Antibody/Cell Binding and Magnetic Transport in a
Microfluidic Device
A Thesis
Presented in Partial Fulfillment of the Requirements for the Degree Master
of Science in the Graduate School of The Ohio State University
By
Shauna Adams, B.S.
Graduate Program in Mechanical Engineering
The Ohio State University
2013
Master’s Examination Committee:
A.T. Conlisk, Advisor
Derek Hansford
Shaurya Prakash
Abstract
The advancement of micro-total analysis systems is increasing the ability to perform
multiple functions in one microfluidic device. These systems have several advantages in
biomedical applications, including lower equipment and personnel costs, reduced power
requirements, faster separations, and smaller sample and reagent volume requirements.
Because of this, there has been a growing interest in the study of particle motion in microand nanochannels. The Automated Cell to Biomolecule Analysis (ACBA) device labels,
sorts, and analyzes cancer cells in an enriched blood sample. Understanding transport
methods and the behaviour of particles in microfluidic lab-on-a-chip or micro-total analysis
systems is important to the advancement of system applications.
Stochastic equations are used to characterize magnetic microbead motion, and the effect
of the Stokes drag force on magnetic microbeads is analyzed. Stochastic differential equations specifically the Langevin and Fokker-Planck equations are used to characterize the
probability density distribution of a microbead in a microfluidic channel. The results show
that due to the size of the microbead these equations the binding probability of a magnetic
microbead to a circulating tumor cell. However the results do indicate that under different
conditions the probability equation can be used if the Peclet number, P e = 10 − 100.
The capture area of a magnetic microbead with and without a cell attached to it is
analyzed to see if the effect of the change of Stokes drag force on the magnetic microbead
effects the magnetic capture area. It is concluded that the added drag from the attached
ii
circulating tumor cell does in fact affect the capture area of the magnetic microbead in the
presence of a magnetic with a constant magnetic field.
iii
Acknowledgments
First and foremost I will like to thank God, my spiritual source of encouragement,
strength, and perseverance.
I will also like to offer my sincere thanks to my advisor A. T. Conlisk for his support and criticism. I thank him for inviting me to join his research team and providing
me with the opportunity to work on a project that will effect the lives of many people to
come. His patience, encouragement, and willingness to explore new concepts has made
this experience one of absolute growth. I will also like thank my thesis committee members, Professor Hansford and Professor Prakash for their feedback, suggestions, and time
throughout these two years. Thank you to my team members, Cong Zhang , Harvey Zambrano, and Zhizi(Jessica) Peng in the Computational Nano and Microfluidics Laboratory,
especially Cong for all of your help, insight, and willingness to explain a topic over and
over until I understood it.
I want to thank my family and friends for everything they have done and said over the
past two years. I would especially like to thank my mom, Delores, and my sister, Imani for
pushing and motivating me along this journey. I would also like to acknowledge my dad,
Melvin Sr. who was always one of my greatest cheerleaders.
Lastly, thank you Sandia National Laboratories for providing me with this opportunity
to obtain my master’s degree on your time and money. Thank you to the Nanoscale Science
and Engineering Center for the research resources. Thank you to GEM.
iv
Vita
June 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phoebus High School
May 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S Mechanical Engineering
North Carolina Agricultural and Technical State University
Greensboro, NC
May 2011 to Present . . . . . . . . . . . . . . . . . . . . . . . . . Sandia National Laboratories
Technical Engineer
May 2011 to Present . . . . . . . . . . . . . . . . . . . . . . . . . Sandia National Laboratories
Masters Fellowship Program and GEM
Fellow
September 2011 to Present . . . . . . . . . . . . . . . . . . . Graduate Research Fellow
Computational Nano and Microfluidics
Laboratory
Columbus, OH
Publications
Adams, Shauna, Zhang, Cong, Zambrano, Harvey, Conlisk, A.T. January 2013. Antibodyantigen Binding in a Flow-through Microfluidic Device, talk presented at the 51st AIAA
Aerospace Sciences Conference, Grapevine, TX, Chapter DOI: 10.2514/6.2013-1114.
Fields of Study
Major Field: Mechanical Engineering
Studies in Microfluidics: A. T. Conlisk
v
Table of Contents
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
1.
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
2.
Microfluidics . . . . . . . . . . . . . . . . . . . . .
Electrolyte Solutions and the Electric Double Layer
Poiseuille Flow . . . . . . . . . . . . . . . . . . . .
Electrokinetic Phenomena . . . . . . . . . . . . . .
Automated Cell to Biomolecule Analysis System . .
Magnetically Induced Flow . . . . . . . . . . . . .
Biophysics of the Biological Cell . . . . . . . . . .
Present Work . . . . . . . . . . . . . . . . . . . . .
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Electroosmosis and Electrophoresis . . . . . . . . . . . . . . . . . . . . . . . 20
2.1
2.2
2.3
2.4
Introduction . .
Electroosmosis
Electrophoresis
Summary . . .
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20
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3.
Stochastic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1
3.2
3.3
4.
4.4
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35
36
38
39
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. 44
Introduction . . . . . . . . . . . . .
Motion of a Circulating Tumor Cell
Binding Probability . . . . . . . . .
4.3.1 Results . . . . . . . . . . .
Summary . . . . . . . . . . . . . .
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Magnetic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.1
5.2
5.3
5.4
5.5
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Binding Probability of Magnetically Labeled Antibody Binding and Circulating Cancer Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1
4.2
4.3
5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Markov Process . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . .
3.1.4 Derivation of the Fokker-Planck Equation . . . . . . . . . . . .
Fokker-Planck Equation Solved for Binding Probability . . . . . . . .
3.2.1 The Wiener Process . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Non-Dimensional Transition Probability Density and Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Definition of the Magnetic Field . . . . . . . . . . . . . . . . . .
5.2.2 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Electric and Magnetic Dipoles . . . . . . . . . . . . . . . . . . .
5.2.4 The Force Induced by a Magnetic Material on a Magnetic Particle
Magnetically Induced Cell Transport . . . . . . . . . . . . . . . . . . .
Characterization of the Magnetic Field . . . . . . . . . . . . . . . . . .
5.4.1 Trapping Efficiency of a Magnetic Bead . . . . . . . . . . . . .
5.4.2 Trapping Efficiency of a Magnetic Bead Bound to a Cell . . . . .
5.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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58
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Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
vii
Appendices
76
A.
MATLAB Calculation of Important Constants . . . . . . . . . . . . . . . . . . 76
B.
MATLAB Code for Transition Probability Density and Probability Density . . 79
C.
MATLAB Code for Magnetic Bead Capture at Different Alpha and y Positions
83
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
viii
List of Tables
Table
Page
1.1
Summary of important parameters in the mixing and labeling stage of the
ACBA System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2
Summary of breast cells, cancerous and non-cancerous . . . . . . . . . . . 17
3.1
Parameters for calculating . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1
Test parameters for binding probability for ACBA system. . . . . . . . . . 50
ix
List of Figures
Figure
1.1
Page
This lab-on-a-chip device was created to detect HIV infected cells from a
blood sample. Research being conducted at UC Davis by Prof. Alexander
Revzin. Photo from http://www.bme.ucdavis.edu/articles/2010/08/12/alexrevzin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The electric double layer (EDL) is made up of the diffuse layer and the
Stern layer. The Stern layer is comprised of only counter-ions of the wall
charge. This layer is immobile.The diffuse layer, though majority counterions also contains some co-ions. These co-ions have the same charge of the
wall. It is these mobile ions in the diffuse layer that move in the presence
of an electric field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Schematic of 2-D Poiseuille flow in a rectangular channel using equation
(1.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.4
Overall schematic of the entire ACBA device. . . . . . . . . . . . . . . . .
9
1.5
CAD representation of entire ACBA System. Stage 1 is where magnetic
beads coated in antibodies bind with circulating tumor cells through antibody/antigen binding as they are mixed together using chaotic mixing.
Stage 2 captures and transports the magnetically labelled cells and magnetic beads using magnet disk arrays controlled by external electromagnets.
