Summary

Clinical Pharmacokinetics 9 (Suppl. I): 49-60 (1984) 0312-5963/84/0001-0049/$06.00/0 © ADIS Press Limited. All rights reserved.

Ultrafiltration vs Equilibrium Dialysis for Determination of Free Fraction

William F. Bowers, Scott Fulton and Judith Thompson Amicon Corporation, Danvers

The analytical basis of membrane separation in determining plasma free drug con­centrations is developed from consideration of the generalised ideal mass action and con­servation laws applied to thermodynamic states of differing component volumes. The in­tuitively surprising theoretical independence of free drug concentration with changing fractional volume of protein predicts the equivalence affinal free drug concentration after uniform pre-dilution with that after equilibrium dialysis against the same volume ofbuffer. Similarly. the ultra filtrate free concentration of drugs in ideal plasma binding equilibrium is theoretically constant. regardless of the extent of reduction in fractional volume of bind­ing protein during filtration.

The pi(fa/ls in calculating free and bound fraction and concentration from equilibrium dialysis are reviewed and expressions presented to correct the major artefacts. The dilution behaviour of drugs is predicted from the normalised mass action model. Practical aspects of l'alidating ultrafiltration free drug levels using equilibrium dialysis are outlined.

Although membrane separation methods are widely employed for determining the biologically available fraction of drugs, hormones or other li­gands which reversibly bind to plasma protein, the analytical foundation and consequences of the physical separation have not always been fully ap­preciated. A number of reviews have appeared dealing with methods for protein-ligand binding study (Bush and Alvin, 1973; Chignell, 1972; Craig, 1965; Greenberg, 1944; Kurz et aI., 1977; Stein­hardt and Reynolds, 1969). It is our purpose to, firstly, consider the predictions of the ideal mass action law for equilibrium states reflecting the changed volume occupied by free drug and plasma protein which result from membrane separation. Secondly, detailed expressions are presented for consistent calculation of free fraction and concen-

tration from equilibrium dialysis data. Thirdly, the dilution behaviour of constant and variably bound drugs is illustrated, and related back to the predic­tion of the generalised mass law. Finally, a scheme for meaningful comparison of ultrafiltration and dialysis is presented in the context of validating routine clinical free drug assays based on ultra­filtration.

1. Ideal Binding Equilibrium: Effect of Change in Component Volume 1.1 General Description

In the simplest model for drug binding to plasma protein considered in this section, we postulate a single drug in 'ideal' equilibrium with a single plasma protein binding site, implying that molar

Ultrafiltration vs Equilibrium Dialysis

concentrations are essentially equivalent to ther­modynamic activities. For protein, the ideal re­quires that affinity and total binding capacity stay constant as drug binds or protein concentration varies, excluding complications such as allosteric interaction or masking of the binding site caused by protein self-association. Where the ideal model applies, the effect of volume change on drug bind­ing will be predictable from the laws of mass action and conservation of mass. Similarly, we initially postulate idealised semipermeable membrane sep­arations without complications due to nonspecific adsorption, Donnan equilibria, hindered passage of free drug, or leakage of protein.

In starting plasma, the drug binding equilibrium is

Cu + Cpu = Cb (Eq. 1).

Free protein site concentration can be rewritten as

Cpu = Cp - Cb (Eq. 2)

and drug protein complex as

Cb=C-Cu (Eq. 3).

Writing the mass action law in terms of the asso­ciation constant,

Cb K = (Eq.4)

Cu' Cpu

we can substitute equations 2 and 3 to obtain the familiar quadratic equation for starting plasma free drug concentration:

Cb K =

Cu' (Cp - Cb)

(C - Cu) (Eq. 5) =

Cu • (Cp - C + Cu)

K· Cu2 + (K • Cp - K • C + 1) • Cu - C = 0 (Eq. 6).

50

Given a volume of plasma, V, equation 6 defines an initial plasma equilibrium thermodynamic state.

