Numerical modelling and analysis of drug release from viscoelastic polymers : Modelagem e análise numérica da dispersão de medicamentos em polímeros viscoelásticos

Texto

(1)UNIVERSIDADE ESTADUAL DE CAMPINAS Instituto de Matemática, Estatística e Computação Científica. JÚLIA SILVA SILVEIRA BORGES. Numerical Modelling and Analysis of Drug Release from Viscoelastic Polymers. Modelagem e Análise Numérica da Dispersão de Medicamentos em Polímeros Viscoelásticos. Campinas 2019.

(2) Júlia Silva Silveira Borges. Numerical Modelling and Analysis of Drug Release from Viscoelastic Polymers Modelagem e Análise Numérica da Dispersão de Medicamentos em Polímeros Viscoelásticos. Tese apresentada ao Instituto de Matemática, Estatística e Computação Científica da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Doutora em Matemática Aplicada. Thesis presented to the Institute of Mathematics, Statistics and Scientific Computing of the University of Campinas in partial fulfillment of the requirements for the degree of Doctor in Applied Mathematics.. Supervisor: Giuseppe Romanazzi. Este exemplar corresponde à versão final da Tese defendida pela aluna Júlia Silva Silveira Borges e orientada pelo Prof. Dr. Giuseppe Romanazzi.. Campinas 2019.

(3) Ficha catalográfica Universidade Estadual de Campinas Biblioteca do Instituto de Matemática, Estatística e Computação Científica Ana Regina Machado - CRB 8/5467. B644n. Borges, Júlia Silva Silveira, 1986BorNumerical modelling and analysis of drug release from viscoelastic polymers / Júlia Silva Silveira Borges. – Campinas, SP : [s.n.], 2019. BorOrientador: Giuseppe Romanazzi. BorTese (doutorado) – Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica. Bor1. Análise numérica. 2. Diferenças finitas. 3. Liberação controlada de fármacos. 4. Materiais viscoelásticos. 5. Equações integro-diferenciais. I. Romanazzi, Giuseppe, 1976-. II. Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. III. Título.. Informações para Biblioteca Digital Título em outro idioma: Modelagem e análise numérica da dispersão de medicamentos em polímeros viscoelásticos Palavras-chave em inglês: Numerical analysis Finite differences Controlled drug delivery Viscoelastic materials Integro-differential equations Área de concentração: Matemática Aplicada Titulação: Doutora em Matemática Aplicada Banca examinadora: Giuseppe Romanazzi [Orientador] Eduardo Cardoso de Abreu Maicon Ribeiro Correa Elías Alfredo Gudiño Rojas José Augusto Mendes Ferreira Data de defesa: 06-12-2019 Programa de Pós-Graduação: Matemática Aplicada Identificação e informações acadêmicas do(a) aluno(a) - ORCID do autor: https://orcid.org/0000-0001-7077-9783 - Currículo Lattes do autor: http://lattes.cnpq.br/2114532510960735. Powered by TCPDF (www.tcpdf.org).

(4) Tese de Doutorado defendida em 06 de dezembro de 2019 e aprovada pela banca examinadora composta pelos Profs. Drs.. Prof(a). Dr(a). GIUSEPPE ROMANAZZI. Prof(a). Dr(a). EDUARDO CARDOSO DE ABREU. Prof(a). Dr(a). MAICON RIBEIRO CORREA. Prof(a). Dr(a). ELÍAS ALFREDO GUDIÑO ROJAS. Prof(a). Dr(a). JOSÉ AUGUSTO MENDES FERREIRA. A Ata da Defesa, assinada pelos membros da Comissão Examinadora, consta no SIGA/Sistema de Fluxo de Dissertação/Tese e na Secretaria de Pós-Graduação do Instituto de Matemática, Estatística e Computação Científica..

(5) Dedicated to my parents Sílvia and Edison..

(6) Acknowledgements First, I thank God for giving me the opportunity to develop this work and for becoming it possible. I wish to express my deepest gratitude to my parents Sílvia and Edison, whom I dedicated this work, and to my sister Lívia. I thank for being the basis of my life and all my achievements. I also thank for supporting me to move forward, for trusting me and making me believe in my potential and the importance of this struggle. I also wish to express my gratitude to my boyfriend Diego for his affection, dedication and patience in difficult times. I thank for the advices and for believing in the conclusion of this work. I wish to thank the friends I met at the doctorate for the pleasant journey we did together. I thank Geisel for the help and care in all moments. I also thank Geovan for the help and support. I thank with gratitude to Thiane for helping with precious mathematical teachings and for the support and encouragement in the difficult times. I sincerely thank Rodrigo, Mária and Moyses for the encouragement. I am grateful for valuable support and help mainly in the early stages of this work. I wish sincerely to thank my advisor, Professor Giuseppe, and to demonstrate my appreciation for his excellent advice, dedication, patience and attention to my study, which proved to be monumental for the success of this work. I am also grateful for the advices for my improvement as a teacher and researcher. I wish to thank Professor Eduardo and Professor Maicon from Unicamp, Professor Elias from UFPR and Professor José Augusto from University of Coimbra for the encouragement, advices and suggestions that was essential for the improvement of this thesis. I would like to recognize the invaluable assistance and scientific contributions of Professor Giuseppe, Professor José Augusto and Professor Eduardo resulting in the submission of an article. Finally, I thank to UFSCar and to CCN - Lagoa do Sino campus for authorizing and supporting my temporary absence that was necessary for the development of this work..

(7) “Our duty is to be useful, not according to our desires but according to our powers.” Henri Frédéric Amiel.

(8) Resumo Uma modelagem e um método numérico para liberação de drogas de polímeros viscoelásticos são apresentados neste trabalho. Consideramos um modelo conhecido que simula a entrada de solvente, a expansão do polímero, a dissolução e difusão do medicamento. Esse problema é modelado por um sistema integro-diferencial e o método usado para aproximar as soluções é um método Implícito-Explícito a Diferenças Finitas. Simulações numéricas ilustram os resultados obtidos. Uma simplificação deste modelo é considerada para desenvolver um método numérico no qual a estabilidade e a convergência possam ser analisadas. Um estudo teórico para estabilidade e ordem de convergência de erros é apresentado e simulações numéricas que confirmam a ordem dos erros estão incluídas. Outro modelo para a dissolução da droga sólida é considerado e a influência deste no comportamento das concentrações é analisada. Palavras-chave: Análise numérica, Diferenças finitas, Liberação controlada de fármacos, Materiais viscoelásticos, Equações integro-diferenciais..

(9) Abstract A modelling and a numerical method for the drug delivery from viscoelastic polymers is presented in this work. We consider a known model that simulates the solvent entrance, the swelling polymer, the drug dissolution and diffusion. This problem is modelled by an integro-differential system and the method used to approximate the solutions is an Implicit-Explicit Finite Differences Method. Numerical simulations illustrate the obtained results. A simplification of this model is considered to develop a numerical method where stability and convergence can be analysed. A theoretical study for stability and error convergence order are presented and numerical simulations that confirm the error order are included. Another model for solid drug dissolution is considered and the influence of this in the concentrations behaviour is analysed. Keywords: Numerical analysis, Finite differences, Drug delivery devices, Viscoelastic materials, Integrodifferential equations..

(10) Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Mathematical Model of Drug Release . . . . . . . . . . . . . . . . . . . 1.1 Description of physical problem . . . . . . . . . . . . . . . . . . . . . . 1.2 Solvent Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Solid Drug Concentration . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Dissolved Drug Concentration . . . . . . . . . . . . . . . . . . . . . . . 1.6 Drug Release differential model in 3-D, boundary and initial conditions 1.7 Drug Release Differential Model in Spherical Coordinates . . . . . . . . 1.8 Swelling Front and Radius Growth . . . . . . . . . . . . . . . . . . . . 1.9 Numerical Method and Algorithm . . . . . . . . . . . . . . . . . . . . . 2 Model and Method Used . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Numerical Discretization and Discrete Operators . . . . . . . . . . . . . 2.2 Numerical Results of Convergence Order . . . . . . . . . . . . . . . . . 3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Constant Coefficients and Uniform Grid Case . . . . . . . . . . . . . . 3.2 Non Constant Coefficients and Non Uniform Grid . . . . . . . . . . . . 3.2.1 Solvent Concentration . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Dissolved and Solid Drug Concentration . . . . . . . . . . . . . 3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Solid Drug Dissolution: a new model . . . . . . . . . . . . . . . . . . . . 5 Final Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Scientific Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Index. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. 12 14 14 15 17 19 21 22 23 24 25 29 30 34 38 40 42 43 47 52 59 65 65 65 66. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68. BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX A Spherical Coordinates . . . . . . . . . . . . . . . . APPENDIX B Spaces and Norms . . . . . . . . . . . . . . . . . . APPENDIX C Preliminary Results for Stability and Convergence APPENDIX D Estimatives for Local Trucation Error . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 69 72 76 77 82.

(11) D.1 Solvent Concentration . . . . . . . . . . . . D.2 Dissolved Drug Concentration . . . . . . . . APPENDIX E Proof of Convergence Theorems E.1 Solvent Concentration Convergence . . . . . E.2 Drug Concentration Convergence . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 82 83 85 85 88.

