## Abstract

Direct interstitial infusion is a technique capable of delivering agents over both small and large dimensions of brain tissue. However, at a sufficiently high volumetric inflow rate, backflow along the catheter shaft may occur and compromise delivery. A scaling relationship for the finite backflow distance along this catheter in pure gray matter (*x*
_{m}) has been determined from a mathematical model based on Stokes flow, Darcy flow in porous media, and elastic deformation of the brain tissue:*x*
_{m} = constant*Q*
_{o}
^{3}
*R*
^{4}
*r*
_{c}
^{4}
*G*
^{−3}μ^{−1}(*Q*
_{o} = volumetric inflow rate, *R* = tissue hydraulic resistance, *r*
_{c} = catheter radius, *G* = shear modulus, and μ = viscosity). This implies that backflow is minimized by the use of small diameter catheters and that a fixed (minimal) backflow distance may be maintained by offsetting an increase in flow rate with a similar decrease in catheter radius. Generally, backflow is avoided in rat gray matter with a 32-gauge catheter operating below 0.5 μl/min. An extension of the scaling relationship to include brain size in the resistance term leads to the finding that absolute backflow distance obtained with a given catheter and inflow rate is weakly affected by the depth of catheter tip placement and, thus, brain size. Finally, an extension of the model to describe catheter passage through a white matter layer before terminating in the gray has been shown to account for observed percentages of albumin in the corpus callosum after a 4-μl infusion of the compound to rat striatum over a range of volumetric inflow rates.

- mathematical model
- intracerebral drug delivery

direct interstitial infusion into the brain parenchyma at high volumetric flow rates has recently been shown to allow delivery of agents into large tissue volumes at relatively constant concentration (7, 8, 17, 19, 22). The ability to evenly dose tissues over a broad distance scale encourages application of this delivery technique to a variety of clinical problems, including the administration of neurotrophic factors for the treatment of neurodegenerative disease, agents for targeted lesioning of specific sites in the brain, and cytokines and protein toxins for tumor therapy (2, 14, 18, 29). In turn, to successfully dose the associated tissue targets, control over the volume and mass of agent infused is a necessary prerequisite, as is the ability to predict the subsequent distribution in tissue.

Previous work has identified several factors that govern drug delivery by direct interstitial infusion. These include the tissue infused (gray or white matter), the size of the infusate molecule, tissue binding, metabolism, and microvascular permeability of the agent and the volumetric flow rate and duration of infusion (7, 22). With large-volume infusions, the boundary contours of the infused brain regions also play a major role in determining the final distribution of agent (16, 23). Large spread of agent can be achieved in both white and gray matter, but white matter exhibits a more anisotropic distribution of agent and a greater ability to accommodate infusate delivery (7,17). A relatively uniform protein distribution in excess of 1 cm length along the infused fiber tracts of the corona radiata has been reported for an infusion of 75 μl of transferrin at 1.1 μl/min into the brain of the cat (7), and relatively uniform distributions of^{111}In-labeled*n*-diethylenetriaminepentaacetic acid-apotransferrin have been reported over the 1.8- to 2.9-cm dimensions of the macaque centrum semiovale when it was infused with 10 ml infusate at 1.9 μl/min (17). Other studies have shown that slowly metabolized macromolecules are associated with greater penetration depth and flatter profiles than are more permeable small molecules infused under similar conditions (7, 19). For isotropic gray matter, simple mathematical models were developed that allowed prediction of time-dependent concentration profiles from the extracellular fraction, metabolic and tissue transport parameters of the infused substance, and volumetric inflow parameters (22).

At low flow rates, the aforementioned factors are the primary determinants of delivery to the tissue, and the mass of an agent delivered to the tissue can be assumed to be equal to the mass in the infused solution. However, at sufficiently high flow rates, solution flows back up the catheter shaft, leaking to the surface and reducing delivery to the tissue itself. This raises new issues of control over delivered mass and requires one to consider additional factors related to backflow, including the interactions among catheter diameter, depth of catheter insertion, volumetric inflow rate, and tissue mechanics. Cserr and Berman (10) and Szentistvanyi et al. (26) experimentally addressed some of these factors in their studies of macromolecular clearance from gray matter, but they ultimately confined their attention to a single flow rate and provided no means to extrapolate from one set of successful experimental parameters to another. Anecdotal reports abound regarding leakback along catheter tracks in various brain regions but they lack quantitation.

Accordingly, we developed a mathematical model that yields a simple relationship among catheter diameter, volumetric inflow rate, and the parameters describing hydraulic flow through and deformation of the tissue. To establish a nondimensional parameter grouping that captures the essential interplay among these parameters yet remains simple enough to be used as an approximate guide for experimental design, the theoretical model has been kept as uncomplicated as possible. Here we report the development of the model and its nondimensional parameter result as well as preliminary experimental confirmation of its ability to account for the mass recovery from infused gray matter.

