SECTION 1. INTRODUCTION

1.1 Introduction

The average American life span has almost tripled within the past two centuries to 76 years and has brought about the need to deliver more medication into the human body. The current methods of delivery are intravenous, tablet(oral), and transdermal drug delivery(TDD) systems. The fastest and most controllable method of delivering medication into the human body is an intravenous needle. Noninvasive methods of drug delivery are the oral tablet, and the transdermal drug delivery(TDD) system. The TDD system is the easiest and most preferred system of delivery. This system uses diffusion gradients to force the transport of medication to take place.

The human body is a Bio-Mechanical-Electro-Chemical machine. To accurately describe any process that occurs in the body, we must completely describe all of the parts from all of the contributing sciences individually and then integrate them to fit the realized experience. The area that we will deal with in this project is Bio-Mechanics. The Electro-Chemical area is relevant but exceeds the time limit for this project. The scope of this project is narrowed down to the study of transport of molecules by the mechanism of diffusion, and the influence of local environmental conditions on various transport processes.

There are advantages and disadvantages to all of the forms of drug delivery listed above. The intravenous method is the best for direct delivery and control of rate of delivery, but it is the worst because it can transmit foreign organisms into the body (highly invasive). The tablet is best for convenience, but its shortcomings are the sudden high rate of release and fast depletion of its concentration. Because the tablet cannot be placed next to the site of interest, most of the medicine is passed through the intestines or flushed out by the kidneys before it is even used.

The TDD system is best known for its non-invasiveness and its placement over the target region. However, it has a long list of areas where improvement is needed. These areas include: slow delivery rates, use of diffusion gradients as the main mechanism for transport, over-doses and under-doses, biological tissue irritation, and its dependence with various non linear biological processes. The rate of initial discharge and the depletion of concentration of medication as seen in Figure 1.2.1.

Figure 1.2.1 The concentration of medication in stomach and blood as a function of time, with conventional dosage form.

1.3 Definition of the Problem

The TDD system was chosen as the research topic. The other two methods of delivery are the extremes, one being completely invasive and the other uncontrolled and diluted. The TDD system is somewhere in between, where the risk of transmitting a foreign organism is close to none and the rate of delivery is somewhat controllable. To be able to improve on the design of the TDD system model, we need to study the current model. This model is not well published due to proprietary interests of competing pharmaceutical companies.

The scope of this project is first to study the diffusion model and then add onto that the other governing mechanisms of transport. Diffusion is defined as the process whereby a material moves from a region of higher concentration to a region of lower concentration, Ref (1). Diffusion in this model will take place starting from the drug reservoir, moving through several layers of filters, and then working its way into the skin. From this point on, the mechanism of transport gets very complex. The research literature and the time limits for this project did not allow for detailed modeling of all processes, although they will be discussed in the future work section of this report. In nature, diffusion occurs in a three-dimensional world. To reduce the computer processing time, a two-dimensional representation is used.

The problem is to define both when the diffusion model is applicable and when it is not, and where the other processes dominate or compete with diffusion in biological tissue. There are several different mass transport process occurring simultaneously where one process may be several orders of magnitude larger than the others.

SECTION 2. THE ANALYTICAL TRANSDERMAL DRUG DELIVERY MODEL

2.1 Formation of a Diffusion Model

To describe diffusion, one must consider the governing equations, then the different factors that go into controlling mass transport within the human body. The governing equations for diffusion are Fick's first and second law, Ref (2).

Fick's first law can be derived from the Generic Transport Equation (GTE). Fick's first law is derived below starting with the GTE equation:

      [1]

• BETA is the correction coefficient which accounts for the semi-permeability of the membrane and is equal to 1 for full permeability,
• RHO is the density of the molecule in mass per volume,
• ALPHA is the kinematic mass transport coefficient which is equal to
  
  , where
  • D_p is the permeability (i.e., if D_p=1, then the path offers no resistance, or D_p=0 if the path is fully blocked),
  • l (lower case L) is the mean free path of the molecule,
  • MW is the molecular weight of the molecule,
  • R is the universal gas constant, and
  • T is the local temperature,
  • n exponent on the MW term is set equal to one for this diffusion problem but can also be defined as follows: for mass transport, n=1, for momentum transport, n=1/2, and for energy transport, n=0
  • THETA is the intensive potential, the driving force equal which is calcualted from
    