(12) 12. Introduction Polymeric platforms are commonly used to carry and deliver drugs based on their optimum drug release properties. They represent one of the main devices used for active and passive target drug delivery (GRASSI et al., 2006). Polymer matrices can control the amounts of drug released in the desired target site maintaining the drug agent at the required level for a prolonged time (SILVA; FERNANDES; BAPTISTA, 2014). Mathematical modelling of drug release from delivery platforms is useful to describe the physical and chemical mechanisms that control the drug delivery. It allows also to simulate the behaviour of prototype drug devices without any invasive human or animal tests, and then it can speed up the medical product development (SIEPMANN; SIEPMANN, 2012). There are many mechanisms that can be considered as fundamental for drug delivery such as dissolution, drug diffusion, matrix swelling, erosion, recrystallization, initial drug distribution on matrix, geometry, etc (GRASSI; GRASSI, 2014). In this work we focus on mathematical modelling and numerical simulation of drug dissolution, diffusion and swelling of polymeric matrices carrying drugs. We consider a drug that is disposed in a polymeric platform that enters in contact with a solvent fluid. As the solvent molecules are absorbed, the polymer deformes leading to an expansion of its volume and to the appearance of a stress gradient. Such phenomena regulate the process of dissolution of the drug stored in the polymeric matrices. Once the drug molecules are dissolved, they diffuse through the platform and surround the outer environment. This problem is mathematically modelled by an integro-differential system with temporal and spatial derivatives. The dissolution of the solid drug is described by an ordinary differential equation with a sink term that becomes a source term in the coupled differential equations that describe the temporal and spatial variation of the dissolved drug. The behaviour of the dissolved drug inside and outside the polymer is modelled by diffusion equations. The absorption of solvent is also described by a diffusion equation that has Fickian and non-Fickian flux terms. The non-Fickian term is due to the stress of the polymer that is opposite to the solvent entrance. The Fickian term is due to the solvent diffusion inside the polymer. The method used to approximate the solution is based on an Implicit-Explicit Finite Differences Method (IMEX). In addition, a complete Finite Element analysis is performed to study the stability, convergence and error estimates of the numerical method..

(13) Introduction. 13. Overview The drug release problem is detailed in Chapter 1 where we describe the differential equations that model the concentration dynamics inside and outside the polymeric domain. We discuss the initial and the boundary conditions that are based on the symmetry of the domain with respect the origin. Also we model the polymer moving boundary and describe the method used to approximate the solution. In Chapter 2, we consider a simplification of this model in order to develop a numerical method whose stability and convergence can be analysed. Here we also describe the discrete operators and the complete numerical method used. Finally in this chapter we show numerically that the method is supraconvergent in the sense of finite differences. The numerical analysis of the drug release problem is done in Chapter 3. Therein we consider a class of problems that includes the drug release model in a fixed domain and we analyse the stability of the numerical method and some error estimates for the solution are established as well as the convergence order. In this chapter we also prove the supraconvergence of our method. Furthemore, we illustrate the theoretical results established considering some problems with known solution. In Chapter 4 we study a different solid drug dissolution model whose sink term is also directly proportional to the solid drug concentration differently to that used in Chapter 2. A comparision of the concentration behaviour with respect the original dissolution model is also discussed in Chapter 4. Finally, in Chapter 5, we present and discuss some conclusions about the contributions of this thesis and give some perspectives of future works in this area of drug release modelling and numerical analysis..

(14) 14. 1 Mathematical Model of Drug Release This chapter describes the physical phenomena of the drug release process in polymeric platforms, see Section 1.1. It also presents the mathematical model that describes the behaviour of the solvent and drug concentrations, see Sections 1.2 - 1.8. The time variation equations for the concentrations, the initial and boundary conditions are those used in (FERREIRA et al., 2018). In Section 1.9 we present an optimized method in the sense of computational time execution with operators of second order in space to solve such complex problem.. 1.1 Description of physical problem We consider a spherical polymer of radius R containing a drug that is immersed ¯ The phenomena regulating the dissolution in a spherical environment Ωe , of fixed radius, R. of the drug and its diffusion in Ωe follows these steps: 1. solvent molecules are absorbed as a consequence of its concentration variation (solvent absorption); 2. the polymer swells and a pressure gradient arises (swelling); 3. a dissolution process occurs due to the contact of the solid drug with the absorbed solvent molecules (dissolution); 4. the molecules of the dissolved drug diffuse through the platform and continue to diffuse in the external surrounding (diffusion). We denote by Ωptq, for t P r0, T s, the spherical polymeric domain at instant t with radius rptq. In this domain we consider the following concentrations: solvent c` px, tq, dissolved drug cd px, tq and solid (undissolved) drug cs px, tq. In the external medium, outside the swelling platform, Ωc,e ptq model the dissolved drug concentration, cde px, tq.. ΩezΩptq we.

(15) Chapter 1. Mathematical Model of Drug Release. 15. Figure 1 – Polymer containing solid drug (red sphere) immersed in a liquid solvent (outer sphere). The possible swelling is drawn by dashed line.. Figure 1 illustrates the domains introduced above. The small solid sphere represents the polymer containing the solid drug at the initial time. This is the initial spherical domain Ωp0q whose radius is R0 . Such sphere is immersed in the solvent liquid ¯ The dashed line represents the possible represented by the outer sphere, Ωe of radius R. swelling of the polymer and refers to the variable domain Ωptq with radius rptq.. The concentration difference inside and outside the polymer leads to the absorption of the solvent molecules. The phenomena involved in this process are described in the sections below.. 1.2 Solvent Concentration To describe the time variation of solvent concentration, we suppose that the solvent mass flux, J` px, tq has a Fickian contribution J`, F px, tq and a non Fickian contribution J`, nF px, tq. We have J` px, tq J`, F px, tq where. and. J`, nF px, tq, with x P Ωptq, t ¡ 0,. (1.1). J`, F px, tq D` ∇c` px, tq. (1.2). J`, nF px, tq Dv ∇σ px, tq.. (1.3). In this way, the time variation of solvent concentration is given by:. Bc` ∇ pD ∇c q ` ` Bt. ∇ pDv ∇σ q.. (1.4). The Fickian contribution is related to the solvent concentration variation and to its diffusion inside the polymer, while the non-Fickian is associated to the internal stress.

(16) Chapter 1. Mathematical Model of Drug Release. 16. gradient of the polymeric chains ∇σ that is in opposition to the fluid entrance. The fluid entrance leads to the polymer swelling that induces a stress relaxation. In the previous equations, D` and Dv denote the diffusion and viscoelastic diffusion coefficients, respectively. The diffusion solvent coefficient D` D` pc` q is of Fujita type, (FUJITA, 1961), and depends on the solvent concentration c` , on the exterior solvent concentration cext and on the diffusion coefficient of the liquid solvent Deq` . We have, . D`. Deq` exp β`. . 1. c` cext. ,. (1.5). where β` is a dimensionless constant, (FERREIRA et al., 2015), (FERREIRA et al., 2018), (VOLPERT; NEPOMNYASHCHY; KANEVSKY, 2018). The viscoelastic diffusion coefficient Dv Dv pc` q has the expression 2. Dv. R8ˆµ c`,. (1.6). where R is the radius of the domain Ωp0q, and µ ˆ is a viscoelasticity coefficient (FERREIRA et al., 2015). Figure 2 illustrates the evolution of solvent liquid concentration in the variable domain for different time values. The method used to obtain these numerical results is described with details in Section 1.9.. Figure 2 – Solvent Concentration c` in the polymer at distance r P r0, rpT qs from its centre, with initial radius R0 103 m and cext 755.74 kg {m3 . Concentration at different times T (in seconds) are depicted.. It is possible to notice that for initial times the solvent concentration is zero in most of the interval around the origin that is the centre of the polymer. In addition, the concentration has a considerable variation over a small spatial interval next to the moving.

(17) Chapter 1. Mathematical Model of Drug Release. 17. boundary, giving rise to a stiff problem at initial times that will be further discussed and analysed in Section 2.1. Going forward in time the spatial derivatives become smooth and the solvent concentration at the origin assumes values different from zero. It is also possible to verify that at the moving boundary, the concentration of the solvent increases with time, approaching the value of the exterior concentration, cext .. 1.3 Stress and Strain To define the stress gradient we start by describing (modelling) the strain, ε. Let V0 be the volume of the polymeric sphere with radius r0 that after a swelling by the absorption of solvent becomes a sphere of volume Vn with radius rn . Consider V` the volume of the absorbed solvent and Vd the volume of the dissolved drug that diffuses into the external domain. Thus, we have Vn V0 V` Vd . Since Vd V0 V` we can consider the approximation below:. V0. Vn. r. . . V` .. The Cauchy strain tensor ε defined by ε 3 Vn 4π. 1 3. . . 3 pV0 4π. 1. V` q. r r0 , where r0 r0. . . 3 V0 4π. 1 3. and. 3. . Hence, we have ε. . 1. V` V0. 1 3. 1.. (1.7). Consider m` the mass of the absorbed solvent. Hence, m` V` ρ` , where ρ` represents the density of solvent. On the other hand, the solvent concentration can be m` V` c` written as c` and consequently we have . V0 V` V0 ρ ` c` Using the last relation and (1.7) we have the strain εpc` q definition: εpc` q . . ρ`. ρ ` c`. 1 3. 1.. (1.8). Exists different models that describe the relation stress-strain in a viscoelastic medium. We chose the Generalized Maxwell-Wiechert model that can describe well the swelling of polymers carried with drugs and have good fitting properties for its parameters..

(18) Chapter 1. Mathematical Model of Drug Release. 18. The viscoelasticity modulus E ptq described by a generalized Maxwell-Wiechert Model is . m ¸. t Ej exp τj j 1. E ptq E0. (1.9). ,. 1, , m, are the Young modules of the polymer, τj Eµj are the j relaxation times and µj are the viscosities associated to the j th spring-dashpot component. where Ej , for j. of the Maxwell-Wiechert model represented in Figure 3, (SHAW; WHITEMAN, 2000), (FIGUEIREDO et al., 2019).. E1. E2. E3. Em. E0 µ1. µ2. µ3. µm. Figure 3 – Maxwell-Wiechert model.. The stress σ depends on the solvent concentration and arises from the solvent permeation with concentration c` in the polymer matrix. The stress-strain relation in the Maxwell-Wiechert model can be written in the integral form given by the Boltzman integral σ ptq . »t 0. E pt sq. Bεpc`q ds, Bs. where εpc` q is the Cauchy strain tensor defined in (1.8).. (1.10). In the Figure 4 we observe the behaviour of the stress σ in r0, rptqs and for different values of time. Note that r is the distance of the considered point in the sphere with respect the centre..