Stage 3 uses an aqueous solution and oil to encapsulate individual cells for
mRNA analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6
A magnetic field is created when an unpaired electron is orbiting the nucleus of an atom. When an electron is paired no magnetic field is created. The direction of spinning of the elctron determines the direction if
the magnetic field. Photo from http://www.ndt-ed.org/EducationResources
/HighSchool/Magnetism/reviewatom.htm . . . . . . . . . . . . . . . . . . 13
1.2
1.3
x
1.7
Size comparison chart of circulating tumor cells to red and white blood
cells. Though actual cells range in size, CTC are normally smaller than the
actual tumor cells. Picture recreated from (Kim et al., 2012) . . . . . . . . 16
1.8
Stained MCF-10 breast cells. Photos from Sung et al. (2009) and http://www.
mcb.arizona.edu/azcc/confocal/examples.htm . . . . . . . . . . . . . . . . 17
1.9
Stained MCF-7 breast cancer cells. Photos from (Ehrhart et al., 2008) and
http://www.rndsystems.com . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.10 MDA-MB-231 breast cancer cells. Photos from atcc.org . . . . . . . . . . 18
2.1
Velocity profile of electroosmotic flow. A curved velocity profile exist
within the electric double layer to account for the no-slip boundary condition and the change in potential. In the bulk fluid the fluid is charge
neutral so the potential is equal to 0. This creates a vertical velocity profile
within the bulk fluid. The bulk fluid maintains the velocity of the flow at
the EDL boundary to satisfy the no slip condition. . . . . . . . . . . . . . . 23
2.2
Electrophoresis takes place when an electrically charged particle is placed
in an electric field. The particle moves in the field. This can take place in a
stationary fluid or relative to the fluid bulk velocity. . . . . . . . . . . . . . 24
3.1
Brownian motion of the antibody that collides with the fluid molecules. . . 30
3.2
Representation of the antibody position from x0 to x from time t0 to t. . . . 39
3.3
The transition probability, P, shows the probability density of particle being
at position x at time t in an unbounded region relative to its initial position
x = x0 at t = t0 , D =1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4
The probability density of an particle being at position x at time t in an
unbounded region, D = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5
The transition probability shows the probability density of a particle being
at position x at time t in a region with no flux boundary on one end given
its initial position x0 = 0 at t0 = 0, D= 0.01. . . . . . . . . . . . . . . . . . 42
3.6
The probability density of a particle being at position x at time t in a region
with no flux boundary at one end for D=0.01. . . . . . . . . . . . . . . . . 43
xi
4.1
Due to the size of the circulating tumor cell (yc ∼ h), the velocity of the cell
can be determined by using the average velocity equation for the Poiseuille
flow of the bulk fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2
The limits a and b defined in the probability equation to find the antibody/bead to CTC/antigen binding probability when a circulating tumor
cell is at position, x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3
The probability density curves from equation (3.39) at different time instances for a 161 node simulation with the conditions described in Table
4.1 including a Peclet number, P e = 3.36 × 106 . . . . . . . . . . . . . . . 50
4.4
These plots show the dimensionless probability density, W ∗ , versus the
dimensionless position, x∗ , at different dimensionless time, t∗ , values for
different Peclet numbers. a) Pe = 10−2 , b) Pe = 10−1 , c) Pe = 1, d) Pe = 10,
e) Pe = 102 , f) Pe = 103 , g) Pe = 104 , h) Pe = 105 . . . . . . . . . . . . . . . 52
5.1
Dipole diagram of magnetic disk in microfluidic system. . . . . . . . . . . 55
5.2
Geometry for calculating the electric potential field due to a dipole at the
point P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3
Geometry for calculating the magnetic field due to a magnetic dipole at the
point P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.4
Electromagnets and solenoid control magnetic fields on magnetic disk. Picture from (Henighan et al., 2010) from NSEC Faculty Dr. Sooryakumar’s
group here at OSU, who investigates the use of “magnetic tweezers” in
microfluidic systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.5
Particle moving along magnetic disk array in a process known as “magnetic
tweezers”. Picture from (Yellen et al., 2007) at Duke University. . . . . . . 64
5.6
The x ranges for a particle’s motion at y = 1 and at y = 0 when y0 = −1,
γ = 0.33, β = −1, and α = −0.5. . . . . . . . . . . . . . . . . . . . . . . 69
5.7
The x range for a particle’s motion at y = 1, y0 = −1, γ = 0.33, β = −1,
and α = −0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
xii
5.8
The x range for a particle’s motion at y = 1, y0 = −1, γ = 0.33, β = −1,
and α = −0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.9
The x range for a particle’s motion at y = 1, y0 = −1, γ = 0.33, β = −1,
and α = −5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
xiii
Chapter 1: Introduction
1.1
Microfluidics
The desire to study and design small scale fluid systems has increased over the last
twenty-five years. With the advancement of Micro-Electro Mechanical Systems (MEMS)
taking off in the early to mid 1980’s, the questions started to emerge on the ability and
value of shrinking fluid systems.
The application of small scale micro and nano fluid systems is vastly diverse and can be
used over a scope of fields. In the medical community lab-on-a-chip technology allows lab
diagnostic processes that took days to process, several hundreds of thousands of dollars,
and many pieces of equipment now can be performed in a system barely visible to the
human eye in minutes or hours and at a cost more accessible to a larger number of people.
The design of microvalves, micropumps, micromixers, and other fluid mechanical systems are allowing engineers and scientist to create systems very similar to their macro-scale
counterparts. Due to the complexity of many micro and nano-fluidic systems and their
uses, molecular biology, chemistry, materials, and engineering knowledge are all being
combined. From water purification systems, to artificial organs, and bio-chemical detection systems the possibilities and capabilities of the uses and applications of micro and
1
Figure 1.1: This lab-on-a-chip device was created to detect HIV infected cells from a blood
sample. Research being conducted at UC Davis by Prof. Alexander Revzin. Photo from
http://www.bme.ucdavis.edu/articles/2010/08/12/alex-revzin
nano-scale fluid systems is nearly limitless (Nguyen & Wereley, 2002). This thesis will describe and analyze different particle transport processes within a lab-on-a-chip microfluidic
device that is used to detect cancer from a person’s blood sample.
A microfluidic system is defined as any system where a critical dimension is either
on the micro scale or between a range of 1-100 µm. Macro systems most commonly are
pressure driven whereas in micro systems electric and magnetic transport is most commonly used. As shown in Figure 1.1 as the critical dimensions decrease on the micro and
nano scale the pressure drop needed to sustain a reasonable flow rate drastically increases,
whereas the voltage needed to sustain a reasonable flow rate slightly increases as the critical
dimensions decrease (Conlisk, 2013).
2
It should be mentioned here that the equations that govern the behaviour of fluids are
still applicable on the micro scale and down to about a minimum dimension of six nanometers on the nano scale (Zhu et al., 2005). This means that energy, momentum, and mass
are still conserved and the continuity and Navier-Stokes equations can be used.
1.2
Electrolyte Solutions and the Electric Double Layer
Aqueous electrolyte solutions are the primary solutions used in many microfluidic devices. These solutions have the ability to conduct electricity thus making them ideal for
electrically induced flow explained later in the chapter. Some examples of commonly used
electrolyte solutions are sodium chloride (NaCl) and water and potassium chloride (KCl)
and water. Both solutions are considered strong electrolytes and good conductors of electricity (Wright, 2007).
An electrolyte solution is created when ionic salt is added to a solvent, in many cases
water. Due to thermodynamic interactions of the molecules the salt dissociates in the solvent. This process creates positive and negative ions in the solvent; in the case of sodium
chloride the sodium and the chloride disassociates into monovalent ions N a+ and Cl− .
The valence, z, of the ion indicates how many electrons an ion has gained or lost and is
normally identified by the whole number in front of the + or - sign. The valence in N a+
is positive one and the valence in Cl− is negative one (Wright, 2007). For example NaCl
completely dissociates in water according to the reaction
N aCl → N a+ + Cl−
(1.1)
In this reaction equation the sodium, N a, molecule has given its one electron in its outer
most shell to chloride, Cl, thus denoting the + sign and chloride received an electron in its
3
outter most shell thus denoting the − sign. The sodium ion is now positively charged and
the chloride ion is now negatively charged thus making the creation of an electric double
layer possible.
The electric double layer (EDL) develops in a channel when a dissociated electrolyte
solution makes contact with the charged walls of the channel. A thin layer of oppositely
charged ions called counter-ions, from the electrolyte solution, will attract to the wall surface. This layer of ions is called the Stern layer and is assumed to be immobile at all times.
The diffuse layer, also known as the Gouy-Chapman diffuse layer, is the layer right above
the Stern layer (Devasenathipathy & Santiago, 2005). This layer has both co-ions, ions that
have the same charge as the channel wall, and counter-ions. The counter-ions outnumber
the co-ions in this layer making the overall layer the charge of the counter-ions. It is this
layer that becomes mobile in the presence of an electric field (Devasenathipathy & Santiago, 2005). The Stern layer and the diffuse layer make up the EDL. The remainder of the
ions are located in the bulk fluid. Figure 1.2 describes the different layers.
The primary length scale of the electric double layer is calculate by,
λ=
√
e RT
1
FI 2
(1.2)
and λ is commonly known as the Debye length (Conlisk, 2013). In this equation e is
J
the electrical permittivity of the medium, R is the universal gas constant (8.314 mol·K
), T
C
is the temperature, F is Faraday’s constant (96, 485 mol
), and I is the ionic strength. The
P 2
ionic strength is defined by, I =
zi ci where, zi is the valence of species i and, ci is the
i
concentration of the electrolyte species at a specific location. This equation is only true
when the species are monovalent and the concentrations are equal (Conlisk, 2013).
4
Figure 1.2: The electric double layer (EDL) is made up of the diffuse layer and the Stern
layer. The Stern layer is comprised of only counter-ions of the wall charge. This layer
is immobile.The diffuse layer, though majority counter-ions also contains some co-ions.
These co-ions have the same charge of the wall. It is these mobile ions in the diffuse layer
that move in the presence of an electric field.