To calculate the effect on free drug concentra­tion of a change in volume occupied by drug and/ or protein in a new thermodynamic state i, we in­voke conservation of mass about protein, equating initial and final amounts:

Cp ' V = CPi • Vdpi (Eq. 7)

CPi = Cp ' V/Vdpi (Eq. 8)

and conservation of mass about free and bound drug

(Eq, 9)

Substituting into equation 5:

K= C' V(Vdpi - CUi' Vdu;/Vdpi

CUi' (Cp ' V/Vdpi - C· V/Vdpi + CUi' Vdu;/Vdpi)

(Eq. 11).

Multiplying through by V dpilV and rearranging gives:

K • Di • CUi2 + (K • Cp - K • C + D i) • CUi - C = 0 (Eq. 12).

Equation 12 is the generalised expression for free drug concentration in any state i of differing com­ponent volume, which may be evaluated by sub­stitution of the dilution factor reflecting the new volume of distribution of free drug. Surprisingly, the free concentration of drug in state i given by equation 12 is independent of the protein volume of distribution in state i.

1.2 Uniform Dilution

After dilution, which we designate state 1, pro­tein and free drug equilibrate to occupy the same increased volume, Vdp1 = VdUI' Substituting into equation 12:

Ultrafiltration vs Equilibrium Dialysis

K· DI • CUl 2 + (K· Cp - K· C + D I) • CUI - C = 0 (Eq. 13).

Since Dl is larger than 1, and all the other terms in equation 13 are the same as in equation 6, we see that CUI must be less than Cu. Equation 13 may be readily solved for the decreased post­dilutional concentration of free drug.

1.3 Equilibrium Dialysis (Differential Dilution)

In equilibrium dialysis, which we designate state 2, the plasma retentate equilibrates across a semi­permeable membrane with dialysate buffer. Pro­tein osmotic pressure generally causes dialysis buffer to enter the retentate, while free drug diffuses into the dialysate buffer and comes to occupy the com­bined volume of the plasma and dialysate. As in state 1, we define a dilution factor for free drug as D2 = Vdu2/V and again substitute into equation 12 to obtain the equation of state 2:

K • D1 • CU12 + (K • Cp - K • C + D 2) • CU2 - C = 0 (Eq. 14).

Comparing the equations of states 1 (equation 13) and 2 (equation 14), we see that provided D, = D2, then CUl = CU2. Ideal uniform dilution and equilibrium dialysis against the same volume of buffer are thermodynamically equivalent states in terms of free drug concentration. If ideal protein binding is maintained, the mass law predicts that the free drug concentration after dilution depends only upon the free drug dilution factor, the total plasma concentrations of drug and protein, and the association constant, but is entirely independent of the protein dilution factor and resulting protein volume of distribution. Calculation of the diluted free concentration requires solution of equation 13 or 14, because free drug variation with dilution is non-linear and dependent on binding parameters.

The equivalence of uniform and differential di­lution of free drug also provides an analogy for understanding the equilibrium basis of ideal ultra­filtration.

Glossary of Symbols"

C Plasma molar concentration of total drug

Cu Plasma molar concentration of unbound drug

Cb Plasma molar concentration of drug bound to a single site

51

Cp Plasma molar concentration of total protein binding Site

Cpu Plasma molar concentration of free protein binding site

K Equilibrium constant (L/mol)

V Volume of undiluted plasma

Vdp; Volume of distribution of protein (in vitro)

Vdu; Volume of distribution of unbound drug (in vitro)

D; Dilution factor for unbound drug (VdulfV)

CU2 Retentate concentration of unbound drug

C2 Relentate concentration of total drug

fu Plasma fraction of unbound drug (Cu/C)

fU2 Retentate fraction of unbound drug (Cu2lC2)

[DPMl Plasma concentration of tracer

[DPMol Dialysate concentration of tracer (no Donnan effect)

