Background: The delivery of drugs through dermal layers known as transdermal drug delivery is one of the important contributions in medical practice. It represents an alternative to oral drug delivery and is poised to provide a substitute to hypodermic injections too. The diffusion of drug from the source to the target site is therefore an important issue to address.Aim: This paper is an attempt to establish a mathematical model for the diffusion of drugs through the transdermal drug delivery system. The model identifies the pattern of drug diffusion in human body and its effective absorption rates at various compartments of skin and sub-cutaneous tissues.Methods: The finite element method has been used to obtain the solution of the mass diffusion equation with appropriate boundary conditions. The tissue absorption rate of drug has been taken as the decreasing function of drug concentration from the skin surface towards the target site. The concentration at nodal points has been calculated which in turn determines the drug absorption at various layers.Conclusion: The drug concentration at the nodal points of different dermal layers has been computed and the graphs were plotted between drug concentration and thickness of dermal layers using MATLAB software. It has been observed that due to dense network of connective tissues in dermal and sub-dermal parts, the drug absorption is maximum as compared to cutaneous tissues.Keywords: Transdermal drug delivery; Diffusion equation; Finite element metho

Khanday, MA

Rafiq, A

Alexandria Journal of Medicine

English

AbstractBackgroundThe delivery of drugs through dermal layers known as transdermal drug delivery is one of the important contributions in medical practice. It represents an alternative to oral drug delivery and is poised to provide a substitute to hypodermic injections too. The diffusion of drug from the source to the target site is therefore an important issue to address.AimThis paper is an attempt to establish a mathematical model for the diffusion of drugs through the transdermal drug delivery system. The model identifies the pattern of drug diffusion in human body and its effective absorption rates at various compartments of skin and sub-cutaneous tissues.MethodsThe finite element method has been used to obtain the solution of the mass diffusion equation with appropriate boundary conditions. The tissue absorption rate of drug has been taken as the decreasing function of drug concentration from the skin surface towards the target site. The concentration at nodal points has been calculated which in turn determines the drug absorption at various layers.ConclusionThe drug concentration at the nodal points of different dermal layers has been computed and the graphs were plotted between drug concentration and thickness of dermal layers using MATLAB software. It has been observed that due to dense network of connective tissues in dermal and sub-dermal parts, the drug absorption is maximum as compared to cutaneous tissues

Khanday, M.A.

Rafiq, Aasma

Elsevier - Publisher Connector 

Variational finite element method to study the absorption rate of drug at various compartments through transdermal drug delivery system 

Background: The delivery of drugs through dermal layers known as transdermal drug delivery is one of the important contributions in medical practice. It represents an alternative to oral drug delivery and is poised to provide a substitute to hypodermic injections too. The diffusion of drug from the source to the target site is therefore an important issue to address.
Aim: This paper is an attempt to establish a mathematical model for the diffusion of drugs through the transdermal drug delivery system. The model identifies the pattern of drug diffusion in human body and its effective absorption rates at various compartments of skin and sub-cutaneous tissues.
Methods: The finite element method has been used to obtain the solution of the mass diffusion equation with appropriate boundary conditions. The tissue absorption rate of drug has been taken as the decreasing function of drug concentration from the skin surface towards the target site. The concentration at nodal points has been calculated which in turn determines the drug absorption at various layers.
Conclusion: The drug concentration at the nodal points of different dermal layers has been computed and the graphs were plotted between drug concentration and thickness of dermal layers using MATLAB software. It has been observed that due to dense network of connective tissues in dermal and sub-dermal parts, the drug absorption is maximum as compared to cutaneous tissues

M.A. Khanday

Aasma Rafiq

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Variational finite element method to study the absorption rate of drug at various compartments through transdermal drug delivery system

