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Introduction to Drug Delivery Changjiang Zhang, Whensheng Shen, Jun Zhang Laboratory for High Performance Scientific Computing and Computer Simulation Department of Computer Science University of Kentucky Why simulate Drug Delivery (1) Allow physicians and scientists to develop and optimize therapeutic approaches systematically (2) Reduce the need for time-consuming and often inconclusive trial-and-error animal testing (3) Open a new frontier of new drug delivery devices, drugs, and delivery designs Drug Delivery by Diffusion Diffusion, is spontaneous spreading of particles or molecules, heat, momentum, or light. Diffusion is one type of transport phenomenon. Diffusion is the movement of particles from higher chemical potential to lower chemical potential (chemical potential can be represented by a change in concentration). It is a physical process rather than a chemical reaction. No net energy expenditure is required. In cell biology, diffusion is often described as a form of passive transport, by which substances cross membranes. Examples of diffusion Schematic drawing of the effects of diffusion through a semipermeable membrane Examples of diffusion Food Coloring diffusing through water at about 2x real-time. The cup on the left contains hot water, while the cup on the right contains cold water. Fick's laws of diffusion Diffusion is governed by Fick's laws. Fick's first law used in steady state diffusion, i.e., when the concentration within the diffusion volume does not change with respect to time (Jin=Jout). J D x Where J is the diffusion flux D is the diffusion coefficient ɸ is the concentration x is the position Fick's laws of diffusion (cont.) Fick's second law used in non-steady or continually changing state diffusion, i.e., when the concentration within the diffusion volume changes with respect to time. 2 D t x 2 Where : ɸ is the concentration t is time D is the diffusion coefficient x is the position For 3D diffusion, the Second Fick's Law looks like: D 2 t If D is not a constant, but depends upon the coordinate and/or concentration, the Second Fick's Law looks like: D t Fick's laws of diffusion (cont.) The D at different temperatures is often found to be well predicted by the Arrhenius Equation: E A 0 D D e RT Where: D is the diffusion coefficient D0 is the maximum diffusion coefficient at infinite temperature EA is the activation energy for diffusion T is the temperature in units of kelvins R is the gas constant Remark: (1) The activation energy in chemistry and biology is the threshold energy, or the energy that must be overcome in order for a chemical reaction to occur. Activation energy may otherwise be denoted as the minimum energy necessary for a specific chemical reaction to occur. (2) The gas constant is a physical constant used in equations of state to relate various groups of state functions to one another. Background -- The Arrhenius Equation Common sense and chemical intuition suggest that the higher the temperature, the faster a given chemical reaction will proceed. Quantitatively this relationship between the rate a reaction proceeds and its temperature is determined by the Arrhenius Equation. At higher temperatures, the probability that two molecules will collide is higher. This higher collision rate results in a higher kinetic energy, which has an effect on the activation energy of the reaction. The activation energy is the amount of energy required to ensure that a reaction happens. This calculator calculates the effect of temperature on reaction rates using the Arrhenius equation. Ea k A e RT where k is the rate coefficient, A is a constant, Ea is the activation energy, R is the universal gas constant, and T is the temperature (in Kelvin). R has the value of 8.314 x 10-3 kJ mol-1K-1 Methods of Drug Delivery Injection, ingestion, and inhalation have been the mode of traditional drug delivery. Various new methods are being developed. The general idea with all of these approaches is the notion of self-administered, targeted, sustained release with strengthened bioavailability. Drug delivery can be classified into three groups: (1) Pharmacological Method; (2) Endogenous Transport; (3) Invasive Methods Invasive methods of Drug Delivery (1) Transdermal drug delivery From an outside source, through the stratum corneum into the viable epidermis, dermis followed by distribution into the blood capillaries and the lymphatic system. Since there is no evidence of active transport in the skin, drugs are thought to permeate through the skin tissue by a passive diffusion process and are usually assumed to obey Fick’s laws of diffusion. Three possible pathways of transport across the skin have been recognized: a) transcellular -- through the cells; b) intercellular -- between the cells; c) transappendageal -- through appendages such as hair follicles and sweat glands. Advantages: a) avoiding the gastro-intestinal tract; b) avoiding first-pass metabolism in the liver; c) increasing patient compliance. Transdermal