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COMPLEX FUNCTIONS OF THE CENTRAL NERVOUS SYSTEM, SLEEP AND LOCOMOTION
1Division of Bioengineering and Physical Science, Office of Research Services, 2Pain and Neurosensory Mechanisms Branch, National Institute of Dental and Craniofacial Research, and 3Surgical Neurology Branch, National Institute of Neurological Disorders and Stroke, National Institutes of Health, Bethesda, Maryland 20892
Submitted 7 August 2002 ; accepted in final form 4 March 2003
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONREFERENCES |
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60% ablation of target neurons in lamina I/II). Computations
revealed that SP-DT' treatment was favored by a stable SP analog
(half-life 60 min), high infusate concentration (385 nM), and careful
catheter placement (adjacent to target lamina I/II). Sensitivity of cytotoxic
regions to NK1 receptor density and white matter protease activity
was also established. These data suggest that intraparenchymal infusions can
be useful for treatment of localized chronic pain. convection-enhanced delivery; intraparenchymal infusions; pain therapy; pharmacodynamic model; convection; finite-element method
and C fibers and, possibly, A
fibers
(5,
15,
28,
35,
52). In this study, we
investigate the transport and drug delivery of SP-DT' (47,000 mol
wt), a COOH-terminal truncated diphtheria toxin that retains the transmembrane
and catalytic domains of diphtheria toxin but replaces the native binding
domain with SP (15) or an
analog of SP [(pGlu5, MePhe8,
MeGly9)-substance P-(5–11) (DiMeC7)] stabilized against
peptidase activity (14). Previous animal studies investigating NK1R-targeted neurotoxins have used intrathecal delivery to circumvent the blood-brain barrier. In the rat, intrathecal infusion of these compounds has been shown to significantly reduce hyperalgesia (5, 28). However, a major challenge remains in achieving this effect at the larger scale of primates and humans. In these species, the principal cell targets, laminae I and II of the dorsal horn, lie deeper in the spinal cord and are not in direct contact with the intrathecal cerebrospinal fluid (CSF), as in the rat. Consequently, the neurotoxins must traverse a much greater distance within the primate or human spinal cord tissue and, given the slow rate of macromolecular diffusion in tissue (43), be substantially more exposed to the degradative action of endogenous peptidases. As a result, these therapeutic molecules are unlikely to reach their target cells in sufficient concentration to be efficacious, and the need for a different mode of drug delivery is indicated. In addition, intrathecal infusion results in bilateral treatment. However, many conditions are localized, and, in such cases, a more discrete application would be therapeutically advantageous.
An alternative to intrathecal delivery is direct interstitial infusion or convection-enhanced delivery (CED) (7, 25, 32). By this method, an agent is infused directly into the nervous tissue via a cannula and is transported through the interstitial space primarily by convection, or bulk flow, allowing for greater volume transmission and control over exposure (7, 24, 25, 32). To anticipate the tissue distribution of administered agents by infusion into the central nervous system (CNS), mathematical models of CED have been developed. Early models accounted for the distribution into gray matter regions of the brain, characterized by isotropic transport and mechanical properties (3, 4, 32). More recently, a three-dimensional image-based finite-element model of CED into the spinal cord has been developed (44) that captures transport in gray and anisotropic white matter regions and accounts for the distribution of albumin observed in rat spinal cord.
Interstitial transport is anisotropic within white matter (7, 41, 42) because of preferential diffusion and fluid flow in the interstitial space parallel to axonal fibers. Taking advantage of this low-resistance pathway, Lonser et al. (26) and Wood and colleagues (54) showed that direct infusion into the white matter dorsal column provides a basis for safe distribution of macromolecules over significant volumes of the spinal cord. In the present study, we extend the anisotropic CED model (44) to describe SP-DT' transport when the toxin is infused into the white matter dorsal column of the human spinal cord. The objective of an effective chronic pain therapy is to reach nociceptive neurons located in laminae I, II, and X in concentrations adequate to induce cytotoxicity while, at the same time, avoiding the destruction of anterior motor neurons or other cells within the spinal cord that express the NK1R (1, 6). To address these constraints, the model incorporated primary transport considerations, including regional anatomic boundaries, interstitial fluid flow, enzymatic degradation, and binding of SP-DT' to target neurons. Ultimately, cytosolic diphtheria toxin blocks cellular protein synthesis and kills the cell. By coupling intracellular trafficking kinetics (45) to the model, cytosolic concentrations of toxin were determined which provided a basis for predicting regions of cell death.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONREFERENCES |
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Interstitial fluid flow. To describe fluid flow in the spinal cord, the tissue was treated as a porous medium composed of an interstitial volume phase through which bulk flow occurs and a noninterstitial "solid" phase composed of intercellular matrix, cells, white matter fibers, and blood vessels. Fluid and solute transport through porous media is described by a continuum model accounting for differential mass and momentum balances. For steady-state flow conditions, conservation of fluid mass is expressed by the continuity relation
 | (1) |
where 
· is the divergence operator and 
is the
volume-averaged fluid velocity. The contribution of fluid sources and sinks
would require the inclusion of additional pressure-dependent terms
proportional to the difference in interstitial and capillary pressure.