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Alexandria Journal of Medicine (2015) 51, 219–223HO ST E D  BYAlexandria University Faculty of MedicineAlexandria Journal of Medicinehttp://www.elsevier.com/locate/ajmeVariational finite element method to studythe absorption rate of drug at various compartmentsthrough transdermal drug delivery system* Corresponding author.E-mail addresses: khanday@gmail.com (M.A. Khanday), rafiqaas-ma@gmail.com (A. Rafiq).Peer review under responsibility of Alexandria University Faculty ofMedicine.http://dx.doi.org/10.1016/j.ajme.2014.09.0012090-5068 ª 2014 Alexandria University Faculty of Medicine. Production and hosting by Elsevier B.V. All rights reserved.M.A. Khanday *, Aasma RafiqDepartment of Mathematics, University of Kashmir, Hazratbal Srinagar 190006, Jammu and Kashmir, IndiaReceived 5 July 2014; accepted 9 September 2014Available online 25 November 2014KEYWORDSTransdermal drug delivery;Diffusion equation;Finite element methodAbstract Background: The delivery of drugs through dermal layers known as transdermal drugdelivery is one of the important contributions in medical practice. It represents an alternative to oraldrug delivery and is poised to provide a substitute to hypodermic injections too. The diffusion ofdrug from the source to the target site is therefore an important issue to address.Aim: This paper is an attempt to establish a mathematical model for the diffusion of drugs throughthe transdermal drug delivery system. The model identifies the pattern of drug diffusion in humanbody and its effective absorption rates at various compartments of skin and sub-cutaneous tissues.Methods: The finite element method has been used to obtain the solution of the mass diffusionequation with appropriate boundary conditions. The tissue absorption rate of drug has been takenas the decreasing function of drug concentration from the skin surface towards the target site. Theconcentration at nodal points has been calculated which in turn determines the drug absorption atvarious layers.Conclusion: The drug concentration at the nodal points of different dermal layers has been com-puted and the graphs were plotted between drug concentration and thickness of dermal layers usingMATLAB software. It has been observed that due to dense network of connective tissues in dermaland sub-dermal parts, the drug absorption is maximum as compared to cutaneous tissues.ª 2014 Alexandria University Faculty of Medicine. Production and hosting by Elsevier B.V. All rightsreserved.1. IntroductionIn medical sciences, various attempts have been made to searchnew drugs and to find the improved ways of drug delivery inthe body. Oral drug mechanism delivery like pills has beenconsidered as one of the most appropriate methods of drugadministration for decades worldwide. There are many drugswhich cannot be taken orally so they are injected into the bodyby making use of hypodermic needles. However, the hypoder-220 M.A. Khanday, A. Rafiqmic injections have many disadvantages such as the presenceof pain, the appearance of infections and the requirement formedical expertise to complete the process.1 For the administra-tion of drug and its absorption to target the desired compart-ment of the biological tissue; it is one of the challengingproblems to control the adverse effects of the drug deliverythrough in-vivo tissues. Thus, it is desirable to address thisissue by using an appropriate method of drug administrationthrough dermal regions of the human body with minimal sideeffects. Transdermal drug delivery (TDD) is an alternate routeof drug administration to pills and injections. This methodoperates by delivery of drugs into the human body acrossthe skin and subcutaneous layers. TDD system has the abilityto overcome the problems associated with the traditional deliv-ery of drugs.Transdermal drug delivery system has been in existencesince last thirty