[DPMool Dialysate concentration of tracer (significant Donnan effect)

[DPMR] Retentate concentration of tracer (no osmotic dilution)

[DPMcb21 Retentate concentration of bound tracer (no osmotic dilution)

[DPMRO] Retentate concentration of osmotically diluted tracer

[DPMcu21 Retentate concentration of unbound tracer

EDR Effective Donnan ratio

Dp2 Dilution factor for retentate protein (Vdp2fV = Cp/CP2)

Subscript i Thermodynamic state after change in volume of distribution of unbound drug and/or protein

Subscript 1

Subscript 2

Subscript 3

Thermodynamic state after ideal uniform dilution

Thermodynamic state of plasma at dialysis equilibrium

Thermodynamic state after ideal ultrafiltration

• From the recently proposed general nomenclature for phar­macokinetics of Rowland and Tucker (1980).

Ultrafiltration vs Equilibrium Dialysis

1.4 Ultrafiltration(Differential Concentration)

In ultrafiltration, a pressure gradient forces plasma water containing free drug through a perm­selective membrane. The simplest model assumes complete selectivity of the membrane. The com­bined volume of ultrafiltrate plus retentate equals the (undiluted) initial plasma volume, so Vdu3 = V and D3 = I. Substituting into equation 12 for free drug concentration in the post-ultrafiltration thermodynamic state 3, comprising retentate plus ultrafiltrate,

K· Cul + (K· Cp - K· C + 1)· CU3 - C = 0 (Eq. 15).

Equations 6 for starting plasma and 15 for post­ultrafiltration equilibrium are, in fact, identical, predicting that ultra filtrate and retentate free drug concentration will be constant, regardless of the ex­tent of ideal ultrafiltration and the degree of con­centration of retained binding protein. Although the theoretical invariance of free ligand with changing protein volume of distribution was first stated in general form over 20 years ago (Gilbert and Jenkins, 1959) and later derived from different directions for ultrafiltration (Sophianopoulos et al., 1978; Whitlam and Brown, 1981), this point has escaped universal appreciation (e.g. Christensen et al., 1980; Jung et al., 1981; Paxton and Donald, 1980). Perhaps the most dramatic experimental demonstration of equilibrium ultrafiltration is the work of Paulus (1969), who found that aspartoki­nase substrate binding parameters obtained by ultrafiltration to apparent dryness and counting the fraction bound on the membrane agreed with re­sults by an equilibrium gel filtration method.

An alternative intuitive explanation of the effect of component volume change can be found by re­examination of the mass law

Cb K = (Eq. 16).

Cu Cpu

Any dilutional change decreasing the concentra­tion of free drug, Cu, will result in a dissociation

52

of Cb to Cpu; plus CUi, to maintain a constant value of K. Dissociation releases additional free drug, re­sulting in re-establishment of equilibrium at a new intermediate value of CUi and a new lower value of Cbi/CpUi.

In ideal ultrafiltration, however, free drug vol­ume remains constant, while the concentrations of bound drug and free protein site increase at exactly the same rate as total protein. Thus, the retentate ratio Cbi/CpUi remains constant for any volume of ultrafiltrate, requiring no change in Cu to maintain K constant. Plasma free drug fraction is therefore theoretically obtainable directly from the ratio of ultrafiltrate free concentration to the total drug concentration in starting plasma.

1.5 Multiple Competing Ligands and Multiple Classes of Sites

The principles and predictions deloped for com­ponent volume effects on the free concentration of a single drug binding to a single non-interacting protein site are equally valid for the more general situation described, for example, by m classes of independent sites (k) in reversible equilibrium with n different competing drugs, metabolites or endog­enous ligands (;) [Klotz and Hunston, i 975; Mun­son and Rodbard, 1980; Robbins and Johnson, 1979]. In this case, a set of simultaneous equations of state for the plasma concentration of each ligand can be written

"' ikK· JCu ,C = ,Cu + L: kCp· ------

k=1 n

I + L: ikK· JCu j=1

(Eq. 17).

