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Mathematical Models for Topical and Transdermal Drug Products

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Topical Drug Bioavailability, Bioequivalence, and Penetration

Abstract

Mathematical models of epidermal and dermal transport, which includes transport of a solute through vehicle and various layers of the skin, metabolism in the skin and its subsequent distribution and clearance into systemic circulation from underlying tissues, play an essential role in development of topical and transdermal drug products and are reviewed in this chapter.

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References

  1. Roberts MS, Anissimov YG. Mathematical models in percutaneous absorption. In: Bronaugh RL, Maibach HI, Editors. Percutaneous absorption drugs—cosmetics—mechanisms—methodology. New York: Marcel Dekker; 2005. pp. 1–44.

    Google Scholar 

  2. Roberts MS, Walters KA. The relationship between structure and barrier function of skin. In: Roberts MS, Walters KA, editors. Dermal absorption and toxicity assessment. New-York: Marcel Dekker; 1998. pp. 1–42.

    Google Scholar 

  3. Scheuplein RJ. A personal view of skin permeation. Skin Pharmacol Physiol. 2013;26:199–212.

    CAS  PubMed  Google Scholar 

  4. Magnusson BM, et al. Molecular size as the main determinant of solute maximum flux across the skin. J Invest Dermatol. 2004;122(4):993–9.

    CAS  PubMed  Google Scholar 

  5. Sloan KB, Wasdo SC, Rautio J. Design for optimized topical delivery: prodrugs and a paradigm change. Pharm Res. 2006;23(12):2729–47.

    CAS  PubMed  Google Scholar 

  6. Roberts MS, Cross SE, Pellett MA. Skin transport. In: Walters KA, Editor. Dermatological and transdermal formulations. New York: Marcel Dekker; 2002. pp. 89–195.

    Google Scholar 

  7. Kuswahyuning R, Roberts MS. Concentration dependency in nicotine skin penetration flux from aqueous solutions reflects vehicle induced changes in nicotine stratum corneum retention. Pharm Res. 2014;31:1501–11.

    CAS  PubMed  Google Scholar 

  8. Bunge AL, Persichetti JM, Payan JP. Explaining skin permeation of 2-butoxyethanol from neat and aqueous solutions. Int J Pharm. 2012;435(1):50–62.

    CAS  PubMed  Google Scholar 

  9. Frasch HF, et al. Dermal permeation of 2-hydroxypropyl acrylate, a model water-miscible compound: effects of concentration, thermodynamic activity and skin hydration. Int J Pharm. 2014;460(1/2):240–7.

    CAS  PubMed  Google Scholar 

  10. Crank J. The mathematics of diffusion. 2 ed. Oxford: Clarendon Press; 1975.

    Google Scholar 

  11. Yano Y, et al. Two-compartment dispersion model for analysis of organ perfusion system of drugs by fast inverse Laplace transform (FILT). J Pharmacokinet Biopharm. 1989;17(2):179–202.

    CAS  PubMed  Google Scholar 

  12. Yano Y, Yamaoka K, Tanaka H. A nonlinear least squares program, MULTI(FILT), based on fast inverse Laplace transform for microcomputers. Chem Pharm Bull (Tokyo). 1989;37(4):1035–8.

    CAS  Google Scholar 

  13. Purves RD. Accuracy of numerical inversion of Laplace transforms for pharmacokinetic parameter estimation. J Pharm Sci. 1995;84(1):71–4.

    CAS  PubMed  Google Scholar 

  14. Anissimov YG, Watkinson A. Modelling skin penetration using the Laplace transform technique. Skin Pharmacol Physiol. 2013;26(4/6):286–94.

    CAS  PubMed  Google Scholar 

  15. Pirot F, et al. Characterization of the permeability barrier of human skin in vivo. Proc Natl Acad Sci. 1997;94(4):1562–7.

    Google Scholar 

  16. Frasch HF, Barbero AM. The transient dermal exposure: theory and experimental examples using skin and silicone membranes. J Pharm Sci. 2008;97(4):1578–92.

    CAS  PubMed  Google Scholar 

  17. Siddiqui O, Roberts MS, Polack AE. Percutaneous absorption of steroids: relative contributions of epidermal penetration and dermal clearance. J Pharmacokinet Biopharm. 1989;17(4):405–24.

    CAS  PubMed  Google Scholar 

  18. Roberts MS. Structure-permeability considerations in percutaneous absorption. In: Scott RC, et al., editors. Prediction of percutaneous penetration. London: IBC Technical Services; 1991. pp. 210–28.