5
1.3
Poiseuille Flow
A common way to transport fluid in a fluidic system is to use a pressure difference
across the system. Fluid travels from highest pressure to lowest pressure which makes
transport method ideal. The movement of fluid using a pressure gradient is known as pressure driven flow or Poiseuille flow. In a system where fluid is viscous and incompressible
the Navier Stokes equation can be used to characterize the fluid flow. The Navier Stokes
equations in rectangular coordinates for a channel where W >> h can be written as,
2
∂ux
∂ux
∂p
∂ ux ∂ 2 ux
∂ux
+ ux
+ vy
=−
+µ
+
ρ
∂t
∂x
∂y
∂x
∂x2
∂y 2
2
∂uy
∂ uy ∂ 2 uy
∂uy
∂uy
∂p
ρ
+ uy
+ uy
+µ
+
=−
∂t
∂x
∂y
∂y
∂x2
∂y 2
(1.3)
(1.4)
Under the following assumptions equations (1.3) and (1.4) can be simplified a great deal.
It is assumed that the fluid is only moving in the x direction as a function of y, the fluid
flow is fully developed, and the fluid is in a steady state. The coordinate reference can be
viewed in Figure 1.3. The Navier-Stokes equations reduce to
2 ∂ ux
∂p
+µ
0=−
∂x
∂y 2
∂p
0=−
∂y
(1.5)
(1.6)
∂p
Since − ∂y
= 0 the pressure drop is only dependent on x. The velocity can be calculated
using the following boundary conditions, the no-slip boundary condition at the walls and
due to symmetry of the system
dux
dy
= 0 at y=0 and
dp
dx
= − ∆p
. With these conditions in
L
place the velocity becomes,
ux =
1 ∆p 2
y − h2
2µ L
6
(1.7)
Figure 1.3: Schematic of 2-D Poiseuille flow in a rectangular channel using equation (1.3).
here µ is the fluid viscosity, ∆p = p2 − p1 , L is the length L − L0 , and h is half the height
of the channel. This parabolic flow is known as Poiseuille flow and can be seen in Figure
1.3.
The pressure driven volumetric flow rate through a rectangular channel can be calculated by taking the integral of the velocity in terms of the length and width of the channel,
Zh ZW
Q=
ux dzdy
−h 0
Q=−
2W h3 ∆p
3µ L
(1.8)
In the case where L stays the same and h and W decrease it can be seen in equation (1.8)
that in order to keep the same flow rate the pressure will have to significantly increase. It
is because of this fact that alternative methods to induce flow are at times considered for
microfluidic and nanofluidic systems.
7
1.4
Electrokinetic Phenomena
As mentioned in the previous section there are alternate methods to transport fluids
in microfluidic systems. One other method of transporting fluid is through electrokinetic
transport. Electrokinetic transport occurs when an electric field causes or is created by the
motion of a fluid or particle in a fluidic system. There are three main electrokinetic phenomena that occur in microfluidics, they are electroosmosis, electrophoresis, and streaming
potential. Sedimentation potential is also an electrokinetic phenomena, but it will not be
discussed in detail in this thesis (Conlisk, 2013; Devasenathipathy & Santiago, 2005).
The three electrokinetic phenomena may be defined as follows.
• Electroosmosis occurs as a result of an electric field introduction to a system that
causes the counter-ions in the diffuse layer of a microfluidic system to flow in a
certain direction causing bulk motion of the fluid in the channel.
• Streaming potential is caused when a pressure gradient creates an electric field in
electrolyte solutions causing a potential along the charged wall when current is not
introduced.
• Electrophoresis occurs because of the particles suspended in the fluid in the channel.
If an electric field is introduced to charged particles within a solution the particles
move in a certain direction causing fluid flow. This can occur in solutions that are
not electrolyte solutions
These electrokinetic phenomena indicate that in the presence of an electric field, motion
of either a charged particle or the bulk fluid electrolyte can occur. The steaming potential
however creates an electric potential along the surface of the channel wall where there was
8
Figure 1.4: Overall schematic of the entire ACBA device.
not one due to motion of a fluid. Electroosmosis and electrophoresis will be discussed in
the next chapter of the text.
1.5
Automated Cell to Biomolecule Analysis System
The Automated Cell to Biomolecule Analysis (ACBA) device is being developed by
the Nanoscale Science and Engineering Center (NSEC), Center for the Affordable Nanoengieering of Polymeric Biomedical Devices (CANPBD) and is a micro total analysis
system (µ-TSA) system designed to detect and analyze circulating tumor cells (CTCs) in a
blood sample. The ACBA device is made up of four stages: labeling the cancer cells with
a magnetically tagged antibody; separation of these cells bound to the antibody; transport
and analysis. These stages can be seen in Figure 1.4. The fluid flow in the first stage of the
ACBA device is a pressure driven flow controlled by manual dispensing from a syringe.
One focus of this thesis is on the binding of the magnetically labeled antibodies to a circulating tumour cell. Table 1.1 shows the properties of the functions that make up the first
stage of the ACBA system.
Circulating tumor cells are cancer cells found in the blood stream indicating the presence of a tumor in the body. These cells are very rare as their presence is one CTC cell for
every 109 blood cells in a whole blood sample (Nagrath et al., 2007). The ACBA system
9
Figure 1.5: CAD representation of entire ACBA System. Stage 1 is where magnetic beads
coated in antibodies bind with circulating tumor cells through antibody/antigen binding as
they are mixed together using chaotic mixing. Stage 2 captures and transports the magnetically labelled cells and magnetic beads using magnet disk arrays controlled by external
electromagnets. Stage 3 uses an aqueous solution and oil to encapsulate individual cells for
mRNA analysis.
10
MCF-7 Cell
Diameter, d, µm
25
m2
−14
Diffusion Coefficient, D, s
1.67 × 10
Concentration, CC , cells
10
mL
Surface Potential, σ, mV
−20.32
Mass, mC , pg
208
Antibody Coated Magnetic Bead
Diameter, d, µm
2.8
m2
−13
Diffusion Coefficient, D, s
1.49 × 10
beads
Concentration, CA , mL
6 − 7 × 108
Mass, mA , pg
15
Bulk Fluid (PBS) Properties
µL
Flow Rate, Q, min
0.1
Viscosity, µ, P a · s
10−3
System
Inlet Demensions
length × width × height, µm 1370 × 200 × 30
Mixing Channel Dimensions
length × width × height, µm 2500 × 400 × 30
Table 1.1: Summary of important parameters in the mixing and labeling stage of the ACBA
System.
11
is designed to identify these cells using magnetically labeled antibodies that bind to the
cells in the mixing process. The blood samples used in the system are enriched so that the
possible CTC to blood ratio becomes 1:10,000. Chaotic mixing is used in the first stage of
the ACBA system on the two laminar flow streams. Fluid flow folding mixes the solution
containing the magnetically labelled antibodies with the enriched blood sample possibly
containing the CTCs.
Once the antibodies bind with the CTC cell they are detected using magnetic disk located in the second stage of the system. These magnetic disks are controlled by external
electromagnets. The control of these magnetic fields allow control of the magnetic particle
containing the CTC in two main dimensions, left/right and up/down. When these magnets
are turned on they capture the particle in the induced magnetic field pulling the particle
from the natural flow of the system. This process separates the CTCs from the remaining
cells in the system (Chen et al., 2013).
After the cells are captured they are transported along the array of magnetic disks to an
area where they can be transported individually for analysis, this encompasses Stage III of
the ACBA system. Using two-phase flow the CTCs are encapsulated for mRNA analysis.
Micro-RNA are messenger molecules that send genetic information between DNA and
ribosomes in a cell
The last stage of the ACBA system uses electrical systems and biosensing technology
to analyse the messenger RNA, mRNA, of the cell to correctly identify the cell based on its
biological fingerprint. An actual diagram of the ACBA system can be seen in Figure 1.5.
The first two stages of this system will be the main focus of this thesis. The fluidics,
physics, and applied mathematics determining the behavior of the fluid and the particles
within the fluid will the analysed and/or modelled in great detail in the following chapters.
12
Figure 1.6: A magnetic field is created when an unpaired electron is orbiting the nucleus of
an atom. When an electron is paired no magnetic field is created. The direction of spinning
of the elctron determines the direction if the magnetic field. Photo from http://www.ndted.org/EducationResources /HighSchool/Magnetism/reviewatom.htm
1.6
Magnetically Induced Flow
The use of magnets and magnetic particles are another way the control the behavior of
the fluid or a particle in the fluid of a microfluidic device. The ACBA system, defined in
Section 1.5, uses both magnetic particles and magnets to control the behavior of selective
particles in the device. Magnets in microfluidic systems serve several different functions.
They can be used to induce a bulk fluid flow, or to capture and transport particles and biological materials within a bulk fluid (Pamme, 2006). Both of these occur when a magnetic
material are in the presence of a magnetic field. The magnetic field is the area around a
magnet where a magnetic force is present. The magnetic field is created by the spinning of
unpaired electrons rotating around the nucleus of the atom as seen in Figure 1.6. The magnetic field is indicated by magnetic field lines that run from the two poles of the magnet.
