4-3: Mass transfer and Drug Delivery
 By the end of the class, you should be able to:
-
1
Understand the molecular basis of diffusion
Describe the modes of drug transport in the body
Apply the diffusive flux calculation to design a drug delivery system
Introduction to Mass Transfer
When mass is transferred from one phase to another, or through a single phase, the basic mechanisms are
the same (like the Fourier’s law in heat transfer).
 Mass transfer occurs in chemical manufacturing, food production, energy generation, waste treatment
sectors, etc.
 Mass transfer plays a crucial role in pharmaceutical manufacturing, for the dissolution of active
pharmaceutical ingredients in solvents, ensuring uniformity and efficacy in the final product. Crystallization,
filtration, and drying heavily rely on mass transfer principles to optimize yields and product quality.
 Mass transfer governs the transport of drugs within the human body, including absorption, distribution,
metabolism, and excretion of drugs. E.g., the rate at which a drug is absorbed into the bloodstream from
the gastrointestinal tract is influenced by the diffusion of molecules across biological membranes. This
process is critical for achieving therapeutic concentrations in target tissues.
 Additionally, mass transfer is vital for designing controlled-release formulations that can modulate drug
release rates, enhancing therapeutic outcomes while minimizing side effects.

2
Introduction to Pharmaceuticals
 Pharmaceuticals (medicine, drugs) are substances that are used in medical diagnosis,
cure, treatment and prevention of diseases.
 Roles of chemical engineers / bioengineers / biochemical engineers:
❖ Manufacture of pharmaceuticals: Chemical synthesis, fermentation, processing
❖ Formulation of pharmaceuticals
❖ Quality control of pharmaceuticals
❖ Drug delivery
❖ Tissue engineering / regenerative medicine
3
Pharmaceutical industry
 Pharma revenues worldwide totaled 1.5 trillion
US$ in 2022 (Similar with GDP of Korea,
Indonesia, Spain).

Huge growth potential (aging population, fast
developing countries e.g. China and India)
 World’s leading pharma companies
USA: Pfizer, Johnson and Johnson, Merck, Lilly
 UK: GlaxoSmithKline, AstraZeneca, Unilever
 Switzerland: Roche, Novartis
 Germany: Bayer, Boehringer Ingelheim
 France: Sanofi-Aventis

Visualizing the World’s Biggest Pharmaceutical Companies
https://www.statista.com/topics/1764/global-pharmaceutical-industry/#topicOverview
4
http://www.phrma.org/
Transport of drug in the body
http://www.timedomaincvd.com/CVD_Fundamentals/xprt/xprt_conv_diff.html
5
http://www.bio.miami.edu/tom/courses/bil360/bil360s12.html
Molecular diffusion from a microscopic
and macroscopic point of view (wiki).
Diffusion
 Diffusion is the movement of molecules from high concentration to
low concentration through thermal motion
 Mathematically, diffusion can be described by Fick’s Law:
J x = −D
C
x
Concentration gradient
 The flux (in the x-direction) Jx measures how many molecules pass a plane perpendicular
to the x-direction per unit time per unit area
 The diffusion coefficient D (dimension: length2/time) measures the rate of the spreading
of molecules, or how far the molecules travel in each time step in the random walk
6
Diffusion coefficients
 Diffusion coefficients in water are around 10-5 cm2/s for small molecules, and 10-6 cm2/s
for globular proteins.
 The diffusion coefficients of spherical particles much larger than the solvent (e.g.
proteins, polymer particles) can be estimated by the Stokes-Einstein equation:
k BT
D=
6r
kB = Boltzmann’s constant (1.38 × 10-23 J K-1)
T = Absolute temperature (K)
 = Viscosity of medium (Pa s), for water m = 0.001 Pa s (at 20°C)
r = Radius of spherical particle
 The diffusion length scale and time scale is related through the diffusion coefficient:
t D = L2 / D
7
L = approximate displacement of diffusing molecule after time tD
Example 1: Relevant length scale for diffusion
 Diffusion coefficient of protein in water is ~ 10-6 cm2/s. How far can a protein travel in
100 seconds by diffusion?
t D = L2 / D
𝐿=
𝐷𝑡𝐷 =
10−6 cm2 s−1 100s = 0.01cm = 0.1 mm = 100 μ𝑚
 Using diffusion length as your argument, within what distance are tissues vascularized
by blood capillaries?
The main limitation in engineering in vitro tissues is the lack of a sufficient
blood vessel system — the vascularization.
Vascularization is the key challenge in tissue engineering
8
Diffusion in drug delivery
 Diffusion in “bulk” medium
Biological medium (e.g. bodily fluid) / Engineered medium (e.g. polymeric particle)
 Modeled by diffusion coefficient of the drug in the medium

