Journal of Neuroscience Methods, 1 (1979) 165--178 © Elsevier/North-Holland Biomedical Press

165

THE PHYSICS OF IONTOPHORETIC PIPETTES

ROBERT D. PURVES

Department of Anatomy and Embryology, University College London, Gower Street, London WCIE 6BT (England) (Received March 22nd, 1979) (Accepted April 20th, 1979)

A theoretical study has been made of the release of drugs from iontophoretic pipettes. The chief predictions are: (1) in the steady state, release becomes linear with current and independent of pipette geometry when the ejecting voltage exceeds approximately 100 mV; (2) release rises relatively slowly during an ejecting pulse to approach its steady state value. At the end of the pulse, release falls abruptly. This asymmetry of the release curve is greatly accentuated by the prior application of a retaining current. It provides an explanation for the characteristic time course of response commonly seen when drugs are applied iontophoretically to central neurones; (3) the above effects occur on a time scale which depends strongly on the geometry of the pipette. Release from pipettes of large tip diameter and nearly cylindrical bore, such as those generally used in the central nervous system, should be thousands of times slower than from pipettes of small tip diameter and large taper angle. Experimental observations of the release of an ionized fluorescent compound quantitatively confirm prediction (1) and qualitatively confirm predictions (2) and (3).

INTRODUCTION

The iontophoretic method for the microapplication of drugs and neurotransmitters is widely used to provide rapid, but somewhat imprecise, control of drug concentrations in the vicinity of receptive cell surfaces (Curtis, 1964; Kelly, 1975). The ideal iontophoretic pipette should display the followingproperties. (1) In the absence of ejecting current there should be no resting efflux. (2) On the application of current the efflux should jump instantaneously to a value proportional to the magnitude of the current. (3) At the end of the ejecting period the efflux should drop instantaneously to zero. The time-dependent behaviour of real pipettes differs from that of the ideal, sometimes markedly so (Clarke et al., 1973; Bradshaw et al., 1973). The purpose of this paper is to examine the mechanism of departures from ideal behaviour, and to make practical recommendations for their minimisation. The central assumption will be that the movement of ions within a pipette is governed by gradients of concentration and potential, and is described by the Nernst--Planck equation. In an earlier paper (Purves, 1977b) an expres-

166

sion for the steady state release as a function of current was derived. Further aspects of the steady state behaviour will be developed here, but the most interesting findings result from an extension of the theory to treat timedependent release. Some of the predictions have been verified experimentally by direct observation of pipettes containing an ionized fluorescent compound, quinacrine. T HE OR ETI C AL RESULTS

The micropipette will be treated throughout as a cone of included angle 20, truncated at its tip by the sphere r = r0 (Fig. 1A). 0 is understood to be measured in radians, but numerical values will be given in degrees. The tip diameter is 2a ~ 2ro0 for small 0. Fluxes within the pipette are radially directed with respect to the point r = 0. The appropriate form of the Nernst--Planck equation (Purves, 1977b; Hill-Smith and Purves, 1978) is: 1 aC 1 D at

a2Cl ar 2

aCli~ ar \r

I

) r > r0, t > 0

2zFl~Co~r2

(1)

in which C1 is concentration of the ion of interest, D is its diffusion coefficient, t is time, I the applied current through the pipette, z the charge number, F is Faraday's constant, and ~ = 27r(1 -- cos 0) ~ 7r02 is the solid angle subtended by the cone. An assumption implicit in Eqn. (1) is that the conductivity everywhere in the pipette has a constant value o, given for an ideal electrolyte by the Nernst--Einstein equation o = 2 z 2 F 2 D C o / R T where Co is the concentration of the initial filling solution. This condition will be

1

2

r 2/a

4

5

Fig. 1. A (left): geometry of pipette. B: steady state concentration profile within pipette. Numbers against curves are values of normalized current expressed as qF/qD. Positive values indicate retaining current and negative values indicate ejecting current. The horizontal axis represents normalized distance along the pipette, expressed as r/ro ~ rO/a.

167

satisfied if the medium outside the pipette is an electrolyte whose initial concentration is also Co, and if all ions present have the same diffusion coefficient and absolute charge number.