### Glossary

- e
_{ij} - Strain tensor element
*G*- Shear modulus of gray brain tissue
*h*- Thickness of annulus at
*x*(=*h*_{o}at*x*= 0) - κ
_{g}, κ_{w} - Hydraulic conductivity of gray (g) and white (w) brain tissue (interstitial)
*K*- Elastic constant equal to 2
*G*/*r*_{c} *L*- Equivalent cylindrical radius of a brain hemisphere
- λ
- Lamé constant
- μ
- Viscosity of water at 37°C
*N*_{p}- Scaling constant for pressure
*N*_{x}- Scaling constant for axial distance
*v*- Poisson ratio
- ω
- Radial displacement vector at
*r*[ω(*r*_{c}) =*h*(*x*)] - p
- Hydrostatic pressure at
*x*(= p_{i}− p_{o}) - φ
- Extracellular volume fraction of brain tissue
*Q*_{o}- Volumetric infusion rate
*Q*- Volumetric flow rate across an annular cross-section at
*x* *Q*_{s}- Volumetric flow rate across hemispheric surface closing annulus at catheter tip
*r*- Radial position in tissue (
*r*= 0 at catheter axis,*r*=*r*_{c}at catheter surface,*r*≈*L*at brain surface) *r*_{c}- Radius of infusion catheter
*r*_{tiss}- Radial extent from catheter of albumin deposition in rat brain autoradiograms
*R*- Hydraulic resistance of porous tissue
*S*- Outer surface area of annulus
- ς
- One-third thickness of rat corpus callosum
- ς
_{ij} - Stress tensor element
*u*(=_{x}*u*)- Axial velocity of fluid in annulus at
*x*,*y* - ν
- Radial velocity of fluid through porous tissue at
*r* *x*- Axial position along catheter (
*x*= 0 at tip) *x*_{m}- Axial length of annulus, i.e., backflow distance
*y*- Radial position across annulus (
*y*= 0 at exterior catheter wall)

## THEORY

Previous theory of high flow infusion into the brain interstitium has been based on the assumption that delivery occurs from a point or small spherical source centered on the catheter tip (22). We now alter that description to provide for the possibility that the hydrostatic pressure imposed on the tissue by the infusate will cause the tissue to move back from the surface of the catheter, opening an annular space extending along a portion of the catheter length. If sufficiently long, this annulus becomes an extended source of infusate and distorts the spherical symmetry of the infusate distribution expected for point dosing of an isotropic tissue such as gray matter. In the extreme, this annulus may extend to the brain surface and allow loss of infusate directly to the cerebrospinal fluid.

Our primary goal is to develop a mathematical model of incipient backflow along the catheter shaft, deriving from it a simple algebraic relationship that allows examination of the trade-off between catheter diameter and inflow rate that maintains this incipient condition and prevents loss of infusate from the targeted tissue mass to adjacent structures or across the brain surface. Thus the theory is tailored to best apply to that situation when the annular space has extended along the catheter shaft for a distance considerably longer than the catheter diameter but short of the actual length from catheter tip to brain boundary.

*Formulation*. To keep the tissue mechanics as simple as possible, we approximate the tissue behavior as simple elastic expansion directed radially outward from the catheter surface in a nearly semi-infinite isotropic gray matter volume surrounding the catheter (Fig. 1). (Later we modify the equations to account for the situation when the catheter also passes perpendicularly through an intermediate white matter band.) Because the width of the annular ring about the catheter is expected to be small relative to the catheter diameter, we ignore tissue strain along the direction of the catheter axis. We also ignore mechanical coupling to the region beyond the catheter tip, but note that this region may act as a sink for infusate.

The mathematical description consists of three principal elements: two equations describing fluid flow in the annulus and into the porous tissue surrounding it and a third equation describing deformation of the brain tissue under the applied fluid pressure. Fluid flow in the annulus is given by the Stokes equation, with inertial and external force terms neglected
Equation 1where*x* is the distance from catheter tip along the catheter axis, *y* is the radial distance from catheter surface directed along the normal to the surface, p(*x*) is the hydrostatic pressure at axial position*x* relative to outside cerebrospinal fluid pressure, μ is viscosity, and*u _{x}
* is the fluid velocity component parallel to the catheter. The Reynolds number for flow in the annulus at a nominal flow of 0.5 μl/min is 0.2 given a narrow annulus width of 0.0007 cm. In accordance with the usual assumptions of lubrication theory (4), the contribution of terms other than that involving the

*u*velocity component are small and can henceforth be neglected. Likewise pressure change over the small range of

_{x}*y*can be neglected, and the pressure can be approximated as a function of

*x*only. The transient associated with establishment of a pseudomomentum balance on the fluid within the annulus is very short and hence the left-hand side of the Stokes equation is zero.

The volumetric flow *Q* through the annulus at position *x* is given by
Equation 2 where*h*(*x*) is the local thickness of the annular ring, assumed to be small relative to the catheter radius*r*
_{c}. The ratio of*h*(*x*)/*r*
_{c}is largest at the catheter tip and is <0.08 at all flow rates investigated, a value consistent with the use of lubrication theory.

Fluid flow across the outer annular boundary into tissue is based on Darcy’s law of flow in porous media. For a brain volume of fixed size, this may be expressed in terms of the local mass balance on the fluid in the annulus and a tissue resistance*R* dependent on the hydraulic conductivity of the tissue (appendix
), i.e.
Equation 3where d*Q*(*x*)/d*x*accounts for the fluid loss into tissue interstitial space at any axial position *x* > 0.

This formulation assumes pure radial flow from the annulus into the tissue and ignores any axially directed flow in the porous medium. This approximation is best at high infusion rates when the length of the backflow annulus is large relative to the catheter diameter and the axial pressure gradient is small relative to the radial one. At flow rates of 0.5 μl/min or above (regardless of catheter diameter from 22 to 32 gauge), the ratio of axial to radial pressure gradients is estimated to be only 0.2 to 0.3 or less and the pure radial approximation can still be invoked. However at lower flow rates of interest (0.10–0.50 μl/min), the gradient ratio may be as large as 0.6. The use of *Eq*.*
3
* with resistance based solely on outward radial flow (appendix ) would then be inaccurate. However, because axial flow can be considered an additional path for fluid flow that lowers overall hydrodynamic resistance, the approximate formula*Eq*. *
3
*can still be used if the resistance were lowered to empirically account for axial flow. On the basis of the 0.6 gradient ratio, we estimate that the corrected resistance is not lower than 60% of the pure radial flow value. Thus, to maintain mathematical simplicity, we have retained the form of *Eq*.*
3
* over the entire volumetric flow rate range of 0.10 μl/min and above, with the caveat that the*R* parameter value eventually substituted into it will be only ∼80% of the value computed for pure radial flow. This should assure that *R*errors do not exceed ±20%, whereas catheter diameters and volumetric flow rates are varied over 4- to 10-fold ranges. (As shall be seen, backflow distances are nearly proportional to*R*, and thus they will reflect similar uncertainty.)

