Stability Analysis and Analytical Solution of a Nonlinear Model for Controlled Drug Release: Travelling Wave Fronts

Transcription

1 Stability Analysis an Analytical Solution of a Nonlinear Moel for Controlle Drug Release: Travelling Wave Fronts Chontita Rattanaul, an Yongwimon Lenbury Abstract In this paper, the process of rug issolution an release from a planar matri is investigate base on two couple nonlinear partial ifferential equations propose by Göran Frenning in 3. In the moelling the process rug asorption has been isregare, assuming concentration-inepenent iffusion coefficients, using perfect sin conitions, an specializing to a planar geometry, The concentration profile of the mobile, or iffusing, the resulting moel is rather comple an has been investigate only numerically an only approimate solution have been possible. In this paper it is shown that an analytical solution can be obtaine eactly in the form of a travelling wave front. We escribe the metho for fining the analytical solutions using the travelling wave coorinate when the wave is assume to be moving at constant spee. The moel system of partial ifferential equations is transforme into two couple orinary ifferential equations, which is analyse interms of the stability of its steay state. Analytical solutions are erive in three possible cases, giving travelling wave solutions. We then iscuss a comparison between the eact solutions obtaine here an the analytical short-time approimation as well as the curves obtaine from the moifie Higuchi formula reporte by Frenning in 3. Keywors Delay ifferential equations; omega limit set; persistence; signal transuction; stability. O I. INTRODUCTION NE of the most important means of rug-elivery is the utilization of the matri systems. The objectives of controlle elivery systems are to reuce the frequency an/or to increase the effectiveness of the rug through rug localization at the target site, in orer to reuce the ose require to provie uniform rug elivery. Controlle rug elivery requires the nowlege of both physical an polymer sciences so that it is possible to prouce well characterize an reproucible osage forms, which Manuscript May 7, : Revise version receive May 7,. This wor was supporte in part by the Centre of Ecellence in Mathematics, CHE, Thailan, an Mahiol University. C. Rattanaul is with the Department of Mathematics, Faculty of Science, Mahiol University, Rama 6 Roa, Bango an the Centre of Ecellence in Mathematics, CHE, 38 Si Ayutthaya Roa, Bango, Thailan ( chontita.rat@mahiol.ac.th). Y. Lenbury is with the Department of Mathematics, Faculty of Science, Mahiol University, Rama 6 Roa, Bango an the Centre of Ecellence in Mathematics, CHE, 38 Si Ayutthaya Roa, Bango, Thailan (corresponing author, phone: ; fa: ; yongwimon.len@mahiol.ac.th). control rug entry into the boy accoring the specifications of the require rug elivery profile [l]. It is well ocumente that in this type of rug elivery, the rate of rug release is principally controlle by the elivery system itself, though it may be influence by surrouning conitions, such as ph, enzymes, ions, motility an physiological conitions []. Mathematical moeling an computation has also become an important tool in the esign of pharmaceutical proucts ealing with controlle-release mechanism. As eplaine in [3], when rug release from a matri is controlle by iffusion through the polymeric matri, its release inetics obey Fic's st an n laws [3]: J = DC () c t = DC () where J represents the iffusional flu of the rug; D the is iffusional coefficient; C is the concentration of the rug; an is the istance of iffusion. When rug release is ominate by surface erosion, Hopfenberg s equation has been foun to give goo preiction of rug elivery in spherical, cylinrical, an planar geometrical forms [4]. Accoring to [3], we may istinguish between two conceptually