However, such terms were neglected in this model, because tissues of the CNS
are devoid of an active lymphatic system
(9), characterized by low rates
of fluid transfer across the capillary walls at the pressures encountered
during interstitial infusion at a moderate flow rate
(32), subject to negligibly
low rates of water formation by metabolism
(22), and because only
isotonic solutions of SP-DT' were considered. The momentum equation is
expressed by Darcy's law
 | (2) |
where 
is the hydraulic conductivity tensor, is the gradient
operator, and p is the interstitial fluid pressure. is a measure of
the conductance of the material to fluid flow and is dependent on pore
geometry and the structure of the interstitial matrix. is isotropic in
gray matter (12,
25) but highly anisotropic in
white matter because of the presence of the aligned nerve fibers
(7,
26). White matter hydraulic
anisotropy is quantified by large values of
wm-z/wm-x,
which is the ratio of the hydraulic conductivity of white matter in the
craniocaudal direction, parallel to the aligned fibers, to the hydraulic
conductivity of white matter in the transverse direction
(Table 1).
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The model was limited to a description of flow through a rigid porous medium. Only low volumetric flow rates were considered, and elastic expansion effects were accounted for by increasing the interstitial volume fraction (
) from the normal value of 0.20 to 0.26
(12,
40,
44). In the present study,
and were derived from data specific for an infusion rate of 0.1
µl/min (26,
44,
54). Boundary conditions were
such that a volumetric infusion rate of 0.1 µl/min was applied at the
infusion site boundary, and zero fluid pressure conditions were applied at the
external spinal cord-CSF boundaries. Along the symmetrical plane (z =
0), symmetry boundary conditions were assumed: ·
= 0, where
is the unit normal vector to the
surface. At the far end of the tissue (z = 20 mm), fluid pressure was
approximated as zero, because the vast majority of fluid is routed into the
surrounding CSF before it reaches this point. (Applying = 0
along this boundary resulted in negligible changes in the velocity
profile.)
SP receptor-mediated delivery. As the SP-DT' complex moves through the extracellular space, it is subject to protein degradation and cellular interaction. Data of Garland et al. (16), Bowden et al. (8), and Grady et al. (18) were used to develop a simple kinetic model (Fig. 2) of SP binding and intracellular trafficking (45). This kinetic model incorporates the principal trafficking steps, including surface binding between SP and NK1R, clathrin-mediated endocytosis followed by spatial translation to a perinuclear endosome, where SP is sorted from its receptor, rapid degradation of SP once sorted to the late endosomes/early lysosomes, and return of sorted receptor to plasma membrane via recycling endosomes. In addition, the model follows previous research allowing for a very low rate of transfer of the diphtheria toxin moiety from early sorting endosomes to cytosol (57). Incorporation of this trafficking model into the present model allowed for estimation of the cytosolic diphtheria toxin concentration, which is ultimately responsible for cell kill.
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Transport equations were derived by applying mass conservation to SP-DT' and the NK1R in their various compartments: SPe (SP-DT' in the extracellular space), B (bound SP-DT' on the cell surface), R (free NK1R on the cell surface), SPs (early/sorting endosomal SP-DT'), Rs (NK1R in early/sorting endosomes), Rpost (NK1R in recycling endosomes), and Ccyt (SP-DT'/diphtheria toxin in cytosol). The relation governing interstitial transport of SPe is (all concentrations expressed on a tissue averaged basis)
 | (3) |
where t is time and D is the effective diffusivity tensor of
the interstitium and is the interstitial volume fraction. Other
constants are km (rate of degradation by peptidases) and
k1 and k2 (rates of SP-DT'
association and dissociation from the receptor). The first two terms on the
right-hand side of Eq. 3 correspond to diffusive and convective
transport, respectively. Competitive binding with native SP was neglected
because of the low endogenous levels present [0.4 nM, calculated from
measures of capsaicin-evoked release of SP-like immunoreactivity in guinea pig
dorsal horn sections from Geppetti et al.