In addition to radial flow across the annulus, fluid may also permeate into the tissue lying in the semi-infinite volume distal to the catheter tip. To estimate this permeation rate, we model the surface available for this transport as a hemisphere centered on the catheter tip and require that the same flux exists across a surface element of the outside annular surface as across the hemisphere at*x* = 0, i.e., the fluid velocity is continuous at the junction between annulus and hemisphere. From*Eq*. *
3
*and the definition of d*Q* in terms of the average fluid velocity across the annular face,ν̅, we obtain at*x* = 0
Equation 4after noting that d*S* for the approximately cylindrical outer annular surface is 2π[*r*
_{c} +*h*(0)]d*x*. From this, we obtain the expression for
(0). Integrating this across the hemispheric area, the volumetric fluid loss there,*Q*
_{s}, is found to be
Equation 5To account approximately for the formation of the annulus, we treat the tissue at each axial d*x* element as a thick elastic cylinder expanding radially outward under the annular pressure p(*x*) to a distance*h*(*x*) according to
Equation 6where*G* is the shear modulus of gray matter and *K* is defined by the second equality. *Equation6
* follows from the theory for infinitesimal strain applied to a cylinder, in which the elements of the Almansi strain tensor, a differential force balance, the general Hookian stress-strain relationship, and a gray matter Poisson’s ratio nearing 0.5 are combined to yield the expression shown (appendix ) (3, 27). The time for elastic deformation to establish itself in a fluid-filled porous medium composed of incompressible elements is defined as the consolidation time. *Equation6
* is a steady-state approximation in which it is assumed that the consolidation times are short relative to the infusion periods under consideration. This is consistent with estimates of these consolidation times, computed as (2*L*/π)^{2}(1 −2*v*)/[2*G*φκ_{g}(1 − *v*)] from a previous infusion analysis (3), that range from 1.5 min to instantaneous, depending on the choice of Poisson ratio from 0.35 to 0.5. Only with the Poisson ratio chosen as 0.35 (15) and at the highest flow rate modeled by us do the consolidation and infusion times become similar. (*v* Is Poisson’s ratio, and the other parameters are defined in Table 1.)

The Stokes, Darcy flow, and tissue deformation equations (*Eqs*.*
1
*, *
3
*, and *
6
*) may be solved together with no-slip boundary conditions at *y* = 0 and*h*(*x*) to yield the following expression for pressure as a function of axial position (appendix )
Equation 7aThe complete expression for p(*x*) consists of *Eq*.*
7a
* plus the two additional boundary conditions
Equation 7b
Equation 7cwhere*h*
_{o} ≡*h*(0) and*Q*
_{o} is the volumetric flow rate delivered by the catheter; p(0) is the pressure at the catheter tip and is approximately equal to the fluid pressure at the pump. *Equation7b
* is the elastic expansion of the tissue at the catheter tip in terms of the tip displacement parameter*h*
_{o}.*Equation7c
* follows from the conservation of fluid mass over the entire axial length of the system, i.e., equating the total inlet flow rate from the catheter to the sum of the flow rates across the annular wall and hemispheric tip
Equation 8The integral in this expression has a finite*x*-range under the assumption (later confirmed) that the annulus is finite and terminates at a distance*x*
_{m} from the catheter tip [i.e., both*h*(*x*) and*Q*(*x*) are zero at*x*
_{m}].*Q*(0) is the net volumetric flow into the annulus after allowing for fluid movement into tissue distal to the catheter tip. From *Eq*.*
2
* and the solution to the Stokes*Eq*.*
1
*, the quantity*Q*(0) may in turn be related back to p(0) and *h*
_{o}according to (appendix ,*Eq*.*
EC3
*)
Equation 9When this expression is combined with *Eq*.*
5
* and substituted into*Eq*.*
8
*,*Eq*.*
7c
* results.

*Equation 7*, *a–c*, constitutes a model for backflow accompanying infusion into gray matter only (a gray matter model). These equations may also be modified to account for a slightly more complicated model in which the catheter is assumed to pass through an intermediate white matter layer lying normal to the axis of the catheter. This situation is encountered, for example, with infusions into the rat striatum in which a catheter enters the brain at the dorsal cortical surface and then passes downward through the cortex (gray) and corpus callosum (white matter band lying above the caudate) before terminating in the caudate nucleus (gray) itself (Fig.2). In this layered case, the tissue resistance and elasticity constant become functions of axial position,*R*(*x*) and*K*(*x*), to reflect the different tissue properties in the gray and white matter regions. When the *x*-dependence of these two parameters is taken into account, *Eq*.*7*, *a–c*, changes only slightly to yield a layered model alternative consisting of *Eq*. *7*′,*a–c*
Equation 7d
Equation 7e
Equation 7f
*R*(*x*) and *K*(*x*) functionality must be specified for a particular pattern of layers.