ifferent scenarios, epening on the relative magnitues of the rug concentration an solubility in the matri. If the rug concentration is sufficiently low so that all rugs can be issolve, an the issolution process procees rapily enough, we may easily etermine the release rate [3]. In this type of release, all rugs may be assume to be completely issolve in before any release has occurre, an the rug concentration in the matri can therefore be solve from the heat conuction equation [3]. In the secon, more general, type of rug release from a matri, where the rug concentrations are higher, or the solubility is low, two forms of rugs, namely the soli an issolve forms, coeist, in the matri an the process becomes istinctly more comple. For this more general situation when rug loaing is much higher than rug solubility ( C C ), the moel propose by Higuchi [5] has s shown to perform well for planar matri uner the perfect sin assumption, although it was originally formulate for rug release from ointment bases containing rugs in suspension. Issue 3, Volume 6, 35

2 The original Higuchi moel has been subject to a great eal of generalizations an improvements [6-]. Recently, a similar moel has been stuie by Frenning an Strømme [3], who investigate the problem of rug release from spherical pellet units into the finite volume of issolution meium, uner the assumption that some of the issolve rug coul become immobilize by absorption to the pellet constituents. In [4], this moel was reformulate by isregaring rug absorption, an assuming that the iffusion coefficient is concentrationinepenent. Using perfect sin conitions, an analytical short-time approimation to the solution was erive [4]. It is the purpose of this paper to show that an eact solution can be obtaine analytically in the form of a travelling wave front. We escribe the metho for fining the analytical solutions using the travelling wave coorinate in the situation that the wave is presumably moving at a constant spee. The analytical solutions are then iscusse in comparison to the analytical short-time approimation erive in [4]. II. REFERENCED MODEL In [4], a planar matri system whose normal is in the irection was investigate. The assumptions are that the lateral imensions of the system are much larger than its thicness L, so that the process of rug release coul be effectively consiere to be one-imensional. The bounary at = is assume to be impenetrable to the rug, while the matri is in contact with the liqui at = L. The perfect sin conition assumes that the matri is in contact with a well-mie issolution meium, the volume of which is sufficiently large so that it can be assure that the rug concentration is virtually zero at all times. In orer to simplify the analysis, it was assume by Frenning [4] that liqui absorption occurs at a much faster rate than rug issolution an subsequent release. Thus, the matri which contains finely isperse soli rug is fully wette in the initial state. Also, in the initial state, entire rug is present in the soli form. Letting ( t, ) be rug concentration in the issolve phase an s( t, ) the concentration of rug in the soli phase, it is then possible to escribe rug issolution an release by the following equations [4]. t = st (3) / s = s 3 ( εc ) (4) t s where t is the time, c s the saturation constant, ε the porosity, D the rug iffusivity, the issolution rate. The initial conitions are (, ) = s(, ) = which means that all rug is assume to be present in the soli form in the initial state [4]. Similarly, the bounary conitions are = = s( t, L ) = which follow from the assumption that the interface at = is impenetrable to the rug, while the rug concentration at = L is ept at zero as a consequence of the sin conition. III. MODEL ANALYSIS 3. Traveling wave coorinate We introuce the travelling wave coorinate z = t, where is the constant velocity at