This field can span the length of an entire system or be contained and controlled within a
small area.
13
There are two types of magnets. Those who produce their own magnetic fields and those
that produce magnetic fields as a result of an electric current. The latter is known most commonly as an electromagnet. The former can be either a permanent or temporary magnet.
Both types of magnets are used in microfluidic systems for a variety of functions. Magnets used to control magnetic material or magnetized fluids are utilized as pumps, switches,
valves, mixers, trappers, and sorters in microfluidic systems as detailed by Pamme (Pamme,
2006). Magnets can be very beneficial in systems specialized for isolation processes and
systems designed to use fluids with different electrical properties. The electrical properties
such as the conductivity of a fluid are not affected by the presence of an magnetic field.
Magnet use in microfluidic systems can be either active or passive. An active magnetic
system is attached directly to the microfluidic device and all the equipment needed to produce a magnetic field is attached to the system. This system either has permanent magnets
or electromagnets microfabricated into the design of the system. A passive magnet requires
the use of external magnets to control the magnetic particles in the fluid system.
Similar to electrically induced flow, magnetically induced flow takes place with the
introduction of a magnetic field to a bulk fluid concentrated with nanoparticles. The motion
of the bulk fluid caused by the direction of the magnetic fields is described as Hartmann
flow and will be discussed later in this thesis.
Magnetic particles have been used to sort, transport, and identify objects in a microfluidic system (Yellen et al., 2005). This method’s advantage over its counterparts of electrically and optically induced flow is that it does not harm cells and other living objects.
Other methods have the ability to heat up the system causing the living objects and cells to
become damaged or even die (Yellen et al., 2005).
14
1.7
Biophysics of the Biological Cell
The ACBA device is specifically designed to detect and analyze cancer cells found in
a blood sample. Cancer cells like many other cells are very complex. Cancer is the title
given to over a 100 diseases where mutations in a cell causes a cell to divert from its usual
proliferation and survival cycle (Alberts et al., 1998; Peng, 2011). The mutation causes
the cell to no longer follows the life cycle pattern it was designed to follow and instead
the gene that prompts cell production in an existing cell can become too active producing
cells in excess; this gene is called an oncogene (Alberts et al., 1998). The regeneration
process naturally established to replenish cells as other cells die is the catalyst for cancer
cell growth.
Cancer cells produce other cancer cells during cell division. The mutation found in the
original cell is often carried over to its progeny or child cells. A tumor forms when the
mutated cells cluster together to form a mass. If this mass is found in the part of the tissue
where it is suppose to be found it is benign or non cancerous to the living being. These
cells can spread to other parts of the body using the circulatory and lymphatic systems as
their carrier. It is at this point that the cancer cells become circulating tumour cells (CTCs)
(Mavroudis, 2010; Alberts et al., 1998). CTCs are the most dangerous of the mutated
cells since they have the ability to invade tissue where they are not suppose to be found and
continue to regenerate and form tumors. These cells prevent healthy cells around them from
producing identical healthy cells, and they destroy neighboring tissue. A size comparison
of CTCs to blood cells is in Figure 1.7. CTCs are normally smaller than the actual tumor
cells (Kim et al., 2012).
All cells have individual markers called antigens. Proteins called antibodies are produced by B cells, a class of white blood cells, and are designed to bind to antigen to identify
15
Figure 1.7: Size comparison chart of circulating tumor cells to red and white blood cells.
Though actual cells range in size, CTC are normally smaller than the actual tumor cells.
Picture recreated from (Kim et al., 2012)
a cell for destruction when the immune system considers it a foreign object (Alberts et al.,
1998). Each antigen has its own distinctive antibody. This means that an antibody will only
bind to an antigen it is designed to bind with. For this reason antibodies are widely used in
many microfluidic tagging or labelling applications.
The antibodies used for the ACBA microfluidic device are designed to detect three types
of human mammary epithelial cells. Each are a different type of breast cell line that is either
cancerous or non-cancerous. They include the MCF-10a, MCF-7, and the MDA-MB-231
breast cell line and breast cancer cell lines respectively (Peng, 2011). The MCF-10 cell
line are normal non-cancerous breast epithelial cells. They average in size between 20 30µm in diameter and have a surface potential of around -31.16 mV (Peng, 2011; Zhang
et al., 2009). One of the most tested and utilized breast cancer cell lines is the MCF-7
cancer cells. They are spherical in shape, range in size between 10 - 50 µm in diameter and
have a surface charge potential around -20.32 mV (Peng, 2011; Zhang et al., 2009). These
16
Figure 1.8: Stained MCF-10 breast cells. Photos from Sung et al. (2009) and http://www.
mcb.arizona.edu/azcc/confocal/examples.htm
cell characteristics are used for analysis throughout this thesis. The MDA-MB-231 breast
cancer line is a progressive strains of cancer. These cells are rod-like in shape and have a
surface potential between -24 to -31 mV (Peng, 2011; Trickler et al., 2008). Figures 1.8 1.10 provide visual representations of these different cells.
Table 1.2: Summary of breast cells, cancerous and non-cancerous
Cell
Size(µm)
Surface Potential (mV )
MCF - 10
20 − 30 (diameter)
−31.16
MCF - 7
10 − 15 (diameter)
−20.32
MDA-MB-231 10 × 70 − 90 (width × length)
−24 to −31
1.8
Present Work
This thesis will describe the different fluid and particle transport modes in microfluidic
devices including pressure driven flow, electroosmosis and electophoresis. The binding behavior of antibody coated magnetic beads and circulating tumor cells will then be analyzed.
17
Figure 1.9: Stained MCF-7 breast cancer cells. Photos from (Ehrhart et al., 2008) and
http://www.rndsystems.com
Figure 1.10: MDA-MB-231 breast cancer cells. Photos from atcc.org
18
Lastly, the trapping ability of a magnetic bead and a magnetic bead with cell attached will
be studied in detail.
In chapter 2 two types of electrokinetic forms of transport will be looked at in detail.
The velocity equations for electrophoretic and electroosmotic flow will be dervived, and
examples of how these two phenomena are used in microfluidic applications are then discussed.
In chapter 3 stochastic motion is analyzed in great detail. More specifically the Langevin
and Fokker-Plank stochastic equations are derived. The Langevin equation describes the
random motion of the magnetic bead, and the Fokker-Planck equation describes the transition probability density distribution of the magnetic bead along the length of the channel.
A specific form of the Fokker-Plank equation known as the Wiener Process is then applied
as a simplified model to describe the probability density of a simplified 1-D problem undergoing Brownian motion. In chapter 4 the probability density equation derived in the
previous chapter will be used along with the known motion of the larger circulating tumor
cell to predict the probability of a bead and cell binding at different times in the system.
In chapter 5 an analysis of magnetic fields will be conducted and the capture area of the
magnetic bead with and without a cell attached by an array of magnets will be looked at.
The concept of “magnetic tweezer” will also be discussed.
This thesis focuses on the use of stochastic differential equations to predict particle
binding in a microfluidic system as well as the study of the capture efficiency of a magnetic
bead with and without attached cells are both added knowledge in the field of microfluidics.
19
Chapter 2: Electroosmosis and Electrophoresis
2.1
Introduction
While the focus of this thesis is on the ACBA system which uses pressure driven flow,
electrokinetic phenomena are important for similar applications. As discussed in the previous chapter electrokinetic techniques are used in a variety of microfluidic systems. They
are used to pump and mix fluids as well as separate particles and microorganisms in electrolyte and neutral solutions (Haeberle & Zengerle, 2007; Nguyen & Wereley, 2002; Mitra
& Chakraborty, 2012; Conlisk, 2013). Electrokinetic techniques are ideal to use in microfluidic lab on a chip applications because they scale down well so their results are not
greatly affected as a system is miniaturized (Conlisk, 2013). They are also fairly easy in
integrate into micron-sized systems (Nguyen & Wereley, 2002). As noted in the previous
chapter there are three main types of electrokinetic phenomena found in microfluidic devices electroosmosis, electrophoresis, and streaming potential (Nguyen & Wereley, 2002).
In this chapter eletroosmosis and electrophoresis will be discussed in greater detail. The
disadvantage to the use of electrokinetic techniques is that the electric fields either used or
generated in the processes can negatively affect biomaterials in the case of lab-on-a-chip
applications, especially if heat is produced in excess (Yellen et al., 2005).
20
2.2
Electroosmosis
Electroosmosis is used to pump electrically charged fluids in several lab-on-a-chip applications (Conlisk, 2013). The ability to move electrolyte solutions using an electric field
is advantageous with the limited space afforded on all lab on a chip platforms. As mentioned above, the creation of the electric double layer (EDL) when electrolyte solutions and
naturally charged surfaces like silca and glass combine make it possible for electroosmotic
flow to occur (Conlisk, 2013; Nguyen & Wereley, 2002).
Electroosmotic flow is caused by movement of ions in the diffuse layer of the EDL in
the presence of an electric field. Due to the electroneutrality of the bulk fluid the velocity
of the bulk fluid is determined by at the boundary between the EDL and the bulk fliud.