    Google Scholar 

  19. Anissimov YG, Roberts MS. Diffusion modeling of percutaneous absorption kinetics: 1. Effects of flow rate, receptor sampling rate and viable epidermal resistance for a constant donor concentration. J Pharm Sci. 1999;88(11):1201–9.

    CAS  PubMed  Google Scholar 

  20. Parry GE, et al. Percutaneous-absorption of benzoic-acid across human skin .1. invitro experiments and mathematical-modeling. Pharm Res. 1990;7(3):230–6.

    CAS  PubMed  Google Scholar 

  21. Scheuplein RJ, Blank IH. Mechanism of percutaneous absorption IV. Penetration of non electrolytes (alcohols) from aqueous solutions and from pure liquids. J Invest Dermatol. 1973;60:286–96.

    CAS  PubMed  Google Scholar 

  22. Roberts MS, et al. The percutaneous absorption of phenolic compounds: the mechanism of diffusion across the stratum corneum. J Pharm Pharmacol. 1978;30(8):486–90.

    CAS  PubMed  Google Scholar 

  23. Guy RH, Hadgraft J. Physicochemical interpretation of the pharmacokinetics of percutaneous-absorption. J Pharmacokinetics Biopharmaceutics. 1983;11(2): pp. 189–203.

    Google Scholar 

  24. Kubota K, Ishizaki T. A theoretical consideration of percutaneous drug absorption. J Pharmacokinet Biopharm. 1985;13(1):55–72.

    CAS  PubMed  Google Scholar 

  25. Cooper ER, Berner B. Finite dose pharmacokinetics of skin penetration. J Pharm Sci. Oct 1985.74:1100–2.

    CAS  PubMed  Google Scholar 

  26. Anissimov YG, Roberts MS. Diffusion modeling of percutaneous absorption kinetics: 2. Finite vehicle volume and solvent deposited solids. J Pharm Sci. 2001;90(4):504–20.

    CAS  PubMed  Google Scholar 

  27. Kasting GB. Kinetics of finite dose absorption through skin 1. Vanillylnonanamide. J Pharm Sci. 2001;90(2):202–12.

    CAS  PubMed  Google Scholar 

  28. Scheuplein RJ, Ross LW. Mechanism of percutaneous absorption. V. Percutaneous absorption of solvent deposited solids. J Invest Dermatol. 1974;62(4):353–60.

    CAS  PubMed  Google Scholar 

  29. Kakemi K, et al. Model studies on percutaneous absorption and transport in the ointment. I. Theoretical aspects. Chem Pharm Bull Tokyo. 1975;23(9):2109–13.

    CAS  PubMed  Google Scholar 

  30. Guy RH, Hadgraft J. A theoretical description relating skin penetration to the thickness of the applied medicament. Int J Pharm. 1980;6(3/4):321–32.

    Google Scholar 

  31. Hadgraft J. The epidermal reservoir: a theoretical approach. Int J Pharm. 1979;2:265–74.

    Google Scholar 

  32. Cleek RL, Bunge AL. A new method for estimating dermal absorption from chemical exposure .1. General approach. Pharm Res. 1993;10(4):497–506.

    CAS  PubMed  Google Scholar 

  33. Bunge AL, Cleek RL. A new method for estimating dermal absorption from chemical exposure: 2 Effect of molecular weight and octanol–water partitioning. Pharm Res. 1995;12(1):88–95.

    CAS  PubMed  Google Scholar 

  34. Seko N, et al. Theoretical analysis of the effect of cutaneous metabolism on skin permeation of parabens based on a two-layer skin diffusion/metabolism model. Biol Pharm Bull. 1999;22(3):281–7.

    CAS  PubMed  Google Scholar 

  35. Scheuplein RJ, Morgan LJ. “Bound water” in keratin membranes measured by a microbalance technique. Nature. 1967;214(87):456–8.

    CAS  PubMed  Google Scholar 

  36. Roberts MS, Triggs EJ, Anderson RA. Permeability of solutes through biological membranes measured by a desorption technique. Nature. 1975;275:225–7.