which the wave is assume to be moving. By substituting z in (3) an (4), we obtain a couple system of orinary ifferential equations with respect to z: v = D + s (5) / s = s 3 ( γ ) (6) where () enotes the erivative with respect to z, stans for, an γ stans for ε cs. Integrating (5) an combining with (6) lea us to a single secon-orer ifferential equation terms of as / 3 D + v + ( v + D ) ( γ ) = (7) / 3 Now, by letting X =, an Y = X, we can write (7) as X = Y (8) / 3 Y = ( vx + DY ) ( X γ ) Y (9) / 3 D D 3. Dynamical analysis Before we erive the analytical solution, a stability analysis may be carrie out on the moel system written in the form of equations (3) an (4). The system possess only one nontrivial equilibrium state in the feasible region, namely ( X, Y ) = ( γ,). The Jacobian matri of the system (8) (9) about its steay state ( γ,) is /3 /3 J = ( DY + X) ( X γ) ( X γ) + /3 /3 /3 /3 D 3 D( DY X) 3 ( DY X) D + + At ( X, Y ) = ( γ,) the Jacobian becomes J( γ,) = / 3 γ D D whose eigenvalues are / 3 γ λ, = ± ( ) + 4 D D D which are real an opposite in signs. Therefore, the steay state ( X, Y ) = ( γ,) is a sale point. Issue 3, Volume 6, 35

3 Y = Y becoming unboune as time passes. The corresponing plots of the concentration of issolve rug an its rate of change as functions of z for the same parameter values as in Fig. - are seen in Fig. 3 an 4 respectively X = X Fig. Phase portrait of the system (8)-(9) in the case <. Here D =.8, =, v =, γ =. z Fig. 3 Plot of, the issolve rug concentration, as a function of z for the same parameter values as in Fig. -. Y Y X Fig Computer simulation of the system (8)-(9), showing the solution trajectory in the ( X, Y ) -plane in the case that D =.8, v =, =, γ =, X () =, Y () =.. In what follows, we are able to erive analytical solutions, in terms of the travelling wave coorinate, in 3 of the cases. Fig. shows the solution trajectory in the phase plane starting from the initial point at ( X, Y ) = (,.) an z Fig. 4 Plot of as a function of z for the same parameter values as in Fig. -. Issue 3, Volume 6, 353

4 IV. ANALYTICAL SOLUTION In orer to erive an analytic solution for the moel equation (5), we first let 3 / C = + D () We will see a solution of the form m n = AC + BC () so that we have 3 / C C = + D () Also, Eq. () gives 3 / D m n = C ( AC + BC ) (3) Substituting () an (3) into (7), we obtain 3 / 3/ D m n C C + C γ C ( AC BC ) / = Rearranging the above equation yiels 3 / 5 / AD m C C γ + + C C + C / 3 5 / 3 5 / 3 BD 5 / 3 C n+ + = (4) On inspection, we observe that we may fin eact solutions in the following two possible cases. Case : m =, n = For these values of m an n, Eq. (4) becomes 3 / 5 / AD C C + γ C C + C / 3 5 / 3 5 / 3 BD / + 5 / 3 C = (5) By letting AD = γ (6) equation (5) reuces to 3 / 5 / BD / C C C + C = (7) 5 / 3 5 / 3 If we let BD α = >, β = < (8) 5 / 3 BD 3 we can write Eq. (7) as C = β + α C which can be easily solve, yieling C = t an( αβ z + αk ) (9) α where K is the constant of integration. Substituting (7) into (3), rearranging an using (6), one obtains = ( tan 3/ ( αβ z+ αk) + cot / ( αβ z+ αk) 3/ ) + γ () α So that we have satisfie., z = = the following equation must be 3/ / 3 / tan αk + cot αk = γα () We can then obtain the level of rug in soli form by integrating (5) so that Fig. 5 Travelling wave solution in Case for the concentration of rug in the ilute form, given by (3), plotte as a function of for ifferent time. 3 / s = C + l or, 3 / s = tan ( αβ z + αk) + l () 3 / α where l is the constant of integration. Thus, a travelling wave solution is given as = ( tan 3/ ( αβ( t) + αk) + cot / ( αβ( t) + αk) 3/ ) α + γ (3) 3/ s = tan ( αβ ( t) + αk) + l (4) 3/ α where α, β, A, an B are given by (6) an (8), an (). Issue 3, Volume 6, 354

