Transdermal iontophoresis of rotigotine: influence of concentration, temperature and current density in human skin in vitro.

Iontophoretic transport of rotigotine across human stratum corneum (HSC) was studied in vitro in side by side diffusion cells according to the following protocol: 6 h of passive diffusion, 9 h of iontophoresis followed by 5 h of passive diffusion. A current density of 0.5 mA cm(-2) was applied. The parameters studied were the influence of the rotigotine concentration in donor phase and the influence of the molecular weight of the co-ions. To this end, Na(+) was replaced by tetra ethyl ammonium (TEA(+)) or tetra butyl ammonium (TBA(+)) (both at pH 5 and 6). In addition, the influence of the acceptor phase temperature (32 degrees C versus room temperature), the replacement of HSC by dermatomed human skin (DHS), and the relation between drug transport and current density were examined. The estimated steady-state flux (Flux(ss)) gradually increased with the drug concentration in the donor phase in a linear manner. The flux was also linearly correlated with the applied current density providing a convenient approach to individual dose titration. The use of TEA(+) as co-ion increased the rotigotine iontophoretic flux significantly, while TBA(+) did not. Replacing HSC by DHS reduced the iontophoretic rotigotine transport, while an increase in temperature to 32 degrees C increased the rotigotine flux. The maximum Flux(ss) achieved was around 80 nmol cm(-2) h(-1) indicating that by means of iontophoresis, a therapeutic level of rotigotine might be achieved with a reasonable patch size.     

Controlled transdermal iontophoresis by ion-exchange fiber.

The objective of this study was to assess the transdermal delivery of drugs using iontophoresis with cation- and anion-exchange fibers as controlled drug delivery vehicles. Complexation of charged model drugs with the ion-exchange fibers was studied as a method to achieve controlled transdermal drug delivery. Drug release from the cation-exchange fiber into a physiological saline was dependent on the lipophilicity of the drug. The release rates of lipophilic tacrine and propranolol were significantly slower than that of hydrophilic nadolol. Permeation of tacrine across the skin was directly related to the iontophoretic current density and drug concentration used. Anion-exchange fiber was tested with anionic sodium salicylate. The iontophoretic flux enhancement of sodium salicylate from the fiber was substantial. As the drug has to be released from the ion-exchange fiber before permeating across the skin, a clear reduction in the drug fluxes from the cationic and anionic fibers were observed compared to the respective fluxes of the drugs in solution. Overall, the ion-exchange fibers act as a drug reservoir, controlling the release and iontophoretic transdermal delivery of the drug.     

Optimization of iontophoretic transdermal delivery of a peptide and a non-peptide drug

In vitro iontophoretic transdermal delivery (ITD) of a tripeptide, enalaprilat (EP) and a non-peptide, cromolyn sodium (CS), across frozen hairless guinea pig (HGP) skin were investigated. Parameters for optimization of ITD included the influence of ionic strength (μ), buffer type and size, drug loading in the donor and the effect of pH. Drug permeation into the receptor compartment was monitored using HPLC assay methods developed for the study. An optimum μ of 6.66 mM in acetate buffer was found necessary for efficient ITD of CS. An exponential decrease in the flux of CS was observed with an increasing μ. Buffer ions larger than acetate ions inhibited the transport of CS ions. With an increase in the donor concentration of CS, a hyperbolic relationship for the increase in flux was observed. For EP, permeation was not detectable when μ was increased to greater than 31 mM in phosphate-buffered solution. With an increase in pH above the pK_a1 (3.55) for EP, a linear decrease in flux was observed. Higher drug loading of EP in the donor compartment provided better permeation. Effect of freezing of HGP skin on the iontophoretic delivery of EP and CS was also evaluated. Flux values for either of the drugs studied were similar when frozen or fresh skins were used. Reversibility studies indicated that no gross current induced permeation changes occurred with the HGP skin. Passive permeation of either of the drugs investigated was negligible.     

Controlled transport of timolol maleate through artificial membranes under passive and iontophoretic conditions.