drug delivery tends to give a controlled, sustained release of durg. (2) Drug delivery to arterial wall Through catheter-mediated endovascular application or through surgically implanted perivascular release devices. Invasive methods (1) Offer a direct way of administering therapeutic drug into targeted regions via a catheter. (2) The efficacy is measured in terms of penetration depth and treatment volume. (3) Diffusive transport depends on the free concentration gradient, the diffusivity, tortuosity of the tissue and molecular weight of the drug. Simulation of Drug Delivery In order to simulate drug diffusion and deposition, we need to know: (1) Geometry of the Target (2) Mathematical Model (3) Computational Approach How to get Geometry of the target This needs an accurate geometric reconstruction of target inner structure from patient-specific imaging data from MRI or CT , etc. The geometry information is composed of volumes and boundary surfaces. Do triangulation also known as grid generation or meshing. Normally 3D is used or 2D if the geometry is symmetric or simplifying the simulation. Mathematical Model Commonly used Mathematical Models or Equations: (1) Fick’s laws of diffusion (2) Incompressible Navier-Stokes Equations (3) Convection-Diffusion Equations, also known as Transport Equations (4) Convection-Diffusion-Reaction Model (5) Chemical Kinetics Computational Approach (1) Analytical solution (2) Numerical solution Some cases Case One: The Use of Mathematical Modeling and Simulation Tools to study transdermal Drug Delivery Systems L. Simon, M. Fernandes, N.W. Loney New Jersey Institute of Technology, Otto H. York Department of Chemical Engineering University Heights, Newwark, NJ 07102 Case Two: A multiple-pathway model for the diffusion of drugs in skin A. J. Lee, J. R. King, T.G. Rogers Department of Theoretical Mechanics, University of Nottingham, UK Case Three: Drug delivery into the human Brain A. A. Linninger, etc. Dept. of Chemical Engineering and Bioengineer, Universty of Illinois at Chicago, Chicago, IL 60607, USA Case Four: Computer Simulations of fibroblast growth factor transport in capillary W. Shen, C. Zhang, J. Zhang, etc. University of Kentucky, Lexington, KY Background Knowledge Skin Structure Case One: The Use of Mathematical Modeling and Simulation Tools to study trans-dermal Drug Delivery Systems A closed-form mathematical solution was obtained for in vitro skin permeation of a drug dissolved in a vehicle. The solution to the mathematical model, which was described by Fickian diffusion equations and appropriate boundary conditions, was derived using Laplace transform methods. The Residue Theorem was applied to invert the equations from the Laplace domain into the time domain. The closed-form solution, obtained for the present percutaneous drug-delivery model, can be readily applied to many drug/vehicle systems to predict drug- release profiles, reducing the cost associated with extensive experimental procedures. Results showed that both axial and temporal variations in the concentration were significant in the skin. The time required for all of the drug to penetrate through the skin is less for a small dose than for a large dose. Case One: The Use of Mathematical Modeling and Simulation Tools to study trans-dermal Drug Delivery Systems The transport of drugs, in a vehicle and in the skin, is governed by Fick’s second law of diffusion. Here are some equations and boundary conditions used: C1 2C1 D1 la x 0 (1) t x 2 C2 2C2 D2 0 x lb (2) t x 2 Where D1 and D2 are the drug-diffusion coefficients in the vehicle and in the skin, respectively. The skin and membrane thickness are represented by la and lb, respectively. The initial conditions are: C1 x,0 C1, 0 la x 0 (3) C2 x,0 0 0 x lb (4) Where C1,0 is the initial drug concentration in the vehicle. The boundary initial conditions are: C1 la , t (5) 0 x C1 D1 0, t D2 C2 0, t (6) x x C1 0, t k mC2 0, t (7) C2 lb , t 0 (8) Where km is the vehicle-skin partition coefficient for the drug. Case One: The Use of Mathematical Modeling and Simulation Tools to study trans-dermal Drug Delivery Systems They obtained an analytical solution by applying Normalization, Laplace transform method and using Residue Theorem. Here is their results: C1 C2 D2 x x U1 U2 t Mt – the cumulative amount of drug that is delivered C1, 0 C1, 0 l 2b la lb through the skin (1) The drug concentration remains almost constant along the vehicle layer. (2) Both axial and temporal variations of concentration are significant in the membrane. (3) The thinner application is depleted more quickly than the thicker application. Case Two: A multiple-pathway model for the diffusion of drugs in skin A mathematical model for the diffusion of drugs in skin is presented. The penetration of the drug by both trans-cellular and intercellular pathways, as well as its interchange between these pathways, is considered. A pharmacologically motivated asymptotic limit is identified and analysed to obtain an analytical expression for the flux of drug to the blood