5 s C = (7) Jz + K where K is the constant of integration. Upon substitution (7) into (3) an using (4) we are le to AD 3 / = ( Jz + K) + γ (8) So that we have, z = = we nee the following equation to be satisfie. AD γ 3/ = (9) K Using (5) then gives s = + l (3) 3 / ( Jz + K) where l is the constant of integration. Thus, a travelling wave solution to (3)-(4) is given as AD 3/ = ( J ( t) + K) + γ (3) s = + l (3) 3 / ( J ( t) + K) where α, β, A, an B are given by (4), (6), an (9) Fig. 6 Travelling wave solution in Case for the concentration of rug s in the soli form, given by (4), plotte as a function of for ifferent time t. The analytical travelling wave solution given by (3)-(4) is shown in Fig. 5 an 6 to move in the irection of ecreasing as time elapses. Here, π α =, K =, l =, =, =, an γ = Case : m =, n = In this case, equation (4) becomes 3 / 5 / C C + γ C C / 3 5 / 3 AD 5 / BD + C + C = (3) 5 / 3 5 / 3 If we let BD = γ (4) then (3) reuces to 3 / AD 5 / C C C = 5 / 3 5 / 3 (5) or C = JC, where ( AD ) J = (6) 5 / 3 3 Solving (6), we obtain Fig. 7 Wave front solution in Case for the concentration of rug in the issolve form, given by (3), plotte as a function of for ifferent time t. Issue 3, Volume 6, 355

6 s Fig. 8 Wave front solution in Case for the concentration of rug s in the soli form, given by (3), plotte as a function of for ifferent time t. The graphs of the wave front solution given in (3)-(3) are shown in Fig. (7)-(8) to move in the irection of ecreasing as time elapses. Here, D =., K =.4, v =, =, γ =.5, an l =. 3 Case 3: m =, n = n +. For convenience, we let 3a 3b A =, B = in (), an rop the prime on the eponent n. Thus, we are now looing for a solution in the form 3 3 / n / = ( ac + bc + ) (33) such that n C = AC + B C (34) Differentiating (33) an using (34), we obtain 3 3 / n / n = ac + b( n + ) C ( A C + BC ) (35) Substituting (33) an (35) in (7), choosing A = a, an B = b, an rearranging, we are le to 3D 3 b 3D 3a a ab + + ab( n + ) C + + C D D n+ / 3/ 3D n / γ + b ( n + ) C + C / 3 3aD 5/ 3bD n+ 3/ C C (36) 5/ 3 5/ 3 + = 5/ 3 Upon observation, we see that if we let n = then Eq. (36) can be written as 9 3 γ 3bD 9 3 3/ abd + b C + + C+ a D+ a C /3 5/3 4 v v 4 3aD 5/ + C = (37) 5/3 5/3 v v Thus, if we let γ 3bD γ + =, or b = / 3 5 / 3 v v 3D (38) 3aD =, or a = 5 / 3 5 / 3 v v 3D then (37) will be satisfie. With the above choice in (38), we are in the case where v =, an (34) becomes C = ac + bc / which can be easily solve to yiel 3 / 3 az + C( z) = b e / 3 a Substituting (39) into (), we obtain (39) z 3/ 3D + D = C = γ e D which can be irectly solve, using an integrating factor, with, z = = to yiel 3 z / D+ z / D = γ ze + γ e Therefore, we have erive an analytical solution in terms of the travelling wave coorinate as follows. 3 ( ) ( ) / t D = γ t e e γ (4) an, similarly to the previous cases, 3 ( t) / D+ s = γ De + l (4) l being a constant of integration. The graphs of the wave front solution given in (4)-(4) are shown in Fig. (9)-() with =, 3 + γ =, D = 5 e, =., an l = γ + De. Issue 3, Volume 6, 356

7 V. DISCUSSION To illustrate that the analytic solution can allow us to investigate the impact of ifferent values of the system s physical properties, such as the saturation constant, the porosity, the rug iffusivity, or the issolution rate, on the waveform structures, we show in Fig. the plot of the travelling wave front of concentration of rug in issolve form for a ifferent set of parametric values, namely, γ =, D =, =., an =. Fig. 9 Wave front solution in Case 3 for the concentration of rug in the issolve form, given by (4), plotte as a function of for ifferent time t. s Fig. Wave front solution in Case 3 for the concentration of rug in the issolve form, given by (4), plotte as a function of for ifferent time t: γ =, D =, =., =, an. z = = Fig. Wave front solution in Case 3 for the concentration of rug in the issolve form, given by (4), plotte as a function of for ifferent time t. Finally, to put