    , where
    • [S] is the concentration of species S in units of mass per volume

      Combining equations yields a relationship that defines the diffusivity, D, of species S as
      

      

      and redefine variables as follows,

         and   

      [2]

Fick's first law quantifies the amount of matter that is being moved from a point of higher concentration to that of a lower one. The units of mass flux, J, are mass per area per time, the diffusivity coefficient, D, is area per time, and the concentration gradient (grad of C) is mass per volume per unit length.

      [3]

Just as a multiplexer has many inputs and only one output, this transport model has many input variables and one response. This response determines the model's efficiency, speed, order of magnitude, influence over its competition, bio-availibility, metabolic activity, reservoir concentration, and others of lesser importance to the transport taking place. For example, when the heart rate increases, the blood flow rate increases, and, in turn helps to increase the rate of diffusion near the capillary walls due to lower pressures and lower concentrations in that area. However, when the heart rate increases, the rate of heat flow out of the blood also increases, which deters the diffusion process due to the counter convection of water moving radially out of the skin. When the magnitude of the velocity and the pressure are high, the concentration of the diffusing species builds up and the gradient which powers diffusion is nearly zero as can be seen in Fig 2.2.2.

Figure 2.2.2 Shows the influence of built up concentration due to high velocities or high pressure.

Just as the human body operates well within its normal operating envelope, mass transport, which is an integral part of human processes, works well within its ranges.

The scale of matter is very important when considering the diffusion model. On the atomic scale, there are 3 atoms in an H₂O molecule with a molecular weight of 18, compared to a cell with an average of 10E-14 atoms.

The size of the diffusing species must be equal to or smaller than the largest width of the pathway. There are two forms of diffusion: vacancy, and interstitial diffusion. Vacancy diffusion occurs when there is a vacancy of the same size or when another atom of the same size moves and leaves a vacancy. Vacancy diffusion occurs when you have impure material. Interstitial diffusion occurs when an atom of the smaller size moves in-between larger atoms. Interstitial diffusion will occur in materials without impurities, and in mass transport processes within the human body.

The order of magnitude of the rate of transport will be related to the scale of the species being diffused and to the scale of the species that it is being diffused in, as seen in Figure 2.3.1. For example, the size of a gas molecule compared to the size of a polymer is very small. So, the diffusive characteristics will be inversely proportional. The species A (a gas) will easily diffuse through species B (a film) with a path width greater than or equal to the width of a molecule of species A. Similarly, species C (a polymer) will not be able to diffuse through a path width smaller than itself.

A    B

Figure 2.3.1 A and B. Influence of relative size/scale on diffusion.

The relative mobility and geometry of the drug molecule will influence its ability to maneuver around obstacles as shown in Figure 2.3.2 A, B, and C. Rate limiting membranes (Figure 2.3.2 D) use this mechanism for slowing down the rate of transport. A particular molecule will have to vibrate and rotate (A->B->C) before it orients itself in the correct position to pass through the membrane. The rate of transport will be slower as the membrane geometry gets more complex.

   D

Figure 2.3.2 A, B, C, and D. Influence of Molecular mobility

- Relative convection is the magnitude and direction of the local flow relative to the concentration gradients. An example of this is cells absorbing or expelling water. If the cells are absorbing water then the cell walls will absorb the medication into the cell, increasing diffusion of medication into the local site. When the cells are expelling water, the excess water will be pumped out of the local area due to extra-cellular fluid exceeding capacity (i.e. super saturating), this will aid the transport of medication away from the local area by lowering the local concentration and increasing the diffusion gradient. However, If this occurs at the target area, the medication will not be able to move into the cell.

- The heat dissipating mechanism regulates diffusion in the same way as the relative convection mechanism. This relation is due to the fact that the body regulates its temperature by regulating blood flow in specific areas. For example, if the body temperature is higher than normal (37 degrees Celsius), the total percent by volume of blood heading out to the limbs will increase. The sweat glands will absorb the warm water out of the blood and move it to the surface of the skin. The water on the surface of the skin will evaporate, lowering the surface temperature. This process not only drives the heat radially outward, but it also interferes with any foreign material trying to work its way in.

