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A mathematical model for potassium diffusion in dentinal tubules
Archs oral Bid. Vol. 39, Suppl., p. 145% 1994 Copyright 0 1994 Else&r Science Ltd Printed in Great Britain. All rights reserved 0003-9969/94 $7.00 + 0.00
Pergamon
A MATHEMATICAL MODEL FOR POTASSIUM DIFFUSION IN DENTINAL TUBULES W. J. STEAD,’
P. B. WARREN,’
R. ORCHARDSON’
and A. J. ROBERTS’
‘Unilever Dental Research, Port Sunlight Laboratories, Wirral, England and *Institute of Physiology, University of Glasgow, Glasgow, Scotland Key words:
potassium,
diffusion,
dentinal
Potassium salts are used as the active agent in dentifrices to relieve dentinal hypersensitivity. The potassium ions are thought to exert their effect by inactivating nerves located at the inner ends of dentinal tubules or in the subjacent pulp, but it is not clear whether topically applied potassium ions can diffuse along the dentinal tubules in sufficient quantities to affect the excitability of intradental nerves (Orchardson and Peacock, 1994; this issue). The model described here considered the diffusion of potassium ions against an outward convective flow of fluid along a conical tubule 2 mm long and tapering from a diameter of 2 pm at the pulpal end to 1 pm at the oral end. The tubule contained an odontoblast process that extended 700pm from the pulp and occupied 80% of the tubule cross-sectional area. The rate of fluid flow along individual tubules was calculated from permeability data on dentine slices and the numbers and diameters of tubules (Fogel, Marshall and Pashley, 1988). The calculated flow rates were 0.4-l.Opm/s at the oral end of the tubule and are similar to the in tlizw flow rates (1.4 f 1.2 pm/s) reported by Vongsavan and Matthews (1992). Ion movements between the pulp and the fluids in the tubule were governed by a permeability barrier representing the tightly packed odontoblast layer. This barrier may
0
5
10
15
20
25
30
tubules,
hypersensitive
dentine.
be freely permeable (permeability, k = so), impermeable (k = 0) or partially permeable (k = 1, 10) to potassium ions. The pulp was taken as an infinite source of fluid and a sink for potassium ions. The potassium ion concentration in the pulp was assumed to be 4mM, and at the oral end of the tubule the salivary potassium concentration was taken as IO-20 mM. The nerve terminal was assumed to be located at the pulpal end of the tubule near the permeability barrier. The resting concentration of potassium in the tubule was calculated (Probstein. 1989) for different fluid-flow velocities and salivary K+ concentrations using several different barrier permeabilities. The effect of applying a I-min pulse of 500 mM potassium, to mimic the use of a desensitizing dentifrice, was then considered. The barrier permeability, (k) proved to be a key parameter in determining the potassium ion concentration at the nerve terminal, both in the steady state and after a pulse of potassium. After a K + pulse, at low or moderate barrier permeabilities (k = O-10) enough potassium diffused to the inner end of the tubule to reach concentrations (> 16 mM) necessary for blocking conduction in nerve fibres, but a highly permeable barrier (k = co) caused most of the potassium to be lost to the pulp (Fig. 1). The elevations in [K+] reached maximum values 5-7 min after the application of the high-potassium pulse, and declined thereafter towards resting levels. When the outward fluid flow rate was increased to 4 pm/s, the amount of potassium diffusing inwards was reduced, attaining a maximum of only 5 mM following a high potassium pulse. In hypersensitive teeth, the pulp pressure and permeability of the odontoblast layer may be affected by the state of the tooth pulp and these variables in turn may have a profound effect on intratubular potassium concentrations attained during application of K-containing preparations. REFERENCES
Time (min)
Fogel H. M., Marshall F. J. and Pashley D. H. (1988) J. dent. Res. 67, 1381-1385. Orchardson R. and Peacock J. M. (1994) Archs oral Biol. 39 (SuppI.), 81SA36S. Probstein R. F. (1989) Physicochemical Hydrodynamics: An Introduction. Butterworth, Boston. Vongsavan N. and Matthews B. (1992) Archs oral Biol. 37, 175-185.
Fig. 1.The time course of changes in [K + ] at the pulpal end of a 2-mm tubule, following a 1-min application of 500 mM K+ to the oral end of the tubule. The fluid flow rate was 0.4 pm/s, and the initial salivary [K+] was 20 mM. The four traces represent different values for the barrier permeability: O-impermeable (k = 0); O-freely permeable (k = ru); partially permeable: A (k = 1) and l (k = 10).