    Google Scholar 

  37. Scheuplein RJ, Blank IH. Permeability of the skin. Physiol Rev. 1971;51(4):702–47.

    CAS  PubMed  Google Scholar 

  38. Mitragotri S, et al. Mathematical models of skin permeability: an overview. Int J Pharm. 2011;418:115–29.

    CAS  PubMed  Google Scholar 

  39. Anissimov YG, Roberts MS. Diffusion modeling of percutaneous absorption kinetics: 4. Effects of a slow equilibration process within stratum corneum on absorption and desorption kinetics. J Pharm Sci. 2009;98:772–81.

    CAS  PubMed  Google Scholar 

  40. Watkinson AC, et al. Computer simulation of penetrant concentration-depth profiles in the stratum corneum. Int J Pharm. 1992;87:175–82.

    Google Scholar 

  41. Mueller B, Anissimov YG, Roberts MS. Unexpected clobetasol propionate profile in human stratum corneum after topical application in vitro. Pharm Res. 2003;20(11):1835–7.

    CAS  PubMed  Google Scholar 

  42. Anissimov YG, Roberts MS. Diffusion modeling of percutaneous absorption kinetics: 3. Variable diffusion and partition coefficients, consequences for stratum corneum depth profiles and desorption kinetics. J Pharm Sci. 2004;93(2):470–87.

    CAS  PubMed  Google Scholar 

  43. Hadgraft J. Calculations of drug release from controlled release devices: the slab. Int J Pharm. 1979;2:177–94.

    CAS  Google Scholar 

  44. Chandrasekaran SK, Bayne W, Shaw JE. Pharmacokinetics of drug permeation through human skin. J Pharm Sci, 1978;67:1370–4.

    Google Scholar 

  45. Iordanskii AL, et al. Modeling of the drug delivery from a hydrophilic transdermal therapeutic system across polymer membrane. Eur J Pharm Biopharm. 2000;49(3):287–93.

    CAS  PubMed  Google Scholar 

  46. Higuchi WI. Diffusional models useful in biopharmaceutics. J Pharm Sci. 1967;56:315–24.

    CAS  Google Scholar 

  47. Riegelman S. Pharmacokinetics: pharmacokinetic factors affecting epidermal penetration and percutaneous adsorption. Clin Pharmacol Ther. 1974;16(5 Part 2):873–83.

    CAS  PubMed  Google Scholar 

  48. Wallace SM, Barnett G. Pharmacokinetic analysis of percutaneous absorption: evidence of parallel penetration pathways for methotrexate. J Pharmacokinet Biopharm. 1978;6(4):315–25.

    CAS  PubMed  Google Scholar 

  49. Roberts MS, Anissimov YG, Gonsalvez RA. Mathematical models in percutaneous absorption. In: Bronaugh RL Maibach HI, editors. Percutaneous absorption drugs—cosmetics—mechanisms—methodology. New York: Marcel Dekker; 1999. pp. 3–55.

    Google Scholar 

  50. McCarley KD, Bunge AL. Physiologically relevant two-compartment pharmacokinetic models for skin. J Pharm Sci. 2000;89(9):1212–35.

    CAS  PubMed  Google Scholar 

  51. McCarley KD, Bunge AL. Physiologically relevant one-compartment pharmacokinetic models for skin. 1. Development of models. J Pharm Sci. 1998;87(4):470–81.

    CAS  PubMed  Google Scholar 

  52. Zatz J. Simulation studies of skin permeation. J Soc Cosm Chem. 1992;43:37–48.

    Google Scholar 

  53. Anissimov YG. Mathematical models for different exposure conditions. In: Roberts MS, Walters KA, Editors. Dermal absorption and toxicity assessment. New York: Informa Healthcare; 2008, pp. 271–86.

    Google Scholar 

  54. Roberts MS, Anderson RA, Swarbrick J. Permeability of human epidermis to phenolic compounds. J Pharm Pharmacol. 1977;29(11):677–83.

    CAS  PubMed  Google Scholar 

  55. Wu MS. Determination of concentration dependent water diffusivity in a keratinous membrane. J Pharm Sci. 1983;72:1421–3.

    CAS  PubMed  Google Scholar 

  56. Gienger G, Knoch A, Merkle HP. Modeling and numerical computation of drug transport in laminates—model case evaluation of transdermal delivery system. J Pharm Sci. 1986;75(1):9–15.

    CAS  PubMed  Google Scholar 

  57. Higuchi WI, Higuchi T. Theoretical analysis of diffusion movement through heterogeneous barriers. J Am Pharm Assoc Sci Ed. 1960;49:598–606.