*Parameter values*. The constant parameters of the models appear in Table 1 and include those that describe the properties of gray and white matter as well as a representative choice of catheter diameter. Values were drawn from the literature, except for the gray matter hydraulic conductivity. Because of the wide range of literature values for this constant, its value was chosen so that the backflow distance computed from the gray matter model for a flow rate of 0.5 μl/min and the other parameters of Table1 was just less than the distance to the ventral side of the white matter layer. This is in accordance with recent experimental observation in which infused albumin just remained confined to the caudate and not to the corpus callosum at this flow rate (9). That 0.5 μl/min was a threshold rate for backflow in this experiment was indicated by the observation that some but not all animals exhibited the very beginnings of detectable albumin in the corpus callosum. Parameters used with the layered model apply to the rat brain, assuming a dorsal-ventral (D-V)-oriented catheter with tip located in the caudate center at coordinates D-V 5.0 mm, medial-lateral (M-L) 2.5 mm, and anterior-posterior (A-P) 0.5 mm, a white matter layer from 0.22 to 0.27 cm, and an overlying gray layer (24). The hydraulic conductivity function of the layered model is assumed to be the gray matter value with a superimposed normal distribution centered on the white matter layer with a 3ς equal to the anatomical width of the layer and an average conductivity over this layer of sixfold the gray matter conductivity (Fig. 2). For simplicity and because no elastic moduli for white matter in intact brain are available, we have used the gray matter shear modulus for both gray and white matter.

*Numerical evaluation of annular pressure.* Numerical evaluations of the pressure profiles that develop about an infusion catheter were obtained as a function of flow rate for both the gray matter and layered models. The formulae for the pressure in the annulus,*Eq*. *7*or *7*′, were solved numerically using the adaptive NDSolve algorithm in Mathematica v.2.2 (30). A unique finding of these calculations was that, for these choices of parameters and *h*
_{o}values of a few percent of*r*
_{c}, the solution to *Eq*.*7* or*7*′ was always a p(*x*) profile that terminated at a finite value of *x*, *x*
_{m}. This was a key result, because it suggested the approach for determining*x*
_{m} and examining its dependence on experimental variables such as catheter (needle) diameter and volumetric inflow rate. The choice of*h*
_{o} is not arbitrary, however, because*x*
_{m} is coupled to this *h*
_{o} choice through the conservation of fluid mass relationship in*Eq*.*
8
*, i.e., after substitution of*Eq*. *
3
*for d*Q*/d*x*and *Eq*.*
5
* for*Q*
_{s} in*Eq*. *
8
*and allowing for the possible axial dependence of tissue resistance and elasticity, through
Equation 10As a consequence, the final solution for p(*x*) was obtained by iterating between the input parameter*h*
_{o} and the*x*
_{m} value until the unique {*h*
_{o}, *x*
_{m}} pair was found that satisfied the *Eq*.*
10
* conservation of mass.

## RESULTS

*Backflow distances*. Backflow distances associated with a 32-gauge catheter and a set of volumetric inflow rates ranging from 0.03 to 5.00 μl/min were first computed from the gray matter model (*Eq*.*7* using the parameters of Table 1). The results appear in the first two columns of Table2. Backflow distances computed over this span of flow range from <1 mm to almost 1 cm. Because this range of distances brackets the sizes of the rat caudate and many of the smaller human nuclei, there is a clear indication from the Table 2 data that volumetric inflow rates must be carefully chosen to avoid the losses or maldistribution associated with backflows that reach the outer boundary of the targeted nucleus. For example, with a centered 32-gauge catheter, confinement of infused solute to the rat caudate (radius of 0.22 cm) requires that flow rates not exceed 0.5 μl/min, whereas confinement to the thinner parietal cortex of the rat (half thickness of 0.1 cm) requires that the flow rate not exceed 0.2 μl/min. Figure3 presents the backflow distances computed for pure rat gray matter over a range of catheter radii and volumetric flow rates.

To investigate the effects that an intervening white matter layer has on backflow, backflow distances were computed for the same 32-gauge catheter and flow rate range as in Table 2 using the layered model of*Eq*.*7*′ and the rat-specific parameters of Table 1. The backflow distances derived from this model appear in *column 3* of Table 2. At flow rates of 0.5 μl/min or less, backflow distances are the same as obtained from the gray matter model. However, substantial differences in these distances develop between the models at higher flow rates. This occurs because, in the layered model, fluid can more easily be conducted into the white matter layer (corpus callosum) lying between 0.22 and 0.27 cm, leaving, at a given flow rate, less fluid available to fill and extend the annulus. Thus, with a conductive white layer present, a flow rate of 1 μl/min leads to a backflow that terminates in the white matter rather than extending into the overlying cortex. The white matter layer also affects the magnitude of the flow rate that would drive infusate all the way to the rat brain surface. With the corpus callosum present, nearly twice the volumetric flow rate is required for flow to reach the surface of the rat cortex.

*Comparison with experimental observation.* Preliminary validation of the backflow model and its parameters was accomplished by demonstrating its ability to account for the recently observed percentage of caudate-infused mass that flows back into the corpus callosum as a function of flow rate and catheter size (9).

Briefly, this experimental study involved the constant volumetric infusion of 4 μl of osmotically balanced PBS-buffered^{14}C-radiolabeled albumin into the rat caudate (D-V 5.0 mm, M-L 2.5 mm, and A-P 0.5 mm) through catheters of 22, 28, or 32 gauge. After the infusion was completed, the cannula was withdrawn at a rate of 1 mm/min. The animal was then immediately killed and the brain was immediately removed and frozen in isopentane at −70°C and sectioned into 20-μm-thick slices. Quantitative autoradiography was performed, and the distribution of the compound in the gray and overlying white matter regions was determined from the superposition of adjacent sections developed for histology and autoradiography using the National Institutes of Health program Image.