years. Transdermal therapeutic systems aredefined as self contained, continuous dosage forms which whenapplied to the skin deliver the drug through it at a controlledrate to the systemic circulation. TDD is a feasible administra-tion route for potent and low molecular weight therapeuticagents. The choice of therapeutic agent is determined by anumber of factors which includes the physio-co-chemical prop-erties of the drug and its interactions with the membrane. Thisis due to the barrier function of the skin represented by theouter layer- the stratum corneum, which generally allows diffu-sion of only small molecular weight solutes. To overcome thisdiffusion limitation at the stratum corneum, various methodshave been developed to enhance drug delivery across stratumcorneum which includes chemical enhancers or physical enhan-cer technique e.g. iontophoresis and ultrasound.The transport of drug through the device and skin followsthe Fickian diffusion process by means of a simple homoge-neous membrane. Kalia and Guy2 have established the math-ematical transport models describing the release of drug fromvarious delivery systems and formulations based on the solu-tion of Fick’s second law. In these models, skin was simplytreated as a boundary condition and the main attention wasfocussed on the drug transport, also analytical expressionswere found for the amount of drug released from the device.Qaliaf et al.3 developed a mathematical model and carriednumerical simulations to describe the pharmacokinetics ofdrug penetration into the skin from microneedle array. Missel4used FEM for modelling of diffusion and partitioning in bio-logical systems. One-dimensional mathematical models weredeveloped by Lee et al.5,6 which considers a single componentdrug penetration via both intercellular and transcellular path-ways of the skin. The skin was divided into stratum corneumand viable epidermal layers and diffusion of drug from theporous device into the multiple pathway model of skin was cal-culated by finite difference approximations. Saxena, Pardasaniand their co-workers7–9 had carried out a notable research forthe study of heat distribution in biological tissues. Khanday10studied the temperature distribution in multi-layered humanskin and sub-cutaneous (SST) by assuming the thermal con-ductivity as a function of temperature. Khanday et al.11–14studied the diffusion of heat and mass in biological tissuesparticularly dermal regions and human head. Sharma andSaxena15 used FEM to study the drug distribution in TDD A physical process in which ion flow diffusively in a medium drivenby an applied electric field.systems in steady state case by making use of quadratic shapefunction. Most of the models have taken into considerationonly the drug transport in the devices and did not discussthe flow of drug in the skin and sub-cutaneous tissues. In thispaper, a variational finite element approach with linear shapefunction is used to obtain the solution of diffusion equationtaking into account the role of diffusion constant and the rateconstants. The method has been used because of its applicabil-ity over the irregular geometries. Since the human skin hasirregular geometry, therefore this method guarantees a reason-able outcome as compared to the other numerical methods.2. Formulation of modelThe transport of drugs through transdermal system is gov-erned by the differential equation given in Crank16@@xD@C@x  AðCÞ  B ¼ @C@tð1Þwhere C;Dand t denote the drug concentration, mass diffusiv-ity and time. Also, AðCÞ is the absorption rate of drug by thetissues, which is a function of concentration and B is the rate ofdrug intake by the blood.Consider the domain consisting of three layers viz., epider-mis, dermis and subcutaneous region with outer and innerboundaries as a0 and a3. The concentration of the drug atthe outer skin surface (epidermis) is known as C0 and it hasnegligible concentration