(19) Chapter 1. Mathematical Model of Drug Release. 19. Figure 4 – Polymer stress σ at distance r P r0, rpT qs from its centre, with initial radius R0 103 m and cext 755.74 kg {m3 . Polymer stress at different times T (in seconds) are depicted.. We note that at beginning of the solvent absorption, a large value of stress gradient is presented close to the polymer boundary. This gradient is in opposition with Bσ is negative and Bc` is positive, see the evolution equation respect the solvent entrance: Br Br of c` in (1.20) and figures 2 and 4. A stress relaxation is observed in Figure 4. Using the definition (1.10) of the stress σ ptq and by the integration by parts we have. ˆ pc` ptqq σ ptq Eε m ¸. E ptqεpc` p0qq . m ¸ Ej. »t 0. kerpt sqεpc` psqqds,. Ej and kerpsq τ exp τs . Now by using the hypothesis that j j 0 j 1 j c` px, 0q 0 at initial time, we can rewrite the solvent concentration equation (1.4) as follows. where Eˆ. B c` Bt. ∇. . . ˆ v pc` qε1 pc` q ∇c` D` pc` q ED . ∇ Dv pc` q. »t 0. kerpt sqε1 pc` psqq∇c` psqds .. (1.11). 1.4 Solid Drug Concentration It is known that when the solid drug, cs , present in the polymeric matrix comes into contact with a solvent fluid, the dissolution process initiates. In this section we describe the mathematical model of such dissolution process. It is important to emphasize that drug dissolution is different from drug release. According (SIEPMANN; SIEPMANN, 2013) the drug release process is a more complex phenomenon that includes the drug dissolution as.

(20) Chapter 1. Mathematical Model of Drug Release. 20. one of its stages. In this article, the authors also deal with the physical phenomena involved in the dissolution process. The dissolution rate defined as the solid concentration variation over time generally depends on the difference between drug solubility and dissolved drug concentration over time. This model was initially presented in (NOYES; WHITNEY, 1897) and the Noyes-Whitney equation. Bcs K pc c q, sol d Bt where csol is the solubility and K is a dissolution constant, has been used in several works, (MCGINTY; PONTRELLI, 2015). In this work we use, as done in (ROTHSTEIN; FEDERSPIEL; LITTLE, 2009), a more complete Noyes-Whitney equation that consider a time variation of solid drug concentration directly proportional to the solvent concentration c` : f pcs , cd , c` q H pcs q kd. csol cd c` , csol. (1.12). where H pcs q is the Heaviside function based on the cs values, kd is the drug dissolution rate (FERREIRA et al., 2018). Also, we suppose that csol is constant in time (CONT et al., 2014). The Figure 5 illustrates the behavior of the solid (undissolved) drug.. Figure 5 – Solid Drug Concentration cs in the polymer at distance r P r0, rpT qs from its centre, with initial radius R0 103 m and initial concentration c0 1 kg {m3 at different times T (in seconds).. We observe that as the solid drug dissolves, there is a decrease of such concentration throughout the domain starting from the boundary, reaching values close to zero for longer times in all the domain..

(21) Chapter 1. Mathematical Model of Drug Release. 21. Other Noyes-Whitney equations can be used to model the solid drug dissolution. For instance, we can suppose that the rate of time variation of solid drug concentration is also proportional to some power of solid drug, cβs . This modelling is discussed in Chapter 4.. 1.5 Dissolved Drug Concentration To model the time-space variation of dissolved drug concentration, we consider that drug diffuses and solid drug dissolves proportionally to the term f defined in (1.12). Hence,. Bcd ∇ pD ∇c q d d Bt. f pcs , cd , c` q.. (1.13). The diffusion coefficient in the above equation is of Fujita-type and depends on the external solvent concentration, cext , on the current solvent concentration and on the diffusion coefficient of the dissolved drug, Dde : . Dd. Dde exp βd. . 1. c` cext. ,. (1.14). where βd is a dimensionless constant. In the external domain, Ωc,e ptq we model the exterior concentration of dissolved drug, cde . Here we do not have dissolution, then only the diffusion phenomenon is modelled. Since the solvent concentration in Ωc,e is constant, equal to cext , the diffusion coefficient is constant and equal to Dde , see (1.14). Thus, we have:. Bcde ∇ pD ∇c q, de de Bt. in Ωc,e ptq, t P p0, T s.. In Figure 6 we illustrate the behavior of the dissolved drug inside, r and outside, r ¡ rpT q, the swelling polymer. (1.15). rpT q,.

(22) Chapter 1. Mathematical Model of Drug Release. 22. Figure 6 – Dissolved Drug Concentration inside and outside the polymer at different times T (in seconds).. It is possible to notice that for initial times, the dissolved drug concentration is high near the moving boundary and has values close to zero in the outer domain and near the centre of the sphere. This is because dissolution takes place at beginning only close to the polymer boundary in contact with the solvent. As t increases, we notice a diffusion of the dissolved drug inside the polymer. Thus, the dissolved drug concentration increases close the origin and near the moving boundary.. 1.6 Drug Release differential model in 3-D, boundary and initial conditions We can assert finally that the evolution of the solvent and drug concentrations c` , cd , cs and cde for t P p0, T s are described by the following system of partial differential equations:. B ∇ pD ∇c q ∇ pD ∇σq ` ` v B Bcd ∇ pD ∇c q f pc , c , c q d d s d ` ' Bt ' ' ' ' ' % B cs f pc , c , c q s d ` Bt $ c` ' ' ' ' t ' ' &. Bcde ∇ pD ∇c q, de de Bt. in Ωptq,. in Ωc,e ptq.. (1.16). (1.17). Together with the above equations we consider the following boundary conditions in the moving front B Ωptq:.

(23) Chapter 1. Mathematical Model of Drug Release. J` pc` q η. Jd pcd q η cd cs. 23. αpc` cextq, Jdepcdeq η, cde, 0,. (1.18). where α is the polymer permeability, η is the exterior unitary normal to Ωptq and the fluxes are given by . ˆ 1 pc` q ∇c` Dv D` Dv Eε. J` pc` q. Jd pcd q Dd ∇cd , Jde pcde q Dde ∇cde .. »t 0. ker pt sqε1 pc` psqq∇c` psq ds,. We also consider that Ωe is isolated. In B Ωc,e zB Ωptq we have then Jde pcde q η. 0,. (1.19). where η is the exterior unitary normal to Ωe . Moreover we consider the initial conditions:. c` px, 0q 0, cs px, 0q c0 , cd px, 0q 0,. x P Ωp0q,. cde px, 0q 0, c` px, 0q cext , x P Ωe zΩp0q.. 1.7 Drug Release Differential Model in Spherical Coordinates In this section we present the model in spherical coordinates that describes the drug release inside Ωptq and outside the polymer Ωe . We suppose that the drug release problem (1.16) - (1.17) is symmetric with respect the centre of the polymeric sphere. Thus we can rewrite the differential system (1.16) - (1.17) in spherical coordinates, see Appendix A. Then the problem becomes one dimensional in space, depending only on the distance r with respect the centre of the sphere. Inside the sphere Ωptq we have $ c` ' ' ' ' ' t ' ' ' &. B 1 B r2 D pc pr, tqq Bc` ` ` B r2 B r Br Bcd 1 B r2D pc pr, tqq Bcd. d ` ' B t r2 B r Br ' ' ' ' ' ' Bc ' % s f pcs , cd , c` q, Bt. B σ Dv pc` pr, tqq pr, tq , Br. f pcs , cd , c` q,. (1.20).

(24) Chapter 1. Mathematical Model of Drug Release. for all r have. 24. P p0, rptqq with t P p0, T s. Furthermore, for all r P prptq, R¯ q and t P p0, T s, we Bcde 1 B r2D Bcde . de B t r2 B r Br. (1.21). At the moving interface rptq from equation (1.18) we obtain J` prptq, tq η. αpc`prptq, tq cextq, Jd prptq, tq η Jde prptq, tq η, cd prptq, tq cde prptq, tq, cs prptq, tq 0,. (1.22). with J` prptq, tq η. Jd prptq, tq η Jde prptq, tq η. pD` Dv Eˆ ε1pc`prptq, tqqq BBcr` prptq, tq »t Dv kerpt sqε1pc`prpsq, sqq BBcr` prpsq, sq ds, 0 B cd Dd Br prptq, tq, Dde BBcrde prptq, tq.. (1.23). We suppose that the system (1.20) - (1.21) is isolated that means no absorption of drug is ¯ we have for t ¡ 0 that Jde pR, ¯ tq 0. considered outside the sphere Ωe . Hence, at r R For the symmetry with respect the centre of the polymeric sphere, we have for t ¡ 0. Bc` p0, tq Bcd p0, tq 0 Br Br. in the center of polymer r. 0. The initial conditions used are. c` pr, 0q 0, cd pr, 0q 0, cs pr, 0q c0 , r cde pr, 0q 0, c` pr, 0q cext ,. P p0, R0q, ¯ q, r P pR0 , R. where c0 is the initial constant concentration of the solid drug.. 1.8 Swelling Front and Radius Growth To describe the moving front, B Ωptq, we need to model rptq, the radius of Ωptq at instant t. To define the radius growth we use instead the mass conservation. In (FERREIRA et al., 2018), the authors prove that there is mass conservation only if r1 ptq α, where α is the permeability. Furthermore, the radius growth can be written as 1 D` pc` q BBcr` r1 ptq ρ`. Dv pc` q BBσr. 1. . c` ρ`. . cd ρd. 1 D ρd d. pc`q BBcr. d. ,. (1.24).