Equation 17 is readily solved for the free and bound concentrations of each ligand using standard iter­ative numerical methods.

2. Determination 0/ Dialysable Plasma Free Drug 2.1 Dialysis Retentate Free Fraction

Although equilibrium dialysis is considered by many to be the preferred method for determining

Ultrafiltration vs Equilibrium Dialysis

drug-protein binding, the literature abounds with examples of incorrect calculation of dialysis data, and a number of critical papers have appeared dealing with various aspects of this thorny analyt­ical problem (Behm and Wagner, 1979; Builder and Segel, 1978; Chadwick et aI., 1979; Kinget et aI., 1979; Kurz et aI., 1977; MacMahon et ai., 1984; McNamara and Bogardus, 1982; Nimmo et aI., 1977; Selikson et at, 1980; Smith and Jubiz, 1980; Steinhardt and Reynolds, 1969; Van der Giesen and Wilting, 1982). Common conceptual errors include failure to appreciate the dilutional shift in equilib­rium caused by diffusion of free drug into the di­alysate which increases the retentate bound frac­tion of variably bound drug, and failure to consistently define the post-dialysis equilibrium system with respect to free and total amounts of drug.

Practical problems often arise in dealing with significant Donnan effects, osmotic dilution of re­tentate, nonspecific adsorption of fee drug to the dialysis apparatus, binding of buffer ions, attain­ment of dialysis equilibrium, control of retentate pH, tracer impu.rity, and protein leakage into the dialysate.

In the calculation scheme suggested here, loss of drug by nonspecific adsorption is explicitly ac­counted for by determining levels in both dialysate and retentate. For variably bound drugs the reten­tate free fraction so obtained will reflect the actual equilibrium total drug level in the retentate after loss of free drug to the dialysate and to nonspecific adsorption, if present. Free fraction of the variably bound drug in starting plasma must then be cal­culated from empirical binding parameters for the plasma sample obtained at varying total drug con­centrations (e.g. Behm and Wagner, 1979; Monot et ai., 1983) or preferably at varying dialysate volumes to account for endogenous binding in­hibitors or competing ligands in the plasma sample (Sjoholm et aI., 1976; Smith and Jubiz, 1980).

Significant overestimation in free fraction and concentration can result from even slight leakage of protein into the dialysate. For a 90% bound drug, 1 % leakage of binding protein will cause a 9% over-

53

estimate of free drug concentration. Absence of protein in the dialysate should be confirmed with a sensitive protein assay.

In general, there will be an increase in the vol­ume of dialysis retentate due to influx of dialysate caused by the colloid osmotic pressure of plasma protein. Use of isotonic or higher concentrations of low molecular weight buffer does not entirely prevent this osmotic dilution, because buffer and salt ions freely equilibrate into the plasma com­partment, and the osmotic pressure due to protein remains to be satisfied. In practice, retentate plasma volume increases until a hydraulic counterpressure in the retentate is achieved which offsets the col­loid osmotic pressure. Variation in the internal volume of knotted dialysis tubing, or in the stretch­ing of the dialysis membrane in multichamber de­vices, results in considerable variation in the final osmotic-hydraulic equilibrium retentate volume. Accurate and precise determination offree fraction by dialysis thus requires correction for this variable dilution of retentate as outlined below.

The Donnan distribution may cause significant error in determination of plasma binding of drugs which are weakly or moderately bound, because the dialysate concentration will not reflect the free con­centration in plasma retentate (Keen, 1966). Use of swamping concentrations of electrolytes to di­minish the Donnan distribution may cause arte­facts due to competitive ion binding (Steinhardt and Reynolds, 1969; Van der Giesen and Wilting, 1982). It is possible to correct for significant Don­nan distribution by calculating the effective Don­nan ratio from drug ionisation and the distribution of a non-binding ion (Boyer et ai., 1947; Keen, 1966; Suter and Rosenbusch, 1977). Alternatively, if di­alysis is performed under essentially physiological conditions, for example using a plasma ultrafiltrate as the dialysate (Van der Giesen and Wilting, 1982), it could be argued that the Donnan distribution in vitro will generally mimic that in vivo across the capillary wall, and the obtained free drug concen­tration will more closely reflect pharmacological availability than does the true plasma free concen­tration.