    Google Scholar 

  58. Chandrasekaran SK, et al. Scopolamine permiation through human skin in vitro. AlChE J. 1976;22:828–32.

    CAS  Google Scholar 

  59. Kubota K, Koyama E, Twizell EH. Dual sorption model for the nonlinear percutaneous permeation kinetics of timolol. J Pharm Sci. 1993;82(12):1205–8.

    CAS  PubMed  Google Scholar 

  60. Ando HY, Ho NFH, Higuchi WI. Skin as an active metabolizing barrier. 1. Theoretical analysis of topical bioavailability. J Pharm Sci. 1977;66:1525–8.

    CAS  PubMed  Google Scholar 

  61. Yu CD, et al. Physical model evaluation of topical prodrug delivery-simultaneous transport and bioconversion of vidarabine-5′-valerate II: parameter determinations. J Pharm Sci. 1979;68(11):1347–57.

    CAS  PubMed  Google Scholar 

  62. Yu CD, et al., Physical model evaluation of topical prodrug delivery-simultaneous transport and bioconversion of vidarabine 5′-valerate. Part 1. Physical model development. Part 2. Parameter determinations. J Pharm Sci. 1979;68:1341–6.

    CAS  PubMed  Google Scholar 

  63. Yu CD, et al., Physical model evaluation of topical prodrug delivery-simultaneous transport and bioconversion of vidarabine 5'-valerate. Part 5. Mechanistic analysis of influence of nonhomogeneous enzyme distributions in hairless mouse skin. J Pharm Sci. 1980;69:775–80.

    CAS  PubMed  Google Scholar 

  64. Fox JL, et al. General physical model for simultaneous diffusion and metabolism in biological-membranes—computational approach for the steady-state case. Int J Pharm. 1979;2(1):41–57.

    CAS  Google Scholar 

  65. Hadgraft J. Theoretical aspects of metabolism in the epidermis. Int J Pharm. 1980;4:229–39.

    CAS  Google Scholar 

  66. Guy RH, Hadgraft J. Percutaneous metabolism with saturable enzyme-kinetics. Int J Pharm. 1982;11(3):187–97.

    CAS  Google Scholar 

  67. Kubota K, Ademola J, Maibach HI. Simultaneous diffusion and metabolism of betamethasone 17-valerate in the living skin equivalent. J Pharm Sci. 1995;84(12):1478–81.

    CAS  PubMed  Google Scholar 

  68. Guy RH, Hadgraft J. Pharmacokinetics of percutaneous absorption with concurrent metabolism. Int J Pharm. 1984;20:43–51.

    CAS  Google Scholar 

  69. Hadgraft J, Wolff HM. In vitro/in vivo correlations in transdermal drug delivery. Dermal Absorption Toxicity Assessment. 1998;91:269–79.

    CAS  Google Scholar 

  70. Robinson PJ. Prediction: Simple risk models and overview of dermal risk assessment. In: Roberts MS, Walters KA, editors. Dermal absorption and toxicity assessment. New York: Marcel Dekker; 1998. pp. 203–29.

    Google Scholar 

  71. Yalkowsky SH, Valvani SC. Solubility and partitioning I: solubility of nonelectrolytes in water. J Pharm Sci. 1980;69(8):912–22.

    CAS  PubMed  Google Scholar 

  72. Reifenrath WG, Robinson PB. In vitro skin evaporation and penetration characteristics of mosquito repellents. J Pharm Sci, 1982;71:1014–8.

    CAS  PubMed  Google Scholar 

  73. Guy RH, Hadgraft J. Percutaneous absorption kinetics of topically applied agents liable to surface loss. J Soc Cosmet Chem. 1984;35:103–13.

    Google Scholar 

  74. Guy RH, Hadgraft J. A theoretical description of the effects of volatility and substantivity on percutaneous-absorption. Int J Pharm. 1984;18(1/2):139–47.

    CAS  Google Scholar 

  75. Saiyasombati P, Kasting GB. Disposition of benzyl alcohol after topical application to human skin in vitro. J Pharm Sci. 2003;92(10):2128–39.

    CAS  PubMed  Google Scholar 

  76. Kasting GB, Miller MA. Kinetics of finite dose absorption through skin 2: volatile compounds. J Pharm Sci. 2006;95(2):268–80.

    CAS  PubMed  Google Scholar 

  77. Miller MA, Bhatt V, Kasting GB. Dose and airflow dependence of benzyl alcohol disposition on skin. J Pharm Sci. 2006;95(2):281–91.