Linearity o f efflux with ejecting current The boundary conditions C1-* Co, r-~ oo and C~ = 0, r = r0 lead to a particularly simple expression for the steady state efflux q (Purves, 1977b): (2)

qF q = exp(qF/qD)

-- 1

where qD =DCo~ro-~ ~rDCoOa is the diffusional leak in the absence of applied current and qf = I/2zF is the Faraday equivalent release (for transport number 0.5). The sign convention is such that I and qF take negative values for ejection and positive values for retention. Equation (2) shows that efflux increases linearly with current only for values of q F / q D m o r e negative than a b o u t --4, since exp(--4) < < 1. This result may be p u t in more useful form if it is expressed in terms of the voltage E applied to the pipette. For constant conductivity o, the resistance of the pipette is given with small error by 1/O~ro ~ 1/IroOa. On substitution, q F / q D = z E F / R T ~ E/(25 mV) for unit charge number. Thus we arrive at the conclusion that steady state release becomes linear with current and independent of pipette geometry only when the ejecting voltage exceeds some 100 mV.

Steady state concentration profile within pipette In the absence of applied current the concentration of drug in the pipette falls sharply towards the tip, owing to diffusion from the small solid angle of the pipette into the large solid angle of the space outside. The customary application of a small retaining current to diminish the spontaneous leak of drug further depletes the tip region. This depletion takes on some importance when the time course of release is considered, since refilling of the depleted region is the cause of poor dynamic behaviour during pulsed ejection (see Kelly, 1975). An expression is derived here for the concentration profile within the pipette. The boundary condition C1 = 0, r = r0 considered above obviously cannot lead to an accurate description of the concentration near the tip. A better approximation is obtained if diffusion and iontophoresis are allowed to proceed in the region outside, as well as inside, the pipette. In an earlier paper (Purves, 1977b) an a t t e m p t was made to justify the simple boundary condition by showing that the resulting efflux (Eqn. (2) above) differed insignificantly from that predicted b y the more complicated model. Unfortunately the justification was faulty, owing to several sign errors. These are corrected in the derivation which follows. If C2 denotes the concentration of drug in the region outside the pipette,

168 then in the steady state we have a pair of differential equations: d2CL+ dC~ (2 dr 2

drr

r

I

dd2C2 C 2+ ( 2 dr 2

d~r

)

2zFD-CoJ/r2 ' = 0, r > r0

I

(3)

)

r + 2zFD-Co~2r 2 = O, r > ro

(4)

in which ~2 = 4w -- ~ ~ 4w is the solid angle of the whole space outside. The differing signs of the second terms in parentheses in Eqns. (3) and (4) arise because current in the positive r direction inside the pipette traverses the region outside in the negative r direction. The solutions to the two equations have to be matched by the requirement of continuity of concentration and diffusional flux: C1 = C2, r = r0 r i D t~ dC1 dr = _r2o D ~2

dC2

, r = ro

The boundary conditions are C~ -~ C0, r -~ oo and C2 -~ 0, r -* oo. The solutions to Eqns. (3) and (4) are found to be:

{ [qF(

02 Co exp ~DD 1 + 4

C,

ro]~ r/j--1

}

=

(5)

Co(exp[ C2-

-- l} (6)

exp[

where qF and qD have the meanings given earlier, and the appropriate substitutions have been made for the solid angles ~ and ~2. The concentration profile C~(r) is plotted in Fig. 1B for a range of values of q v / q D , including both ejection and retention. The included angle 20 at the pipette tip does not appear as a parameter of the curves shown because for 0 ~< 5 °, C l ( r ) is independent of 0 within the accuracy of drawing. The efflux q is given by the sum of diffusional and iontophoretic components: dC1 r=ro q = ~Dr2° dr

IC1 r=r° = kq f \ qF 02_)] 2zFCo exP[~D D (1+

(q)

This is the corrected form of the result given by Purves (1977b). It does not

169

differ importantly from the simple result (Eqn. (2)) if 0 ~< 5°; this finding justifies the simple boundary condition C, = 0, r -- r0, which will be used in the subsequent treatment of time-dependent ejection. R e l e a s e as a f u n c t i o n

of time

Time-dependent solutions to Eqn. (1) were obtained by an explicit finitedifference method (Smith, 1965). The substitutions x = ro/r, T = D t / r ~ and C = C ( x , T ) = C1 transform Eqn. (1) into: x- 4~C_a2C+qF_aC OT

Ox 2

0 0

(8)

q D aX '

whose spatial domain is finite. The boundary conditions become C = 0, x = 1 and C = Co, x = 0. The finite-difference approximation to Eqn. (8) giving the concentration on the (j + 1)th time row is:

Ci,j+a=Ci,j + l i 4 f x 2 f T i = 1 , 2 .... , Y - - 1

--~X~D

Ci-l,j--Ci, j +

2+6X~-D]

i+l,], (9)

where f x = 1 / N is the mesh spacing, and fiT is the time step. The efflux q is given by: aC q l q n = - - -~x x= l

in Eqn. (8). The derivative in this expression was approximated by a 5 point backward difference formula. Initial conditions were set up from the steady state solution of Eqn. (8), namely:

4

C(x) =

(10)

explUF[ -- 1 LqDJ

The solution to the finite difference equation should tend to that of Eqn. (8) as the mesh spacing and the time step tend to zero. Solutions with f x = 1/20 and 6T = 1/1600 differ b y a few percent only from those with f x = 1/40 and 6T = 1/6400, and so the latter intervals were assumed to give sufficient accuracy. Trials with single and double precision variables indicated that round-off error did not affect the results. Some representative c o m p u t e d efflux curves are given in Figs. 2 and 3. A remarkable and unexpected feature is the disparity between onset and offset of ejection; release rises relatively slowly during the pulse to approach the steady state b u t falls abruptly at the end. The prior application of retaining

170 B

4C

40

-4O f

-30

~//~D 20

-40 20

-20

-3

f

0

-20 -lO

-I0

f I

0

2

I

I

4

6

--

0

2

4

6

T

T

Fig. 2. C o m p u t e d t i m e course o f e j e c t i o n f r o m i o n t o p h o r e t i c p i p e t t e d u r i n g pulses of various s t r e n g t h s applied f r o m t i m e T = 0 t o T = 5. N u m b e r s against curves are values of qF/qD d u r i n g t h e pulse. A: r e t a i n i n g c u r r e n t is zero. B: s t e a d y r e t a i n i n g c u r r e n t o f qY/qD = +4 applied b e f o r e a n d a f t e r ejecting pulse.

current slows the onset still further, and increases the disparity. The time scale on which the above effects are established is not apparent from Figs. 2 and 3, because dimensionless units T = D t / r ~ ~ D t 02/a 2 have been used. Some numerical examples show that the time scale depends strongly on the shape of the pipettes. For ions of diffusion coefficient D = 1 pm2/msec and a pipette of internal tip diameter 2a = 0.105 pm and

A

B

40

0

20

0

q'/cl e1 0

c~/°~D 2(:

2

T

4

6

0

2

4

6

T

Fig. 3. A : e f f e c t of various r e t a i n i n g c u r r e n t s o n c o m p u t e d e j e c t i o n d u r i n g a pulse of fixed s t r e n g t h qF/qD = - - 4 0 . N u m b e r s against curves are values o f s t e a d y r e t a i n i n g c u r r e n t applied b e f o r e a n d a f t e r t h e pulse. B: e f f e c t o f r e t a i n i n g c u r r e n t applied at various t i m e s b e f o r e a n e j e c t i o n pulse o f fixed s t r e n g t h qF/qD = - - 2 0 . T h e r e t a i n i n g c u r r e n t o f fixed s t r e n g t h qF/qD = +20 p r e c e d e d e a c h e j e c t i o n pulse b y t h e t i m e s i n d i c a t e d . N o t e t h a t if t h e e j e c t i n g pulse were c o n t i n u e d i n d e f i n i t e l y , all t h e e f f l u x curves in A w o u l d t e n d t o t h e s a m e v a l u e q/qD = 40.0. Similarly in B, all curves w o u l d t e n d t o t h e value q/qD = 20.0.

171

included angle 20 = 6 °, the scale reads in msec. For a coarse pipette typical of those used in the central nervous system, with 2a = 1.1 pm and 20 = 2 °, the scale reads in sec. A pipette fabricated with the most advanced technology (Brown and Flaming, 1977) may have 2a = 0.05 pm ~ind 20 = 9 °. In this case the scale would read in hundreds of psec. MICROSCOPIC VIEWING OF IONTOPHORESIS