A modeling prediction of the fraction of albumin distributed into the gray and white layers was not possible a priori because an accurate determination of the gray matter hydraulic conductivity relative to its white matter counterpart was not available from the literature. However, an approximate value for it was obtained by fitting the backflow model (other parameters in Table 1) to experimental data (9) so that the model reproduced the incipient backflow into the corpus callosum observed at 0.5 μl/min with a 32-gauge catheter.

The fraction of protein mass present in the white matter was then calculated from theory as the fraction of steady-state infusate flow crossing the portion of the annular wall located in the corpus callosum, i.e., *Eq*.*
3
* integrated over 0.22 <*x* < 0.27 and divided by the volumetric flow rate*Q*
_{0}. Theoretical computation and experimental measurement for infusion via a 32-gauge catheter appear as *columns 4* and*5* of Table 2. At lower flow rates, as forced by the parameter fit, the model confines all albumin to the caudate and forces agreement with observation (e.g., Fig.4
*A*). However, at larger flow rates, delivery to the white matter is substantial and the model correctly accounts for the percentage of protein that partitions into the callosum. This also accounts for the typical umbrella-shaped distribution seen experimentally in coronal cross section (e.g., Fig. 4
*B*). Figure 4
*C* presents the computed outward radial volumetric flow (per axial length) across the annular wall as a function of axial position for the 5 μl/min infusion rate of Fig. 4
*B*. When these radial flows are integrated over the infusion time, the volumes of fluid entering the tissue sections (of thickness d*x*) transverse to the catheter axis may be calculated. From these, the radii of the infused sections (*r*
_{tiss}) may be computed, because the fluid is confined to the extracellular volume. Three such radii are shown in Fig. 4
*C*, corresponding to the upper caudate, corpus callosum, and cortex where*r*
_{tiss} = 0.10, 0.21, and 0.06 cm, respectively. The upper caudate and corpus callosum values compare with the M-L spread from the catheter track in Fig.4
*B*. Simulation overpredicts the value of observed M-L spread in the cortex, although it correctly predicts it to be substantially smaller than in the upper caudate. Simultaneous maintenance of the conditions (on a 32-gauge catheter in rat brain) that 0.5 μl/min generates a backflow extending to the caudate/callosum interface, 5.0 μl/min generates a backflow extending toward the cortical surface, and 1.0 to 5.0 μl/min flow rates partition ∼30% of the infused fluid into callosum constrains both the gray and white matter conductivities to a narrow range (∼30%) about the values presented in Table 1.

Of importance, theory also predicts that backflow distances increase with catheter radius (see *Scaling relationship*), and this was confirmed experimentally when overflow into the white matter, absent with a 32-gauge catheter, was observed to occur with both 28- and 22-gauge catheters. Theoretical and experimental values for the percentage deposition into the corpus callosum were in close agreement for a 22-gauge delivery (25 vs. 21 ± 7%), whereas for an intermediate 28-gauge delivery, experimental measurement exceeded theoretical prediction (8% theoretical vs. 35% experimental). Overall, the model was able to account for all but one of six flow-rate/catheter-radius pairs.

*Scaling relationship*. Experimentalists are concerned with the effects of catheter radius and flow rate selections on backflow. They frequently need to know if the backflow effect of a change in flow rate can be offset by a corresponding change in catheter radius. In this situation, backflow distances need not be predicted, they only need to be held fixed while infusion parameters are altered. Such trade-offs do not require solution of the complete backflow model (i.e., *Eq*.*7*) but only identification of a nondimensional (scaling) grouping that scales the distance variable according to the principal parameters of the model. We now derive such a grouping for the backflow distance in gray matter. It is an important result because it provides a simple means for allowing the experimentalist to extrapolate from one set of infusion conditions, known to avoid significant backflow, to another set that similarly avoids backflow.

Our approach is to nondimensionalize the differential pressure equation, *Eq*.*
7a
*, and seek the nondimensional grouping for the *x* variable, because this is the independent variable related to backflow distance. We begin by observing that two scaling constants are required to nondimensionalize *Eq*.*
7a
*, one for pressure (*N*
_{p}) and one for the *x* distance (*N _{x}
*). It is apparent that a typical term such as p′′(

*x*)p

^{2}(

*x*) can be nondimensionalized by multiplying by

*N*

_{p}

^{−3}N

_{x}

^{2}. By multiplying the entire

*Eq*.

*7a*by this factor and, for numerical convenience, requiring that each side be of order unity, we find that the right-hand side of

*Eq*.

*7a*becomes Equation 11Likewise,

*Eq*.

*7c*may be nondimensionalized to yield another relationship involving

*N*

_{p}and

*N*. After substituting p(0)/

_{x}*K*for

*h*

_{o}(from

*Eq*.

*7b*) and multiplying through by p(0)

^{3}, the left-hand side of the equation becomes p′(0)p(0)

^{3}, which is nondimensionalized by

*N*

_{p}

^{−4}

*N*. Multiplying again by this factor leads to a first right-hand term satisfying Equation 12Simultaneous solution of

_{x}*Eqs*.

*11*and

*12*and use of

*Eq*.

*6*lead to scaling factors of Equation 13and Equation 14The backflow distance

*x*

_{m}is scaled by

*N*so that the constant of

_{x}*x*

_{m}/

*N*=

_{x}*constant*is unitless. Hence, if a set of infusion conditions has been established in which backflow is only incipient, the condition that the quantity Equation 15remains constant will allow the experimentalist to extrapolate from these conditions to new ones consistent with maintenance of this incipient backflow and, thus, focal delivery.

Accurate scaling has been confirmed for all pure gray matter computations covering the volumetric inflow range of 0.1–5.0 μl/min and catheter range of 22–32 gauge (0.0356–0.0114 cm), and the *constant* was determined to be 1.336. For the special case of backflow in rat brain gray matter,*x*
_{m} (cm) = 11.41
with *r*
_{c} in centimeters and*Q*
_{o} in microliters per minute.