(17)] compared with the high
SP-DT' concentrations considered for therapy (tissue-averaged values of
20, 100, and 200 nM) and the large pool of NK1R available for
binding (12–23 nM). Equations associated with binding and cellular
trafficking within NK1R-expressing neurons are given by
 | (4) |
 | (5) |
 | (6) |
 | (7) |
 | (8) |
and
 | (9) |
Intracellular trafficking constants are as follows: ki (rate of SP-DT' internalization), ktrans (rate of SP-DT' transfer from endosome to cytosol), ks (rate of SP-DT' and NK1R intracellular sorting), and krcyc (rate of NK1R transfer from the recycling endosomes to the cell surface). Because NK1R are conserved
 | (10) |
where Bmax is the NK1R capacity and varies within different laminae of the spinal cord.
To determine the temporal and spatial distribution of the SP-DT' complex, Eqs. 1–10 were solved over the spinal cord volume. For the initial condition before infusion, concentration of membrane-bound NK1R (R) was set equal to the corresponding Bmax. All
other concentrations were initially zero. Concentration of SPe at
the infusion site boundary was set equal to Cinf where
is the tissue porosity and Cinf is the infusate concentration, a
constant value (77, 385, and 770 nM). Transport of SPe across the
external spinal cord/CSF surfaces was assumed to be dominated by convection,
with negligible loss due to diffusion (-DSPe
· = 0). Along the symmetrical
plane (z = 0), SPe flux normal to the surface was set to
zero (SPe ·
= 0).
Parameter values. Parameters are tabulated in Tables 1 and 2. Tissue porosity and hydraulic conductivity values for gray and white matter were taken from a
previous analysis of albumin infusion in the rat cord
(44). Effective diffusivity in
gray matter (Dgm) was taken from measures of a
similar-molecular-weight protein (i.e., albumin, 66,000 mol wt) in brain
tissue (48), and the tensor
components of the effective diffusivity in white matter
(Dwm-x,
Dwm-y, and
Dwm-z) were estimated from
measurements of anisotropic diffusion in the spinal cord
(41). Rate constants
k1, k2, ki,
ks, and krcyc were taken from previous
analysis (45) of the data of
Garland et al. (16), Bowden et
al. (8), and Grady et al.
(18). The gray matter
degradation rate (km-gm)
was set equivalent to an SP half-life of 60 min
(14). Some sensitivity studies
were also performed using an SP gray matter half-life of 3 min. SPe
degradation rate in white matter interstitium was bracketed by setting it
equal to the CSF degradation rate
(23) or to one-half of the
gray matter rate. The latter is a high-end estimate that accounts for the
absence of enkephalinase activity in white matter
(39) but allows for
degradation by peptidases that have not been quantitatively characterized
(11). Receptor capacities
(Bmax) ranged from zero in the white matter, where no
NK1R are detectable
(13), to tissue-averaged
concentration values of 23 nM, reflective of the receptor densities in laminae
I, II, and X (1). Sensitivity
to changes in receptor capacity was also analyzed.
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Diphtheria toxin cytotoxicity analysis is dependent on the rate of transfer of toxin from the endosome to cytosol, as determined by ktrans. Although this parameter has not been determined specifically for SP-DT', it is known to depend heavily on the toxic moiety and is independent of the cell line involved (57). Accordingly, the value of ktrans in our target neurons has been roughly approximated by the 3.5 x 10-5 min-1 value measured previously for diphtheria toxin conjugated to transferrin (57). (Sensitivity computations show that substantial uncertainty can be tolerated in this parameter, because the steep leading edge of the Ccyt profiles renders the cytotoxic contours insensitive to nearly order-of-magnitude uncertainties in the cytosolic concentrations of the diphtheria toxin moiety.)