    CAS  PubMed  Google Scholar 

  78. Scheuplein RJ. Mechanism of percutaneous absorption. II. transient diffusion and the relative importance of various routes of skin penetration. J Invest Dermatol. 1967;48(1):79–88.

    CAS  PubMed  Google Scholar 

  79. Ghanem AH, et al. The effects of ethanol on the transport of beta-estradiol and other permeants in hairless mouse skin. II. a new quantitative approach. J Controlled Release. 1987;6:75–83.

    CAS  Google Scholar 

  80. Hatanaka T, et al. Prediction of skin permeability of drugs .2. development of composite membrane as a skin alternative. Int J Pharm. 1992;79(1):21–8.

    CAS  Google Scholar 

  81. Tojo K, Chiang CC, Chien YW. Drug permeation across the skin: effect of penetrant hydrophilicity. J Pharm Sci. 1987;76:123–6.

    CAS  PubMed  Google Scholar 

  82. Yamashita F, et al. Analysis of skin penetration enhancement based on a 2-layer skin diffusion-model with polar and nonpolar routes in the stratum-corneum—dose-dependent effect of 1-geranylazacycloheptan-2-one on drugs with different lipophilicities. Biol Pharm Bull. 1993;16(7):690–7.

    CAS  PubMed  Google Scholar 

  83. Edwards DA, Langer R. A linear-theory of transdermal transport phenomena. J Pharm Sci. 1994;83(9):1315–34.

    CAS  PubMed  Google Scholar 

  84. Malkinson FD, Ferguson EH. Percutaneous absorption of hydrocortisone-4-C14 in two human subjects. J Invest Dermatol. 1955;25(5):281–3.

    CAS  PubMed  Google Scholar 

  85. Vickers CF. Stratum corneum reservoir for drugs. Adv Biol Skin. 1972;12:177–89.

    CAS  PubMed  Google Scholar 

  86. Reddy MB, Guy RH, Bunge AL. Does epidermal turnover reduce percutaneous penetration? Pharm Res. 2000;17(11):1414–9.

    CAS  PubMed  Google Scholar 

  87. Roberts MS, Cross SE, Anissimov YG. Factors affecting the formation of a skin reservoir for topically applied solutes: skin pharmacology and applied skin. Physiology. 2004;17:3–16.

    CAS  Google Scholar 

  88. Rohatagi S, et al. Integrated pharmacokinetic and metabolic modeling of selegiline and metabolites after transdermal administration. Biopharm Drug Dispos. 1997;18(7):567–84.

    CAS  PubMed  Google Scholar 

  89. Cross SE, Wu Z, Roberts MS. Effect of perfusion flow rate on the tissue uptake of solutes after dermal application using the rat isolated perfused hindlimb preparation. J Pharm Pharmacol. 1994;46(10):844–50.

    CAS  PubMed  Google Scholar 

  90. Cross SE, Wu Z, Roberts MS. The effect of protein binding on the deep tissue penetration and efflux of dermally applied salicylic acid, lidocaine and diazepam in the perfused rat hindlimb. J Pharmacol Exp Ther. 1996;277(1):366–74.

    CAS  PubMed  Google Scholar 

  91. Williams PL, Carver MP, Riviere JE. A physiologically relevant pharmacokinetic model of xenobiotic percutaneous-absorption utilizing the isolated perfused porcine skin flap. J Pharm Sci. 1990;79(4):305–11.

    CAS  PubMed  Google Scholar 

  92. Reddy MB, McCarley KD, Bunge AL. Physiologically relevant one-compartment pharmacokinetic models for skin. 2. comparison of models when combined with a systemic pharmacokinetic model. J Pharm Sci. 1998;87(4):482–90.

    CAS  PubMed  Google Scholar 

  93. Roberts MS, et al., Modeling of subcutaneous absorption kinetics of infusion solutions in the elderly using technetium. J Pharmacokinet Biopharm, 1997;25:1–21.

    CAS  PubMed  Google Scholar 

  94. Singh P, Roberts MS, Maibach HI. Modeling of plasma-levels of drugs following transdermal iontophoresis. J Controlled Release. 1995;33(2):293–8.

    CAS  Google Scholar 

  95. Imanidis G, et al. Estimation of skin target site acyclovir concentrations following controlled (trans)dermal drug-delivery in topical and systemic treatment of cutaneous hsv-1 infections in hairless mice. Pharm Res. 1994;11(7):1035–41.