In the idealised case of negligible osmotic di-

Ultrafiltration vs Equilibrium Dialysis

lution or Donnan distribution, the retentate and dialysate free concentrations are equal, and the re­tentate free fraction is:

[DPMcu21 = ----------~----

[DPMcb2] + [DPMcU2]

[DPMo]

[DPMR]

2.1.1 Osmotic Dilution of Retentate­Negligible Donnan Effect

(Eq. 18).

Given the difficulty of determining post-dialysis retentate volume, the most accurate method of cor­rection is to calculate the retentate protein dilu­tion, DP2, from the ratio of protein concentration measured in starting plasma to that in an aliquot of retentate. The total drug concentration in reten­tate diluted by dialysate containing free drug is:

1 [DPMRol = [DPMcb21' - + [DPMol (Eq. 19).

Dp2

Solving for bound concentration in undiluted retentate:

[DPMcb11 = {[DPMRol - [DPMolJ • DP2 (Eq. 20).

The free fraction normalised to undiluted retentate:

[DPMo] ~2 = ------------------------~~~~

{[DPMRO] - [DPMo]} • D p2 + [DPMo] (Eq. 21).

In the limit of no dilution, DP2 = 1 and equa­tion 21 reduces to equation 18.

2.1.2 Donna"n Effect - Negligible Osmotic Dilution For ionised drugs with moderate free fraction,

retentate total concentration and free fraction will be different to that for a non-ionised drug of com­parable binding affinity and total plasma level, be­cause the Donnan distribution reciprocally alters the free concentration in retentate and dialysate, requiring different dialysis equilibria in each case to satisfy the mass action law. As discussed above

54

for osmotic dilution and adsorption, exact deter­mination of the plasma free fraction of variably bound drugs subject to significant Donnan distri­bution requires calculation from empirical appar­ent binding parameters obtained at varying total drug levels.

The free fraction reflecting the post-dialysis re­tentate level of ionised drug is obtained using the corrected dialysate concentration:

[DPMcu21 = [DPMool • EDR (Eq. 22)

[DPMooJ • EDR =

[DPMRI (Eq. 23).

2.1.3 Donnan Efj'ecl and Osmotic Dilution Combining equations 21 and 22

[DPMool • EDR fUl = --------------------------------

{[DPMRol - [DPMool • EDRJ ' Dp2 + [DPMool • EDR

(Eq. 24).

2.2 Dialysis Retentate Total Drug Concentration

As in section 2.1, in the idealised case of neg­ligible osmotic dilution, the post-dialysis retentate total concentration is obtained from the drug and tracer levels in starting plasma

[DPMR] . c [DPM]

(Eq. 25).

Equation 25 reflects the actual total drug in the dialysed plasma, incorporating the effects of dif­fusion of free drug into the dialysate chamber, as well as significant nonspecific binding loss and Donnan effects, if present.

2.2.1 Osmotic Dilution of Retentate From equation 24, total concentration normal­

ised to undiluted retentate is:

{[OPMROj - [DPMDDj' EDR} • D p, + [OPMooj' EOR C,= 'C

[DPM)

(Eq. 26).

Ultrafiltration vs Equilibrium Dialysis

If Donnan distribution is negligible, equation 26 is used with EDR set equal to 1.

2.3 Dialysis Retentate Free Drug Concentration

Post-dialysis free drug concentration is subject only to correction for the Donnan effect, if present:

[DPMDDl • EDR

[DPMl

[DPMDl

[DPM] ·c

·C

(Eq. 27).

