((A21))

The above expressions hold for any value of . For the pure sampling mode, equation A20 simplifies to

- ((A22))

The true relative recovery for sampling, *R*, is defined as

- ((A23))

Substituting equation A18 into the above yields the explicit expression

- ((A25))

The axial variation in perfusate concentration is obtained by integrating the perfusate balance (A2) combined with equation A13 from *z* = 0 to *z*,

- ((A26))

An appropriate mean perfusate concentration value is the axial-average,

- ((A27))

The concentrations in the tissue vary in both the *r* and *z* directions. A representative *r-*direction profile is, likewise, obtained by axial-averaging. This mean concentration profile in the trauma layer, *r*_{o} ≤ *r* ≤ *r*_{o} + δ, is

- ((A28))

and in the surrounding tissue, *r* ≥ *r*_{o} + δ, is

- ((A29))

Equating the axial-averaged expressions for the flux into the probe, equations A1 and A13, gives a relationship for calculating the probe interface concentration,

- ((A30))

and the axial-average extracellular concentration at the interface between the two tissue layers is

- ((A31))