(25) Chapter 1. Mathematical Model of Drug Release. 25. where ρ` and ρd represent the solvent and drug densities, respectively. The Figure 7 illustrates the behavior of the radius growth rptq of the polymer in time. This variation of the moving boundary is obtained by using the algorithm described in Section 1.9.. Figure 7 – Radius Growth.. We can observe that the swelling is faster at the initial times and r1 ptq becomes almost constant for larger times.. 1.9 Numerical Method and Algorithm The results presented in the previous sections were obtained by using a modified algorithm with respect that used in (FERREIRA et al., 2018). The code is developed in the software MATLAB. It is an implicit-explicit method discretized in time and space using discrete operators of second order in space and first order in time. Among the modifications done, we changed the discrete operators used in the boundary conditions (1.23) and in the radius growth equation (1.24) by using only second order operators. Furthermore, we developed an algorithm that solves a system for the dissolved and undissolved drug inside and outsite the polymer, see the algorithm below. Finally we accomplished an improvement of the computational time by vectorizing the code. The numerical method for solving (1.20)-(1.21) is 1. Solvent Concentration Dt cn`, i. . 1 1 Mh pri2 ri2 hi 1 2. where D`,ni 1. 1. q. D`,ni 1 1 Dx cn`, i. Mhpri2q. 1. n1 n1 Dv, i 1 Dx σi 1. D`,ni 1 Dx cn`, i. D`pMhpcn`,i 1qq and Dv,ni 1 Dv pMhpc`,ni 1qq;. . n1 n1 Dv, i Dx σi. . ,.

(26) Chapter 1. Mathematical Model of Drug Release. 26. 2. Dissolved Drug Concentration Dt cnd, i. . 1 1 Mh pri2 2 1 ri hi. fp. 1. qDd,ni 1 1 Dxcnd, i 1 Mhpri2qDd,ni 1 Dxcnd, i. . 2. cn , cn q, d, i `, i. cns, i 1 ,. n1 where Dd, i. DdpMhpcn`,i 1qq;. 3. Undissolved Drug Concentration. f pcns, i, cnd, i, cn`, iq;. Dt cns, i. 4. Dissolved Drug Concentration on Exterior Dt cnde, i. . 1 1 D Mh pri2 de ri2 hi 1. n n 2 1 q Dx cde, i 1 Mh pri q Dx cd, i. . .. 2. hi hi 1 In this numerical method, we consider hi ri ri1 , hi 1 and Mh vh pri q 2 2 1 pvhpriq vhpri1qq. Furthermore, Dt denotes the backward Euler operator in time and 2 vh pri q vh pri1 q Dx vh pri q . hi On the moving boundary we use the following discrete operators to approximate the derivatives present in the boundary conditions (1.22). ˜ vh pri q 2hi D hi phi. hi1 hi hi1 vh pri1 q vh pri q hi1 q hi hi1. hi. hi1 phi. hi1 q. vh pri2 q. and D vh pri q . 2hi 1 hi 2 v pr q h i hi 1 phi 1 hi 2 q. hi 1 hi 2 vh pri hi 1 hi 2. 1. q h phhi 1 i 2 i 1. hi. 2. q vhpri 2q.. On the sphere centre (origin of our domain) we use the centred operator Dc vh pri q . vh pri. 1. hi. q vhpri1q , hi. 1. where we consider the fictitious point r1 : r1 with h0. h1.. Bc` , Bcd and Bσ . We note that, by using (1.24), r1 ptq depends on c` , cd , Br Br Br Moreover cd , and cs depend on c` , see (1.14) and the dissolution rate given in (1.12). Since c` depends on the boundary conditions (1.18) and α r1 ptq, then c` depends on r1 ptq. To solve these coupled relations we use the following algorithm: 1. For n 1,. , N. a) Calculate the concentrations using the radius of the previous iteration, rptn1 q:.

(27) Chapter 1. Mathematical Model of Drug Release. 27. i. Calculate c˜` prptn1 q, tn q and σ ˜ prptn1 q, tn q;. ii. Calculate c˜d prptn1 q, tn q, c˜de prptn1 q, tn q and c˜s prptn1 q, tn q using c˜` prptn1 q, tn q; b) Calculate the moving front and the permeability at the current time using c˜` prptn1 q, tn q, c˜d prptn1 q, tn q, and σ ˜ prptn1 q, tn q : i. α r1 ptn q using (1.24);. ii. rptn q using rptn q rptn1 q. α∆t;. c) Update the domain and compute the concentrations using the current domain radius rptn q; i. Calculate c` prptn q, tn q and σ prptn q, tn q;. ii. Calculate cd prptn q, tn q, cde prptn q, tn q and cs prptn q, tn q using c` prptn q, tn q; n n 1; tn tn1 ∆t; Observe that to calculate cn`, i in first equation of the numerical method, it is necessary to calculate previously σin1 . Considering the definitions of σ and viscoelasticity modulus E(t), (1.10) and (1.9) respectively, and using the definition of strain, (1.8), we obtain the equation below: σ prptq, tq εpc` prptq, tqqEˆ. »t 0. E 1 pt sq εpc` prpsq, sqq ds.. So we use cn`, i to calculate σin as follows. σ pri ptn q, tn q. εpc`priptnq, tnqqEˆ » tn. tn1. » tn1 0. E 1 ptn sq εpc` pri psq, sqq ds. E 1 ptn sq εpc` pri psq, sqq ds,. with the integrals that are computed by a composite Trapezoidal Formula. The solutions presented in the figures 2, 4, 5, 6 and 7 were obtained using the algorithm described above. The problem parameters and constants are described in the Table 1 and are the same as those used in (FERREIRA et al., 2018)..

(28) Chapter 1. Mathematical Model of Drug Release. Constant mesh size in radius Constant stepsize in time. 28 105. h pmq. 104. ∆t psq. ρ` pkg {m3 q Densities. Diffusion coefficients of the dissolved drug Diffusion coefficients of the liquid solvent Dimensionless constants on diffusion coefficients Dissolution reaction Exterior radius Exterior solvent concentration Initial solid drug concentration Initial radius. Maxwell-Wiechert model. ρd pkg {m. q ρp pkg {m q Dde pm2 {sq Deq` pm2 {sq. 1400. 3. 1175. Viscosity of a polymer-solvent solution. 2.72 1010 3.74 109. β`. 0.8. βd. 0.5. ¯ pmq R. 4 103. kd ps1 q. 102. cext pkg {m3 q c0 pkg {m. 3. R0 pmq. q. 755.74 1. 103. m. 1. E0 pP aq. 103. E1 pP aq. 103. τ1 psq Solubility. 1000. 3. 250. csol pkg {m. 3. µ ˆ pP a sq. q. 1 106. Table 1 – Constants and parameters of drug release problem..

(29) 29. 2 Model and Method Used In this Chapter we describe the model and the method that is analysed numerically in this thesis to approximate the solution of the drug release process. The model previously presented in Chapter 1 is indeed complex for a numerical analysis. We simplify it in order to develop a stable and convergent numerical method. In the previous model the polymer swelling lead to an enlargement of the computational and physical domain as we advanced in time. A first simplification that helps in the numerical analysis is to consider only a fixed spherical domain with radius R for all time t. Moreover in Chapter 1 we studied the behaviour of the dissolved drug inside and outside the polymer, here we restrict the domain only to the polymer and we suppose at initial time that some solvent molecules are already inside the polymer but without having enhanced the drug dissolution. Based on these hypothesis we consider the following initial conditions for c` , cd and cs : c` pr, 0q . cext 2 r , cs pr, 0q c0 , cd pr, 0q 0, r R2. P Ω,. (2.1). where we consider Ω p0, Rq and c0 the initial solid drug concentration. Note that the initial solvent concentration given in (2.1) is the exterior solvent concentration cext on the boundary r R whereas at origin it is null. Furthermore, c` is supposed quadratic with respect r at initial time. At the origin (centre of the polymeric sphere) we consider the symmetry used in the model described in Chapter 1. Consequently we have . . Bc` Bcd Br r0 Br r0. 0.. (2.2). Furthermore at the boundary r R the solvent concentration is supposed to be equal to the outer concentration cext and we also suppose that the dissolved drug leaves the polymer at the boundary, so we have cd pR, tq 0 for all t. Hence, we consider the following Dirichlet conditions: $ &c R, t `. p q cext, %c pR, tq 0. d. (2.3). No boundary conditions for cs are needed since its equation presents only derivatives with respect time. As in (1.20) the model is symmetric with respect to the origin, then we can.

(30) Chapter 2. Model and Method Used. 30. write the model in spherical coordinates system (see Appendix A): $ c` ' ' ' ' ' t ' ' ' &. B 1 B r2 D pc pr, tqq Bc` ` ` B r2 B r Br Bcd 1 B r2D pc pr, tqq Bcd. d ` ' B t r2 B r Br ' ' ' ' ' ' Bc ' % s f pcs , cd , c` q. Bt. B σ Dv pc` pr, tqq pr, tq , Br. f pcs , cd , c` q,. (2.4). 2.1 Numerical Discretization and Discrete Operators Here we describe the numerical method used to approximate the solution of the drug release problem (2.4) and its discrete spatial operators. R. N ¸. . Let Λ be a sequence of vectors h ph1 , . . . , hN q such that hi hi and hmax. i 1. Ωh. ¡ 0, i 1, . . . , N ,. imax hi . We define the following grids: 1,...,N. tri| i 0, . . . , N ; ri ri1 Ωh. Considering the fictitious point r1. hi , i 1, . . . , N ; r0. 0, rN Ru,. Ωhzt0u.. (2.6). r1 we have h0 h1 and we can define the grid . Ωh. Ωh Y tr1u.. (2.7). . Let vh be a discrete function defined in the set Ωh . We consider hi and define the following discrete operators Dx vh pri q. . Dc vh pri q. . Dx vh pri q. . (2.5). vh pri q vh pri1 q , i 0, . . . , N 1, hi vh pri 1 q vh pri1 q , i 0, . . . , N 1, hi hi 1 vh pri 1 q vh pri q , i 0, . . . , N 1. hi 1. 1 2. hi. hi 2. 1. (2.8). 2. Also, we consider the average operator Mh vh pri q . 1 pvhpriq 2. vh pri1 qq.. (2.9). To approximate the solution of the problem we use the semidiscrete Method of Lines (MOL), where the derivatives are approximated by using the operators (2.8). The.