    CAS  PubMed  Google Scholar 

  96. Tegeder I, et al. Application of microdialysis for the determination of muscle and subcutaneous tissue concentrations after oral and topical ibuprofen administration. Clin Pharmacol Ther. 1999;65(4):357–68.

    CAS  PubMed  Google Scholar 

  97. Cooper ER. Pharmacokinetics of skin penetration. J Pharm Sci. 1976;65(9):1396–7.

    CAS  PubMed  Google Scholar 

  98. Cooper ER, Effect of diffusional lag time on multicompartmental pharmacokinetics for transepidermal infusion. J Pharm Sci. 1979;68:1469–1470.

    CAS  PubMed  Google Scholar 

  99. McDougal JN. Prediction—physiological models. In: Roberts MS, Walters KA, Editors. Dermal absorption and toxicity assessment. New York: Marcel Dekker; 1998. pp. 189–202.

    Google Scholar 

  100. Jepson GW, McDougal JN. Physiologically based modeling of nonsteady state dermal absorption of halogenated methanes from an aqueous solution. Toxicol Appl Pharmacol. 1997;144(2):315–24.

    CAS  PubMed  Google Scholar 

  101. Timchalk C, et al. A physiologically based pharmacokinetic and pharmacodynamic (PBPK/PD) model for the organophosphate insecticide chlorpyrifos in rats and humans. Toxicol Sci. 2002;66(1):34–53.

    CAS  PubMed  Google Scholar 

  102. Poet TS, et al. PBPK modeling of the percutaneous absorption of perchloroethylene from a soil matrix in rats and humans. Toxicol Sci. 2002;67(1):17–31.

    CAS  PubMed  Google Scholar 

  103. Poet TS, et al. Assessment of the percutaneous absorption of trichloroethylene in rats and humans using MS/MS real-time breath analysis and physiologically based pharmacokinetic modeling. Toxicol Sci. 2000;56(1):61–72.

    CAS  PubMed  Google Scholar 

  104. Poet TS, et al. Utility of real time breath analysis and physiologically based pharmacokinetic modeling to determine the percutaneous absorption of methyl chloroform in rats and humans. Toxicol Sci. 2000;54(1):42–51.

    CAS  PubMed  Google Scholar 

  105. Thrall KD, Weitz KK, Woodstock AD. Use of real-time breath analysis and physiologically based pharmacokinetic modeling to evaluate dermal absorption of aqueous toluene in human volunteers. Toxicol Sci. 2002;68(2):280–7.

    CAS  PubMed  Google Scholar 

  106. Qiao GL, et al. Dermatoxicokinetic modeling of p-nitrophenol and its conjugation metabolite in swine following topical and intravenous administration. Toxicol Sci. 2000;54(2):284–94.

    CAS  PubMed  Google Scholar 

  107. Riviere JE, et al. Dermal absorption and distribution of topically dosed jet fuels jet-A, JP-8, and JP-8(100). Toxicol Appl Pharmacol. 1999;160(1):60–75.

    CAS  PubMed  Google Scholar 

  108. Benowitz NL, et al. Stable isotope method for studying transdermal drug absorption: the nicotine patch. Clin Pharmacol Ther. 1991;50(3):286–93.

    CAS  PubMed  Google Scholar 

  109. Welin-Berger K, et al. In vitro-in vivo correlation in man of a topically applied local anesthetic agent using numerical convolution and deconvolution. J Pharm Sci. 2003;92(2):398–406.

    CAS  PubMed  Google Scholar 

  110. Anissimov YG, et al. Mathematical and pharmacokinetic modelling of epidermal and dermal transport processes. Adv Drug Deliv Rev. 2013;65:169–90.

    CAS  PubMed  Google Scholar 

  111. Jepps OG, et al. Modeling the human skin barrier—towards a better understanding of dermal absorption. Adv Drug Deliv Rev. 2013;65:152–68.

    CAS  PubMed  Google Scholar 

  112. Gupta E, Wientjes MG, Au JL. Penetration kinetics of 2',3'-dideoxyinosine in dermis is described by the distributed model. Pharm Res. 1995;12(1):108–12.

    CAS  PubMed  Google Scholar 

  113. Cross SE, et al. Self promotion of deep tissue penetration and distribution of methylsalicylate after topical application. Pharm Res. 1999;16(3):427–33.