This means that to fully understand the velocity profile of electroosmotic flow the velocity
profile has to be broken down into two sections: the velocity profile within the EDL and
velocity profile outside the EDL.
To calculate the velocity of a fluid in electrosomotic flow within a rectangular cross
area when the width of the channel is significantly larger than the height of the channel
some assumptions are made. First, the electric double layer is thin compared to the height
of the total channel. Secondly, the zeta potentials are equal at the walls. We also assume
the fluid flow is steady-state and fully developed and the velocity the velocity profile is
one-dimensional and only a function of y in the x-direction. The velocity can be calculated
from the Navier Stokes equations and Poisson equation, assuming a Boltzmann distribution
within the EDL (Conlisk, 2013; Nguyen & Wereley, 2002; Kirby, 2010).
The Navier Stokes equations are
ρ
∂u
+ u · ∇u
∂t
= −∇P + µ∇2 u + ρe E
21
(2.1)
and the Poisson equation is
e ∇2 φ = −ρe
(2.2)
By inserting Equation (2.2) into Equation (2.1) and applying the assumptions mentioned
above the N-S equation becomes
Ee ∇2 φ = µ∇2 u
(2.3)
Equation (2.3) can be solved by integrating both sides and plugging in the boundary conditions within the EDL, φ = ζ at y = 0 and
du
dy
=
dφ
dy
= 0 at y = ∞, ζ is the potential at the
shear plane between the Stern and diffuse layer. The velocity of the fluid due to the electric
field E within the EDL,is (Nguyen & Wereley, 2002)
ueof EDL =
e E(φ − ζ)
µ
(2.4)
In the bulk fluid φ = 0 due to its neutrality, so to satisfy the no slip condition boundary
condition the constant velocity of the bulk fluid has to be the velocity of the fluid at the
boundary. This means that the bulk fluid velocity outside the EDL is
ueof = −
e Eζ
µ
(2.5)
In the above equations u is the electroosmotic velocity of the fluid within the EDL or in the
bulk fluid, ρE is the electric potential, E is the electric field, µ is the fluid viscosity, and φ
is the potential.
The velocity profile in electroosmotic flow is “plug” like (Devasenathipathy & Santiago, 2005). This means that the majority of the profile for this flow is straight as seen in
Figure 2.1. This is ideal for many visualization applications due to uniformity and consistent distribution of particles throughout the system that is not obtainable with the parabolic
profile produced in pressure driven flow (Weigl et al., 2003).
22
Figure 2.1: Velocity profile of electroosmotic flow. A curved velocity profile exist within
the electric double layer to account for the no-slip boundary condition and the change in
potential. In the bulk fluid the fluid is charge neutral so the potential is equal to 0. This
creates a vertical velocity profile within the bulk fluid. The bulk fluid maintains the velocity
of the flow at the EDL boundary to satisfy the no slip condition.
Electroosmosis has several applications to lab-on-a-chip applications. One of the applications of electroosmotic flow is electroosmotic pumping. Electroosmotic pumping is
controlled by the switching on and off of an electric field. Takamura et al. (2003) performed
work on controlling pumping using low voltage portable sources and multiple sources in
one device (Takamura et al., 2003). Many devices require multiple voltage sources to control the flow of an electrolyte solution in different parts of a microfluidic device. Due to
the plug-like profile of electroosmotic flow Takamura et al. (2003) were able to control to
motion of fluids in a uniform manner. With the use of the low voltage sources embedded
within the system they were able to increase their pumping pressure by decreasing their
channel size.
23
Figure 2.2: Electrophoresis takes place when an electrically charged particle is placed in
an electric field. The particle moves in the field. This can take place in a stationary fluid or
relative to the fluid bulk velocity.
2.3
Electrophoresis
Electrophoresis is used in many lab on a chip applications (Haeberle & Zengerle, 2007;
Nguyen & Wereley, 2002). Since the velocity of a particle is determined by both its size
and charge strength, electrophoresis is ideal for separating different types of particles. It is
used to analyse DNA, tag biomolecules, and manipulate proteins.
Electrophoresis describes the motion of an electrically charged particle in the presence
of an electrical field relative to the motion of the bulk fluid. The bulk solution surrounding
the particle can be either electrically neutral, thus eliminating the creation of an electric
double layer at the walls of the channel, or an electrolyte solution. In the case where the
charged particle is in an electrolyte solution the particle will move relative to the bulk fluid
(Conlisk, 2013). An electric double layer will develop around the surface of the particle,
and the EDL on the charged particle induces the velocity of the particle in a microsystem
(Nguyen & Wereley, 2002; Conlisk, 2013). A representation of this flow can be seen in
Figure 2.2
24
In many cases electrophoresis is used in the analysis phase of many lab-on-a-chip devices. Because electrophoresis is the movement of a charged object relative to its surrounding in the presence if an electric field, its has the ability to control the movement of an object
based on its electrical properties and it is primarily used in separation of bio-material for
chemical and biochemical analysis (Ugaz & Christensen, 2007). There are four different
types of electrophoretic methods that can be used in microfluidic devices. They include
free solution electrophoresis, gel electrophoresis, isoelectric focusing, and micellar electrokinetic chromatography (Ugaz & Christensen, 2007). Free solution electrophoresis uses
the difference in charge strength to separate material in a microfluidic device. The stronger
the charge of the material the faster it will move through the system, thus causing a gradient
of material based on charge strength.
In gel electrophoresis the charged material travel through a gel like medium. This
medium allows the charged objects to not only be separated by charge but also by size. In
this case if two or more objects have similar charges but differ in size, the smaller object
will move faster through the system thus causing a size and charge gradient of particles
in the system. Isoelectric focusing uses a pH gradient to neutralize charged objects as
they travel through the system. Once charged material reach a particular pH the material
becomes charge neutral thus hindering its mobility through the system.
Assuming the Reynolds number is smaller than one and that the Debye length is much
much larger than the particle radius (λ a) the electrical force is equivalent to the Stokes
drag equation
qE = 6πµueph a
25
(2.6)
In equation (2.6), q is the total charge on the particle, E is the electric field, µ is the fluid
viscosity, ueph is the velocity of the particle due to electrophoresis, and a is the radius of
the particle. Solving for the electrophoretic velocity
ueph =
qE
6πµa
(2.7)
The total charge on the sphere is (Conlisk, 2013)
q = 4πe a
2
Z∞
d
dr
2 dφ
r2 2 dr
dr
(2.8)
a
2
= 4πe a ζ
1 1
+
λ a
(2.9)
This equation is derived by EDL calculations on the charged sphere. Inserting equation
(2.8) into equation (2.7) we find
ueph =
E
(4πe aζ)
6πµa
(2.10)
2Ee ζ
3µ
(2.11)
which simplifies to
ueph =
This form of the electrophoretic velocity equation is known as the Debye - Huckel equation
it can be used when λ a (Conlisk, 2013).
In the case where the radius of the particle is much larger than the Debye length (a λ)
the electrophoretic velocity is
ueph =
Ee ζ
µ
(2.12)
this equation is known as the Helmholtz-Smouluchowski equation and is derived using
the same principles to derive the velocity from electroosmotic flow equation (Nguyen &
Wereley, 2002). This equation can best be applied to particles larger than 100 nm in size
(Nguyen & Wereley, 2002)
26
An application of electrophoresis in a system include DNA analysis. Liu & Guttman
(2004) use electrophoresis to analyze DNA on a microchip (Liu & Guttman, 2004). Compared to the conventional method of analyzing DNA which uses different types of systems
and over several different steps, the entire DNA analysis process can be performed on one
microchip. As a result the reaction time between the reactants is faster, less material and
chemicals are used thus reducing cost, and results are obtained quicker. Electrophoresis
allows for different size DNA stands to to separated for analysis (Liu & Guttman, 2004).
2.4
Summary
In this chapter the two main electrokinetic phenomena, electroosmosis and electrophoresis, have been discussed in great detail. The velocity behavior for both phenomena was derived. Electrophoresis involves the velocity of a charged particle in a microfluidic system
in the presences of an electric field. Electroosmosis is the movement of bulk fluid in the
presence of an electric field. The velocity profile for this phenomena produces a vertical
velocity in bulk fluid making it ideal to use in studies involving visual detection. The practical applications of both of these phenomena in existing technologies was then described.
It was noted that electrophoresis and electroosmosis can occur in the same system and also
be combined with other processes.
27
Chapter 3: Stochastic Phenomena
3.1
Introduction
The instantaneous mechanical state of a system consisting of many particles described
by classical mechanics requires only the specification of a set of positions and momenta
of the particles. If the particles in the system are heavy enough compared to surrounding
molecules the classical mechanical approach will provide an accurate description of the
physical state of a system. Then the usual approach to describing the time evolution of
this mechanical state of such many-body system is by use of a coupled set of the Newton’s
equations, which describes the individual dynamics of all the particles in the system. This
approach has led to the development of the widely used numerical technique called molecular dynamics. Molecular dynamics simulations have provided numerous insights into the
behavior of fluidic systems at the nanoscale (Zambrano et al., 2009, 2012). However, in
microfluidic systems, the Brownian motion of particles with size of tens of microns, is
outside the reach of the molecular dynamics technique due to the typical long time and
large spatial scales inherent in microfluidics systems. Moreover, the deterministic problem
for more than a few particles exhibits non-deterministic, chaotic behavior (Conlisk et al.,
1989).