(31) Chapter 2. Model and Method Used. 31. semidiscretization in space of model (2.4) used is 1. c`, i ptq. . 1 1 Mh pri2 2 ri hi 1. 1. q pD`, i. 1. Dx c`, i. 1. ptq. Dv, i. 1. Dx σi. 1. ptqq. . 2. Mhpri2q pD`, i Dxc`, iptq Dv, i Dxσiptqq , 1 1 1 cd, i ptq 2 Mh pri2 1 qDd, i 1 Dx cd, i 1 ptq Mh pri2 qDd, i Dx cd, i ptq ri hi f pcs, i ptq, cd, i ptq, c`, i ptqq, 1 cs, i ptq f pcs, i ptq, cd, i ptq, c`, i ptqq, 1 2. where D`, i. (2.10). D`pMhpc`, iptqqq, Dv, i Dv pMhpc`, iptqqq and Dd, i DdpMhpc`, iptqqq.. The ODE system (2.10) can be written as U 1 ptq F pU ptqq where F is nonlinear since the non-linearity of function f . This system is solved at once: equations are numerically solved together at each time t. In particular, this system is solved by the MatLab’s solver ODE23, a explicit Runge-Kutta method based on the Bogacki–Shampine (2,3) method, (BOGACKI; SHAMPINE, 1989). This method has a smaller order (order 3) then the usual solver ODE45 (based on the explicit Dormand-Prince (4,5) method that has order 5, (DORMAND; PRINCE, 1980)) but is more efficient when is used in problems with moderate stiffness, (BOGACKI; SHAMPINE, 1989), (SHAMPINE; REICHELT, 1997). To analyse the stiffness of the problem U 1 ptq F pU ptqq we use the spectrum of the Jacobian matrix JF . Since the solid drug variation c1s ptq depends on c` and cd but does not explicitly depends on cs ptq, only the first two equations of system (2.10) influence the stiffness. Thus we consider the ODEs system reduced to these two equations V 1 ptq F pV ptqq, with F constrained to the first and second equations of (2.10). Since it is a nonlinear system, due to the nonlinearity of f , we consider a linearization V 1 ptq JF pV p0qqpV ptq V p0qq F pV p0qq, where JF is the Jacobian matrix of F . The stiffness is analysed at initial time t 0, since it is evident that at the beginning of the release process, the rate of concentrations V 1 changes rapidly. This is particularly evident since the polymer swelling is not considered here. We observe that JF pV p0qq is diagonalizable and so the eigenvalues are an appropriate measure of the variation of V 1 ptq respect V ptq. We use then as a stiffness measure, the stiffness radius: stiffr. |λi| , max min |λ | i. where λi are eigenvalues of JF pV p0qq. Table 2 shows stiffr for different spatial meshes including the ones used in our simulations. In simulations presented in this chapter we use meshes with n ¥ 28 subintervals..

(32) Chapter 2. Model and Method Used. 32. n. max |λi |. min |λi |. 22. 4.2653. 6.7314 e 03. 3. 5.8414. 4. 6.8733. 2 2. 4.9489 e 02. 2.2805 e. 01. 6. 9.1220 e. 01. 7. 3.6488 e. 02. 8. 1.4595 e. 03. 9. 5.8381 e. 03. 10. 2.3352 e. 04. 2 2 2 2 2. 2.4980 e 02. 5. 2. stiffr. 4.9494 e 02 4.9489 e 02 4.9487 e 02 4.9486 e 02 4.9486 e 02 4.9486 e 02. 6.3365 e. 02. 2.3384 e. 02. 1.3888 e. 02. 4.6076 e. 02. 1.8432 e. 03. 7.3733 e. 03. 2.9493 e. 04. 1.1797 e. 05. 4.7190 e. 05. Table 2 – Stiffness radius for different grids.. Analysing the stiffness radius we can say that our problem is stiff, so for the previous discussion regarding the MATLAB solvers, we opted to use the ODE23 to approximate the solution of U 1 ptq F pU ptqq. In Table 3 we compare the numerical results obtained at time T 1, running the code for differents grids using solver ODE23 and ODE45. This table shows that ODE23 requires less computational time respect ODE45, that is a consequence of the small number of functions evaluations done in ODE23. ODE23. ODE45. F evaluation. Computational time psq. F evaluation. Computational time psq. 2. 2. 61. 0.64. 91. 0.84. 2. 3. 82. 0.45. 91. 0.80. 2. 4. 88. 0.82. 91. 0.86. 2. 5. 97. 0.64. 97. 0.86. 2. 6. 106. 0.60. 127. 0.96. 2. 7. 250. 1.83. 397. 2.37. 2. 8. 895. 5.16. 1 453. 7.87. 2. 9. 3 508. 20.41. 5 659. 31.88. 10. 13 951. 91.65. 22 303. 142.12. n. 2. Table 3 – Number of evaluations of F pU ptqq and computational time using ODE23 and ODE45.. Now we describe how our numerical method computes the stress and diffusion coefficients. First note that D` , Dd and stress σ depend on solvent concentration c` through a non linear relation, see (1.5), (1.14) and (1.10). In order to reduce such nonlinearity, we use the previous time solvent concentration c` pt ∆tq to compute the stress and diffusion.

(33) Chapter 2. Model and Method Used. coefficients: D`, i D` pMh pc`, i pt ∆tqqq, Dd,i Dd pMh pc`, i pt ∆tqqq, Dv,i Dv pMh pc`, i pt ∆tqqq and σi ∆t is the time step discretization, see (2.10).. 33. σpc`, ipt ∆tqq where. Classical spherical polymeric platforms have dimension R 103 m, see for example (FERREIRA et al., 2018). To avoid a small mesh size in our discretization we rescale the problem using R 1. This means that all quantities, such as D` , Dv and Dd , depending on the dimension of the polymer, are rescaled. We built an uniform grid with 256 subintervals of size h 3.90625 103 and analysed the behaviour of concentrations for different time values, see Figures 8 and 9. Such behaviour is similar to that described and illustrated in the previous chapter.. Figure 8 – Solvent Concentration on Ωh at different times T (in seconds). In Figure 8 we note that near the boundary r R the solvent concentration c` has high derivatives at the beginning and it becomes smooth later on time. This is a sign of a high solvent flow entering in the polymer at the beginning. As it is expected physically, for the solvent diffusion inside a polymer, the solvent concentration is zero in the centre at initial time and increases along the time. It is also possible to note in such figure that c` pR, tq cext for all t, differently to that observed in Figure 2 of Chapter 1. Despite this, in Figure 9 it is possible to notice a roughly similar behaviour for the solid drug dissolution and dissolved drug diffusion of that observed in previous model in Chapter 1. A difference is noted at the boundary where we used cd 0 instead of the Dirichlet and Neumann conditions (1.18). We note a decrease of solid drug concentration reaching values close to zero for longer times and a displacement of the diffusion wave of the dissolved drug inside the polymer. Furthermore, the dissolved drug concentration increases near the origin. At the equilibrium it is expected that c` cext , cd 1 and cs 0..

(34) Chapter 2. Model and Method Used. 34. (a) Dissolved Drug. (b) Undissolved Drug. Figure 9 – Dissolved and Undissolved Drug Concentrations on Ωh at different times T (in seconds). Since we consider a non-zero initial condition for c` , the drug dissolution occurs more rapidly compared to the results obtained in Chapter 1. Moreover, the Dirichlet boundary condition for cd leads to larger derivatives near r R.. 2.2 Numerical Results of Convergence Order Here we analyse the numerical results and the convergence order of finite difference method described in the previous section. Initially we define the supraconvergence concept. A finite difference method is called supraconvergent when it has an higher convergence order than the truncation error order measured pointwise or in the } }8 norm, (KREISS et al., 1986). Thus for example if u is a solution of a differential problem.

(35) Chapter 2. Model and Method Used. 35. and uh is its approximation obtained by some finite difference method having a truncation error Th , such that |Th | Chs , where h is the maximum mesh-size used in the grid. If the error of the method }u uh } converges to zero with order greater than s, then the method is called supraconvergent. In our case we will prove that in a discrete H 1 norm the error goes to zero with order two, see Theorem 3, for any non uniform mesh, even if the truncation error is of order one in the infinity norm, see (D.1) in Appendix D. Then our method is supraconvergent in a discrete H 1 norm. A similar analysis has been done for instance in (BARBEIRO et al., 2018), (FERREIRA; GUNDINO; OLIVEIRA, 2013), (FERREIRA; PINTO, 2013). In the following we prove numerically that our method is supraconvergent. In the next Chapter we will instead prove it theoretically. cext We consider constant diffusion coefficients using c` pr, tq for all r and 2 t. Since the exact solution of the system is not known, we approximate this solution 1 using a tiny mesh size h 17 7.63 106 associated to 131072 subintervals. Let u˜ 2 pc˜`, h, c˜d, h, c˜s, hq be such approximated solution and ui pc`, hi , cd, hi , cs, hi q the approximate solution using differents mesh size hi ¡ h, we define ek,hi c˜k,h ck,hi for k `, d, s, that is calculated in the same points of grids, Ωh and Ωhi . The errors are measure by using the norms }ek,hi }h,8 max }enk,hi }h, for k `, d, s, (2.11). . n 0,...,M. b. }ek,h }h,1 nmax }enk,h }2h }Dxenk,h }2 , 0,...,M i. i. i. for k. `, d,. (2.12). where M is the number of time steps done to approximate the solutions at time T . The rates of convergences are obtained by using the formulas Rateh,8 Rateh,1. }ek,h }h,8. . log }ek,hi 1}h,8 i , for k hi 1 log hi. `, d, s,. . log }ek,hi 1}h,1 i , for k hi 1 log hi. `, d.. }ek,h }h,1. (2.13). Since the discrete norms } }h and } } approximate the continuous L2 - norm, then } }H approximates a norm in C 0 pr0, T s, H 1 r0, Rsq. In Chapter 3 we will define these discrete norms. The Table 4 and Figure 10 illustrate the errors and rates obtained by these approximations..