The passive and iontophoretic permeability of timolol maleate (TM) through porous and dense artificial membranes was investigated in order to select the most optimal membrane for a transdermal drug delivery system. For the meso-porous membranes (pore diameter 2-50 nm), the TM permeability for passive diffusion and iontophoresis was practically the same. For the micro-porous membranes (pore diameter<2 nm), a significant transport contribution of iontophoresis was observed, which was more pronounced when higher current densities were applied. The electrical resistance of all the porous membranes was lower than the electrical resistance of human skin. For dense membranes, passive and iontophoretic TM permeability was significantly lower than for porous membranes and in most cases their electrical resistance was comparable or even higher than the resistance of human skin. For most of the membranes studied the average adsorption of TM at 37 degrees C was low (0.02-0.33 mg/cm(2)) and independent of the TM concentration. For the meso-porous mixed cellulose acetate-cellulose nitrate membrane the TM adsorption was significantly higher and increased with the TM concentration. Based on our results, the optimum membrane for an iontophoretic transdermal TM delivery system is the LFC 1 micro-porous membrane because it mainly controls the TM delivery (TM iontophoretic permeability: 0.86 x 10(-6) cm/s), has very low electrical resistance (0.9-1.5 komega cm(2)) and the TM adsorption to it is low (0.15 mg/cm(2)). The therapeutic plasma TM concentration is achievable by application of this membrane in realistic sizes (5-64 cm(2)) and by application of current densities between 0.13 and 0.5 mA/cm(2).     

The effect of electroporation on iontophoretic transdermal delivery of calcium regulating hormones.

Electrically-assisted delivery by iontophoresis and/or electroporation was used in vitro to deliver the calcium regulating hormones, salmon calcitonin (sCT) and parathyroid hormone (1-34) (PTH) through human epidermis. Such delivery could be useful for chronic treatment of post-menopausal osteoporosis and other clinical indications as a superior alternative to parenteral delivery. sCT (50 microg/ml) or PTH (1-34) (100 microg/ml) formulation was prepared in citrate buffer (pH 4.0 or 5.0, respectively). Epidermis separated from human cadaver skin was used. Iontophoresis was applied using a constant current power source and electroporation with an exponential pulse generator. Silver/silver chloride electrodes were used. A combination of electroporation and iontophoresis resulted in higher transdermal permeation than either one technique alone. Electroporation also shortened the lag time of iontophoretic transdermal delivery of salmon calcitonin. Pulsing at lower voltages followed by iontophoresis did not result in increased transport (over iontophoresis alone), perhaps because the transdermal voltage was very low. The transdermal transport of salmon calcitonin by pulsing with 15 pulses (1 ppm) of 500 V (200 ms) followed by iontophoresis led to a quick input and high flux. The average transdermal voltage was only about 50 V for a 500 V study.     

Iontophoretic delivery of 5-aminolevulinic acid and its methyl ester using a carbopol gel as vehicle.

The aim of this study was to evaluate a Carbopol gel as a vehicle for iontophoretic delivery of 5-aminolevulinic acid (ALA) and its methyl ester (m-ALA). The formulation was characterized rheologically and the passive diffusion of ALA and m-ALA in the gels was measured. Addition of ALA and m-ALA did not change the rheological behavior of the gel and the diffusion coefficients of ALA and m-ALA were 4.4 +/- 1.2 x 10(-6) and 3.08 +/- 0.7 x 10(-7) cm2 s(-1), respectively. The anodal iontophoretic transport of ALA and m-ALA through porcine skin in vitro was followed for 15 h at a constant current of 0.4 mA. When incorporating ALA in the gel, the steady-state was reached in 10-12 h at a flux level of approx. 65 nmol cm(-2) h(-1) compared to 2.5-4 h and a level of approximately 145 nmol cm(-2) h(-1) for m-ALA. The total amount of m-ALA delivered after 15 h of iontophoresis resulted in a six-fold enhancement over ALA delivery. Iontophoretic delivery from the gel formulation seems to be better than, or comparable to, the passive delivery from formulations commonly used clinically, in spite of the 10-20 times lower concentration of the drug in the gel formulation. The skin uptake after iontophoresis for m-ALA showed a nine-fold increase over that of ALA in the stratum corneum (SC).     