There are alternative methods to describe dynamics of particle motion which is in contrast with the deterministic approach. These are called the stochastic methods, which were
28
pioneered by Einstein (1956) and Langevin (1908), among many others. In this approach
a many body system is treated by using the equations of motion describing the dynamics
of only some selected particles moving in the presence of the other particles in the system
which are now regarded as a background or bath whose detailed dynamics is not explicitly
treated. In stochastic methods, the dynamic variables of interest in a many body system,
such as the position and velocity of a particle in a fluid, are discussed in great detail and
some other aspects of the problem are treated by theory of random processes. Methodologies of this type have been used to study the properties of fluidic systems ranging from
atomic and molecular liquids to colloidal suspensions and macroscale fluidic systems.
Brownian motion is an example of a stochastic process (Snook, 2007). In 1827, it was
first described in detail by Robert Brown, a botanist, while observing pollen under a microscope (Coffey et al., 2004). Eighty years later, Einstein combined the Maxwell-Boltzmann
distribution and the idea of random walk, a stochastic principle, to mathematically explain
Brownian motion as a diffusion equation (Coffey et al., 2004). Brown’s discoveries were
the early stages for what is now the Fokker-Planck equation, a partial differential equation
that combines Einstein’s diffusion equation and a deterministic term to obtain the probability density function of a stochastic process over time (Coffey et al., 2004; McKane,
2009). A few years later, Langevin was the first to derive a stochastic differential equation
to describe Brownian motion (Coffey et al., 2004). Called the Langevin equation, he used
Newton’s Law as the foundation of his reasoning (Coffey et al., 2004). It is the principle of
these equations that are the foundation for determining the motion behavior of the antigen
in the ACBA system.
The motion of the antibodies within the ACBA device is assumed to be driven by a
random force, which is the net effect of interaction forces between the antibody and the
29
surrounding fluid molecules. The time evolution of the motion is described probabilistically with a time-dependent random variable (Figure 3.1). The time revolution of the
random variable is the spatial position in one dimension x of the anti-body at different time
instances.
Furthermore, a Markov process is assumed to determine the antibody locations, meaning the particle position x, at time t, is determined by its previous position at x0 , at its
previous time t0 , and does not depend on the positions earlier than time t0 , as long as the
time interval ∆t = t − t0 is much smaller than the characteristic time of the process but
larger than the time interval between two successive collisions of the anti-body with the
surrounding molecules. This is the definition of a Markov process which will discussed in
the next section (McKane, 2009).
Figure 3.1: Brownian motion of the antibody that collides with the fluid molecules.
30
3.1.1
Markov Process
A Markov process is a process in which the present state of a system is determined
only by the state in the immediate past. For a Markov process, the probability density
function(pdf), W , depends only on the previous state For a general ith state then
Z
W (xi+1 , ti+1 ) =
P (xi , ti )P (xi+1 , ti+1 )dxi
(3.1)
where P is the transition probability. The transition probability is the probability that the
antibody will be at location xi+1 at time ti+1 given that it was at xi at a previous time ti .
Now for a Markov chain (McKane, 2009),
Z
P (xi+2 , ti+2 , xi , ti ) =
P (xi+2 , ti+2 |xi+1 , ti+1 ) × P (xi+1 , ti+1 |xi , ti )dxi+1
(3.2)
The variables W and P are both non-negative and if they satisfy (3.1) and (3.2) they define
a Markov process.
In many cases, numerical solutions can be used to model physical processes, as that
time and the dependent variable computed is a discrete set of numbers. A discrete Markov
process is called a Markov chain. In this case the integrals in equations (3.1) and (3.2)
become sums. A stationary process is one in which the trasition probability depends only
on t − t0 .
An example of a Markov chain is a random walk in one-dimension. The simplest
random walk is such that the “walker” must move at every time step. With “n” denoting
the step to emphasize the discrete nature of the Markov chain, the transition probability is
given by
Pnn0

 P
q
=

0
if n = n0 + 1
if n = n0 − 1
otherwise
31
with P + q = 1. If there are boundaries, then boundary conditions are required. If Pn
describes the transition probability at a boundary then Pn = 1 defines an absorbing boundary and Pn = 0 describes a reflecting boundary- meaning the particle can not stay at the
boundary.
In general a Markov chain is defined by the equation
P (n, t + 1) =
X
Pnn0 P (n0 , t)
(3.3)
n0
with P being the transition probability density. At large times assuming that the system
tends to a stationary or steady state
P (n) =
X
Pnn0 P (n0 )
(3.4)
n0
3.1.2
Langevin Equation
In general, particle motion or in this case antibody bead motion is governed by Newton’s Law ma = F and this is given by
dX(t)
d2 X(t)
= Fext (t) + β ū(X(t), t) −
+ R(t)
m
dt2
dt
where X(t) is the location of the particle at time t, and
dX(t)
dt
(3.5)
is the particle velocity up .
β = 6πµa is the Stokes drag coefficient, µ is the viscosity, a is the radius of the particle,
dX(t)
and m is the particle mass. Here Fext (t) is the external body forces, β ū(X(t), t) − dt
is the Stokes Drag force and R(t) is a random force. Here ū(X(t), t) is the local average
velocity of the fluid (Peng, 2011).
1
ū(X(t), t) =
V
Z
u(x, t)δd (x − X(t))dV
(3.6)
This equation including the random force is termed a Langevin equation; if R(t) = 0
the motion of the particle is deterministic. The variable R(t) is independent of x and varies
32
rapidly compared to the variation of x(t) (Coffey et al., 2004). Statistically the average of
all the random forces equal zero, hR(t)i = 0, where hi denote average. This can also be
shown as
1
hR(t)i =
t − t0
Zt
R(t)dt = 0
(3.7)
t0
After initial molecular collisions the R(t) value becomes independent of its previous value
and no correlations exist over time between the values. Using the delta-function to characterize this behavior (Barrat & Hansen, 2003; McKane, 2009)
hR(t)R(t0 )i = 2Dδ(t − t0 ).
where D is the diffusion coefficient,
(3.8)
kT
β
and δ is the Dirac Delta Function in a 2D system.
+ ∞ if x2 + y 2 = 0
δ(x, y) =
(3.9)
0
if x2 + y 2 6= 0
and
Z∞ Z∞
δ(x, y)dxdy = 1
(3.10)
−∞ −∞
The Langevin equation assumes the time scale of the collisions between the particle and
surrounding molecules is much shorter than the time scale for the velocity of the particle
(Barrat & Hansen, 2003). A deterministic analysis of particle motion is only valid if the
mass of the particle is large so that any oscillations due to variations in temperature are
negligible. The thermal velocity of the particle is given by
r
kT
vth =
m
and thus if the particle is “small” the thermal velocity is large and the deterministic approach is not valid. Here k is the Boltzmann constant, T is the thermal temperature of the
fluid, and m is the mass of the particles. To insure that the mean energy of the particle is
correct (i.e. E = 21 kT ), a random force is required.
33
The system of particles in the present case is that consisting of the water molecules, the
ions in the PBS solution, the antibodies and the biological cell. The sizes of the antibody
and the cell are provided in Table 1.5.
If we choose the velocity scale U0 as the average bulk flow velocity in a channel, then
the dimensionless velocity is given by ū∗ =
ū(X,t)
.
U0
The dimensionless particle displace-
ment in each of the coordinate directions is defined as X∗ = X/L where L is the channel
length. The dimensionless time is t ∗ = t/t0 with scale t0 = L/U0 .
Table 3.1: Parameters for calculating .
m
1.5 × 10−17 kg
U0
2.1 × 10−4
m
s
β
2.6 × 10−8
Ns
m
L
2400 × 10−6 m 2.2 × 10−4
We may also define the force scale βU0 = 6πµaU0 . Therefore all the forces now can
be represented by a dimensionless force ratio in the form of F∗ =
F
.
βU0
Equation (3.5) can
dX(t)
X(t)
= ū(X(t), t) −
+ Fext + R(t)
dt2
dt
(3.11)
be rewritten in a dimensionless form (the * is dropped) as
2d
where 2 =
mU0
.
βL
2
The values are in Table 3.1. Typically the coefficient ∼ 10−4 and so
the acceleration term is usually negligible. This means that the particle paths, apart from a
very short initial time period are governed by a first order differential equation
dX(t)
= ū(X(t), t) + Fext + R(t)
dt
(3.12)
which is subject to initial conditions on each particle. If there are N particles, there are 3N
spatial variables corresponding to the positions of each particle in three-dimensional space.
34
All of these variables are random variables because of the presence of the random force
R(t).
3.1.3
Fokker-Planck Equation
The Fokker-Planck equation has been applied to several biological systems in literature.