- Water aids diffusion indirectly by reducing the viscosity of the fluid medium. The lower viscosity allows the concentration of the drug to become diluted in this area. The high concentration of drug at the patch site and the low concentration in this area generate a concentration gradient directed inwards.

- The metabolic activity rate (MAR) of the local area will control how fast certain reactions happen. A major foe of diffusion is the active carrier transport mechanism. The active transport mechanism uses carrier molecules to carry molecules that would normally not be able to reach their target sites in time for the organism to survive just by diffusion. An example of a carrier molecule is the hemoglobin molecule, which binds to it four oxygen molecules. Without the Hemoglobin molecule the blood would only be able to supply our body with one sixty-fourth the amount of oxygen that it receives due to Hemoglobin. Molecular binding sites is a major topic for research in the biomedical field, namely, how to gain access to binding mechanisms on other floater molecules to get a free ride to the target. The MAR is too fast for any diffusion to occur. Diffusion is a time dependent mechanism. Two species have to sit next to each other long enough for diffusion to take place. But if the carrier molecules bind with and carry away the second species, then diffusion cannot occur.

- As the density of the material increases, its volume for a constant mass decreases, therefore shrinking the porous pathways that allow diffusion.

- The pH concentration of the fluid medium controls the speed of the binding reactions that occur in it. This is done by the secondary bonds and Vander Waals effects. One OH- will have to repel another OH-, or high pH will not repel its neighbor polymer chain as much. Normal pH for the human body is 7.4.

Figure 2.3.3 Influence of pH concentration on binding site shape.

The pH controls the shape of a binding site, so that if the pH is just slightly off, the binding site is distorted, and the correct binding will not occur as seen in Fig. 2.3.3.

Due to constraints in time and available literature on modeling diffusion that includes active transport bio-mechanisms, the next section will numerically model diffusion only as it is influenced by structure. How various active transport mechanisms contribute to the average diffusion coefficient is currently not known. Hence approximations are made for the diffusion coefficient used to calcuate concentration gradients in the next section.

SECTION 3. THE NUMERICAL DIFFUSION MODEL

3.1 Program information

Since there is no exact solution for this second order non-homogeneous equation, a FORTRAN program is used to approximate a solution to this problem. Although diffusion is inherently a three-dimensional phenomena, only the two-dimensional non-steady state Laplace's equation is chosen over the three-dimensional equation to reduce the computational work load. To reduce the three-dimensional equation to a two-dimensional equation, the concentration is restricted to vary only in the x and y directions, and is held constant in the z direction. The invariance in the z direction simplifies the three-dimensional equation to the following two-dimensional equation.

            [4]

The program developed here, DIFFUSON.FOR, was modified from one that calculated the velocity profile around a grid (x and y) in time (t). The original program was written by Dr. Ronald Kriz to solve the same type of equation(a second order, non-homogeneous) to find the velocity profiles around obstacles within the grid. DIFFUSON.FOR was modified to solve equation [3] with boundary and initial conditions. This modified program was used as a tool to generate data output for specific materials, and initial and boundary conditions for MSE2094 (Analytical Methods in Materials Engineering, taught by Dr. Kriz in Spring semester 1995). More detailed information on the method used to numerically solve equation [3] can be found in References 5 and 6.

The user has control over all of the variables through the input file INPUT.INP, which defines for the program the following variables:

-N = the number of intervals in one section of the grid, currently set to 99 intervals.
-LVERT = the number of sections on the vertical side of the grid, currently set equal to 1.
-LHORZ = the number of sections on the horizontal side of the grid, currently set equal to 1.
-W = convergence factor for successive over-relaxation (sleep) method, currently set equal to 1.85.
-SIGMA = the constant in the Laplace equation. For our case it is the material diffusivity, set equal to 1.6*10E-5. This controls the speed of the rate of diffusion.
-THETA = for guaranteed unconditional stability, 1/2 < THETA < 1, currently set equal to .75.
-PAR = this parameter controls the stability of the convergence of the calculation, currently set equal to delta t times N squared. See Appendix A for details about this parameter.
-TS = total number of time steps, currently set equal to 1000 steps.
-ITMAX = The largest number of iterations for convergence for each time step, currently set equal to 100 iterations.
-ERRSOR = the percent error to expect in the final solution, currently set equal to .001.
-CONINI = the initial concentration for the whole grid, currently set equal to 25.0.
-CONOBS = the fixed concentration of the obstacles. Work is in progress to make this a variable (function of time or temp, etc.), currently set equal to 25.0.
-MEDCON = the fixed concentration of medication on the surface. Work is in progress to make this a variable (function of time), currently set equal to 120.0.