Many workers have measured iontophoretic release into small volumes of solution by subsequent assay of the drug content (see Kelly, 1975). 'Realtime' assays give more immediate information a b o u t the rate of iontophoretic release. For example, Kusano et al. (1975) measured the emission of light by aequorin following iontophoresis of calcium ions, and Dionne (1976) described a fast microassay in which an ion-sensitive microelectrode was used to monitor the concentration of cholinergic agonists. These methods do not, however, reveal the processes occurring within the tip of the iontophoretic pipette. Direct observation of the release of a fluorescent c o m p o u n d , described here, gives a novel and vivid impression of the physics of iontophoretic release and allows some of the predictions made earlier to be tested. Pipettes were filled with quinacrine HC1 50 mM (Sigma) and m o u n t e d with their tips in saline, under a cover-glass, on the stage of a Zeiss Photomicroscope III fitted with an epifluorescent condenser IIIRS. Fluorescence excited by violet light was viewed with 40X planapochromat and 100× 'Neofluar' oil-immersion objectives. The current through the pipettes (resistance 20--200 M ~ ) was controlled by a high voltage Howland current p u m p (Fig. 4). Outward currents of 2 nA or more produced an easily visible efflux of fluorophore (Fig. 5). Retaining currents depleted the terminal region of the pipette (Fig. 5) and delayed the onset of efflux during a subsequent ejecting pulse. During this lag time, the diffuse boundary between fluorescent and non-fluorescent regions of the pipette could be seen advancing towards the tip. The speed of progression of the boundary depended on the magnitude of the ejecting current. Ejection from fine-tipped pipettes of high resistance was seen to be much more rapid than from coarse pipettes of low resistance. A difference between the rates of onset and offset of ejection, predicted from the theoretical results given earlier, was immediately apparent, and was most marked when large retaining currents were used. Semiquantitative estimates of the efflux were made b y monitoring the o u t p u t from the microscope's built-in photomultiplier tube. The speed of response of the recording m e t h o d was restricted by the rate of diffusion of quinacrine into, and o u t of, the receptive field o f the photomultiplier. When a 100X off-immersion objective was used, the receptive field was a b o u t 4 # m in horizontal extent and responses peaking 30 msec after a brief ejection pulse could be recorded (Fig. 6A). Most recordings were made with a 40X

172 ~+120 25k 1W

Ip

t

+120 22p;

20M

1.5. "gv

33k lW

o-12o-IS I 100k 1M

120k 20M

33k tW

-12o

~ l ~

l SIC-,OUT

i

~°k G

IN ~lOk

Fig. 4. Circuit diagram of high-voltage current pump. The unit can apply at least 50 V to a pipette. All operational amplifiers are types CA3140, NPN transistors are types BF258, PNP transistors are types MPSU60, and unmarked diodes are general purpose silicon types. Amplifier A1 with associated circuitry forms a high-voltage unity gain follower, and the circuit block which includes amplifier A2 given a non-inverting gain of 10 with an output swing of at least 100 V. Adjustable negative capacitance is provided via R3. Retaining current of ± 25 nA is set by R4, and pulsed ejection is governed by a command signal (15 V max.) giving approximately 50 nA output per volt input. The wiring at the non-inverting input of A1 must be kept as short as practicable, any lengths greater than a few cm being screened with a driven shield as shown. To adjust the circuit, turn preset control R1 to minimum resistance, turn the retaining current off, and connect a high value resistor (at least 20 M ~ ) to the micropipette terminal. Connect the differential inputs of a digital voltmeter between TP A and the free end of the resistor. With a jumper lead, connect the free end of the resistor alternately to ground and to the +15 V line, noting the voltmeter reading each time. Adjust R1 so that both readings are the same, within 1 mV. Monitor the voltage between TP A and ground, with the micropipette terminal alternately grounded directly and grounded via the high value resistor. Adjust preset control R2 for m i n i m u m difference ( 10 l° ~ and the leakage current is less than 100 pA. o b j e c t i v e ; t h e larger r e c e p t i v e f i e l d gave c o n s i d e r a b l y s l o w e r r e s p o n s e s . Efflux should become linear with current when the ejecting voltage exceeds 100 mV. This prediction was tested by sweeping the current through a 50 M~ pipette from --10 nA to +30 nA over 5 min. The knee of the

m

|

J

Fig. 5. Three fluorescence photomicrographs of a pipette containing quinacrine 50 mM. Left: retaining (inward) current of 8 nA. Centre: no retaining current. Right: ejecting (outward) current of 20 nA. Pipette resistance 44 M~. Calibration bar: 10 ~m.

A

B

f

C

I

j

-0.SV

j

l

I

f f ~

~f

I 0

I

J

J 0.5V

Fig. 6. Oscilloscope records of photomultiplier output during ejection of quinacrine from micropipettes. Vertical scale is arbitrary though proportional to photomultiplier current. A: superimposed responses to 200 nA, 4 msec pulses from 130 M ~ pipette. Time marker shows 10 msec intervals. B: steady-state relation between voltage applied to 50 M ~ pipette (horizontal axis) and fluorescence intensity. The ramp shown was used for calibration. C: pulses of 40 nA, 50 msec applied every 5 sec. Retaining current of 20 nA applied for 2 sec at the time indicated by the bottom trace. Note that the subsequent ejection response was completely suppressed.