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Pergamon
A MATHEMATICAL MODEL FOR POTASSIUM DIFFUSION IN DENTINAL TUBULES W. J. STEAD,’
P. B. WARREN,’
R. ORCHARDSON’
and A. J. ROBERTS’
‘Unilever Dental Research, Port Sunlight Laboratories, Wirral, England and *Institute of Physiology, University of Glasgow, Glasgow, Scotland Key words:
potassium,
diffusion,
dentinal
Potassium salts are used as the active agent in dentifrices to relieve dentinal hypersensitivity. The potassium ions are thought to exert their effect by inactivating nerves located at the inner ends of dentinal tubules or in the subjacent pulp, but it is not clear whether topically applied potassium ions can diffuse along the dentinal tubules in sufficient quantities to affect the excitability of intradental nerves (Orchardson and Peacock, 1994; this issue). The model described here considered the diffusion of potassium ions against an outward convective flow of fluid along a conical tubule 2 mm long and tapering from a diameter of 2 pm at the pulpal end to 1 pm at the oral end. The tubule contained an odontoblast process that extended 700pm from the pulp and occupied 80% of the tubule cross-sectional area. The rate of fluid flow along individual tubules was calculated from permeability data on dentine slices and the numbers and diameters of tubules (Fogel, Marshall and Pashley, 1988). The calculated flow rates were 0.4-l.Opm/s at the oral end of the tubule and are similar to the in tlizw flow rates (1.4 f 1.2 pm/s) reported by Vongsavan and Matthews (1992). Ion movements between the pulp and the fluids in the tubule were governed by a permeability barrier representing the tightly packed odontoblast layer. This barrier may
0
5
10
15
20
25
30
tubules,
hypersensitive
dentine.
be freely permeable (permeability, k = so), impermeable (k = 0) or partially permeable (k = 1, 10) to potassium ions. The pulp was taken as an infinite source of fluid and a sink for potassium ions. The potassium ion concentration in the pulp was assumed to be 4mM, and at the oral end of the tubule the salivary potassium concentration was taken as IO-20 mM. The nerve terminal was assumed to be located at the pulpal end of the tubule near the permeability barrier. The resting concentration of potassium in the tubule was calculated (Probstein. 1989) for different fluid-flow velocities and salivary K+ concentrations using several different barrier permeabilities. The effect of applying a I-min pulse of 500 mM potassium, to mimic the use of a desensitizing dentifrice, was then considered. The barrier permeability, (k) proved to be a key parameter in determining the potassium ion concentration at the nerve terminal, both in the steady state and after a pulse of potassium. After a K + pulse, at low or moderate barrier permeabilities (k = O-10) enough potassium diffused to the inner end of the tubule to reach concentrations (> 16 mM) necessary for blocking conduction in nerve fibres, but a highly permeable barrier (k = co) caused most of the potassium to be lost to the pulp (Fig. 1). The elevations in [K+] reached maximum values 5-7 min after the application of the high-potassium pulse, and declined thereafter towards resting levels. When the outward fluid flow rate was increased to 4 pm/s, the amount of potassium diffusing inwards was reduced, attaining a maximum of only 5 mM following a high potassium pulse. In hypersensitive teeth, the pulp pressure and permeability of the odontoblast layer may be affected by the state of the tooth pulp and these variables in turn may have a profound effect on intratubular potassium concentrations attained during application of K-containing preparations. REFERENCES
Time (min)
Fogel H. M., Marshall F. J. and Pashley D. H. (1988) J. dent. Res. 67, 1381-1385. Orchardson R. and Peacock J. M. (1994) Archs oral Biol. 39 (SuppI.), 81SA36S. Probstein R. F. (1989) Physicochemical Hydrodynamics: An Introduction. Butterworth, Boston. Vongsavan N. and Matthews B. (1992) Archs oral Biol. 37, 175-185.
Fig. 1.The time course of changes in [K + ] at the pulpal end of a 2-mm tubule, following a 1-min application of 500 mM K+ to the oral end of the tubule. The fluid flow rate was 0.4 pm/s, and the initial salivary [K+] was 20 mM. The four traces represent different values for the barrier permeability: O-impermeable (k = 0); O-freely permeable (k = ru); partially permeable: A (k = 1) and l (k = 10).
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