at steady state. Relevant model data is discussed, and some numerical results are presented Case Two: A multiple-pathway model for the diffusion of drugs in skin A Schematic representation of the transcellular and intercellular routes through the skin layers Case Two: A multiple-pathway model for the diffusion of drugs in skin Mathematical Model Fick’s second law is used, and also drug binding is considered. The model incorporates the metabolism of the drug using irreversible first-order enzyme kinetics. So, their model looks like: 2 D f I , T t x 2 Where I and T are intercellular and transcellular concentrations of drug. f(I,T) is a irreversible first-order enzyme kinetics equation. They obtained analytical solution and also some numerical solution. Case Three: Drug delivery into the human Brain Geometry construction: (1) Get MR data (GE medical systems); (2) Reconstruction, convert 3D image data into a precise patient- specific object-oriented model of the brain geometry; (3) Grid generation, segregates the geometric objects into a fine mesh Case Three: Drug delivery into the human Brain Mathematical Model: C C ( Ci ) ( uCi ) ( vCi ) ( D( x, y) i ) ( D( x, y) i ) Ri (1) t x y x x y y In (1), Ci is the concentration of species i in the bulk. u and v are the fluid velocity components, D(x,y) is direction-dependent diffusion coefficient of the drug. The reaction terms, Ri, account for the metabolic uptake of the drug. N Ri M i R ˆ i ,r r 1 is the Arrhenius molar rate of creation or destruction of the species i in reaction r, Mi is its molecular weight. Case Three: Drug delivery into the human Brain Mathematical Model (cont.) Continuity of the bulk flow u v 0 (2) x y X-momentum for fluid in porous medium u u P 2u 2u u v u 2 2 s x x (3) x y x y y-momentum for fluid in porous medium v v P 2v 2v u v u 2 2 s y x x y y y (4) 1 1 s x u u u s x v v v 2 2 In eqs. (2)~(4), u and v are the flow velocity, ρ ( ρ = constant) and μ are the infusate density and viscosity, α is the viscous resistance term equal to the inverse of permeability and β is the inertial resistance parameter. The unknowns in eqs. (1)~(4) are the species concentration field, Ci(x,y), the flow velocities u and v, as well as the hydrostatic pressure field P. Case Three: Drug delivery into the human Brain Computational Approach The transport and reaction system for drug infusion consists of a set of coupled, non-linear PDEs (1) –(4). Using finite volume (FV) approach to solve these equations and get the simulated results. Some boundary conditions and parameters used: Case Three: Drug delivery into the human Brain Some Results (1) Steady State Drug infusion without metabolic uptake Velocity magnitude at different mass flow rates (m), from a to c convection increase Drug concentration at different mass flow rates (m), from a to c convection increase Case Three: Drug delivery into the human Brain (2) 2D patient specific drug-delivery into the human brain Diffusion (left-case), low flow micro-infusion (middle-case) and high flow micro- infusion of the drug into the anisotropic 2D human brain Case Four: Computer Simulations of fibroblast growth factor transport in capillary FiberCell System Case Four: Computer Simulations of fibroblast growth factor transport in capillary Current study – using convection-diffusion-reaction model to simulate fibroblast growth factor (FGF-2) binding to cell surface molecules of receptor and heparan sulfate proteoglycan and MAP (mitogen-activated protein) kinase signaling under flow condition. This model includes three parts: (1) The flow of media using incompressible Navier-Stokes equation (2) The transport of FGF-2 using a convection-diffusion transport equation (3) The local binding and signaling by chemical kinetics Case Four: Computer Simulations of fibroblast growth factor transport in capillary Due to the particular geometry of FiberCell System, it is convenient to write the governing equations in cylindrical coordinates. Assume the flow in the vascular system is axisymmetrical and laminar, a 3D problem can be reduced to 2D. Here are the incompressible Navier- Stokes Equations: The mass conservation equation: u u v (1) 0 r r x The radial momentum equation: u u u p 1 u v g r r rr rx (2) t r x r r r x The axial momentum equation: v v v p 1 u v g x r rx xx (3) t r x x r r x Case Four: Computer Simulations of fibroblast growth factor transport in capillary In Eqs. (1) ~ (3), is the density , u is the radial velocity, v is the axial velocity, p is the dynamic pressure, gr and gx the radial and axial components of gravity respectively, and the stress tensors are u rr 2 r u v rx x r v xx 2 x For a Newtonian incompressible flow, the viscosity is a constant. Case Four: Computer Simulations of fibroblast growth factor transport in capillary Result – Media flow in artificial capillary Visualization of laminar flow in part of the artificial capillary within x = 0 ~0.005m (a) Flood plot of velocity u (b) Vector plot of velocities u and v Thank You!

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