Transdermal delivery of regular insulin to chronic diabetic rats: effect of skin preparation and electrical enhancement

The efficacy of passive transdermal versus electrically-enhanced, or iontophoretic delivery of insulin was studied. The effect of skin pre-treatment on iontophoretic delivery of insulin was also investigated. Rectangular pulses of 0.25 mA/cm² current amplitude, 2 kHz frequency, and 50% duty cycle were used as anodal stimulation for electrically enhanced transdermal delivery of insulin. Twenty (20) BB/Wor chronic diabetic adult male rats were shaved 48 h prior to the study, and some experimental groups had hair stubble removed with a depilatory lotion. The iontophoretic drug-containing electrode was filled with 3 ml of porcine regular insulin (100 IU/ml) which had been adjusted to an acidic pH of 3.68 using 0.1 M HCl. The iontophoretic electrodes were then adhered to the abdomen of the alert rat. Results of the iontophoretic procedure were quantified by monitoring changes in blood glucose levels. When insulin was placed on the shaved skin, blood glucose levels fell in the chronic diabetic rat. In general, glucose levels fell more quickly and more profoundly using an iontophoretic enhancement of transdermal insulin delivery. However, some skin preparations facilitated movement of insulin more efficiently. The most profound effect of lowered blood glucose occurred when a depilatory lotion was used on the day of the study in conjunction with iontophoresis, where blood glucose levels fell by 61% after 1 h of iontophoretic treatment. Results indicate that insulin was delivered passively at therapeutic levels when the skin had been treated with the depilatory lotion on the same day as the study, as measured through a reduction in blood glucose levels of 29% after 1 h of passive delivery. When the depilatory lotion was used 24 h prior to iontophoresis, blood glucose remained near initial blood glucose levels. In the groups that did not have the depilatory lotion applied, blood glucose levels fell by 8% after 1 h of iontophoretic insulin delivery. The experimental evidence indicates a substantial increase in the penetration of insulin with the same-day application of a depilatory lotion in conjunction with iontophoretic enhancement.     

In vitro and in vivo iontophoresis of a tripeptide across nude rat skin

The iontophoretic delivery across nude rat skin of a tripeptide (Threonme-Lysine-Proline), which is positively charged at pH 7.4, has been measured in vitro and in vivo. The peptide was delivered from a poloxamer gel which demonstrated thermoreversible properties: the gel is a liquid at room temperature, a feature which facilitates preparation, whereas, at 37°C, the gel solidifies and provides a welldefined delivery system. The in vitro studies showed that: (a) iontophoresis significantly enhanced peptide delivery compared to passive transport; (b) peptide delivery was directly proportional to the applied continuous direct current density over the range 0.18–0.36 mA/cm²; (c) following 6 h of iontophoresis, minimal degradation of the peptide in either the donor or receptor phases of the diffusion cell (by electrolysis and/or metabolism) was observed; (d) exposure of the skin to direct current prior to the application (without current) of the tripeptide lowered the barrier function to the passive transport of Thr-Lys-Pro. In vivo, a single intravenous injection of radiolabeled tripeptide was rapidly eliminated (primarily in the urine) with an apparent half-life of less than 30 min. When the peptide was delivered by iontophoresis, considerable enhancement over passive diffusion was again achieved; the cumulative delivery of tripeptide in vivo was predictable from the in vitro results. In addition, and once more in parallel to the in vitro findings, pre-exposure of the skin to the iontophoretic current lowered barrier function to the subsequent passive delivery of the peptide. Skin sectioning and radioassay demonstrated that, relative to passive diffusion, iontophoresis produced (i) greater localization of the peptide in the skin, and (ii) delivery of peptide to deeper layers of the skin. Finally, histological and electron microscopical examination or rat skin following 3 hours of iontophoresis (at 0.36 mA/ cm²) in vivo did not reveal any gross morphological changes.     

In vitro transdermal iontophoretic transport of timolol maleate: effect of age and species.

Transdermal iontophoresis would be a promising method for the systemic delivery of water soluble and ionic drugs of relatively high molecular size, including peptides. In the present study, the effect of biological parameters such as age of the animal and species variation (rat, rabbit, mouse, guinea pig and human) on the transdermal iontophoretic transport was studied using timolol maleate (TM) as a model drug. The iontophoretic transport of TM across the skins obtained from the rats of different age groups was found to be similar. The results of the present study suggest that the age of the animal (Wistar rats: 1-8 months) did not appear to influence the transdermal iontophoretic transport of TM significantly. The amount of TM transported during iontophoresis (2 h) was significantly different among the different skin species. But the total amount of TM transported up to 24 h (2 h iontophoresis+22 h post-iontophoretic passive diffusion) was not significantly different among the different species studied. The present study provides further evidence that iontophoresis technique reduces the interspecies differences in the transdermal permeation of drugs, which is normally observed in passive diffusion of drugs. However, it must be noted that excised skins have been used in the present study to investigate the role of age and species variation on the iontophoretic transport of TM. The influence of these parameters under in vivo conditions might be different considering the physiological differences in different species and in the animals of different age groups.     