3. Free Drug Variation with Dilution 3.1 Model Systems

By way of illustration, the effect of ideal uni­form dilution and equilibrium dialysis on the free concentration and fraction of phenytoin is simu­lated in figure 1, using equation 12 and reported values for phenytoin-albumin apparent binding parameters (LeComte et ai., 1979). As seen in fig­ure la, the predicted plasma phenytoin free frac­tion on the ordinate varies inversely with protein concentration, but is invariant with plasma total drug concentration over a range exceeding that seen clinically. Similarly, the predicted dialysis retentate free fraction, fU2' is constant with decreasing total retentate drug concentration caused by increasing dialysate volume dilution of free drug, represented by movement to the right along the solid lines. This again is consistent with the well-known independ­ence of phenytoin free fraction with variation of total drug concentration within the therapeutic range (LeComte et ai., 1979; Lunde et ai., 1970; Porter and Layzer, 1975). Conversely, the ideal mass law model predicts that fu 1, the uniformly di­luted phenytoin free fraction, will increase nearly proportionally with dilution in the therapeutic range, represented by movement to the right along the broken lines in figure lao This free fraction is equivalent to that obtained in equilibrium dialysis referred to total moles of drug in dialysate plus re-

55

tentate (Oppenheimer and Surks, 1964; Vlahos et ai., 1982). The predicted behaviour offu1 with uni­form dilution is also in good agreemept with ob­served behaviour of this drug (Lunde et ai., 1970). The variation of free drug concentration with di-

O.B - - - fU1

--1"2

0.6

Cp (mmOI~) __ c: 0.4 0 ---'fi

2.5 ~ " ~ 0.2 u. 5

a

O.B

c: 0

~ 0.6

" u c: 8 " ~ 0.4

CUi '0

" CU ~ E C; 0 ... -C-z 0.2

b 0.8

0.-1

-

Phenytoin

/' /

/

Unbound solute

0.6 0.4

/ I / I

/ I // I

/ I // ,I

/

/' /'

/ /

0.2

Fig. 1. Dilution effect on plasma free phenytoin (after LeComte et aI., 1979). Simulations using equation 12 with K = 745 L/mol, n = 8 and albumin concentrations of 4.35g/100ml (Cp = 5 mmol/ L) or 2.18g/100ml (Cp = 2.5 mmol/L). Identical results obtained with total plasma drug concentrations from 0.5 to 25 I'g/ml (C = 2 to 100 I'mol/L). [0; = dilution factor for unbound drug.] (a) Effect of increasing dialysate volume on retentate free frac­tion, assuming retentate volume normalised to starting plasma volume according to equation 21. Also, effect of increasing buffer volume in uniform dilution on diluted free fraction assuming ideal mass law behaviour, or the equivalent dialysis free fraction re­ferred to total drug in dialysate plus retentate. (b) Effect of dilution on free concentration compared with start­ing plasma.

Ultrafiltration vs Equilibrium Dialysis

lution for the 2 albumin total site concentrations of figure la is simulated in figure 1 b, expressed in normalised form as a fraction of the plasma free concentration. As drug binding increases with in­creasing protein concentration, 'buffering' ofthe free concentration to dilution increases and the free concentration curve shows greater departure from the limiting linear dilution behaviour of a non­bound solute (indicated by the dotted line).

The dilution behaviour of disopyramide, which exhibits variable binding in the therapeutic range, is in contrast to that of phenytoin. Scatchard ana­lysis of plasma disopyramide binding indicates the presence of at least 1 high affinity, low capacity class of binding sites, as well as at least 1 lower affinity, less saturable class of site (Hinderling et aI., 1974). The 2-site apparent binding parameters of Hinderling and co-workers obtained with pooled plasma were used with equations 8 and 10 substi­tuted into equation 17 to predict the behaviour of disopyramide (fig. 2). The intermediate pair of curves in figure 2a, which roughly correspond to the therapeutic range for disopyramide, show the significant variability of disopyramide retentate free fraction with dilution, emphasising the need for empirical model-fitting and back-calculation to ac­curately determine the plasma free traction from dialysis data for variably bound drugs. Figure 2b shows that free disopyramide concentration vari­ation with dilution over the entire concentration range studied by Hinderling et al. (1974) is more nearly linear than that seen for phenytoin, reflect­ing the lower degree of plasma binding of this drug.