(36) Chapter 2. Model and Method Used. hi (approx.) 1.222 104. 6.1035 105 3.0518 105 1.5226 105. }e`,h }h,8 2.9006 106 7.1437 107 1.7225 107 3.4489 108 i. 36. Rate c`. 2.0216 2.0521 2.3203. }ed,h }h,8 4.0202 108 2.1156 108 5.4619 109 1.1068 109 i. Table 4 – Norm of the errors }ek,hi }h,8 for k rates.. Rate cd. 1.9536 2.3030 2.2823. }es,h }h,8 1.4737 1012 9.8206 1013 3.0113 1013 6.2931 1014 i. Rate cs. 0.5856 1.7054 2.2585. `, d, s and the corresponding convergence. (a) Solvent. (b) Dissolved Drug. (c) Undissolved Drug. Figure 10 – Plot of the logarithmic of the errors and corresponding convergence rate: errors (in blue), line with slope equal to 2 (in red).. Table 4 illustrates the errors and the rate of convergence of errors in norm }ek,hi }h,8, for k `, d, s. In Figure 10 is possible to notice that the error decreases with a second order of convergence. We also compute the convergence order and rates in norm }ek,hi }h,1 , for k `, d. These results are presented on Table 5 and Figure 11. We obtain numerically with these norms also in C 0 pr0, T s, H 1 r0, Rsq a second order of convergence..

(37) Chapter 2. Model and Method Used. hi (approx.) 1.222 104. 6.1035 105 3.0518 105 1.5226 105. 37. }e`,h }h,1 1.8792 102 4.8104 103 1.2743 103 2.5708 104 i. Rate c`. 1.9659 1.9165 2.3093. Table 5 – Norm of the errors }ek,hi }h,1 for k rates.. (a) Solvent. }ed,h }h,1 4.5144 104 3.7053 104 1.0199 104 2.0806 105 i. Rate cd. 0.2849 1.8612 2.2933. `, d and the corresponding convergence. (b) Dissolved Drug. Figure 11 – Plot of the logarithmic of the errors and corresponding convergence rate: errors (in blue), line with slope equal to 2 (in red).. The theoretical justification of the numerical results obtained here will be presented in the next chapter..

(38) 38. 3 Numerical Analysis In this chapter we analyse the stability and convergence of the method for drug release presented in the previous chapter where it has been illustrated numerically to be second order convergent in L2 and H 1 . Here we demonstrate in particular the stability and convergence for this method when it is applied for a wide class of one- dimentional problems defined in p0, Rq p0, T s $ c` ' ' ' ' t ' ' ' ' ' &. B B apc q Bc` B » t qpt, s, c psq, c ptqq Bc` psqds , ` ` B Bx ` Bx Bx 0 Bx Bcd B dpc q Bcd f pc , c , c q, ` ap s d ` ' B t B x B x ' ' ' ' ' ' ' ' % B cs f pc , c , c q. ap s d ` Bt We consider in (3.1) the following boundary conditions for all t P p0, T s $ & B c` p0, tq B cd p0, tq 0, Bx Bx % c` pR, tq cext , cd pR, tq 0,. (3.1). (3.2). and the initial conditions: c` px, 0q c`0 pxq,. cd px, 0q cd0 pxq,. cs px, 0q cs0 pxq,. (3.3). for given functions c`0 pxq, cd0 pxq and cs0 pxq.. On (3.1) we consider that functions apz q, q pt, s, z, y q and dpz q satisfy the hypothesis:. ¤ apzq ¤ M, |a1pzq| ¤ M, z P IR, Bq Bq |qpt, s, z, yq| ¤ M, Bz pt, s, z, yq ¤ M, By pt, s, z, yq ¤ M , for pt, s, z, y q P r0, T s2 IR IR.. H1 : 0 a0 H2 :. H3 : 0 d0. ¤ dpzq ¤ Md, |d 1pzq| ¤ Md, z P IR,. where a0 , M, d0 and Md are positive constants. Notice that for (1.11) and (1.13) the problem (2.4) described in Chapter ˆ v pz qε1 pz q, 2 satisfies the hypothesis H1 , H2 and H3 where we use apz q D` pz q ED q pt, s, z, y q Dv py qker pt sqε1 pz q and dpz q Dd pz q. Furthermore, we assume that.

(39) Chapter 3. Numerical Analysis. 39. f pcs ptq, cd ptq, c` ptqq defined in (1.12) is replaced by fap pcs ptq, cd ptq, c` ptqq that is obtained replacing the Heaviside function H pcs q by its regularization Hk pcs q Hk pcs q :. 1. e2Kcs px,tq. 1. H pcspx, tqq.. Also let suppose that the regularization satisfies the hypothesis: H4 : |Hk | ¤ Mr and |Hk1 | ¤ Mr in IR, where Mr is a positive constant. For the stability analysis and to prove supraconvergence of our method we need to compute the error made in the approximations using a suitable discrete norm. In (FERREIRA; GUNDINO; OLIVEIRA, 2013) and (FERREIRA; PINTO; ROMANAZZI, 2012) a similar numerical analysis for a single integro-differential equation has been presented. Here instead we focus on a coupled system where solvent, dissolved and undissolved drugs are analysed. We start by defining the used discrete spaces similar to that used in (2.5)-(2.7) of Chapter 2: Ωh Ωh. . Ωh. txi | i 0, . . . , N, xi xi1 Ωhzt0u, Ω h Y t x 1 u.. hi , i 1, . . . , N, x0. 0, xN Ru,. Hence, we define the following grid function sets:. Vh,0 Wh Uh. tvh : Ωh Ñ R | vhpxN q 0u, twh : Ωh Ñ Ru, tuh : Ωh Ñ Ru.. (3.4). In Wh we define the inner product:. pwh, qhqh h21 whpx0qqhpx0q for wh , qh. . N ¸1. . hi 1{2 wh pxi qqh pxi q. i 1. hN wh pxN qqh pxN q, 2. (3.5). P Wh. In Uh we use the inner product: puh, vhq . N ¸. . hi uh pxi qvh pxi q,. (3.6). i 1. for uh , vh P Uh . We also consider the semi-norms } }h and } } associated to these inner we define the following norm: products (3.5), (3.6) and then for uh : r0, T s Ñ Vh,0 d. }uh}H . ". max. Pr. t 0, T. s. }uhptq}. »t. 2 h. }Dxuhpsq}. 2. 0. *. ds .. (3.7).

(40) Chapter 3. Numerical Analysis. 40. The operators Dx , Dx and Dc used in our numerical method are defined in (2.8). In the Section 3.1 we demonstrate the stability of the method applied to the solvent concentration equation in (3.1) where constant diffusion coefficients and uniform spatial mesh are used. In Section 3.2 we generalize the problem to a non uniform mesh with non constant coefficients and analyse the stability. There, we also prove the second order convergence of the numerical method for functions in C 4 and show numerically that a such order of convergence is valid for a wider class of problems that are in C 3 and in H 3 , see Section 3.3.. 3.1 Constant Coefficients and Uniform Grid Case In this section we establish the stability of the method for solvent concentration equation with constant diffusion coefficients and uniform grid, that is we use hi h for ˆ and q ps, t, z, y q kerpt sq Dv , all i. In the first equation of (3.1) we consider apz q D ˆ and Dv are constant. where D Considering uniform grid, we define the following discrete operator for second order derivatives: vh pxi. ∆x vh pxi q . 1. q 2vhpxiq. vh pxi1 q. h2. , i 0, . . . , N. 1.. (3.8). Note that we have Dx pDx vh pxi qq ∆x vh pxi q. Denoted by c`,h ptq the semi-discrete approximation for c` ptq, we have the following initial boundary value problem $ ' ' dc`,h ' ' t ' ' ' dt ' ' ' ' ' &. q Dˆ ∆xc`,hptq. p. »t. Dv 0. kerpt sq ∆x c`,h psqds. in Ωh txN u p0, T s,. (3.9). ' ' ' ' ' ' Dc c`,h x0 , t 0, c`,h xN , t ' ' ' ' ' ' % c 0 given.. p. `,h. q. p. q cext, t P p0, T s,. pq. The next theorem discusses an estimate for the solution of the semi-discrete system (3.9) with respect to the H norm defined in (3.7). Theorem 1. Let c`, h P C 1 pr0, T s, Wh q an approximate solution of the semi-discrete system (3.9). Then, there are positive constants C1 and C2 such that. }c`, hptq}. »t. 2 h 0. }Dx c`, hpsq}2 ds ¤ C1}c`, hp0q}2h eC. 2. t. , t P r0, T s.. (3.10).