Optimization of aminolevulinic acid delivery by iontophoresis.

The objective was to optimize aminolevulinic acid (ALA) electrotransport into and through the skin by adjustment of formulation composition and ionic strength. ALA delivery was investigated as a function of the polarity and concentrations of drug and background electrolyte in the donor solution. The anodal iontophoretic flux of ALA from a 10% solution was compared with the drug's passive flux from the same formulation to which 5% dimethyl sulphoxide (DMSO) had been added. Iontophoresis of the predominantly zwitterionic ALA from the anode is more efficient than that from the cathode. It was possible, though, to increase the electrotransport of ALA by simultaneously delivering the drug from both anode and cathode. Reduction of NaCl concentration in the anode led to a 3- to 4-fold increase in ALA flux. Transport of ALA across the skin and the amount of prodrug delivered into the skin (SC and [epidermis+dermis]) were approximately 4-fold greater with iontophoresis as compared to the passive application of the DMSO formulation. In conclusion: (a) electroosmosis from the anode is enhanced when the background electrolyte concentration is lowered; and (b) low-level iontophoresis enhances ALA transport across and, more importantly, into the [epidermis+dermis] than a simple formulation incorporating DMSO.     

Pharmacokinetics of methotrexate in rabbit skin and plasma after iv-bolus and iontophoretic administrations.

The pharmacokinetics of methotrexate (MTX) in rabbit's skin and plasma after iv-bolus and iontophoretic delivery at different current densities was studied. Linear microdialysis probes were introduced into the upper dorsal shaved skin of tranquilized rabbits. Commercially available patches were used to deliver MTX for 1 h at different current densities (100, 200, and 300 microA/cm2) on different occasions. Iv-boluses (10 mg/kg) of MTX were also administered. Retrodialysis was performed at the end of the experiments to estimate probe recovery. Plasma and microdialysis samples were analyzed using a validated HPLC assay. Following iv-bolus, MTX showed a bi-exponential decay both in plasma and in skin. Cmax in skin occurred with a delay of 22 min compared with plasma. No quantifiable concentration of MTX was detected in the skin on passive drug delivery. Systemic exposure to MTX (AUC) and Cmax increased linearly with current density. Nevertheless, exposure to MTX in the skin did not increase linearly with current density, whereas Cmax did. In conclusion, iontophoresis remarkably improved the dermal delivery of MTX over passive diffusion. However, total exposure did not increase with current density in the skin, suggesting that for local applications lower current densities may achieve the same effects with minimal systemic exposure.     

Transdermal delivery of timolol by electroporation through human skin.

The purpose was to achieve therapeutic fluxes of timolol by transdermal delivery using skin electroporation. The transdermal transport of timolol through human stratum corneum was studied in three compartment diffusion cells. The electrodes, buffer composition and pulse conditions were optimized. Timolol maleate concentration in the donor compartment was 40 mg/ml. Square wave pulses were applied. Electroporation enhanced the transdermal transport of timolol by 1-2 orders of magnitude as compared to passive diffusion. Even though the current application lasted for only 10 s, the transdermal transport remained high after pulsing for at least 6 h. Higher fluxes were obtained with Pt electrodes close to the skin and a phosphate buffer. 10 pulses of 400 V-10 ms were more efficient than 10 low voltage-long duration pulses. Therapeutic fluxes of timolol (>50 microg/cm(2) per h) through human stratum corneum were achieved by electroporation.     