The scaling factor of *Eq*.*
13
* may also be modified slightly to take into account the effect of brain size on backflow. To do so, an additional multiplier of *G* must be introduced and *R* must be reexpressed to exhibit its dependence on brain dimension. These two changes will account both for the increased resistance to porous flow and the greater difficulty of expanding the annulus that occurs when brain size increases. The dependence of *R* on the approximate radius of brain volume *L*is given by *Eq*.*
EA4
* in appendix. The dependence of the tissue displacement on*L* may be inferred from the first right-hand term of *Eq*.*
EB9
* in appendix (the second term is negligibly small), where*G* appears multiplied by (1 −
*L*
^{2}). The result of substituting *Eq*.*
EA4
* for*R* and introducing the (1 −
*L*
^{2}) multiplier leads to an expanded*N _{x}
*
Equation 16Of the two terms involving

*L*, the log term is numerically more significant. Because this term arose from size effects on tissue resistance, whereas the other arose from elasticity considerations, its numerical dominance indicates that the primary effect of increased brain size during infusion is an increased resistance to porous tissue flow. For fixed volumetric inflow and catheter diameter,

*N*, and thus the backflow distance

_{x}*x*

_{m}to which it is proportional, increase by only 46% when the brain dimension

*L*increases 10-fold. Hence we conclude that the absolute backflow distance is little affected by increasing brain size. In turn, this implies that the same magnitude of backflow would be encountered by someone infusing the rat caudate as infusing one of the smaller deep nuclei of the human brain.

## DISCUSSION

The preliminary experimental pattern of protein mass deposition that we observed in rat gray matter infusions is consistent with a model of backflow that involves formation of a narrow annular manifold around the catheter that then partitions infused mass into surrounding gray and white matter regions. At the lower flow rates (0.1–0.5 μl/min), confinement of mass to the caudate alone is consistent with backflow distances that do not exceed 0.22 cm (Table 2). At higher flow rates (0.5–2 μl/min), the annulus extends beyond the boundary of the caudate and terminates in the white matter of the corpus callosum; at the highest flow rates (>5 μl/min), the annulus may extend across both the callosum and the overlying cortex, allowing infusate to reach the surface of the brain. Computations show that the pressure in the annulus diminishes gradually with distance from the catheter tip so that relative partitioning into the gray and white matter volumes is largely controlled by the combination of the regional hydraulic conductivity and regional annular surface area. The significance of these observations is that selection of too large a flow rate to a gray matter target will lead to diminished delivery and internal leakage to nearby white matter regions and, perhaps, to external leakage as well.

The findings of this study provide a means to ensure that infused drugs do not immediately escape an intended target tissue in brain by retrograde flow along the catheter track during the infusion. Were such escape allowed, not only would the intended target be underexposed but more distant sensitive targets might be exposed unintentionally. The guidelines of this study can be used to ensure that the infusate carrier fluid be initially confined to a focal spherical region within a gray matter target. Whether the drug solute also remains within this spherical target depends primarily on its capillary permeability and rate of degradation (22). Macromolecules are characterized by very slow rates of microvascular clearance and, given sufficient time and no significant cellular uptake and degradation, will eventually diffuse or advect out of an initially infused region and still dose distant sites, although generally at much lower rates and with a different spatial distribution than would occur if substantial backflow were present. However, either for macromolecules that are rapidly degraded or for smaller molecules that are rapidly cleared by microvascular transport or degradation, initial focal (spherical) delivery may be maintained indefinitely because the infusate concentration may be selected to guarantee that the drug solute is cleared below its minimum active concentration before reaching the boundary of the infused space.

A recent example of such focal delivery is the targeted lesioning of the globus pallidus interna (Gpi) by the small molecular weight excitotoxin, quinolinic acid, in the treatment of Parkinson’s disease (20). In this particular case, a catheter size and volumetric infusion rate are chosen to avoid backflow according to the considerations of this study, then the infusion volume is selected so that it just fills the greatest inscribed sphere of the Gpi target. With the use of a previously developed radial-symmetric infusion model (22) and parameters determined from microdialysis (5), the infusion concentration is selected so that diffusion-advection and clearance of quinolinic acid do not allow the minimum active concentration to be attained outside the infused sphere of the Gpi. Hence the neighboring globus pallidus externa has been shown to be unaffected during a quinolinic acid infusion in which 85% of the Gpi is lesioned.

The theory that we have developed has also allowed us to develop a simple scaling relationship between catheter diameter, volumetric inflow rate of infusate, mechanical properties of gray matter, and the extent of backflow (*Eq*.*
15
*). Determination of this algebraic quantity was the primary goal of our work because it provides an investigator with a means to extrapolate from one set of experimental parameters, known to be consistent with minor or negligible backflow, to another set associated with the same minor backflow. The theory behind this scaling relationship has been kept approximate enough to allow straightforward derivation, yet sufficiently inclusive to capture the major transport and mechanical effects involved. The scaling relationship itself, *Eq*.*
15
*, is simple enough to be taken into the laboratory and used immediately for experimental design.

An important aspect of this expression is the relationship it gives between catheter radius and the volumetric inflow rate of infusate. For a fixed backflow distance in gray matter, these parameters satisfy the condition that*r*
_{c}
^{4}
*Q*
_{o}
^{3}must remain constant. This implies, qualitatively, that backflow is minimized by the use of small catheter diameters and, quantitatively, that a fractional reduction of needle diameter is offset by nearly the same fractional increase in flow rate. Hence, for example, if an acceptable backflow distance can be obtained with a 32-gauge needle (radius of 0.0114 cm) at 0.10 μl/min, then the same distance should be obtained with a larger 20-gauge needle (radius of 0.0451 cm) at only 0.016 μl/min. Such trade-offs may be important in selecting catheter sizes that allow infusions to occur in convenient time frames. In our experimentation, we have found that a small-diameter needle practical for implementation is a 32-gauge stainless steel needle, although, for deep penetration into brain, it needs to be sleeved with 26-gauge tubing 1 cm above the tip to prevent bending that can interfere with stereotactic placement.