(41) Chapter 3. Numerical Analysis. 41. Proof: Let c`, h be an approximate solution for the system (3.9). We have:. p c1. `,h. . ptq, c`,hptqqh . 1d } c`,h ptq}2h 2 dt. »t. ˆ x c`, h ptq D∆. . Dv 0. kerpt sq∆x c`, h psq ds, c`,h ptq. ˆ p∆x c`, h ptq, c`,h ptqqh D » t. kerpt sq∆x c`, h psq ds, c`,h ptq. Dv 0. , h. . h. Using the property (C.6) in Appendix C and considering the Neumann condition in x0 of system (3.9), we have: 1d }c`,hptq}2h 2 dt. ˆ }Dx c`,h ptq}2 D. Dv. »t 0. kerpt sq pDx c`,h psq, Dx c`,h ptqqh ds.. Since kerpsq P L2 pr0, T sq and using Cauchy-Schwarz inequality and Lemma 1 in Appendix C we have » t . . ˆ }Dx c`,h ptq}2 D. 1 d }c`,h ptq}2h 2 dt 1 d }c`,h ptq}2h 2 dt. ˆ }Dx c`,h ptq}2 D ˆ }Dx c`,h ptq}2 D. ¤. Dv }Dx c`,h ptq}. 1 d }c`,h ptq}2h 2 dt. ˆ }Dx c`,h ptq}2 D. ¤. ε21 }Dx c`,h ptq}2. ¤. 2ε21. d }c`,h ptq}2h dt. ¤. kerpt sqpDx c`,h psq, Dx c`,h ptqq ds. 1 d }c`,h ptq}2h 2 dt. Dv . ¤. ˆ }Dx c`,h ptq} 2D. 2. 0. »t. Dv 0. kerpt sq}Dx c`,h psq} »t 0. }Dx c`,h ptq}. kerpt sq}Dx c`,h psq} ds. Dv2 4ε21. » t 0. 2. kerpt sq}Dx c`,h psq} ds. }Dx c`,h ptqh }. Dv2 }ker}2L2 2ε21. »t. ¤ }c`,h p0q}. Dv2 }ker}2L2 2ε21. »t»s. 2. ds. 0. }Dx c`,h psq}2 ds,. Hence,. }c`,h ptq}. 2 h. ˆ ε21 q 2pD. Considering C˜1. »t 0. ker}L Dv }2ε 2 2. pDˆ ε21q ¡ 0, we have }c`,h ptq}2h. »t 0. }Dx c`,h psq} 2. 2. 2. ds. and γ. 1. 2 h. 1 }c`,h p0q}2h γ. ¤. 1 }c`,h p0q}2h γ. »t. 2 h. y ptq and considering α . ¤. 1 }c`,hp0q}2h, K γ. we obtain. }c`,hptq}. »t. 2 h 0. 0. }Dx c`,h pµq}2 dµds.. mint1, 2pDˆ ε21qu, where ε1 in chosen such that. }Dx c`,h psq}2 ds ¤. Finally, we get for y ptq }c`,h ptq}. 0. 0. » » C˜1 t s }Dx c`,h pµq}2 dµ ds γ 0 0 » » s C˜1 t }Dx c`,h pµq}2 dµ γ 0 0. }c`,h psq}2h. ds.. }Dxc`,hpsq}2 ds the following relation. 1 }c`,hp0q}2h γ. » C˜1 t y psq ds γ 0. ˜. Cγ1 , β 1 and t0 0 in Lemma 3 in Appendix C. }Dxc`,hpsq}2 ds ¤. C1 }c`,h p0q}2h eC2 t ,.

(42) Chapter 3. Numerical Analysis. for all t P r0, T s, where C1. 42. γ1. ˜. and C2. Cγ1 . . This theorem can be used to establish the stability of the method for the solvent concentration equation. In fact, let c`,h and c˜`,h be solutions of (3.9) with different initial conditions, c`,h p0q and c˜`,h p0q. Therefore »t. c1. ˆ `,h ptq D∆x c`,h. Dv. c˜1. Dv. `,h. 0. »t. ˆ x c˜`,h ptq D∆. 0. kerpt sq∆x c`,h psq ds, kerpt sq∆x c˜`,h psq ds,. and so defining w`,h c`,h c˜`,h and proceeding as in the proof of the Theorem 1, it is possible to find positive constants C1 and C2 , such that »t. }w`,hptq}. 2 h 0. }Dxw`,hpsq}2 ds ¤. C1 }w`,h p0q}2h eC2 T , @t P r0, T s.. (3.11). Since for (3.11) }c`,h c˜`,h }H is bounded by }pc`,h c˜`,h qp0q}h , we can conclude the stability of the method.. 3.2 Non Constant Coefficients and Non Uniform Grid In this section we study the stability and convergence of the method that solves the full concentrations problem (3.1) in c` , cd , cs in the generalized case when non constant diffusion coefficients and non uniform grids are used. As done previously, c`,h ptq represents the semi-discrete approximation of c` ptq that is now defined as the solution of the following initial boundary value problem $ ' ' dc`,h ' ' t ' ' dt ' ' ' ' ' ' ' ' ' ' &. . p q Dx apMhc`,hptqqDxc`,hptq D. » t. x. 0. q pt, s, Mh c`,h psq, Mh c`,h ptqqDx c`,h psqds. t up. s. in Ωh xN 0, T , ' ' ' ' ' ' ' ' ' ' Dc c`,h x0 , t 0, c`,h xN , t cext , t ' ' ' ' ' ' %. p. q. p. q. c`,h p0q given,. where the average Mh operator is defined in (2.9).. (3.12). P p0, T s,.

(43) Chapter 3. Numerical Analysis. 43. The semi-discrete approximations cd,h , cs,h for the dissolved and undissolved drugs cd and cs , are solutions of the differential problems $ dcd,h ' ' t ' ' dt ' ' ' ' ' &. p q DxpDdpMhc`,hptqqDxcd,hptqq. f pcs,h ptq, cd,h ptq, c`,h ptqq. in Ωh txN u p0, T s,. (3.13). ' ' ' Dc cd,h x0 , t ' ' ' ' ' ' %. p. q 0, cd,hpxN , tq 0, t P p0, T s,. cd,h p0q given,. and. $ dcs,h ' & t. p q f pcs,hptq, cd,hptq, c`,hptqq in Ωh txN u p0, T s,. dt. ' %. (3.14). cs,h p0q given.. 3.2.1 Solvent Concentration In order to treat the homogeneous Neumann boundary conditions at x 0, a new boundary finite difference operator needs to be introduced ˜ c,a vh px0 q 1 a pMh vh px1 qq Dx vh px1 q D 2. where vh. a pMh vh px0 qq Dx vh px0 q ,. (3.15). P Vh,0 and a : IR Ñ IR. In what follows we consider the set. Λ. #. h ph1 ,. . N ¸. hmax , hN q | hi ¡ 0, i 1, . . . , N, R hi , hmin i1. ¤C. +. ,. (3.16). with C positive constant. The next theorem proves the stability of solvent concentration. Theorem 2. Assume that H1 , H2 are valid and suppose that c`,h and c˜`,h are two solutions , we of (3.12) with different initial conditions. Considering w`,h ptq c`,h ptq c˜`,h ptq P Vh,0 have for ε 0. }w`,hptq}. 2 h. 2pa0 . 2. 1 2 M ¤ e 2. q. » t 1 M2 2. »t. e. »t s. }Dxc`,hpµq}28dµ. 0. }Dxc`,hpsq}8ds 2. 0. }Dxw`,hpsq}2 ds. (3.17). }w`,hp0q}2h, t P r0, T s.. and, to simplify the proof, we consider (3.12) Proof: Let wh ptq c`,h ptq c˜`,h ptq P Vh,0 without the integral term even if a similar proof is valid when the integral term is considered. Using Proposition 5 of Appendix C and adding and subtracting the term.

(44) Chapter 3. Numerical Analysis. 44. papMh c˜`,hptqq Dxc`,hptq, Dxw`,hptqq. , it can be shown that }w`,h ptq}2h is solution of the. initial value problem $ 1d ' ' w`,h t ' ' 2 dt ' ' ' ' ' &. }. ' ' ' ' ' ' ' ' ' %. p q}2h. papMhc˜`,hptqqDxw`,hptq, Dxw`,hptqq . p apMhc`,hptqq apMhc˜`,hptqq D˜ c,ac`,hpx0, tqw`,hpx0, tq. w`,h p0q. Dx c`,h ptq, Dx w`,h ptqq. ˜ c,a c˜`,h px0 , tqw`,h px0 , tq, t P p0, T s, D. c`,hp0q c˜`,hp0q,. (3.18) ˜ c,a is defined in (3.15). Adding and subtracting where the finite difference operator D apMh c`,h px0 qq Dx c`,h px1 q in (3.15) and using the Mean Value Theorem it is possible to show that ˜ c,a c`,h px0 , tq 1 a1 pη qh1 Dc c`,h px0 , tqDx c`,h px1 , tq D 2. apMh c`,h px0 , tqqDc c`,h px0 , tq,. where η is in the interval defined by Mh c`,h px0 , tq and Mh c`,h px1 , tq. Since Dc c`,h px0 , tq 0, we obtain ˜ c,a c`,h px0 , tq 0, D and analogously, we have. (3.19). ˜ c,a c˜`,h px0 , tq 0. D. (3.20). ?. Using the definition of } } and Mh we can prove that }Mh w`,h ptq} ¤ 2}w`,h ptq}h , then by the hypothesis H1 and Lemma 1 of Appendix C, it can be shown the next estimates . . apMh c`,h ptqq apMh c˜`,h ptqq Dx c`,h ptq, Dx w`,h ptq. ? ¤ M 2}w`,hptq}h}Dxc`,hptq}8}Dxw`,hptq}. ¤ 212 M 2}w`,hptq}2h}Dxc`,hptq}28. (3.21). 2 }Dx w`,h ptq}2 ,. where 0. In (3.21), the notation }Dx c`,h ptq}8 max |Dx c`,h pxi , tq| is used. Then, i1,...,N considering (3.19), (3.20) and (3.21) in (3.18) and the assumption H1 , we obtain for t P p0, T s. 1 2 M2 d }w`,hptq}2he dt 2pa0 2 q. »t 0. »t 0. }Dxc`,hpsq}28ds 1 2 M2 e . »s 0. }Dxc`,hpµq}28dµ. }Dxw`,hpsq}2 ds ¤ 0.. (3.22).

