# Drug Delivery and Diffusion

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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
RT

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             RT

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.
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
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
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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