174

A

B

__J , i

i

i m

I

Fig. 7. E f f e c t o f r e t a i n i n g c u r r e n t o n t i m e course o f release b y 20 n A pulse f r o m 70 M ~ p i p e t t e . A : r e t a i n i n g c u r r e n t zero. B: r e t a i n i n g c u r r e n t of 20 n A applied 10 sec b e f o r e pulse, a n d i m m e d i a t e l y a f t e r it. T i m e m a r k e r : 1 sec intervals. T h e t i m e r e s o l u t i o n is i n s u f f i c i e n t t o s h o w t h e small a m o u n t of a s y m m e t r y in A, b u t in B t h e o n s e t is clearly slower t h a n offset.

recorded signal (Fig. 6B) occurred below 100 mV, confirming the theory. Similar results were obtained with three other pipettes, whose resistances were 22 M~2, 27 M~2 and 130 M~2. The adverse effects of retaining current on pulsed ejection are illustrated in Fig. 6C. In Fig. 7 two responses are shown, recorded without (Fig. 7A) and with (Fig. 7B) a prior period of retaining current. Disparity between the rates of onset and offset is not evident in Fig. 7A because of the limited speed of the recording method, b u t in Fig. 7B the disparity has been accentuated b y the retaining current and is clearly visible. DISCUSSION

Iontophoresis has been practised successfully for many years with a theoretical foundation no more elaborate than the equation q = n I / z F . This relation, usually, b u t incorrectly 1 referred to as Faraday's Law, gives the efflux q in terms of the applied current I and transport number n, where n is determined empirically. Although this simple equation provides an adequate guide in some experimental circumstances, it does not take account of the efflux which persists in the absence of applied current and which is due to diffusion and bulk flow. Quantitative expressions for the efflux resulting from the latter t w o transport processes were derived by Krnjevi~ et al. (1963) in their important theoretical and experimental study of iontophoresis. However, Krnjevid et al. (1963) assumed that the total efflux was simply the sum of the individually determined components. This is not so. Electro1 Faraday's observations related chiefly t o electrolysis. T h e a b o v e e q u a t i o n w o u l d be b e t t e r n a m e d H i t t o r f ' s Law, a f t e r t h e G e r m a n c h e m i s t w h o gave t h e first essentially c o r r e c t a c c o u n t o f e l e c t r o p h o r e t i c t r a n s p o r t o f ions in s o l u t i o n (see P a r t i n g t o n , 1964).

175 phoresis and diffusion interact in a way described by the Nernst--Planck equation. The appropriate solutions give an expression (Purves, 1977b; Eqn. (2) of this paper) which predicts a curvilinear relation between current and efflux; for sufficiently large ejecting currents this expression tends to the simple Hittorf relation above. The chief new finding of the present study, relating to the steady state, is that release should be linear with current provided that the ejecting voltage applied to the pipette is greater than 100 mV. Fortunately, this value is likely to be exceeded in most practical cases, except when p o t e n t c o m p o u n d s are applied from pipettes of low resistance. Thus, the usual assumption of linearity will not normally be misleading, if election is continued for long

enough to approach steady state. Different considerations apply in the treatment of time-dependent ejection and retention. Iontophoretic experiments on isolated tissues often use, with evident repeatability, ejecting pulses whose duration is measured in milliseconds (Dennis et al., 1971; Dreyer and Peper, 1974). On the other hand, measurements of the release of radiolabelled substances have shown time-dependent effects extending over seconds or minutes (Clarke et al., 1973; Bradshaw et al., 1973). These differing views of the time-dependence of ejection are reconciled by the present finding that the speed of ejection depends strongly on the geometry of the pipette. Release from a pipette of small tip diameter and large included angle may be three or more orders of magnitude faster than from the pipettes customarily used in the central nervous system, which have a relatively wide bore and are nearly cylindrical in profile. A second factor which influences the speed of ejection is the magnitude of the retaining current. Large retaining currents deplete a substantial length at the tip of drug ions. The subsequent refilling of this region during ejection greatly slows the efflux. Pipettes of small tip diameter require less retaining current to suppress spontaneous release, and thereby gain perhaps another order of magnitude in speed over wide-tipped pipettes. These conclusions a b o u t pipette geometry and the speed of release have to some extent already been reached by intuitive or empirical means (for example Dreyer and Peper, 1974). It seemed desirable to put them on a firmer physical basis, since iontophoresis is now being used in a rigorously quantitative way (Kuffler and Yoshikami, 1975; Dreyer et al., 1978; Steinbach and Stevens, 1978), and elaborate kinetic analysis is possible (Purves, 1977a). The time