3.2 Normalised Binding Model

The constant or variable behaviour with vary­ing total concentration of the free fraction of any drug can be more easily understood in terms of a normalised form of the mass action law (Rosner, 1975). This is obtained by rearranging equation 6 in terms of the bound drug fraction rather than the free drug concentration:

(Cb/C)' - ! I + C,,/C • [I + (1/C p • K)ll • Cb/C + Cp/C = 0

(Eq. 28).

56

For a single drug binding to a single class of site,

fu = 1 - CblC (Eq. 29)

and the behaviour of the free fraction for that interaction can be predicted from the ratio CplC and the product Cp ' K, (fig. 3). Uniform dilution can be visualised as vertical movement along a line

0.8

c: :8 g 0.6 Q) g 8 '" ~ 0.4 Cu, "0

'" c;u ~ ~ C,

Unbound solute z 0.2

-C-

b 08 0.6 0.4 0.2

D-' ,

Fig. 2. Simulations for the effect of dilution on free disopyr­amide (after Hinderling et aI., 1974). ,K = 1 X 106 L/mol; ,Cp = 1.6 I-'mol/L; 2K = 8.44 x 103 L/mol; 2Cp = 22.5 Ilmol/L. (a) Effect of increasing dilution on retentate free fraction. (b) Effect of dilution on normalised free concentration.

Ultrafiltration vs Equilibrium Dialysis

parallel to the ordinate, since the ratio Cp/C re­mains constant while Cp' K decreases. For all plasma free fractions less than unity, uniform di­lution will thus cause a dramatic increase in the free fraction. Conversely, the differential dilution of free drug only in equilibrium dialysis, with the free fraction being defined in terms of the un­changed volume of retentate only, can be viewed as movement to the right along a given function line, where the value of Cp • K remains constant while Cp/C increases as total drug, C, decreases due to the loss of free drug into the dialysate. In this case the retentate free fraction will decrease as the

~ ~r-. 1.0

0.8 \1 \~

0.6

0.4

c: ofl g Q)

~ 0.2 u.

57

site to total drug ratio increases up to a limiting value for Cp/C of about 20 for systems with Cp • K products of up to about 100. For very high affinity or high capacity systems with Cp' K products well over 100, the limiting value for the free fraction will be reached at 0 for any value of Cp/C greater than I.

Looking back at the model systems in section 3.1, for phenytoin, Cp ' K is in the range 1.8 to 3.8, and Cp/C is greater than 25 for any value of plasma total phenytoin (C) up to 0.1 mmol/L (25 ~g/ml), approaching the upper limit of clinical practice. Referring to figure 3, the normalised model cor-

Cp·K

r-t-. 0.1

\

~ fU2 (0) -r ~r

~ 10' ~

-3 -2 -1 o 2 3 4

log (Cp/C)

Fig. 3. Variation of free drug fraction with Cp • K and CplC (after Rosner, 1975).

Ultrafiltration vs Equilibrium Dialysis

rectly predicts the constancy of the phenytoin free drug fraction with dialysate dilution or total plasma concentration. Similarly, the disopyramide system at the low end of the therapeutic total plasma drug concentration scale has 1 interaction with a Cp • K of 1.6 and CplC of 0.2 and a second site class with a Cp ' K of 0.19 and a CplC of 2.8. While binding to both classes of sites will vary with total drug level according to figure 3, clearly the former low capacity, high affinity interaction is nearly satu­rated in starting plasma, and this site will cause a significant decrease in free fraction with increasing dialysate volume, moving to the right down the steeply descending portion of the binding curve for this site.