    CAS  PubMed  Google Scholar 

  114. Kretsos K, Kasting GB, Nitsche JM. Distributed diffusion-clearance model for transient drug distribution within the skin. J Pharm Sci. 2004;93(11):2820-35.

    CAS  PubMed  Google Scholar 

  115. Anissimov YG, Roberts MS. Modelling dermal drug distribution after topical application in human. Pharm Res. 2011;28(9):2119–29.

    CAS  PubMed  Google Scholar 

  116. Schaefer H, Stuttgen G. Absolute concentrations of an antimycotic agent, econazole, in the human skin after local application. Arzneimittelforschung. 1976;26(3):432–5.

    CAS  PubMed  Google Scholar 

  117. Schaefer H, Zesch A. Penetration of vitamin A acid into human skin. Acta Derm Venereol Suppl (Stockh). 1975;74:50–5.

    CAS  Google Scholar 

  118. Schaefer H, Zesch A, Stuttgen G. Penetration, permeation, and absorption of triamcinolone acetonide in normal and psoriatic skin. Arch Dermatol Res. 1977;258(3):241–9.

    CAS  PubMed  Google Scholar 

  119. Schaefer H, et al. Quantitative determination of percutaneous absorption of radiolabeled drugs in vitro and in vivo by human skin. Curr Probl Dermatol. 1978;7:80–94.

    CAS  PubMed  Google Scholar 

  120. Zesch A, Schaefer H. [Penetration of radioactive hydrocortisone in human skin from various ointment bases. II. In vivo-experiments (author’s transl)]. Arch Dermatol Forsch. 1975;252(4):245–56.

    CAS  PubMed  Google Scholar 

  121. Dancik Y, et al. Convective transport of highly plasma protein bound drugs facilitates direct penetration into deep tissues after topical application. Brit J Clin Pharmacol. 2012;73(4):564–78.

    CAS  Google Scholar 

  122. Singh P, Maibach HI, Roberts MS. Site of effects. Dermal absorption toxicity assessment. 1998;91:353–70.

    CAS  Google Scholar 

  123. McNeill SC, Potts RO, Francoeur ML. Local enhanced topical delivery (LETD) of drugs: does it truly exist? Pharm Res. 1992;9(11):1422–7.

    CAS  PubMed  Google Scholar 

  124. Singh P, Roberts MS. Blood-flow measurements in skin and underlying tissues by microsphere method—application to dermal pharmacokinetics of polar nonelectrolytes. J Pharm Sci. 1993;82(9):873–9.

    CAS  PubMed  Google Scholar 

  125. Singh P, Roberts MS. Dermal and underlying tissue pharmacokinetics of salicylic acid after topical application. J Pharmacokinet Biopharm. 1993;21(4):337–73.

    CAS  PubMed  Google Scholar 

  126. Roberts MS, Cross SE. A physiological pharmacokinetic model for solute disposition in tissues below a topical application site. Pharm Res. 1999;16(9):1392–8.

    CAS  PubMed  Google Scholar 

  127. Nakayama K, et al. Estimation of intradermal disposition kinetics of drugs: I. analysis by compartment model with contralateral tissues. Pharm Res. 1999;16(2):302–8.

    CAS  PubMed  Google Scholar 

  128. Higaki K, et al. Estimation of intradermal disposition kinetics of drugs: II. factors determining penetration of drugs from viable skin to muscular layer. Int J Pharm. 2002;239(1/2):129–41.

    CAS  PubMed  Google Scholar 

  129. Singh P, Roberts MS. Effects of vasoconstriction on dermal pharmacokinetics and local tissue distribution of compounds. J Pharm Sci. 1994;83(6):783–91.

    CAS  PubMed  Google Scholar 

  130. Cross SE, et al., Is there tissue penetration after application of topical salicylate formulations? Lancet, 1997;350:636.

    CAS  PubMed  Google Scholar 

  131. Muller M, et al. Diclofenac concentrations in defined tissue layers after topical administration. Clin Pharmacol Ther. 1997;62(3):293–9.

    CAS  PubMed  Google Scholar 

  132. Beastall J, et al. The influence of urea on percutaneous-absorption. Pharm Res. 1986;3(5):294–7.

    CAS  PubMed  Google Scholar 

  133. Demana PH, et al., Evaluation of the proposed FDA pilot dose response methodology for topical corticosteroid bioequivalence testing. Pharm Res, 1997;14:303–8.