beyond the subcutaneous tissue asthe drug is absorbed through the subcutaneous tissue to reachthe target site. Hence at the innermost boundary the concen-tration is C3 ¼ 0.By the law of mass action, the absorption rate of drug bythe tissue decreases with changing time, so we can say thatabsorption rate is decreasing function of drug concentration.Hence, we assumeAiðCðiÞÞ ¼ expðkiCðiÞÞ; ði ¼ 1; 2; 3Þ ð2Þwhere kið 1Þ represents the rate constants. The drug distribu-tion has a mild barrier at the interfaces due to non-homoge-neous character of tissues in various layers of dermalregions. So,CðiÞ ¼ qiCðiþ1Þjx¼aiði¼1;2Þ ð3ÞHere xandqi ði ¼ 1; 2Þ represents the depth below the skinsurface and the skin partition coefficients for the drug at therespective interfaces respectively. qi  1 for most of thedrugs17.The solution CðxÞ of the problem can be approximated bypolynomial functions of x of higher degrees. Since the distancebetween the skin layers is of minute length, the higher ordershape functions may not contribute a significant change inthe drug diffusion, therefore the concentration of drug in eachregion is approximated by a linear polynomial asCðiÞ ¼ /i þ uix; ði ¼ 1; 2; 3Þ ð4Þ3. Mathematical methodThe solution of such type of problem has been carried out byvarious researchers. The numerical solution of the boundary Let F ðsÞandGðsÞ be two polynomials in s where the degree of F ðsÞ islower than that of GðsÞ and if GðsÞ has n distinct rootsai ði ¼ 1; 2; . . . ; nÞ, thenL1FðsÞGðsÞ ; t ¼Xni¼1FðaiÞG0ðaiÞeai tVariational finite element method to study the absorption rate of drug 221value problem is important for various situations due to thenon-availability of the analytical and exact solutions. But,the finite element method is treated to be one of the most reli-able methods in numerical systems. The method has advantageover other methods due to position dependent properties ofparameters and their flexibility. Moreover, the method is appli-cable to understand the feasible diffusion in irregular geome-tries. The variational integralI ¼ZFðC;C0; xÞdx ð5Þin optimum form is equivalent to Euler-Lagrange differentialequation given by Myer18@F@C ddx@F@C0 ¼ 0; where C0 ¼ @C@xð6ÞOn comparing Eq. (1) with Euler-Lagrange Eq. (6), we obtainthe variational integralI ¼Z a30D2@C@x 2þ ð1þ BÞC k2C2 þ 12@C2@t" #dx ð7ÞThe variational integral on each of the sub domains are givenasI1 ¼12D1ðC1  C0Þ2a1þ ð1þ B1ÞðC1 þ C0Þa1" k1a13C20 þ C21 þ C0C1 þ a132C0 _C0 þ 2C1 _C1 þ C1 _C0 þ C0 _C1 ið8ÞIi ¼12Diai  ai1Ci Ci1qi1 þ ð1þ BiÞCi1qi1þ Ci ai  ai1ð Þ ki3ai  ai1ð Þ Ci1Ciqi1þ C2i þCi1qi1 2" #þ ai  ai13_Ci1Ciqi1þ Ci1_Ciqi1þ 2Ci _Ci þCi1 _Ci1qi1 	ði ¼ 2; 3Þ ð9ÞNow, these integrals Ii are assembled to obtain I ¼P3i¼1Iiand I is optimised with respect to the nodal valuesCi ði ¼ 1; 2Þ to get the system of equationsa1C1 þ a2C2 þ b1 _C1 þ b2 _C2 ¼ c1 ð10Þa2C1 þ a3C2 þ b2 _C1 þ b3 _C2 ¼ c2 ð11Þwhere the coefficients are given in the Appendix.The Eqs. (10) and (11) in the matrix form can be written asACþ B _C ¼ Ywhere A ¼a1 a2a2 a3 ; B ¼b1 b2b2 b3 and Y ¼c1c2 ; C ¼C1C2  ð12ÞCase:I (Steady State)Solving equation AC ¼ Y, we get the values of the concen-tration of the drug at the nodes a1 anda2 as C1ða1; 0ÞandC2ða2; 0Þ which in turn on substitution to Eq. (4) are used todetermine the values of CðiÞ ði ¼ 1; 2; 3Þ.Case:II (Unsteady State)Applying the Laplace transform to the Eq. (12), we haveL ACþ B _C ¼LðYÞor; ALðCÞ þ B sLðCÞ  Cðx; 0Þð Þ ¼LðYÞi:e:; ðAþ BsÞLðCÞ ¼LðYÞ þ BCðx; 0Þwhich gives ða1 þ b1sÞ bC1 þ ða2 þ b2sÞ bC2 ¼ c1s þ b1C1ða1; 0Þþ b2C2ða2; 0Þ ð13Þand ða2 þ b2sÞ bC1 þ ða3 þ b3sÞ bC2 ¼ c2s þ b2C1ða1; 0Þþ b3C2ða2; 0Þ ð14Þwhere ˆ represents Laplace transform.On solving Eqs. (13) and (14), we havebC1 ¼ ða3 þ sb3Þðm1 þ c1s Þ  ða2 þ sb2Þðm2 þ c2s Þðs aÞðs bÞ ð15Þand bC2 ¼ ða1 þ sb1Þðm2 þ c2s Þ  ða2 þ sb2Þðm1 þ c1s Þðs aÞðs bÞ ð16ÞApplying the inverse Laplace transform and making use ofHeaviside