Distribution of khellin in excised human skin following iontophoresis and passive dermal transport

The aim of this work was to study in vitro khellin distribution into human skin after passive or iontophoretic transport. The experiments were performed on excised human skin, using vertical Franz-type diffusion cells. The effects of current application and reservoir pH were studied. At the end of the experiments the skin was sliced thinly and the drug was extracted and analyzed by HPLC. The results showed that khellin is able to penetrate through stratum corneum, to reach basal epidermis and upper dermis. The application time proved to be an important parameter. Current application (30 min; 0.5 mA/cm²), with a donor at pH 7.0, favored khellin accumulation even if the drug is not ionized. On the contrary, the use of a formulation at pH 3.2 inhibited drug accumulation. Leaving the drug reservoir in contact with the skin for 30 min after current application led to a dramatic increase of khellin concentration. A combination of dermal iontophoresis and passive diffusion is then a useful technique to govern khellin distribution in the skin.     

Transdermal iontophoretic delivery of apomorphine in patients improved by surfactant formulation pretreatment.

The objective of the present study is to evaluate the efficacy and the safety of transdermal iontophoretic delivery of R-apomorphine, a potent dopamine agonist, in combination with surfactant pretreatment in patients with advanced Parkinson's disease. Iontophoresis patches were applied in 16 patients for 3.5 h, with 0.5 h of passive delivery followed by 3 h of current application at a current density of 250 microA/cm2. Eight of these patients were treated with a surfactant formulation prior to iontophoresis. The pharmacokinetics, pharmacodynamic effects, systemic and local side effects of R-apomorphine were assessed. The plasma concentration vs. time profiles upon iontophoresis of R-apomorphine were described successfully by a novel pharmacokinetic model. The model suggests that only 1.9% of the dose that has been released from the patch accumulated in the skin. The patients treated with the surfactant formulations showed a statistically significant increase of bioavailability (from 10.6+/-0.8% to 13.2+/-1.4%) and of the steady state input rate (from 75.3+/-6.6 to 98.3+/-12.1 nmol/cm2 h) compared to the control patients (iontophoresis without absorption enhancers). In five out of eight patients in the study group and in three out of eight patients in the control group, clinical improvement was observed.     

In vitro acyclovir distribution in human skin layers after transdermal iontophoresis

The purpose of the present work was to study the in vitro distribution of acyclovir in human skin layers after iontophoresis, applied in order to increase the amount of drug in the basal epidermis, site of Herpes simplex infections. Experiments were done with Franz diffusion cells applying, as donor, acyclovir solutions (pH values: 3.0 and 7.4) or a commercial cream. Quantification of drug at different skin depths was performed by horizontal slicing of frozen skin, and drug extraction and analysis by high-performance liquid chromatography. Seven h of transdermal iontophoresis (0.5 mA cm⁻²) induced an accumulation of acyclovir in epidermis and dermis ranging from 80 to 150 μg cm⁻³, characterized by homogeneous distribution of the drug in skin layers. After short current application time (30 min) however, the concentration profile of drug in skin was not significantly different from that obtained after seven h of passive diffusion, employing pH 3.0 donor solution. After 30 min of iontophoresis, the acyclovir reservoir on the skin was maintained for up to five h producing a dramatic increase of drug concentration in skin, evening out over 80 μg cm⁻³ until a depth of 300 μm. Acyclovir can be accumulated at target site more quickly and maintained at higher level through application of a iontophoretic pulse and by keeping the drug reservoir on skin.     

Gelatin-stabilised microemulsion-based organogels: rheology and application in iontophoretic transdermal drug delivery

Gelatin-containing microemulsion-based organogels (MBGs) have been formulated using pharmaceutically acceptable surfactants and oils such as Tween 85 and isopropyl myristate. MBG formulations were subject to rheological study and their utility in transdermal drug delivery examined. Unlike most organogels, MBGs are electrically conducting and have been successfully employed in this study for the iontophoretic delivery of a model drug through excised pig skin. Iontophoresis using MBGs gave substantially higher release rates for sodium salicylate compared to passive diffusion, and fluxes were proportional to the drug loading and the current density. MBGs provide a convenient means of immobilising the drug and are rheologically similar to their hydrogel counterparts at comparable gelatin concentrations. MBGs also appear to offer improved microbial resistance in comparison to aqueous solution or hydrogels.     

In vitro iontophoresis of R-apomorphine across human stratum corneum. Structure-transport relationship of penetration enhancement.