    CAS  PubMed  Google Scholar 

  134. Singh GJ, et al. Evaluation of the proposed FDA pilot dose-response methodology for topical corticosteroid bioequivalence testing [letter]. Pharm Res, 1998;15(1):4–7 (Biol Sciences Journal holding: [v.1](1984)-12(1995);13(1996)).

    CAS  PubMed  Google Scholar 

  135. Smith EW, Haigh JM, Walker RB. Analysis of chromameter results obtained from corticosteroid-induced skin blanching. Part 1. manipulation of data. Pharm Res. 1998;15:280–5.

    CAS  PubMed  Google Scholar 

  136. Cordero JA, et al. In vitro based index of topical anti-inflammatory activity to compare a series of NSAIDs. Eur J Pharm Biopharm. 2001;51(2):135–42.

    CAS  PubMed  Google Scholar 

  137. Kasting GB. Theoretical-models for iontophoretic delivery. Adv Drug Delivery Rev. 1992;9(2/3):177–99.

    CAS  Google Scholar 

  138. Roberts MS, Lai PM, Anissimov YG. Epidermal iontophoresis: I. development of the ionic mobility-pore model. Pharm Res. 1998;15(10):1569–78.

    CAS  PubMed  Google Scholar 

  139. Cross SE, Roberts MS. Importance of dermal blood supply and epidermis on the transdermal iontophoretic delivery of monovalent cations. J Pharm Sci. 1995;84(5):584–92.

    CAS  PubMed  Google Scholar 

  140. Mitragotri S, Blankschtein D, Langer R. An explanation for the variation of the sonophoretic transdermal transport enhancement from drug to drug. J Pharm Sci. 1997;86(10):1190–2.

    CAS  PubMed  Google Scholar 

  141. Tezel A, et al. Frequency dependence of sonophoresis. Pharm Res. 2001;18(12):1694–700.

    CAS  PubMed  Google Scholar 

  142. Tezel A, Sens A, Mitragotri S. Description of transdermal transport of hydrophilic solutes during low-frequency sonophoresis based on a modified porous pathway model. J Pharm Sci. 2003;92(2):381–93.

    CAS  PubMed  Google Scholar 

  143. Jiang R, et al. Absorption of sunscreens across human skin: an evaluation of commercial products for children and adults. Br J Clin Pharmacol. 1999;48(4):635–7.

    PubMed Central  CAS  PubMed  Google Scholar 

  144. Kubota K, Sznitowska M, Maibach HI. Percutaneous-absorption—a single-layer model. J Pharm Sci. 1993;82(5):450–6.

    CAS  PubMed  Google Scholar 

  145. Beckett AH, Taylor DC, Gorrod JW. Comparison of oral and percutaneous routes in man for systemic administration of ephedrines. J Pharm Pharmacol. 1972;24:65–70.

    Google Scholar 

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Acknowledgments

We thank the NH&MRC of Australia PAH Research Foundation and the NSW and Qld Lions Medical Research Foundation for the support of this work.

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Authors and Affiliations

  1. Queensland Micro- and Nanotechnology Centre, Griffith University, 4111, Brisbane, QLD, Australia

    Yuri G. Anissimov

  2. School of Biomolecular and Physical Sciences, Griffith University, 4222, Gold Coast, QLD, Australia

    Yuri G. Anissimov

  3. Therapeutics Research Unit, School of Medicine, The University of Queensland, Princess Alexandra Hospital, 4102, Woolloongabba, QLD, Australia

    Michael S. Roberts

  4. School of Pharmacy & Medical Sciences, University of South Australia, 5001, Adelaide, SA, Australia

    Michael S. Roberts

Authors
  1. Yuri G. Anissimov
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  2. Michael S. Roberts
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Corresponding author

Correspondence to Yuri G. Anissimov .

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Editors and Affiliations

  1. Pharmaceutical Consultant, North Potomac, Maryland, USA

    Vinod P. Shah

  2. University of California, San Francisco School of Medicine, Department of Dermatology, San Francisco, California, USA

    Howard I. Maibach

  3. The Defence Science and Technology Laboratory, Biomedical Sciences Department, Salisbury, United Kingdom

    John Jenner

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Anissimov, Y., Roberts, M. (2014). Mathematical Models for Topical and Transdermal Drug Products. In: Shah, V., Maibach, H., Jenner, J. (eds) Topical Drug Bioavailability, Bioequivalence, and Penetration. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1289-6_15

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