The form of the scaling relationship shown in*Eq*.*
13
* exhibits a fourth-power dependence on catheter radius*r*
_{c}, and it is this form that we have used for most calculations. Somewhat more precise calculations involving changes in*r*
_{c} could have been performed by introducing the additional dependence on*r*
_{c} contained within the tissue resistance parameter*R* and exhibited in*Eq*.*
EA4
* of appendix. However, this additional*r*
_{c} dependence is only logarithmic and, thus, small relative to the power dependence. Accordingly, we have opted to keep the scaling relationship in the simpler *R*-dependent form of*Eq*.*
15
* for most applications and have ignored the more expanded form except for calculations involving the effects of brain size.

The theoretical backflow model depends on the parameters of Table 1. Four of these are well known or under direct experimental control, i.e., the viscosity, volumetric inflow rate, catheter radius, and white matter thickness. The shear modulus for gray matter is also reasonably well known and has been reported by several groups (1, 21, 27, 28). A corresponding modulus for white matter is not available and we have initially left it unchanged from the gray matter value.

The remaining principal parameters are the hydraulic conductivity values for gray and white matter. Tensor-averaged values for white matter (or closely related data) have appeared in the literature (15,23, 25) but gray matter conductivities have only been estimated. These cover a very broad range from 0.001 to 0.5 of the white matter value. We have found, however, that our model suggests an approximate value for the gray interstitial conductivity of ∼1.6 × 10^{−8}cm^{4} ⋅ dyne^{−1} ⋅ s^{−1}(or 3.2 × 10^{−9} on a whole tissue basis with an extracellular fraction of 0.20) because smaller values are predicted to cause a 4-μl infusion at 0.5 μl/min to overflow the rat caudate in disagreement with experimental observation, and larger values do not lead to the prediction of observed backflow at 1 μl/min. Relative to the average white matter conductivity calculated by Kaczmarek et al. (15), our gray matter value is 0.16 of the white matter value and is within the range suggested in the literature. To keep the derivation and form of the scaling relationship as simple as possible, several approximations have been introduced into the flow model that limit its applicability and accuracy. Aside from the assumptions of lubrication theory, the most important of these are neglect of the axial flow within the tissue adjacent to the catheter and the attribution of deformation solely to radial expansion of the annular wall. The inaccuracies introduced by the approximations are, however, greatest when backflow distances are small (i.e., <0.1 cm, a value obtained, for example, near 0.1 μl/min with a 32-gauge needle). Because many candidate targets for drug delivery in the brain are characterized by dimensions greater than this (e.g., the Gpi or the subthalamic nucleus in the human), such uncertainties would play little role in determining whether backflow would lead either to overflow into adjacent white matter or to significant distortion of the spherical distribution expected for a point source. Characterization of the small inaccuracies that remain in our backflow distance estimates will require the introduction of a more detailed model and computation of its flow fields by finite element techniques.

### Perspectives

In all biological research protocols and therapeutic regimens that require targeted delivery of agents to tissue by direct interstitial infusion, there is a recognized need to select the appropriate infusion parameters so that backflow along the catheter track does not cause the infused agents to miss their intended targets. At present, this selection is most often made on the basis of intuition or rules of thumb of a particular laboratory. Such an approach, however, does not consider all the physiological parameters that affect backflow and, especially, how the selection of one infusion parameter may affect the selection of others. The present work has identified the most relevant parameters for maintaining targeted infusion and, most importantly, has related them through a simple scaling relationship that is useful for estimating the trade-off between catheter radius, volumetric inflow rate, and tissue properties under the condition of maintaining a negligible backflow distance. It implies that backflow is minimized by the use of the smallest possible catheter radius, and that flow rate and catheter radius are nearly inversely related. Evaluated for different sized brains, it also states that the backflow distance achieved for a given catheter and flow rate is only weakly dependent on brain size, increasing slightly with increasing brain size. The practical consequence is that infusion parameters chosen to achieve distribution of agent over a desired absolute distance in a small animal brain are predicted to achieve about the same absolute spread in a primate brain, a finding of utility when extending experimental results on drug delivery in rodents to application in the human. The scaling relationship is useful for estimating backflow and onset of surface leakage for the infusion of both small and large molecules. Although penetration into tissue will be greatly different for various agents due to differences in metabolism and capillary permeability, the shape and extent of the tissue surface through which initial distribution of agent occurs will be determined by the fluid flow and elasticity mechanisms of our backflow model. However, the applicability of the scaling relationship to particles such as plasmids, viral vectors, or liposomes is less certain than it is for molecules no larger than antibodies because it is yet to be generally proven that these relatively large particles will not be severely retarded or even jammed in the interstitial matrix, altering the backflow model parameters (especially the hydraulic conductivity) in a nonlinear and solute concentration-dependent manner.