(45) Chapter 3. Numerical Analysis. 45. Consequently,. }w`,hptq}. 2 h. 2pa0 . 2. 1 2 M ¤ e 2. q. » t 1 M2 2. »t 0. e. »t s. }Dxc`,hpµq}28dµ. 0. }Dxc`,hpsq}28ds. }Dxw`,hpsq}2 ds. }w`,hp0q}2h, t P r0, T s. »t. To conclude the stability we need to impose that. }Dxc`,hpsq}28ds is uniformly. bounded with respect h P Λ and t P r0, T s. Such assumption can be avoided since it is a consequence of the convergence properties of the finite difference scheme (3.12), see Corollary 1. This represents a novelty with the respect the classical proof of convergence theorems where instead the stability is used to prove convergence. 0. the e`,h $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' &. In order to prove the convergence, we consider the semi-discrete problem for Rhc`ptq c`,hptq semi-discrete function: . . Dx apMhc`ptqqDxRhc`ptq Dx apMhc`,hptqqDxc`,hptq »t Dx q pt, s, Mh c` psq, Mh c` ptqqDx Rh c` psqds. de`,h ptq dt. »0t. ' ' ' ' ' ' ' ' ' ' Dc e`,h x0 , t ' ' ' ' ' ' ' ' % e 0 given , `,h. p. Dx qpt, s, Mhc`,hpsq, Mhc`,hptqqDxc`,hpsqds 0 Th ptq in Ωh txN u p0, T s,. q Th ptq, 0. e`,h pxN , tq 0, t P p0, T s,. pq. (3.23) where, to simplify, Mh c` ptq Mh Rh c` ptq. Theorem 3 proves the second order convergence of the method for the solvent concentration case. In particular, the theorem proves qq and then it is that our method is second order of convergence in C 0 pr0, T s, H 1 pVh,0 supraconvergent, since the truncation error is only of first order in non-uniform meshes, see (D.1) in Appendix D. Since our method can be seen as a fully discrete piecewise linear finite element method (see for instance (BARBEIRO et al., 2018), (FERREIRA; PINTO, 2013), (BARBEIRO; FERREIRA; PINTO, 2011)), we can say that the method is also superconvergent in the context of piecewise linear finite element method. Theorem 3. Under the assumption H1 , H2 , and assuming that c`. P C 1pr0, T s, C 0r0, Rsq X C 0pr0, T s, C 3r0, Rsq X L2p0, T, C 4r0, Rsq.

(46) Chapter 3. Numerical Analysis. 46. and. P C 1pr0, T s, Vh,0 q, then for the semi-discretization error e`,h ptq Rh c` ptq c`,h ptq, where c` ptq and c`,h ptq c`,h. are defined respectively by the IBVPs (3.1), (3.2), (3.3) and (3.12), there exist positive constants Ci pc` q, i 1, 2, that are h and t independent, such that holds the following. }e`,hptq}. »t. 2 h. }Dxe`,hpsq}2 ds ¤ C1pc`qeC pc qt 2. 0. }e`,hp0q}2h. h4max. `. . , t P r0, T s.. (3.24). As a consequence of this convergence is possible, as already mentioned, to show » t. the boundness of 0. }Dxc`,hpsq}28ds (see inequality (3.28)) and consequently the stability. of the method. These consequences are enunciated in the Corollaries 1 and 2. Corollary 1. Under the assumptions of Theorem 3, if c`,h p0q Rh c` p0q and the sequence of vectors Λ is such that (3.16) holds, then there exist positive constants M`,d , h and t independent, such that »t 0. }Dxc`,hpsq}28ds ¤ M`,d, t P r0, T s, h P Λ.. (3.25). Proof: Let e`,h ptq Rh c` ptq c`,h ptq be the semi-discrete error where Rh denotes the restriction operator. As a consequence of convergence error, the following estimate holds »t. for h P Λ and t P r0, T s.. 0. }Dxe`,hpsq}2 ds ¤ C. . }e`,hp0q}2h. h4max. ,. (3.26). If we assume that the sequence Λ of vectors h ph1 , h2 , . . . , hN q is such that hmax hmin then. »t. (3.27) »t. »t. }Dxc`,hpsq}8ds ¤ 2 }Dxe`,hpsq}8ds 2. 2. 0. ¤ C,. 0. ¤ 2h. »t. 1. If }e`,h p0q}h. a. Op. 0. }Dxe`,hpsq}. 2. min. ¤ 2C. 2. . 0. h3max. 1 hmin. }DxRhc`psq}28 2. ds. }e`,hp0q}2h. »t. 0. }DxRhc`psq}28ds. (3.28). 2}c` }2L2 p0,T,C 1 r0,Rsq .. hmax q, then, using (3.27), (3.28), we conclude »t 0. }Dxc`,hpsq}28ds ¤ M`,d,. for some positive constant M`,d , h and t independent, and provided that c` P L2 p0, T, C 1 r0, Rsq.. (3.29).

(47) Chapter 3. Numerical Analysis. 47. q be solutions of the IBVP (3.12) with initial Corollary 2. Let c`,h , c˜`,h P C 1 pr0, T s, Vh,0 a conditions c`,h p0q and c˜`,h p0q, respectively, such that |Rh c` p0q c`,h p0q| ¤ C hmax and a |Rhc`p0q c˜`,hp0q| ¤ C hmax. Then for w`,hptq c`,hptq c˜`,hptq we conclude »t. }w`,hptq}. 2 h 0. }Dxw`,hpsq}2 ds ¤ C }w`,hp0q}2h, t P r0, T s.. Proof: In equation (3.17) of Theorem 3 we chose ε satisfying a0 ε2 Lemma, see Appendix C, we obtain (3.30).. (3.30). ¡ 0. Using Grönwall’s. 3.2.2 Dissolved and Solid Drug Concentration In this section we demonstrate the stability and convergence of the method for the dissolved and undissolved drugs. We obtain this in a different way with respect to that done for the solvent concentration. We start by prooving the stability and then we demonstrate the convergence. We also use in the proof, the estimatives obtained for the solvent concentration.. q and c in C 1 pr0, T s, W q be solutions of the IBVPs Theorem 4. Let cd,h P C 1 pr0, T s, Vh,0 s,h h (3.13) and (3.14), respectively. If d satisfies H3 and the Heaviside regulatization Hk satisfies the hypothesis H4 , then, under the assumptions of Theorem 3, we have »t. »t. }cd,hptq}2h }cs,hptq}2h. 2d0. Mr kd. »t. 2. Mr kd. e. s. csol. s. }c`,hpµq}8. »t. 0. . »t. 2. Mr kd. e. ¤ }cd,hp0q}2h }cs,hp0q}2h. 3. 2. Mr k d. e 3 csol. 3. 0. }c`,hpµq}8. csol dµ. 0. dµ. }Dxcd,hpsq}2 ds. }c`,hpµq}8. dµ. }c`,hpsq}2hds, t P p0, T s, h P Λ. (3.31). Proof: From (3.13) (3.14), using the assumption H3 , and Proposition 5, we obtain 1 d }cd,hptq}2h 2 dt. }cs,hptq}2h. d0 }Dx cd,h ptq}2. ¤ pfappcs,hptq, cd,hptq, c`,hptqq, cd,hptq cs,hptqqh. (3.32). ˆ c,d cd,h px0 , tqcd,h px0 , tq, t P p0, T s, D ˆ c,d is defined by where the finite difference operator D 1 ˆ dpMh c`,h px1 , tqqDx vh px1 q Dc,d vh px0 q 2. dpMh c`,h px0 , tqqDx vh px0 q ,.

(48) Chapter 3. Numerical Analysis. for vh. 48. P Vh,0 . As Dccd,hpx0, tq 0, it can be shown that ˆ c,d cd,h px0 , tq 0. D. (3.33). Considering the assumption H4 in the first term of the right hand side of (3.32) we obtain successively that. |pfappcs,hptq, cd,hptq, c`,hptqq, cd,hptq cs,hptqqh| . . ¤ Mr kd }c`,hptq}h c1 }c`,hptq}8}cd,hptq}h }cd,hptq}h }cs,hptq}h sol . Mr kd } c`,h ptq}8 ¤ 2 2 3 c }cd,hptq}2h }cs,hptq}2h 12 Mr kd}c`,hptq}2h. sol. (3.34) Taking (3.33), (3.34) in (3.32) we get d }cd,hptq}2h dt. ¤ Mr kd. . 2. }cs,hptq}2h 2d0}Dxcd,hptq}2. } c`,h ptq}8 3 } cd,h ptq}2h }cs,h ptq}2h c sol. Mr kd }c`,h ptq}. 2 h,. t P p0, T s,. (3.35). and consequently. d Mr kd e dt »t. 2d0. e. »t. 3. 2 0. Mr kd. csol. »s. 3. 2. csol. 0. 0. Mr kd. }c`,hpsq}8. »t. Mr kd. e. ds . }c`,hpµq}8. »s. 3. 2. csol. 0. } cd,hptq}. 2 h. dµ. }Dxcd,hpsq}2 ds. }c`,hpµq}8. 0. }cs,hptq}. 2 h. (3.36). dµ. }c`,hpsq}2hds ¤ 0, t P p0, T s,. that leads to (3.31).. Corollary 3. Under the assumptions of Theorems 3 and 4, if }cd,h p0q}2h }cs,h p0q}2h is a bounded, for h P Λ, and |Rh c` p0q c`,h p0q| ¤ C hmax , then there exists a positive constant C, h and t independent, such that. }cd,hptq}. 2 h. and then. }cs,hptq}. »t. 2 h. »t 0. 0. }Dxcd,hpsq}2 ds ¤ C, t P r0, T s, h P Λ,. (3.37). }cd,hpsq}28ds ¤ C, t P r0, T s, h P Λ,. (3.38). }cs,hptq}2h ¤ C, t P r0, T s, h P Λ.. (3.39).