To achieve a therapeutical effect of the anti-Parkinson's drug R-apomorphine via iontophoresis delivery, enhancement strategies in vitro were explored using three structurally related enhancers, lauric acid (LA), dodecyltrimethylammonium bromide (DTAB) and Laureth-3 oxyethylene ether (C(12)EO(3)). Human stratum corneum and shed snake skin were pretreated with 0.15 M each enhancer solution in propylene glycol (PG). Thereafter, passive diffusion, iontophoretic transport and post-iontophoretic passive diffusion were investigated. Compared to the control (PG pretreatment), a slight inhibition on both passive and iontophoretic delivery was observed with cationic surfactant DTAB pretreated stratum corneum. Pretreatment with anionic surfactant LA resulted in a great enhancement on passive delivery, but only a small enhancing effect on the iontophoretic delivery. Unlike the others, the nonionic surfactant C(12)EO(3) substantially increased iontophoretic transport rate of R-apomorphine by 2.3-fold, whereas passive delivery was basically unchanged or slightly affected. The magnitude of enhancing effect of C(12)EO(3) was dependent on the surfactant concentration and the pretreatment duration. Moreover, comparison of transport data through shed snake skin with human stratum corneum indicates that both shunt- and intercellular pathways are involved in the iontophoretic transport of R-apomorphine.     

Electrically-assisted transdermal delivery of buprenorphine.

The objective of this study was to explore the electrically assisted transdermal delivery of buprenorphine. Oral delivery of buprenorphine, a synthetic opiate analgesic, is less efficient due to low absorption and large first-pass metabolism. While transdermal delivery of buprenorphine is expected to avoid the first-pass effect and thereby be more bioavailable, use of electrical enhancement techniques (iontophoresis and/or electroporation) could provide better programmability. Another use of buprenorphine is for opiate addiction therapy. However, a patch type device is subject to potential abuse as it could be removed by the addict. This abuse can be prevented if drug particles are embedded in the skin. The feasibility of doing so was investigated by electro-incorporation. Buprenorphine HCl (1 mg/ml) in citrate buffer (pH 4.0) was delivered in vitro across human epidermis via iontophoresis using a current density of 0.5 mA/cm(2) and silver-silver chloride electrodes. Electroporation pulses were also applied in some experiments. For electro-incorporation, drug microspheres or particles were driven into full thickness human skin by electroporation. It was observed that the passive transdermal flux of buprenorphine HCl was significantly enhanced by iontophoresis under anodic polarity. The effectiveness of electro-incorporation seemed inconclusive, with pressure also playing a potential role. Delivery was observed with electro-incorporation, but the results were statistically not different from the corresponding controls.     

Contributions of electromigration and electroosmosis to peptide iontophoresis across intact and impaired skin.

d-(Arg)-Kyotorphin iontophoresis was investigated across intact and impaired skins in vitro. Iontophoretic flux increased from 68+/-12 to 538+/-116 nmol cm(-2) h(-1) when the peptide concentration in the anodal compartment was raised from 5 to 40 mM. Electromigration was the principal transport mechanism, accounting for approximately 70% of total peptide delivery. Reducing the number of competing ions in the formulation significantly increased iontophoretic flux but did not affect convective solvent flow. The latter was independent of peptide concentration indicating that skin permselectivity was not modified by kyotorphin transport. Total iontophoretic flux was unaffected when the stratum corneum was removed by tape-stripping (146+/-34 versus 150+/-26 nmol cm(-2) h(-1)). However, the contributions of the different transport mechanisms were significantly altered: (i) electromigration decreased, as more of the charge was carried by anions from the sub-dermal milieu; (ii) electroosmosis was absent; and (iii) passive permeation increased significantly. Transport rates across intact and impaired skin barriers were statistically indistinguishable when the donor electrolyte composition was modified; increased competition from anions was mitigated by the decreased Na+ levels in the formulation. Removal of Cl- ions from the receiver phase further increased peptide delivery, and also increased anodal electroosmosis.     

Spatially constrained localized transport regions due to skin electroporation.

Rapid, controlled molecular transport across human skin is of great interest for transdermal drug delivery and minimally invasive chemical sensing. Short, high-voltage pulses have been shown previously to create localized transport regions in the skin. Here, we show that these regions can be constrained to occur at specific sites using electrically insulating masks that restrict the field lines. The increase in total ionic and molecular transport per area was comparable to the levels observed in unconstrained electroporation of human skin. Constraining the area of intervention to encompass small areas of interest, a primary feature in the design of microdevices for transdermal drug delivery, can provide the same levels of flux as the unconstrained case.