A first objective of this work was to relate the physicochemical properties of a cation to its transport number in the single-ion situation. Molecular weight has been identified as an important determinant of iontophoretic flux in the presence of competing co- and counter-ions, and a distinct inverse relationship has been observed 5–10. In contrast with previous work, attention has been focused here on cation transport numbers measured in the single-carrier situation; in this case, only the diffusivities of the cation and the counter-ion determine the transport number [19
]. All data were taken from experiments in which chloride, the principal endogenous counter-ions, was used. Thus, as the diffusivity of the counter-ion was kept constant, the value of toC+
depended solely on the intrinsic properties of the cation. Further, because Cl−
is the ubiquitous counter-ion in vivo, the results were expected to be predictive of behaviour in the “ practical world”.
shows that toC+
decreases rapidly with molecular weight, with transport numbers approaching zero at 400–500Da. It is noted that the toC+
of small inorganic cations (Li+
) were not well-correlated with their formula weights, leading us to examine more suitable molecular descriptors. In the single-ion scenario, with Cl−
as the counter-ion, the cation transport number may be theoretically predicted by equation 3
Cationic transport numbers (Figure 2.a) in the single-carrier situation and mobilities (Figure 2.b) as a function of atomic/molecular weight.
are the diffusivities of C+
in the membrane, respectively. Unfortunately, these values are not easily measured. On the other hand, the Einstein relation (D=uRT/zF) links ionic mobility (u) to the diffusion coefficient [30
]. The mobility of an ion describes its behaviour in a given solvent, under the influence of an electric field, and depends upon the charge and the hydrodynamic radius (i.e., the radius of the ion plus the surrounding sphere of solvent). While mobility generally decreases with molecular weight, lithium (the smallest of the cations considered here) is, in fact, a less mobile species because its hydrodynamic radius is larger – due to its considerable solvation sphere - than that of the other, small inorganic cations studied. This is clearly apparent upon inspection of where ionic mobility, determined from conductivity measurements, is plotted as a function of the molecular weight.
The relationship between the cationic mobility and transport number across the skin as a single carrier was next examined (). An exponential increase of the transport number with mobility towards a plateau or limiting value was observed. The physical significance of this limit is due to counter-ion (chloride) competition. It has been shown [34
], that a total cationic transport number (StoC+
) of 0.65–0.85 is conserved for a series of anodal formulations containing Li+
over a total ionic concentration range from 125 to 200 mM. The best empirical fit to the data in was the following equation:
Figure 3 (a) The important dependence of cation transport number on mobility (u). The data were empirically fitted to the equation: toC+= B + (A−B)(1−e−kx) and the parameters (mean ± SE) estimated by non-linear regression were: (more ...)
= 0.887. This regression predicts a limiting toC+
of 0.76 ± 0.08, a maximum imposed by the competition with chloride to carry current across the skin, and a value in excellent agreement with experimental data [34
]. At the opposite extreme, equation 4
indicates that cations with mobilities less than 0.95.10−4
will carry a negligible amount of current across the skin during transdermal iontophoresis (i.e., the toC+
for these species would be ~ 0). From the combined datasets examined, this would mean that cations having mobilities close to (or less than) that of quinine would be expected to be transported predominantly by electroosmosis, their electromigration contributions being indistinguishable from zero. However, the situation is in fact more complicated because quinine (and propranolol and, to a certain extent, ropinirole) are relatively lipophilic cations known to interact significantly with the net negative charge of the skin and to change its permselective properties [20
]. It is likely, therefore, that the limit on mobility may be somewhat (but not dramatically) lower than that predicted by equation 4
. This hypothesis is discussed and illustrated further below.
provides a different perspective on the same data and plots the transport number as a function of the hydrodynamic radius calculated using the Stokes-Einstein relation [30
]. Broadly speaking, this approach suggests that toC+
becomes negligible for molecules whose hydrodynamic radii exceed about 8 Å. Two further points about should be emphasized: First, the transport numbers of the small inorganic cations now align correctly (compare with ) as described above. Second, it is important to appreciate that this representation constitutes a simple arithmetical transformation of given that the hydrodynamic radius is directly calculated from the mobility assuming that the ions are spherical [30
]. For data-fitting and prediction, mobility, which is experimentally measured is therefore preferred; for illustrative purposes, on the other hand, graphical presentation as a function of hydrodynamic radius has its obvious advantages.
Further analysis of the data compared the cationic transport numbers in the skin to those in aqueous solution which were estimated from the measured mobilities via equation 2
]. shows the resulting correlation. A Pearson correlation test [33
], which assesses the strength of association between the two variables, yields a correlation coefficient (Pearson r) of 0.91; that is, a strong and positive relationship. The coefficient of determination (r2
) is 0.83, indicating that more than 83% of the variance in the two parameters is shared. In short, therefore, there is a strong correlation between ion transport numbers in the skin and those in aqueous solution. A subsequent linear regression analysis of the data was highly significant and yielded the following expression:
Figure 4 Correlation between cation transport numbers in skin with the corresponding values in water. A Pearson test shows the correlation to be strong and positive. Linear regression analysis confirms that the values in the skin are systematically higher than (more ...)
= 0.67. It follows that, in general, cationic transport numbers in the skin are 1.4 times those in water, a clear indication of the skin’s permselectivity. It is noticeable that the three lowest transport numbers in the skin (that is, those of ropinirole, propranolol, and quinine) are overestimated by equation 5
, a finding that may be explained, at least in part, by the interaction of these drugs with the skin and their inhibition of electroosmotic flow [20
Finally, it is appropriate to assess whether the structure-transport relationships developed here, and in related work [27
], are broadly applicable, in particular to compounds from a distinctly different class. To illustrate this point, data on the iontophoretic delivery of cationic dipeptides (charge +2 at pH 7.4) [31
] have been examined. In this case, mobilities were measured by capillary zone electrophoresis (CZE) and iontophoresis was not perfomed under single-carrier conditions. However, the molar fraction of each peptide in the anodal formulation was easily calculable and, using this known value (xp+
) together with the transport number (tp+
) deduced from the measured flux, it was possible to determine the peptide transport number under single-ion conditions (to,p+
Data supporting the validity of this approach have been reported in the literature for small inorganic cations and for drugs such as lidocaine [23
]. The cationic peptides considered here, their molecular weights and measured mobilities, their calculated top+
and the experimentally determined tp+
), and the predicted values of top+
using equation 4
are in ; comparative data for lidocaine assessed under exactly the same conditions as the peptides are also shown. The three peptides indicated were selected because their measured mobilities fell within the range of those exhibited by the cations studied in the present work. The agreement between the predictions of equation 4
and the experimentally deduced values of top+
is excellent, supporting the utility of the structure-transport relationships derived here and in related recent work. Furthermore, it is worth adding that the single-ion carrier transport numbers of three additional cationic peptides, whose mobilities were less than 1×10−4
(that is, outside the range of those used to develop the quantitative relationship in eq.4
), were under-predicted by the model. This finding reflects two clear messages: first, that predictive equations work best only within the “boundary conditions” of the data used for their generation; and, second, that the results for the lipophilic cations identified earlier (in particular, propranolol, quinine and ropinirole) have skewed the predictions of the model to under-estimate transport numbers of larger cations which do not associate appreciably with the skin membrane. Additional work with such species having molecular weights ≥300 Daltons should clearly be undertaken to better characterize the structure-transport relationship at this limit.
Table 3 Mobilities, and experimental and predicted “single-ion” transport numbers, of three cationic dipeptides  across the skin. Data for lidocaine, obtained under identical experimental conditions, are shown for comparison.