our analytical solution in the contet of numerical an approimate solutions reporte in earlier literatures, we refer to the results shown in [5] an [4]. In [4], fractions of release rug calculate numerically using the moel consisting of (3) an (4) is shown for ifferent values of saturation constant c s. The curves are compare with the analytical short-time approimation as well as the curves obtaine from the moifie Higuchi formula. We observe that the Higuchi solution an the analytical short-time approimation are close to the eact solution only uring the eponential phase of rug release, but offer quite a poor estimate of the release rug near the saturation perio. Issue 3, Volume 6, 357

8 Our eact solution is therefore etremely valuable as it can be use to accurately investigate effects of ifferent values of physiological or physical factors in the controlle rug release system, such as ifferent saturation constants, iffusivity, or porosity. These parameters an their impact on rug elivery are of great importance in the esign of pharmaceutical proucts which are well characterize an in reproucible osage forms, capable of controlling rug entry into the boy accoring the specifications of the require rug elivery profile. VI. CONCLUSION Intensive continuing research, both eperimentally an theoretically, has been on going, especially in the last ecae or so, on the topic of controlle-release technology, ue to its crucial role in the formulation of pharmaceutical proucts [- 4, -5]. Here, we have ae to the current nowlege by showing that an eact solution can be erive for the moel of rug release formulate by Frenning [4] where the lateral imensions of the system are assume to be much larger than its thicness L. The bounary at = is assume to be impenetrable to the rug, while the matri is in contact with the liqui at = L. The matri is assume to be in contact with a well-mie issolution meium, the volume of which is large enough to ensure that the rug concentration is virtually zero at all times. The resulting moel is relatively comple, involving a nonlinear fractional power of the epenent variable. Previous wors have been able to obtain numerical solution an, only by restriction their attention to the initial phase of rug release, an analytical approimate solution can be foun. We have relie on the wor in [7-9], using the travelling wave coorinate, an appropriate choice of combination of powers of the solution an its erivative, we have shown that an eact solution for the concentrations of rug in the soli an issolve forms can be foun as functions of the travelling coorinate, while the travelling wave velocity is foun to be negative as epecte. The methoologies for obtaining the eact solution presente in this stuy can be applie to other problems [, ] which amit travelling wave solutions. In the case that the system is escribe by a moel consisting of couple partial ifferential equations, they can be reuce to a single secon orer ifferential equation which is more easily solve. In Fig., we can see that the rug concentration in the issolve form increases (as t tens to infinity) while the wave of the issolve rug concentration wave front is moving at a constant spee of.. It might be wonere whether it woul have been better to simply solve the moel equations for numerical solutions. To answer such a question, the moel system shoul be solve numerically so that the numerical simulation coul be compare with the analytical solution, which is the subject for our future wor. It can be argue at this juncture that the ability to erive the eact solution coul be quite valuable in the investigation of impacts of various conitions on rug elivery, since the ability to accurately preict the outcome of ifferent choices of material, composition, rug properties, an others, on the release mechanism an inetics is of crucial importance in terms of