 Diffusion across a barrier
Biological barriers (e.g. blood vessel wall) / Engineered barriers (e.g. liposome shell)
 Modeled depending on the mechanism of diffusion across barriers

 Simple permeation (passive, down concentration gradients)
 Facilitated diffusion (passive, down concentration gradients, using transport proteins as carriers)
 Active transport (active, energy requiring)
9
Steady-state diffusive flux across a barrier
Consider a barrier as a membrane of thickness L
C
J x = −D
x
𝐿
C’in
C’out
At steady state,
dC
−Jx
= constant =
dx
D
C=
x =0
x=L
−Jx
x + B where B is a constant … . . (1)
D
Equation (1) means concentration varies linearly with x
→ Concentration gradient is linear
10
Boundary conditions:
At x =0, C= C’in …. (2)
At x=L, C=C’out .... (3)
Sub (2) into equation (1)
C’in = 𝐵
Sub (3) into equation (1)
𝐷 (𝐶 ′ 𝑖𝑛 −𝐶 ′ 𝑜𝑢𝑡)
𝐽𝑥 =
𝐿
Permeability
 The steady-state flux across a membrane is given by:
𝐽 = 𝑃(𝐶𝑖𝑛 − 𝐶𝑜𝑢𝑡 )
where the permeability, P, of a membrane is given by:
𝐾𝑚/𝑤 𝐷𝑚
𝑃=
𝐿
where
𝐾𝑚/𝑤 = equilibrium membrane-water partition coefficient of the drug
𝐷𝑚 = diffusion coefficient of the drug in the membrane
𝐿 = thickness of membrane
𝐶
barrier
in
𝐶𝑖𝑛
out
𝐿
𝐾𝑚/𝑤 𝐶𝑖𝑛 =C’in
𝐶𝑜𝑢𝑡
𝐾𝑚/𝑤 𝐶𝑜𝑢𝑡 = C’out
𝑥
𝐶𝑎
𝐾𝑎/𝑏 =
𝐶𝑏
Concentration = 𝐶𝑎
Concentration = 𝐶𝑏
11
Permeability of human cell membrane
Molecular size is a good indicator
of the cell membrane permeability
W. M. Saltzman, “Drug delivery” Oxford University Press (2001)
12
Permeability of human cell membrane
Oil/Water partition coefficient
is a good indicator of the cell
membrane permeability
W. M. Saltzman, “Drug delivery” Oxford University Press (2001)
13
Example 2: Diffusion through cell membrane
(Adapted from Chapter 2 Question 14 from recommended text: Biomedical Engineering by Saltzman)
Estimate the steady-state flux (mg/cm2/s) by diffusion of a steroid through a lipid bilayer membrane.
The diffusion coefficient for steroid in the lipid bilayer is 10-14 cm2/s, and the concentration is 1 ng/ml on the
outside of the membrane (assuming 0 on the inside in the beginning).
State all your assumptions explicitly
 How will the flux change if the steroid is replaced by an antibody (one type of protein drug)? Give a
qualitative answer and provide your reasoning.

If the diameter of the cell is about 10 m and the duration of the observation is 100 s. Is it reasonable to
assume that the concentration of the steroid drug inside the cell is negligible? How to justify your answer?
14
Answer

Assume the partition coefficient between lipid bilayer membrane (m) and surrounding solution(s) is unity.
𝐾𝑚/𝑠 =1 {Accept any reasonable assumption}
Assume steady-state such that the concentration profile in the membrane is linear.
 Assume the bulk concentration, that is the concentration on the outside of the membranes (𝐶1 ) is
maintained constant and the concentration inside the membrane is negligible (𝐶2 ). {We will check this
assumption later}
 Thickness of membrane, L = 4 nm = 4 x 10-7 cm (The acceptable value from 2 to 10 nm. Kuchel, et.al.
Theory and Problems of Biochemistry, 1988).