## Appendix

#### Relationship between hydraulic conductivity and tissue resistance.

The resistance of porous tissue to pressure-driven flow (*R* in*Eq*.*
3
*) is determined by the value of the hydraulic conductivity in Darcy’s law and, to a lesser degree, the dimension of the tissue subject to the flow. An approximate expression for *R* can be derived by idealizing the infused brain tissue as a cylinder of radius*L* containing the catheter of radius*r*
_{c} surrounded by a small annulus of negligible thickness. At the infusion pressures we are considering, little net water loss occurs across the capillary microvasculature of the brain (22) and, for a portion Δ*x* of the cylinder length, we may immediately express the average tissue radial fluid velocity ν(*r*) in terms of the local volumetric flow rate*q*
_{ν} [= (d*Q*/d*x*) ⋅ Δ*x*] and relate it to pressure through Darcy’s law
Equation A1where α =*q*
_{ν} /(2πΔ*x*), and φ is the extracellular fraction in gray matter. This can be integrated to yield
Equation A2The resistance *R* is defined by*Eq*. *
3
*and related to the annular pressure as
Equation A3d*Q*is simply the volume flow rate across the annular wall, 2π*r*
_{c}Δ*x*ν(*r*
_{c}), so −d*Q*/d*x*is 2π*r*
_{c}ν(*r*
_{c}). Thus substituting into *Eq*.*
EA3
* this result for the differential and *Eq*.*
EA2
* with*r* =*r*
_{c} for the pressure, we obtain an expression that yields
Equation A4for the resistance. For the rat brain with a 32-gauge catheter tip centered in the caudate, *L* is on the order of 0.5 cm, *r*
_{c} is 0.01143 cm, and φ is ∼0.2–0.4 (9) (12). Thus the coefficient before the κ term is on the order of unity, ∼1.5–3.

## Appendix

#### Derivation of the linear deformation Eq. 6.

Text *Eq*.*
6
* describes the elastic expansion that a thick cylinder undergoes when subjected to an inner pressure p_{i} and an outer pressure p_{o}. The inner radius is*r*
_{c} and the outer is *L*. Its derivation follows from standard methods (13), briefly presented here. It begins with identification of the strain tensor elements e_{ij} in each of the cylindrical coordinate directions in terms of the displacement vector ω(*r*) describing the radial movement of a point within the solid. For cylindrical symmetry, these are
Equation B1a
Equation B1b
Equation B1cBecause the annulus formed around the catheter is expected to lie close to the cylindrical surface of the catheter, axial strain in the*z* direction is neglected, as with an infinite cylinder, and e_{zz} is set to zero in *Eq*.*
EB1c
*. Because of the symmetry of the system, all shear strain elements, e_{θ}
_{z}, e_{r}
_{θ}, e_{rz}, are also zero.

Next an equilibrium differential force balance on a cylindrical element of tissue is constructed in terms of the stresses operating on each face. The net radial force arising from the stresses on each side of the volume element is
Equation B2An additional radial force arises from the tangential stress component directed along *r*, i.e.
Equation B3At mechanical equilibrium, the forces of*Eqs*.*
EB2
* and *
EB3
* must sum to zero if gravitational forces are neglected
Equation B4To obtain a solution for the displacement vector from*Eqs*.*B1* and*B4*, we relate the stress and strain elements by the general relationship for Hookian solids (see p. 169 of Ref. 13)
Equation B5where λ and *G* are the first and second Lamé constants, *G* being synonymous with the shear modulus. From*Eq*.*
EB5
*, we immediately write expressions for ς_{rr} and ς_{θθ}, recalling that e_{zz} = 0
Equation B6a
Equation B6bFinally, a differential equation in terms of the displacement vector ω(*r*) is obtained when*Eq*.*
6
*, *a*and *b*, together with*Eq*.*B1* for the e_{ij} strain elements, is substituted into the force balance *Eq*.*
EB4
*. The result is
Equation B7This differential equation is associated with two stress boundary conditions derived from *Eq*.*
EB6a
*, accounting for the internal and external pressures, p_{o} and p_{i}
Equation B8aand
Equation B8b
*EquationEB7
* is a simple ordinary differential equation whose solution is ω(*r*) = c_{1}/*r*+ c_{2}
*r*. The constants of integration are determined from*Eq*.*B8*, *a*and *b*. Evaluating ω(*r*) at the inner radius*r*
_{c}, we find that
Equation B9Finally we obtain text *Eq*.*
6
* by observing that*r*
_{c} <<*L* in our application, noting that the ω(*r*
_{c}) displacement vector at the surface is identical to the annular gap distance*h*(*x*) and that, as follows from a Poisson ratio nearing 0.5, λ greatly exceeds *G* so that the second term in*Eq*.*
EB9
* vanishes. Identifying p(*x*) in the text with p_{i} − p_{o} in this appendix,*Eq*.*
EB9
* becomes
Equation 6

## Appendix

#### Derivation of pressure Eq. 7A.

After ignoring all velocity components except*u _{x}
* and defining it more simply as just

*u*,

*Eq*.

*1*may be integrated twice with respect to

*y*to obtain Equation EC1No-slip boundary conditions exist at both the catheter and tissue surfaces of the annulus, i.e. These may be used to evaluate the integration constants in

*Eq*.

*EC1*to yield for the

*y*velocity component in the annulus Equation EC2Integration over this velocity profile according to text

*Eq*.

*2*for volumetric flow in the annulus yields Equation EC3Differentiation of this

*Q*(

*x*) followed by substitution into the Darcy relationship, text

*Eq*.

*3*, yields Equation EC4Dependence on

*h*(

*x*) may be eliminated from this equation by making use of the deformation relationship text

*Eq*.

*6*Equation EC5to yield text

*Eq*.

*7a*.

## Footnotes

Address for reprint requests and other correspondence: P. F. Morrison, Bioengineering and Physical Science Program/ORS Bldg. 13, Rm. 3N17, National Institutes of Health, Bethesda, MD 20892 (E-mail:pmorris{at}box-p.nih.gov).

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- Copyright © 1999 the American Physiological Society