meical treatment an health care purposes. ACKNOWLEDGMENT This research is supporte by Mahiol University an the Centre of Ecellence in Mathematics, CHE, Thailan, an Mahiol University. REFERENCES [] J.E. Mocel, B.C. Lippol, Zero orer rug release from hyrocolloi matrices, Pharm. Res., Vol., 993, pp [] J. Heller, Bioegraable polymers in controlle rug elivery, Crit. Rev. Ther. Drug Carrier Syst., Vol., N, 984, pp [3] J. Cran, The Mathematics of Diffusion, Ofor University Press, Lonon, 975. [4] P. Colombo, R. Bettini, G. Massimo, P.L. Catellani, P. Santi, N.A. Peppas, Drug iffusion front movement is important in rug release control from swellable matri tablets, J. Pham. Sci. Vol.87, 995, pp [5] T. Higuchi, Rate of release of meicaments from ointment bases containing rugs in suspension, J. Pharmac. Sci., 5 (96), [6] T.J. Roseman, W. Higuchi, Release of meroyprogesterone acetate from a silicone polymer, J. Pharm. Sci., Vol. 59, 97, pp [7] K. Tojo, Intrinsic release rate from matri-type rug elivery systems, J. Pharm. Sci., Vol.74, 985, pp [8] D.R. Paul, S.K. McSpaen, Diffusional release of a solute from a polymeric matri, J. Membr. Sci. Vol., 976, pp [9] P.I. Lee, Diffusional release of a solute from a polymeric matriapproimate analytical solutions, J. Membr. Sci., Vol.7, 98, pp [] A.L. Bunge, Release rates from topical formulations containing rugs in suspension, J. Control. Release, Vol.5, 998, pp [] M.J. Abehoaie, Y.-L. Cheng, Diffusional release of a isperse solute from planar an spherical matrices into finite eternal volume, J. Control. Release, Vol.43, 997, pp [] Y. Zhou, X.Y. Wu, Theoretical analyses of isperse-rug release from planar matrices with bounary layer in a finite meium, J. Control. Release, Vol.84,, pp. -3. [3] G. Frenning, M. Strømme, Drug release moele by issolution, iffusion, an immobilization, Int. J. Pharm., Vol.5, 3, pp [4] G. Frenning, Theoretical investigation of rug release from planar matri systems: effect of a finite issolution rate, J. Control. Release, Vol.9, 3, pp [5] D. Sun, V.S. Manoranjan, H.-M. Yin, Numerical solution for a couple parabolic equations arising inuction heating processes, Discrete an Continuous Dynamical System, 7, pp [6] R.L. Buren, J.D. Faires, A.C. Reynols, Numerical Analysis n Eition, Prinle, Weber an Schmit. Inc, USA, 98. [7] M.O. Gomez, C.M. Chang, V.S. Manoranjan Eact solution for contaminant transport with nonlinear sorption, Appl. Math. Lett., Vol.9, 996, pp [8] V.S. Manoranjan, T.B. Stauffer, Eact solution for contaminant transport with inetic langmuir sorption, Water Resources Research, Vol.3, No.3, 996, pp [9] C.M. Chang, V.S. Manoranjan, Travelling wave solutions of a contaminant transport moel with nonlinear sorption, Mathl. Comput. Moelling., Vol.8, No.9, 998, pp. -. [] E. Kreyszing, Avance Engineering Mathematics 8th Eition, John Wiley an Sons. Inc, USA, 3. [] P. Prasertsang, V.S. Manoranjan, Y. Lenbury, Analytical travelling wave solutions of a ental plaque moel with nonlinear sorption, Nonlinear Stuies, Vol. 8, No.,, pp Issue 3, Volume 6, 358

9 [] S.-P. Tang, An eploratory research on the nanovectors for rug elivery an for gene therapy, WSEAS Transactions on Biology an Biomeicine, Volume 5, Issue, 8, pp [3] S.-P. Tang, An eperimental comparing analysis research on the nanovectors for rug elivery an for gene therapy, WSEAS Transactions on Biology an Biomeicine, Vol. 7, No.,, pp. -. [4] T. Schroer., A. Quintilla., J. Setzier., E. Birtaian., W. Wenzel., an S. Brase., Joint eperimental an theortical Investigation of the propensity of peptois as rug carriers. WSEAS Transuctions on Biology an Biomeucine, Vol. 4, Issue, 7, pp [5] T. Hara, S. Iriyama, M. Ohya, Estimation of rug shape in rug elivery system by simulation, Proceeings of the th WSEAS International Conference on Mathematics an Computers in Biology an Chemistry, 9. Issue 3, Volume 6, 359