Expansion theorem, we get the value of Ci ði ¼ 1; 2Þwhich on substitution to Eq. (4) is used to determine the valueof CðiÞ ði ¼ 1; 2; 3Þ at any instant of time. The overall drug con-centration can be formulated by assembling the linear varia-tions of each region. This indicates polygonal variations ofconcentration with discontinuities of flux at the interfaces. Inorder to obtain the better approximations, the Lagrange’sinterpolation method has been used. The method determinesthe drug concentration with continuous flux between the inter-faces and within the region.CðxÞ ¼ H0x3 H1x2 þH2xH3 ð17Þ4. Numerical calculationsTo solve Eq. (12), we make use of the values given in Sharmaand Saxena15 of the constants given in the Table 1.We can assign different values to the constantsai ði ¼ 0; 1; 2; 3Þ and Di ði ¼ 1; 2; 3Þ depending on the sampleof the skin under study. The set of values considered here area0 ¼ 0 lm; a1 ¼ 0:3 lm; a2 ¼ 0:6 lm; a3 ¼ 1 lmD1 ¼ 0:00068 lm2 s1;D2 ¼ 0:0205 lm2 s1;D3 ¼ 0:002 lm2 s1The numerical calculations have been carried out for 5 gramsof drug. The curves have been plotted for CðiÞði ¼ 1; 2; 3Þ versusthickness of skin layers at time t ¼ 0; 5 and 10 sec. The profilesof drug concentration in two different set of values are shownin Figs. 1 and 2.Table 1 Numerical values.q1 q2 B2 ðlm3Þ B3 k1 ðsec1Þ k2 k3Set-I 0.8 0.9 0.002 0.02 0.4 0.5 0.55Set-II 0.98 1 0.1 0.2 0.5 0.7 0.8Figure 1 Concentration profile for Set-I at t ¼ 0; 5 and 10 sec.Figure 2 Concentration profile for Set-II at t ¼ 0; 5 and 10 sec.Figure 3 Modified drug concentration based on Lagrange’sinterpolation method for Set-I.Figure 4 Modified drug concentration based on Lagrange’sinterpolation method for Set-II.222 M.A. Khanday, A. Rafiq5. Discussion and conclusionThe present work is an attempt to estimate the drug concentra-tion in the human dermal layers. Initially, we have calculatedthe drug concentration at the nodal points of different dermallayers and graphs have been plotted using MATLAB softwarebetween concentration and thickness of dermal layers. Fromthe graphs given in Figs. 1 and 2, the drug concentrationprofiles at different time intervals have been estimated. It hasbeen observed that the curves for CðiÞði ¼ 1; 2; 3Þ fall morequickly in Set-II and reach the equilibrium state earlier thanthe Set-I. Analysing the concentration profiles, it seems thatthe concentration decreases with increase in the partitioncoefficient, absorption coefficients and rate constants. Thecurves given in figures reflect the amount of drug absorptionin the region of papillary and reticular layers due to densenetwork of capillary bed.Lagrange’s interpolation formula has been used and thecurve describing the continuous distribution of drug insidethe dermal layers has been obtained and is given in Figs. 3and 4. The results obtained in this paper appear to be realisticand are stronger than the results obtained by Sharma andSaxena15. They had taken the rate of absorption of drug bythe tissues and rate of drug intake by blood as constants whileas in our model, keeping the physiological aspects of the der-mal layers into consideration, the absorption rate has beentaken as an exponentially decreasing function of the concen-Variational finite element method to study the absorption rate of drug 223tration. Also they have studied the drug distribution in TDDsystems for steady state case while the present work is timedependent problem and therefore their model is a special caseof our model. Our work shows a close resemblance and givesmodified outcome than the work carried out by Sharma andSaxena15 for steady state case. Moreover, our model is timedependent and the time plays a key role in drug diffusion.The developed model helps us in predicting the amount ofthe drug concentration in different tissues at different timeintervals. Also, this model may be helpful in analysing the roleof the diffusion constants