Plant et al. (1993) uses the Fokker-Planck equation to describe the probability of a certain
number of bonds at a particular time on a multivalent liposome during dissociation (Plant
et al., 1993). The Fokker-Planck equation is used to model the time-dependent bacteria
population size probability distribution for bacteria when fluctuated doses of immunoglobulin G (IgG) is administered in an effort to study the effect of bacterial infections under
varying IgG subsitution therapies(Figge, 2009). The general Fokker-Planck equation is an
equation of motion for a transition probability function, p that is a function of a collection
of random variables. Consider the Fokker-Planck equation in an N particle system; in general form, corresponding to the N variables x1 , ...xN the Fokker-Planck equation is given
by
"
#
N
N
2
X
X
∂
∂
∂p
= −
Ai (x) +
Bij (x) P
∂t
∂xi
∂xi ∂xj
i=1
i,j=1
(3.13)
where x = [x1 , ..., xN ], is the vector of random variables, in this case, the three-dimensional
location of the N antibodies, Ai is called the drift vector, and Bij is termed the diffusion
tensor,which is non-negative, definite and symmetric, and P is the transition probability of
the system. It is thus seen that the Fokker-Planck equation represents an equation of motion
for the transition probability of stochastically fluctuating quantities. The Fokker-Planck
equation can be derived from the Langevin equation and further details are available in
(Risken, 1984) and (Schuss, 1980). It should be pointed out that equation (3.13) is entirely
35
equivalent to the system of ordinary differential equations (Risken, 1984)
dxi
= Ai (x) f or i = 1, ...N
dt
(3.14)
where N is the total number of antibodies. Comparing this equation with equation 3.12 it
is seen that for a single particle in a single spatial dimension,
A = ū + Fext + R
where ū is the average fluid velocity in the immediate neighborhood of an antibody.
Consider the form of the Fokker-Planck equation for a single random variable, the
position in one dimension, of a single particle (i.e. antibody), x. Then we write the equation
for the transition probability density, P (x, t|x0 , t0 ), that a particle that starts at the point x0
and ends at the point x at time t later as
∂P
∂
1 ∂2
= − (A(x, t)P ) +
[(B(x, t)P )
∂t
∂x
2 ∂x2
(3.15)
Here A and B are defined by
1
A(x, t) = lim
∆t→0 ∆t
Z
1
∆t→0 ∆t
Z
∞
0
(x − x)P (x|x0 , ∆t)dx0
−∞
∞
0
(x − x)2 P (x|x0 , ∆t)dx0
B(x, t) = lim
−∞
where the scalars A and B have replaced the drift vector and the diffusion tensor respectively. Equation (3.15) needs to be solved subject to the initial condition P (x|x0 , 0) =
δ(x − x0 ) and with appropriate boundary conditions.
3.1.4
Derivation of the Fokker-Planck Equation
The Fokker-Planck equation can be derived from the Chapman-Kolmagorov equation
under certain conditions. This derivation comes from (McKane, 2009). A jump moment
36
for a given system is defined as
Z
Mj (x, t, ∆t) =
(ζ − x)j P (ζ, t + ∆t|x, t)
(3.16)
Consider the Chapman-Kolmagorov equation (3.1) in the form
Z
W =
P (x, t + ∆t|x0 , t)P (x0 , t)dx0
(3.17)
This is the continuous version of the equation
P (n3 , t3 |n1 , t1 ) =
X
P (n3 , t3 |n2 , t2 )P (n2 , t2 |n1 , t1 )
(3.18)
n2
Now, the integrand of (3.17), with x0 = x − ∆x can be written as
P (x, t + ∆t|x0 , t)P (x0 , t) = P ([x − ∆x] + ∆x, t + ∆t|x − ∆x, t)P (x − ∆x, t) (3.19)
Recall that a Taylor series in one dimension is defined by
∞
X f n (x)∆xn
∆x3
∆x2
+ f 000 (x)
+··· =
(3.20)
f (x + ∆x) = f (x) + f (x)∆x + f (x)
2
6
n!
n=0
0
00
Thus equation(3.19) is equivalent to the Taylor series
P ([x − ∆x] + ∆x, t + ∆t)P (x − ∆x, t) =
∞
X
(−1)j
j=0
j!
∆xj
jj
(P (x + ∆x, t + ∆t)P (x, t))
δxj
(3.21)
Integrating equation(3.17) over x0 = x − ∆x,
∞
X
(−1)j ∂ j
[Mj (x, t, ∆t)P (x, t)]
P (x, t + ∆t) =
j
j!
∂x
j=0
(3.22)
P (ξ, t|x, t) = δ(ξ − x)
(3.23)
Now,
so that
lim Mj (x, t, ∆t) = 0
∆t→0
37
for
j≥1
(3.24)
so that
Mj (x, t, ∆t) = Dj (x, t)∆t + O(∆t2 )
(3.25)
Thus
P (x, t + ∆t) =
∞
X
(−1)j ∂ j
[Dj (x, t)∆tP (x, t)]
j
j!
∂x
j=0
(3.26)
The j = 0 term is just P (x, t) since Mj (x, t, ∆t) = 1 so that,
∞
X (−1)j ∂ j
∂P
=
[Dj (x, t)P (x, t)]
j
∂t
j!
∂x
j=1
(3.27)
Equation(3.27) is the Kramers-Moyal expansion. In many cases the jump moment may be
neglected for j > 2 and the result of this truncation is the Fokker-Planck equation
∂
1 ∂2
∂P
= − (A(x, t)P (x, t)) +
(B(x, t)P (x, t))
∂t
∂x
2 ∂x2
(3.28)
The jump moments can be calculated by the following procedure. The stochastic variable,
x is the position of the antibody and its average is defined by
Z
hx(t)i =
xP (x, t|x0 , t0 )dx = x0
(3.29)
In particular for any function f
Z
hf (x(t))i =
f (x)P (f (x), t|x0 , t0 )dx
(3.30)
So that Mj (x, t, ∆t) = h(x(t + ∆t) − x(t))j i.
3.2
Fokker-Planck Equation Solved for Binding Probability
The Fokker-Planck equation is used to determine the transition probability of an antibody position at a given time, given its previous position at previous time instance. Assume
the antibody is at position x0 at time t0 , then the transition probability of the antibody at
position x after a time interval at time t is governed by the one-dimensional Fokker-Planck
38
Figure 3.2: Representation of the antibody position from x0 to x from time t0 to t.
equation (Sinaiski & Zaichik, 2008). If the time interval t−t0 = ∆t 1, A and B become
independent of t.
∂
1 ∂2
∂P (x, t | x0 , t0 )
= − (A(x)P (x, t | x0 , t0 )) +
(B(x)P (x, t | x0 , t0 )) (3.31)
∂t
∂x
2 ∂x2
3.2.1
The Wiener Process
As a first approximation and to gain a basic understanding of the transition probability distribution the classical diffusion equation will be solved. Using the initial condition
mentioned in the previous section, the boundary condition P → 0 as x → ±∞, and the
conditions A = 0 and B = 1, equation (3.31) can be rewritten as
∂P
1 ∂ 2P
=
∂t
2 ∂x2
(3.32)
The Fokker-Planck equation under these conditions is known as the Wiener Process (Schuss,
1980). The physical meaning of A and B are clarified by comparing the one-dimensional
Fokker-Plank equation (where A and B become scalars) with the molecular diffusion equation. Thus
B
2
corresponds to the diffusion coefficient, D, and the drift A corresponds to the
average velocity of antibody displacement. The transition probability, P , and the probability density, W , of this form of the equation is (Risken, 1984)
(x−x0 )2
1
−
P (x, t | x0 , t0 ) = p
e 4πD(t−t0 )
2 πD(t − t0 )
39
(3.33)
Figure 3.3: The transition probability, P, shows the probability density of particle being at
position x at time t in an unbounded region relative to its initial position x = x0 at t = t0 ,
D =1.
and
Z+∞
W (x, t) =
P (x, t | x0 , t0 )W0 dx0
(3.34)
−∞
where W0 is the initial distribution of the particles in the system.
Equations (3.33) and (3.34) explain the most basic one dimensional probability density
distribution of a particle experiencing Brownian motion. Figure 3.3 shows the plot of the
transition probability of a particle in an unbounded region. In this case, the initial condition
on the transition probability is W0 =
1
2
within the region [−1, 1], and zero elsewhere.Also,
D = 1. From the plot we can see that the position of the particle is a maximum in a
region about x = 0 and near t = 0. This can be understood because the initial position
of the antibody is x0 = 0 at t0 = 0. The distribution extends to x = ∞ and x = −∞,
40
Figure 3.4: The probability density of an particle being at position x at time t in an unbounded region, D = 1.
which corresponds to an unbounded region. Figure 3.4 shows the probability density of the
antibody after integrating over the space of all initial positions x0 .