1). Apply Fick’s Law
𝐷
𝐿
𝐽 = 𝐾𝑚/𝑠 (𝐶1 − 𝐶2 )
Given D = 10-14cm2/s ; 𝐶1 = 1 ng/ml = 1 ng/cm3 ;
Assume 𝐶2 = 0 in the time frame considered; 𝐾𝑚/𝑠 =1; L = 4 x 10-7 cm
The calculated diffusive flux, J = 2.5 x 10-8 ng/cm2/s
15
Answer
2) Antibody, typically having a molecular weight of 100 kDa, is orders of magnitude larger in size than
steroid (about 400 Da). According to the estimate of Stokes-Einstein equation, diffusion coefficient is
inversely proportional with size. And since flux is proportional to diffusion coefficient, the flux is expected
to be orders of magnitude lower for the antibody.
Recall the following:
k BT
C
Fick’s Law J x = − D
Stokes-Einstein equation D = 6r
x
3). Consider a cell with 10 m diameter placed in the medium.
Surface area (SA) = 4  (radius of cell)2 = ~3 x 10-6 cm2
Volume of cell = 4/3  (radius of cell)3 = ~5 x 10-10 cm3
Amount inside the cell = Flux × surface area x time = 7.5 x 10-12 ng
Over 100 s, C2= Amount inside cell / volume of cell = 0.015 ng/ml
Since C1 =1 ng/ml >> C2, the value of (C1-C2) can be approximated by the
value of (C1-0).
16
Polymer-based controlled drug delivery systems
https://www.youtube.com/watch?v=O3KccVX574s
17
Science 249:1527-1533, 1990
Example 3: Subcutaneous polymeric implant
An anti-AIDS drug is delivered by a subcutaneous (under-theskin) polymeric implant. The drug is dissolved in the polymer
and slowly diffuses into the bloodstream.
 The drug has a half-life of 2 h due to liver clearance (first order
reaction). The implant is a thin disc with radius 1 cm, releasing
drug from both sides of the disc. For the first few weeks, the
flux of drug from the implant is constant at 0.06 mg cm-2 s-1.
1) What is the maximum achievable amount of drug in the blood
stream?
2) How long does it take to reach 99% of this level after
implantation?

18
Answer
19
ln
Example con’t (determine diffusion from permeation)

3). The disc has polymeric membranes on both sides as a drug reservoir. Drug concentration remains at
10mg/ml inside during the initial phase. Given that the membrane thickness is 1mm. What is the
diffusion coefficient of the drug through the polymeric membranes? State your assumptions.
20
Nanomedicines: More effective and less harmful
Nano size is the scale to cross barriers in human body and interact with DNA or small
proteins at different levels, in blood or within organs, tissues or cells.
 Sustained release
 Minimize the number of injections while keeping the drug concentration in the
therapeutic range
 Targeted delivery
 Cause the drug to accumulate preferentially at the disease site
 Maximize therapeutic effect while minimizing adverse side effects
 Chemical / biological engineers develop materials and carriers which are essential for the
deployment of nanomedicines

https://etp-nanomedicine.eu/about-nanomedicine/what-is-nanomedicine/
21
Engineering Nano-drug carriers: polymeric nanoparticles
The evil twin… virus
Another nanoparticle: Influenza virus
Prabhu, R. et.al. International Journal of Nanomedicine, 2015
22
http://www.sciencedirect.com/science/article/pii/S0025619611603121
COVID-19 vaccines: delivery of mRNA using nanocarriers
Ref: Chung et al. COVID-19 Vaccine Frontrunners and Their Nanotechnology Design, ACS Nano 2020
Manufacturing mRNA vaccine 2’
https://www.youtube.com/watch?v=QAWsINx9C4A
23
Engineered barriers in drug delivery systems
Membranes
Gels
Lipid layers
http://www.sciencedirect.com/science/article/pii/S0301462204001814
24
http://en.wikipedia.org/wiki/File:Liposome.jpg
Example on transdermal delivery using chemical enhancers
To replace needle injection, a chemical enhancer is proposed for delivering drugs through the skin by
increasing the diffusion coefficient across the skin barrier. The inventor claims that the drug can reach the
body within a few minutes after application.
 Consider a skin barrier of thickness of 500 𝜇𝑚, what is the best order-of-magnitude estimate of the
diffusion coefficient of the drug in 𝑐𝑚2 /𝑠?

“Percutaneous Penetration Enhancers Chemical Methods in Penetration Enhancement” by Dragicevic and Maibach, Springer 2015
25
Solution
 The diffusion length scale and time scale is related through the diffusion
coefficient:
tD = L / D
2
L = approximate displacement of diffusing
molecule after time tD
Therefore, an estimate of D can be obtained by (taking “a few” minutes to be 3
minutes):
2
𝐿2
500 𝜇𝑚 2
𝜇𝑚2
𝑐𝑚
𝐷= =
≈ 1400
= 14 × 10−6
𝑡
180 𝑠
𝑠
𝑠
O𝑛 𝑡ℎ𝑒 𝑜𝑟𝑑𝑒𝑟 𝑜𝑓 10−6 𝑐𝑚2 /𝑠
Diffusion coefficients: small molecules in water are around 10-5 cm2/s, and globular proteins 10-6 cm2/s.
26