4. Validation 0/ Ultrajiltrate Free Drug Level 4.1 Donnan Distribution

The presence of a Donnan distribution of micro-ions in ultrafiltration of protein solutions has been clearly documented (Boyer et aI., 1947; Greenberg and Greenberg, 1931-1932; Greenberg and Gunther, 1929-1930). The comments in sec­tion 2.1 on the Donnan distribution in dialysis ap­ply equally to ultrafiltration. Greenberg and Gunther found identical 10% elevations in the chloride content of serum ultrafiltrates and dialy­sates. Furthermore, Greenberg and Greenberg found remarkably constant ion levels in increasing ultrafiltrate volumes from alkali caseinate solu­tions, in contrast to prediction, given the increase in retentate protein concentration which should amplify the Donnan distribution. While the con­stancy of ultrafiltrate ion concentration with in­creasing filtrate volume remains unexplained, ultrafiltration is nevertheless capable of giving free values for plasma ionised drugs which are equiv­alent to dialysis and similarly correctable by the effective Donnan ratio, if desired.

4.2 Membrane Permselectivity

Requirements for protein retention in ultrafil­tration are comparable to those discussed for di-

58

alysis in section 2.1. Furthermore, the kinetic na­ture of ultrafiltration imposes requirements that the membrane pores be large enough to minimise the 'sieve' effect caused by preferential passage of plasma water vs free drug. Use of dialysis tubing for ultrafiltration can cause significant sieve errors depending on drug molecular weight (KUTZ et aI., 1977). Recently available ultrafiltration mem­branes offer significantly improved selectivity.

4.3 Free Drug Recovery From Plasma

In general, the most definitive protocol for ultrafiltration device validation is the sequential method (after Kurz et aI., 1977). In this scheme, patient plasma samples containing drug are first dialysed to equilibrium and the post-dialysis reten­tate free fractions calculated as described in section 2.1. The fraction thus obtained reflects nonspecific loss of drug to the dialysis system, if any, and the altered post-dialysis retentate free fraction of drugs exhibiting concentration-dependent binding or competitive binding with endogenous ligands. The dialysis plasma retentate is then ultrafiltered, and the free fraction calculated from the ultrafiltrate concentration over the dialysis retentate concen­iration. Constant ultrafiltrate free fraction, with in­creasing filtrate volume (identical to the prior di­alysis retentate free fraction) confirms negligible nonspecific loss of free drug to the ultrafiltration device, apparent ideal behaviour of drug binding during filtration, and negligible sieve effect in the presence of polarised retained plasma protein. Pro­tein leakage into dialysate and filtrate should also be routinely monitored by protein assay in device validation studies.

Acknowledgements

We thank Peter Todd and Rudy Haschemeyer (Cor­nell University, Medical College) for developing the non­linear least-squares program used for simulation of di­lution effects in specific site multiple equilibria. We also thank David Kabakoff, Kaye Chegwidden, Maureen Cronin, Jane Ruppel, Anna Jaklitsch, Michael Wandell, and our many other colleagues at Syva Corporation for stimulating discussions and encouragement in the distil­lation of these comments.

Ultrafiltration vs Equilibrium Dialysis

References

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Builder. S.E. and Segel. I.H.: Equilibrium ligand binding assays using labeled substrates: Nature of the errors introduced by radiochemical impurities. Analytical Biochemistry 85: 413-424 (1978).

Bush. M.T. and Alvin. J.D.: Characterization of drug-protein in­teractions by classic methods. Annals of the New York Acad­emy of Sciences 226: 36-43 (1973).

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S9

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Authors' address: Dr William F. Bowers, Amicon Corporation, 17 Cherry Hill Drive, Danvers, Massachusetts 01983 (USA).