Cytotoxic response. Approximate contours of cytotoxicity were predicted by locating the tissue regions where NK1R-expressing cells contained ≥10 molecules of the diphtheria toxin moiety in the cytosol. This threshold number exceeds the single-molecule value reported by
Yamaizumi et al. (55) as
sufficient for a diphtheria toxin fragment A to kill a cell. However, this
threshold is consistent with the number of molecules required to fully deplete
the elongation factor 2 (EF2) pool when the fragment's half-life is taken as
16 h (47), rather than several
days, as reported earlier (33,
56). {The number of toxin
molecules required for depletion of the EF2 pool [N(t)] was
computed from a model of the following form: d[EF2]/dt =
-kN(t)[EF2], where N and [EF2] are the number of
toxin and EF2 molecules per cell, N decays exponentially from its
initial loading value N0 with a 16-h half-life, EF2(0)
2.1 x 106 molecules/cell
(56), and k = 0.001
min-1, corresponding to a toxin destruction rate of
2,000 EF2 molecules per toxin molecule per cell per minute
(33). N0
is the minimal value allowing [EF2] < 1 as t 
.} The
threshold cytosolic concentration for a neuron was thus 10 toxin molecules per
neuronal cell volume of 10.8 pl
(37) or 1.5 pM. Because spinal
cord distributions were computed on a tissue-averaged basis and because this
threshold is approached only in SP receptor-bearing neurons, 1.5 pM was
converted to a tissue-averaged threshold value (Ccyt thres)
according to Ccyt thres = 1.5 pM x (fraction neuronal volume)
x (fraction of NK1R-positive neurons). The fractional
neuronal volume is 0.17
(31); the fraction of
NK1R-positive neurons varies by lamina, with the assumption of the
following values: 0.45, 0.29, and 0.18 in laminae I, IV-VI, and X and 0.11 in
lamina II/III (49). For lamina
I, the primary drug target, Ccyt thres = 0.12 pM. Lower values
apply to the other laminae in accordance with their reduced NK1R
expression, but the cytotoxic contours computed for these laminae differ
little from those computed using the 0.12 pM lamina I threshold because of the
leading-edge steepness of the Ccyt concentration profiles.
Implementing the finite-element method. The equations were solved by a Galerkin finite-element method (FIDAP version 8.60, Fluent, Lebanon, NH) using an iterative segregated approach. Transient solutions were calculated with an implicit time integration scheme (2nd-order trapezoid rule) in
conjunction with an adaptive time-stepping methodology. The spinal cord model
was discretized into finite elements with the use of eight nodal hexahedral
elements. The final mesh for the full three-dimensional model consisted of 1.5
x 106 elements. (Meshes with alternate infusion sites had
similar element counts.) With consideration that the steady-state velocity
field was quickly established and that the presence of infusate solute did not
affect the fluid flow solution, a weakly coupled problem was assumed. Thus the
solution for the steady-state velocity and pressure (Eqs. 1 and
2) was obtained, first, using the full three-dimensional geometry
(
z = 20.0 mm). Concentration distributions of the nonfluid
components were then computed from their mass balances (Eqs.
3–10) while the fluid velocity values were fixed at their
steady-state values. Because of prohibitively long calculation times required
for the 1.5 x 106 element mesh, concentrations were
determined using a truncated mesh representing a thin transverse section at
the site of infusion (z = 0.1 mm, 65,000 elements), at which
level maximum penetration of SP-DT' occurs.
The convergence criterion for velocity and pressure was 10-7, and the concentration convergence criterion was 10-6. Reduction of these criteria by two orders of magnitude did not affect results (negligible change). Mesh independence was
assessed by increasing the number of elements by 30%. The resulting changes in
Ccyt distribution after 2 and 8 h of SP-DT' infusion were
negligible. Computations were conducted on a personal computer workstation
(Dell Precision 620) with dual Pentium III 1.0-GHz processors and 2.0 GB of
random access memory. For calculation of an 8-h infusion, 16 h were
required for steady-state velocity and pressure simulations and 32 h for
albumin transport simulations.
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONREFERENCES |
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z = 20 and 40 mm spinal cord
lengths. These interstitial velocity solutions were used to solve for the
distribution of a nonreactive, nonbinding macromolecule, i.e., albumin
(66,000 mol wt). Resulting simulations showed negligible difference in
craniocaudal distribution of albumin, even after8hof infusion at 0.1
µl/min. The computed craniocaudal distributions of albumin corresponding to
10-, 20-, and 30-µl infusions (1.7, 3.3, and 5 h) were 1.4, 1.8, and 2.2
cm, respectively (using a 0.15Cinj plotting threshold value
and midline cannula placement adjacent to lamina X). When these results are
compared with previous magnetic resonance imaging (MRI) measurements of
gadolinium-labeled albumin infused into the lateral column of primates
(26), simulation results
correspond well to MRI measures for infusions up to 4.2 h (25 µlor
1.6 cm of craniocaudal spread).