and the absorption coefficients indrug delivery system. The study may be very helpful in themedical sciences in various situations.Conflict of interestNone declared.AcknowledgementsThe authors wish to thank the DST, Govt. of India for provid-ing financial support under DST- INSPIRE programme.Appendix A.hi ¼ aiþ1  ai; ði ¼ 0; 1; 2:Þ; gi ¼Dihi;ri ¼ kihi; ði ¼ 1; 2; 3Þqj ¼gjq2j1; nj ¼rjq2j1; pj ¼hjq2j1; ðj ¼ 2; 3Þa1 ¼ 2 g1 þ q2ð Þ 23r1 þ n2ð Þ; a2 ¼1q12g2 þr23 a3 ¼ 2 g2 þ q3ð Þ 23r2 þ n3ð Þ; b1 ¼23h1 þ p2ð Þ;b2 ¼q1p23q1b3 ¼ 2 q1b2 þp33 ; c1 ¼ 2g1 þr13 c1 ¼ 2g1 þp33;c2 ¼n33þ 2q3 q2C3  ð1þ B2Þh2  ð1þ B3Þp3q2mi ¼ biC1ða1; 0Þ þ biþ1C2ða2; 0Þ; ði ¼ 1; 2ÞH0 ¼X3i¼0CiDi; Di ¼ P3i–jj¼0ðai  ajÞ; ði ¼ 0; 1; 2; 3ÞH1 ¼X3i¼0Hi1CiDi; Hi1 ¼X3j¼0j–iaj; ði ¼ 0; 1; 2; 3ÞH2 ¼X3i¼0Hi2CiDi; H02 ¼ a1a2 þ a1a3 þ a2a3;H12 ¼ a0a2 þ a0a3 þ a2a3H22 ¼ a0a1 þ a0a3 þ a1a3; H32 ¼ a0a1 þ a0a2 þ a2a3H3 ¼X3i¼0Hi3CiDi; Hi3 ¼ P3j¼0i–jaj; ði ¼ 0; 1; 2; 3ÞReferences1. Park JH, Allen MG, Prausnitz MR. Biodegradable polymermicroneedles: fabrication, mechanics and transdermal drug deliv-ery. J Controlled Release 2005;104:51–66.2. Kalia YN, Guy RH. Modelling transdermal drug release. AdvDrug Delivery Rev 2001;48:159–72.3. Qaliaf BA, Das DB, Mori D, Cui Z. Modeling transdermaldelivery of high molecular weight drugs from microneedle systems.Philos Trans R Soc A 2007;365:2951–67.4. Missel PJ. Finite element modeling of diffusion and partitioning inbiological systems: the infinite composite medium problem. AnnBiomed Eng 2008;28:1307–17.5. Lee AJ, King JR, Rogers TG. A multiple pathway model for thediffusion of drug in skin. IMA J Math Appl Med Biol1996;13:127–50.6. Lee AJ, King JR, Hibberd S. Mathematical modelling of therelease of drug from porous, non-swelling transdermal drugdelivery devices. IMA J Math Appl Med Biol 1998;15:135–63.7. Khanday MA, Saxena VP. Mathematical study of diffusive fluidtransport and distribution in human dermal regions. Anal TheoryAppl 2010;26(4):350–8.8. Agarwal M, Adlakha N, Pardasani KR. Thermal disturbances indermal regions of human limbs involving metastasis of tumours.Int Math Forum 2010;5(39):1903–14.9. Kumari B, Adlakha N. One dimensional model to study the effectof physical exercise on temperature distribution in peripheralregions of human limbs. Appl Math Sci 2013(27);1335–51.10. Khanday MA. Numerical study of partial differential equations toestimate thermoregulation in human dermal regions for temper-ature dependent thermal conductivity. J Egypt Math Soc Elsevier2013;22(1):152–5.11. Khanday MA, Mir Aijaz, Aasma Rafiq. Mathematical analysis onthe treatment of cancerous tumours using therapy based on localhyperthermia. J Energy Heat Mass Trans 2013;35:295–305.12. Khanday MA, Aasma Rafiq, Mir Aijaz. Mathematical study oftransient heat distribution in human eye using laplace transform.Int J Mod Math Sci 2014;9(2):118–27.13. Khanday MA, Aasma Rafiq, Mir Aijaz. Variational finite elementapproach to estimate the heat distribution in human eye. Bull CalMath Soc 2014;106(2):93–104.14. Khanday MA, Mir Aijaz, Aasma Rafiq. Numerical estimation ofthe fluid distribution pattern in human dermal regions withheterogeneous metabolic fluid generation. J Mech Med Biol2014;15(1):1550001–12.15. Sharma A, Saxena VP. Finite element modeling of drug distribu-tion in transdermal drug delivery system. Ind J Bio-mech2011;26–30.16. Crank J. The mathematics of diffusion. 2nd ed. Oxford UniversityPress; 1975.17. Gupta V, Pagey SS, Saxena VP. Numerical analysis of drugdiffusion in human dermal region with linear shape function.ISOR J Math 2012;4(2):31–6.18. Myer GE. Analytic methods in conduction heat transfer. McGrawHill Company; 1971, pp. 320–428.

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Variational finite element method to study the absorption rate of drug at various compartments through transdermal drug delivery system

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