The output from this program are saved into the file DIFFDATA.OUT. The data in this file are raw and must be processed to be of any value. PV-WAVE, a visualization package, was used for this project in analyzing the output from the FORTRAN program.

During the beginning of this project, the difference between the one-dimensional (one-D) differential equation and the two-dimensional (two-D) differential equation was not well understood. One would have easily replaced the one-D equation with the two-D equation without much thought of the inherent errors. Through the visualization of the data output for the one-D and the two-D input conditions, the difference was observed as in Figure 3.3.1.

Figure 3.3.1 A and B. Shows the difference between the one-D (A) and two-D (B) results.

To calibrate the computer program and its output, Example Problem 5.2 from Callister, "Materials Science and Engineering", Ref (2), was used as reference. Example Problem 5.2 deals with carbon diffusing into steel. The example problem is a one-D diffusion problem, but the computer program was designed around the two-D system. One-D and two-D esults are shown Figure 3.3.1. The left image in Figure 3.3.1 shows an exact one-D solution and the image the right shows diffusion occurring in two-D. The reason for the anisotropic behavior is due to the edge effects. To reduce the impact of edge effects, the width of the grid needs to be increased. If the two-D results shown on the right were stretched out to infinity they would approach the one-D image shown on the left. If the width is great enough the percent error between the exact one-D solution and the two-D numerical approximation at the center point will decrease as seen in Fig. 3.3.2.

Figure 3.3.2 A, B, and C not to scale, shows the importance of the width of the grid.

Image 3.3.2.A is the result of the one-D exact solution, which is independent of the width of the grid. Image 3.3.2.B is the result of a two-D numerical solution, where the grid width is equal to the grid length. Image 3.3.2.C is the result of the same two-D numerical solution, with the grid width ten time the length of the previous grid.

Several different input conditions are tested to show the influence of obstacles in the path of diffusion. The output form these tests is visualized in Fig. 3.3.3.

Figure 3.3.3 Visualization of the influence of obstacles

4.1 Conclusions

The FORTRAN program used a finite-difference method that computed iteration by iteration in a serial order, i.e. compute information at node-one, then node-two and so on. There is a more complex FORTRAN code referred to as FORTRAN-M, which computes the node information for all the nodes simultaneously. The information on this code can be found at Dr. Hienkenschlos'ss Math 4445 Homepage, in the Math Department.

The FORTRAN program described above simple homogeneous diffusion model. When the other models of transport are quantified and can be solved, the resulting data output may be combined with the data output from this process and a morfe complete description of the transport mechanism in the human body can be developed and studied.

Currently, there are only few medications that can be delivered through the use of the TDD system, due to the slow rate of delivery. They are: nitroglycerin(for angina-heart disorders); estradiol(for hormone imbalance); nicotine(to quit smoking); and medications for infants (infant skin is much more permeable than adults).

When all the mechanisms governing the transport of medication through the skin are fully understood, we will be able to generate better medications that will move easily through the barriers of the skin and be delivered to the target location.

Acknowledgments

Thanks goes to the advisors, Dr. Ronald D. Kriz and Dr. Daniel J. Schneck, for their continuous support and patience throughout this project, and for the access and use of the Lab for Scientific Visualization. My appreciation goes out to my classmate and good friend Kenneth Venzant, my roommates Robert Hunter and Dennis Osgood, and Dr. Eric Pappas for their help on the style and structure of this report. Last but not least, thanks to my parents (and my brother, but only for the use of his new computer), who made all of this possible.

REFERENCES