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Cinj
plotting threshold). At the times shown, SPe is confined primarily
to the white matter interstitium, and asymmetry in the distribution was due to
the off-center infusion site. The extent of white matter spread depended on
the level of peptidase activity. For very little activity, equivalent to the
degradation rate in the CSF
(23), the entire extracellular
space of the dorsal column at the level of the cannula tip was filled after 12
h of infusion of 385 nM SP-DT' at 0.1 µl/min
(Fig. 5A). For white
matter activity equal to one-half of the gray matter rate, the maximum spread
did not fill the dorsal column, and spread reached an effective penetration
limit after 8 h of infusion under the same conditions
(Fig. 5B). As
SP-DT' entered the gray matter regions, it was subject to receptor
uptake and degradation, and interstitial convective transport was greatly
reduced. An SPe concentration profile through a section of the
target tissue, lamina I/II, at the level of the cannula tip is shown in
Fig. 6 (8 h of infusion of 385
nM SP-DT' at 0.1 µl/min and white matter peptidase activity of
Fig. 5A). Here,
SPe dropped below the 15% plotting threshold (15 nM,
tissue-averaged value) within 0.01 cm of the gray matter-white matter
interface.
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Intracellular SP-DT'. Simultaneously, concentrations of B, R, SPs, Rs, and Rpost within NK1R-expressing neurons were also determined, and the cross-sectional profiles of these tissue-averaged concentrations through lamina I/II at the level of the cannula tip are presented in Fig. 6 after 8 h of infusion of 385 nM SP-DT' at 0.1 µl/min. As SP-DT' (SPe) enters the laminar region, it binds strongly to cell surface NK1Rs, first to those initially present on the cell surface and then just to those recycled to the surface. This eventually allows the rate of NK1R internalization to balance the recycling rate over the first 0.035 cm from the gray matter-white matter interface, leading to the steady concentrations of surface-bound and endosome-borne NK1R pools (B, Rs, and Rpost), as well as SPs (Fig. 6). Exposed NK1R-expressing neurons extend through nearly 60% of the target lamina I/II at the level of the cannula tip. In unexposed regions, B, SPs, Rs, and Rpost are zero, and the concentration of free receptors on the cell surface (R) equals Bmax. Cytosol concentrations of SP-DT' were also tracked. Because of the low endosomal escape rate (ktrans), these concentrations were orders of magnitude smaller than those of the other cellular compartments (Fig. 7). Higher concentrations of Ccyt in lamina I/II and X tissue reflect the higher NK1R densities and, thus, greater cellular uptake in these regions.
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Cytotoxic regions. A cytosolic toxicity threshold of 0.12 pM was applied to predict spinal cord regions in which NK1R-expressing neurons would be selectively ablated. The advancement of these cytotoxic regions in the transverse plane of the infusion source is presented in
Fig. 8. Unless otherwise noted,
the interstitial white matter degradation rate of SP-DT' in these
simulations was set equal to the CSF degradation rate
(23). The initial phase of
infusion caused SP-DT' to traverse the intermediate white matter
regions, and continued infusion allowed SP-DT' to penetrate farther into
the gray matter regions adjacent to the dorsal column. For the protocol shown
in Fig. 8 (385 nM SP-DT'
with approximately central cannula placement), 33 min (3.3 µl) of
SP-DT' infusion was required for NK1R-expressing neurons at
the nearest lamina to begin reaching cytotoxic levels. With continued
infusion, cytotoxicity was predicted to extend across >60% or 0.43 mm of
lamina I/II (along the cut closest to the infusion site, Figs.
4 and
6) and across the majority of
the lamina X band. The predicted penetration into laminae I, II, and X,
despite the presence of high concentrations of NK1R in these
regions, was not markedly different from penetration into the neighboring
laminae. The rate of transverse laminar cytotoxic penetration began to slow
after 6 h of infusion, and prolongation of the infusion for an additional 6 h
increased the penetration by only 3% along the same lamina I/II cut. Thus
predicted regions of cytotoxicity reached an effective penetration limit in
the transverse direction. At the level of the cannula tip, this limit was
calculated to be 6–8 h (36–48 µl) for the 0.1 µl/min
infusion protocols of this study. At the same time, anterior regions of the
spinal cord were unaffected.
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Continued SP-DT' spread after termination of the interstitial infusion was also analyzed. At 24 h after the baseline 8-h infusion, simulations showed that cytotoxic regions in the transverse plane containing the cannula tip did not change, except in a small region of the substantia gelatinosa on the side most distant from the infusion site (not shown). With shorter infusion periods, predicted cytotoxic regions continued to expand in this same transverse plane after the end of infusion. At 24 h after a 4-h infusion, cytotoxic spread along the lamina I/II cut closest to the infusion site increased by 9% compared with the penetration distance calculated immediately after the end of infusion (Fig. 9).
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Sensitivity of the cytotoxicity zones to changes in infusion protocol was also investigated (Fig. 10). The base case for these computations was cytotoxic response for an infusate concentration of 385 nM, a volumetric infusion rate of 0.1 µl/min over 8 h, degradation rate in gray matter (km-gm) = 0.0116 min-1 (half-life = 60 min), penetration rate into white matter (km-wm) = 0.000052 min-1 (half-life = 9 days), and a central cannula tip location as depicted in Fig. 8. First, increasing or decreasing the infusate concentration from 385 nM to 77 or 770 nM SP-DT' resulted in a corresponding nonlinear increase or decrease in transverse cytotoxic penetration (Fig. 10A). Increasing the concentration to 770 nM increased the penetration along the closest lamina I/II cut by 40%. Penetration to the ventral side of lamina X was also predicted, but it was not associated with significant exposure of the anterior motor regions.
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Second, movement of the infusion site to locations more proximate to the target regions was predicted to have a significant effect. Placement of the cannula tip closer to lamina I/II (x = 0.8 mm) resulted in a 44% increase in cytotoxic penetration along the closest lamina I/II cut
relative to that for the baseline infusion location
(Fig. 10B).
Furthermore, cytotoxic penetration into lamina I/II on the opposite side was
reduced. After the cannula was positioned closer to lamina X
(y = 1.6 mm), SP-DT' was predicted to cross lamina X and
reach the motor neurons in the ventral horn via the white matter of the
ventral column.
Third, use of an SP analog that is resistant to peptidase action provided for a much deeper predicted penetration into the laminae than is achieved with nonstabilized SP. Peptidase activity against neuropeptides is greater in gray than in white matter and, in the case of enkephalinase, nearly undetectable in white matter by immunoautoradiography (39). Accordingly, we determined the advantage of using stabilized SP moieties by contrasting the cytotoxicity penetration depth achieved with such an analog (characteristic peptidase degradation time of 60 min in gray matter, base case) to that achieved by nearly normal SP (characteristic degradation time of 3 min), while maintaining white matter activity at the minimal level typical of degradation in the CSF (23). At the higher gray matter degradation rate, the penetration of cytotoxicity along the lamina I/II cut closest to the cannula tip was predicted to decrease by 40% (Fig. 10C). Because peptidases other than enkephalinase are known to be active against SP in spinal cord slices (11), the possibility exists that white matter activity may be higher than the CSF activity used in our base case. A second simulation was thus performed to determine the effect of white matter peptidase activity on the extent of cytotoxicity. White matter activity was taken at an estimated upper-bound limit of one-half of the gray matter activity against the analog compound (Fig. 10D). With this large increase in the white matter activity (characteristic degradation time of 120 min relative to the base case time of 9 days), the extent of cytotoxicity along the lamina I/II cut closest to the cannula tip decreased by only 28%. The primary peptidase activity controlling the spread of SP analog-mediated toxicity into the laminae thus appears to be that of the gray matter.
Fourth, order-of-magnitude changes of base case laminar NK1R densities (Bmax) significantly affected the expected cytotoxicity distribution in the spinal cord. Although these density parameters have been well measured by autoradiography for normal and altered lumbar spinal cord
after chronic constriction injury of the sciatic nerve
(1), other reports indicate
that binding capacities may be a fewfold different at other spinal cord levels
(10), during polyarthritis
(27), or after hindpaw
inflammation or sciatic denervation
(2). Raising the receptor
concentrations throughout the laminae is predicted to be similar in effect to
a decrease of infusate concentration. Hence, a fivefold increase in
NK1R would decrease the base case cytotoxic region by a degree
similar to the 385 nM to 77 nM shift seen in
Fig. 10A. On the
other hand, a 10-fold decrease in NK1R density is sufficient to
shift the cytotoxic contour from its baseline position to a location well
inside the ventral horn, where motor neurons would be at risk
(Fig. 10E). During
nociception, NK1R levels have been observed to increase
approximately twofold (6,
30). We have estimated that
this would lead to an 20% decrease in dorsal column penetration distance.
In contrast, order-of-magnitude changes in the affinity constant [associated
with peripheral inflammation or chronic constriction injury
(1,
46)] were calculated to have
little effect on the extent of cytotoxicity due to tight binding and the
existence of a sharp moving boundary at the leading edge of the region.
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONREFERENCES |
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For the base case simulation parameters, cytotoxic contours predict selective ablation of over one-half of the target neurons in laminae I and II at the level of the cannula tip after 8 h of 385 nM SP-DT' infusion at 0.1 µl/min. Continually increasing regions of ablation of the target
neurons were predicted with increasing infusate concentration. However, at the
highest doses considered (770 nM), nonspecific binding, which is not accounted
for in the present model, may occur, resulting in nonspecific damage
(29,
52). In previous rat studies
by Wiley and Lappi (52),
intrastriatal injection of SP-saporin at high doses (570 nM) produced
regions of central necrosis at the point of injection. Similar necrosis is
expected with sufficiently high doses of diphtheria toxin.
In the gray matter, a slow SP interstitial degradation rate, consistent with the in vivo degradation rate of the analog DiMeC7 (14), was considered. Such stabilized SP analog conjugates were predicted to penetrate target laminae
60% farther than the penetration distances of more rapidly degraded
native SP conjugates. Thus SP analogs stabilized against peptidase activity
are expected to require a lower dose to be equally as effective as native SP
conjugates. This is consistent with the observations of Wiley and Lappi
(53) and Martin and Sloviter
(29), who reported that
relatively low doses of the [Sar9,
Met(O2)11]-SP conjugate, injected into the rat striatum
and hippocampus, achieved tissue ablation with negligible nonspecific damage.
These findings highlight the limitations of native SP as a targeting moiety
and underscore the need for a degradation-resistant SP analog that reduces the
likelihood of nonspecific neuronal loss by lowering the administered dose
requirement.
For the degradation-resistant SP analogs considered in our simulations, transverse gray matter penetration of SP-DT' was limited by the high binding affinity, i.e., the binding site barrier (21). Evidence of this dynamic was the great sensitivity of cytotoxic spread to NK1R density in tissue. Thus cytotoxic spread may change according to the NK1R distribution, which may vary with spinal cord level (10) or source and duration of pain (2, 27). Transverse cytotoxic spread was also sensitive to cannula placement. By moving the infusion site closer to the target lamina I/II, an asymmetric but more targeted ablation was achieved. Lateralized effects are desirable in all cases where nerve injury or damage affects only one limb. If bilateral treatment is required, the asymmetry in cytotoxic spread suggests that dual infusion from a symmetrically placed cannula adjacent to the corresponding lamina I/II may increase the ablation efficiency. However, care should be taken in cannula placement, inasmuch as more anterior cannula placement resulted in an increased likelihood of exposing motor neurons to the neurotoxin.
After the end of infusion, the SP-DT' remaining in the interstitial space continues to diffuse until it is bound/internalized or degraded by peptidase action. Thus transverse cytotoxic spread was predicted to extend beyond the contours calculated immediately after the end of infusion, at least for shorter infusion times (<6–8hat0.1 µl/min base case). After longer infusions (>8 h), the remaining interstitial SP-DT' cannot penetrate beyond an effective transverse penetration limit, which is determined by the balance of binding and peptidase degradation. As a result, computational analysis of the postinfusional transverse cytotoxic spread may not be necessary after longer infusions.
To be as effective as surgical procedures currently used to alleviate chronic pain, the protein toxin must also spread in the craniocaudal direction to spinal cord regions that are contacted by ascending and descending branches of primary afferent nociceptive neurons
(50). Preliminary studies were
conducted assessing the ability of the spinal cord model to predict
interstitial transport of albumin in the craniocaudal direction. Simulation
results corresponded to MRI measures for infusions up to 25 µl. This
finding corresponds with a previous porous media model of CED in the rat
spinal cord (44), which shows
a tendency to underestimate craniocaudal transport at larger infusion volumes.
Both models predict loss of infusate solute into the adjacent CSF, suggesting
that there may be an additional transport factor affecting the interstitial
convective flow patterns. The pial membrane adherent to the exterior of the
spinal cord may act as a fluid transport barrier, reducing the loss of fluid
into the CSF. Alternatively, hydraulic conductivity may vary spatially within
the white matter, with more localized deformation parallel to the white matter
fibers near the infusion site. However, such transport effects are expected to
have little effect on the convective flow patterns near the cannula tip and
the present transverse transport analysis.
Craniocaudal cytotoxic distribution was calculated for a 25-µl infusion of SP-DT' for a simplified case (assuming symmetry about the medial axis, i.e., the dorsal medial fissure, and central cannula placement, 2.1 mm posterior to lamina X). Simulations measuring the craniocaudal length along which there is cytotoxic penetration into lamina I/II predicted distances of 0.51, 1.36, and 1.43 cm corresponding to 10-, 20-, and 25-µl infusions of 385 nM SP-DT' at 0.1 µl/min (km-wm = 0.000052 min-1). At 24 h after the end of the 25-µl infusion, the craniocaudal length along which there is cytotoxic penetration into lamina I/II was predicted to increase by 34% over that immediately at the end of the infusion to a maximum distance of 1.9 cm. These results suggest that SP-DT' can achieve the rostrocaudal extent necessary for treatment of chronic pain conditions that extend over one or two spinal cord segments.
The hydraulic conductivity anisotropy ratio wm-z/ wm-x, determined in a previous study (44), allows prediction of the
interstitial fluid flow patterns within the white matter of the spinal cord,
provided infusion is limited to a volumetric flow regimen close to 0.1
µl/min. At other flow rates, the anisotropy ratio and tissue porosity may
diverge from the presented values because of tissue deformation effects. It is
plausible, for example, that
wm-z/wm-x
and tissue porosity values may be considerably smaller in the situation of
endogenous flows in normal white matter or, conversely, that these values may
increase with increasing volumetric infusion rate and be associated with
increased convective transport of SP-DT'.
In the determination of cytosol concentrations, uncertainty exists as to what fraction of the residence time in the sorting endosome is available for the transfer of protein toxin to the cytosol (e.g., because of changing pH or changes in budding efficiency during endosomic maturation). Delivery of the toxin to the cytosol is effectively determined by the factor ktrans x (time available for release to the cytosol). The ktrans used in this study was not determined specifically for the SP-DT', because its value has not been measured. We used the value for transferrin-diphtheria toxin determined by Yazdi and Murphy (57), who determined that ktrans depends heavily on the toxic moiety and is independent of the cell line involved. However, model simulations predict that there is no significant effect on the extent of cytotoxicity with nearly an order of magnitude uncertainty in ktrans x (time available for release to the cytosol), because the leading edge of the cytotoxic concentration profile of diphtheria toxin is so steep. However, there may be implications for other protein toxins. Hence, SP coupled to saporin or Pseudomonas exotoxin may have ktrans values that differ greatly from that used for diphtheria toxin, resulting in different cytotoxic contours.
Interstitial infusion of SP-DT' directly into spinal cord tissue has not been undertaken in small or large animal models. Indeed, the results of the present study provide a guideline for future experiments. In particular, further experiments are needed to determine whether partial ablation of lamina
I/II (60%) is sufficient to effectively treat chronic pain. Additional
studies measuring white matter peptidase activity, nonspecific toxicity of
SP-DT', hydraulic conductivity at higher flow rates, and
ktrans would also be useful for further model
development.
Given the complex nature of the tissue response to various infusion parameters, the computational model provides aid in assessing the design and implementation of specific therapeutic protocols. The present analysis predicts that direct infusion of SP-DT' can be used to selectively
ablate NK1-expressing neurons in laminae I, II, and X and that
anterior motor neurons are not at risk with careful consideration of cannula
placement, infusate concentration, and flow rate. Findings applicable to
future SP-DT' infusion designs are as follows: 1) Longer
infusions did not necessarily translate into greater transverse spread into
the regions exhibiting cytotoxic susceptibility. For the infusion conditions
considered in this study, maximal laminar penetration was realized between
6 and 8 h at 0.1 µl/min (36–48 µl). 2) High infusate
concentrations provide maximal targeted ablation. However, nonspecific binding
at very high concentrations should be avoided. 3) SP analogs designed
to be resistant to enzymatic degradation are required, because naturally
occurring SP degrades too rapidly. 4) Placement of the cannula
adjacent to the target lamina I/II region provides for maximum laminar
penetration of that side. However, care should be used in cannula placement,
because more anteromedial placement may result in anterior motor neuron
exposure. 5) Accurate measurement of NK1R density and
peptidase activity in targeted spinal cord tissue is necessary, given the
sensitivity of the cytotoxicity distribution to these parameters.
More generally, the type of model developed here for the spinal cord, with its inclusion of transport anisotropy, realistic anatomy, and pharmacodynamics, may now be extended to the description of drug delivery to other tissues of the CNS. Potential pharmacological agents with distribution that may be so predicted include the neurotrophic factors and excitotoxins for treatment of neurodegenerative disorders, enzymes for deficiency syndromes, genetic vectors, and radioimmunoconjugates for treatment or imaging.
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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