In the microfluidic system of interest, the initial condition is similar to the case for the
unbounded problem; however the boundary conditions change to P → 0 as x → ∞ and
dP
dx
= 0 at x = 0. This means that the probability density flux vanishes for x = 0 to
take into account that the antibodies cannot move backwards in the system because of the
fluid flow. With the application of the new boundary conditions the transition probability
becomes
1
P (x, t | x0 , t0 ) = p
2 πD(t − t0 )
41
(x−x )2
0
− 4πD(t−t
e
0)
(x+x )2
+e
0
− 4πD(t−t
0)
(3.35)
Figure 3.5: The transition probability shows the probability density of a particle being at
position x at time t in a region with no flux boundary on one end given its initial position
x0 = 0 at t0 = 0, D= 0.01.
The integral of equation (3.35) defines the probability distribution for the new boundary
conditions and is given by
Z∞
P (x, t | x0 , t0 )W0 dx0
W (x, t) =
(3.36)
0
=
W0
2
r
t
1+x
−1 + x
√
√
− erf
erf
2Dπ
2 πDt
2 πDt
where the initial condition on the transition probability is W0 = 0.1 within the region [0, 1],
and zero elsewhere.
Figure 3.5 shows the plot of the transition probability of the antibody in a region with
no flux boundary at x = 0 for D = 0.01. From the plot we can see that the position of the
antibody again shows a maximum in a region about x = 0. Figure 3.6 shows the probability
density of the antibody being at position x at time t. Still in this case, the initial condition is
similar to the Wiener process above. The transition probability and the probability density
both increase with the implementation of the new boundary conditions. This makes sense
42
Figure 3.6: The probability density of a particle being at position x at time t in a region
with no flux boundary at one end for D=0.01.
because the particles can only move in one direction thus minimizing the spread of the
particles in the system and increasing the number of particles in the distribution.
3.2.2
Non-Dimensional Transition Probability Density and Probability Density
To apply equations (3.35) and (3.34) to the ACBA system the equations need to be
non-dimensionalized. The non-dimensional parameters are,
x∗ =
x
L
t∗ =
U
t
L
P∗ = PL
and the scaling parameters are L and U . L is the hydraulic diameter which is two times the
height of the binding channel and U is the velocity of the particle located at the center of
the channel.
43
The non-dimensional transition probability density becomes
1
P (x∗ , t∗ | x∗0 , t∗0 ) = q
2 π UDL (t∗ − t∗0 )
to account for the Peclet number, P e =
LU
,
D
√
∗
∗
P (x , t |
x∗0 , t∗0 )L
Pe
= p
2 π(t∗ − t∗0 )
e
−
2
L2 (x∗ −x∗
0)
4π D (t∗ −t∗
0)
UL
−
+e
2
L2 (x∗ +x∗
0)
4π L (t∗ −t∗
0)
UL
!
(3.37)
equation (3.37) can be rewritten as
!
∗
∗ 2
∗
∗ 2
−
e
P e(x −x0 )
4π(t∗ −t∗
0)
−
+e
P e(x +x0 )
4π(t∗ −t∗
0)
(3.38)
with the use of the Peclet number velocity is introduced into the equations. This is beneficial because now the equations account for velocity of the particle in the system. The
non-dimensional probability density is
√
π
W ∗ (x∗ , t∗ ) = W0
2
s
erf
(1 + x∗ )
Pe
4π(t∗ − t∗0 )
s
!
− erf
(x − 1)
!!
Pe
4π(t∗ − t∗0 )
(3.39)
It is derived by taking the integral in terms of x0 of the product of P and the W0 from 0 to
some large value, a0 .
3.3
Summary
In this chapter we have studied stochastic ordinary and partial differential equations.
Two types of stochastic equations studied in this chapter are the Langevin equation and the
Fokker-Planck equation. The Langevin equation identifies the randomness in a system, and
the Fokker-Plank equation provides a transition probability distribution of a random particle
in a system. One form of the Fokker-Planck equation is known as the Wiener Process. The
transition probability was calculated for a distribution of particles under two conditions:
in an unbounded region and in a region with no flux boundary at one end. The no-flux
model describes the conditions in the microfluidic system. The transition probability is
the probability that the antibody will be at location x at time t given that it was at x0 at a
44
previous time t0 . The probability density is the integral of the transition probability over all
possible starting positions x0 for a given antibody.
45
Chapter 4: Binding Probability of Magnetically Labeled Antibody
Binding and Circulating Cancer Cells
4.1
Introduction
In order to bind with the antigens on a cell, the antibodies must reside within a location
very close to the antigen. The probability of this event happening at a certain location at a
certain time is equivalent to the probability of the antibody showing up in any area within
the bead radius distance from the surface of the cell as they both move through the system.
Suppose the cell/antigen is stationary relative to the antibody, then the random motion of
the antibody solely determines the probability of binding.
Using the probability of an antibody being within a certain range of the channel at
the same time the cell is also in the location will provide a rough estimate on the binding
probability. The actual binding between the antibodies and antigen involve biomolecular
interactions at specific binding sites on the circulating tumor cell. This analysis looks at
the probability that the antibody is within the location of the circulating tumor cell so that
binding can occur.
4.2
Motion of a Circulating Tumor Cell
Before using the probability density function to obtain the probability that a particle
will be within a certain area at a certain time the movement of the circulating tumor cell
46
first has to be understood. In this system the CTCs follow a deterministic trajectory. Due
to the size of the cell and its ability to take up the entire height of the microchannel, the
position and velocity of the cell can be evaluated as the average of the bulk fluid velocity.
It has been shown in previous work that the hindered effect caused by the microchannels
walls are negligible in this case (Peng et al., 2012). The velocity of the fluid is given by the
Poiseuille flow as seen in Figure 4.1 and can be derived using the Navier Stokes equations
as
u(y) =
1 dp 2
(y − h2 )
2µ dx
(4.1)
so the average velocity of the cell is
uavg
uavg =
1
=
2yc
Zyc
udy
(4.2)
−yc
1 dp
1
(−h2 + yc2 )
2µ dx
3
(4.3)
If yc ∼ h then
uavg = −
1 dp 2
h
3µ dx
(4.4)
and the position of the cell is
xc = xc (0) + uavg t
In these equations µ is the viscosity of the fluid,
dp
dx
(4.5)
is the pressure gradient, y is the position
in the microchannel, 2h is the total height of the channel, and ±yc are the upper and lower
bounds of the cell position in the vertical direction.
47
Figure 4.1: Due to the size of the circulating tumor cell (yc ∼ h), the velocity of the cell
can be determined by using the average velocity equation for the Poiseuille flow of the bulk
fluid.
Figure 4.2: The limits a and b defined in the probability equation to find the antibody/bead
to CTC/antigen binding probability when a circulating tumor cell is at position, x.
48
4.3
Binding Probability
Using the information from equation (4.6) and the probability density calculated in
equation (3.39) the probability of binding at a certain time can be calculated. When the
center of the cell is at position x at t = t the probability density at that same t = t can be
integrated from x − R − r to x − R. This integral results in the probability that the antibody
bead will be within that specific range behind the cell at that time. The R and r mentioned
above account for the radius (µm) of the cell and the antibody bead respectively. The
pictorial representation of this point can be viewed in Figure 4.2. Analytically the binding
probability is
x−R
ZL
Pbinding@t∗ =t∗ =
W ∗ (x∗ , t∗ )dx∗
(4.6)
x−R−r
L
In this case,
x−R−r
L
and
x−R
L
are the dimensionless limits and W ∗ (x∗ , t∗ ) is the dimension-
less probability density. The binding probability calculated assumes the actual binding will
occur when the cell and the antibody bead make contact. This is a simplified assumption
seeing as though binding will occur only at specific binding sites on the cell that may or
may not be at that location.
To accurately predict the probability of binding in the system the problem has to be
solved in both the dimensional and non-dimensional realm. The dimensional time, t, is
needed to solve equation (4.5) for the physical position of the circulating tumor cell in the
microfluidic system. The position, x, and time, t, are then non-dimensionalized to x∗ and
t∗ using scaling factors L and U = umax , the maximum velocity of the Poiseuille flow.
These non-dimensional values are then used in equation (3.39). This binding probability
is obtained by integrating equation (3.39)over the nondimensional boundaries described
above.
49
12
t*=0.0186
t*=0.559
t*=0.994
10
W*
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
x*
Figure 4.3: The probability density curves from equation (3.39) at different time instances
for a 161 node simulation with the conditions described in Table 4.1 including a Peclet
number, P e = 3.36 × 106 .
4.3.1
Results
Using Mathematica and MATLAB, two analytical and mathematical software packages, and the dimensionless probability density the binding probability is calculated for
specific time and position values using equation (4.6).
The scenario described in Table 4.1 accounts for the first 2400 µm of the ACBA channel. Figure 4.3 shows the probability density curves for different time instances in the large
Peclet number limit.
Table 4.1: Test parameters for binding probability for ACBA system.
L = 2400µm U = 208.3 µm
s
ab = 1.4µm P e = 3.36 × 106
0
0
W (x∗ , t∗ ) = 0.1
In this example binding occurs almost instantaneously for the given initial distribution.
Since the initial condition is W0 = 0.1 for 0 ≤ x∗ ≤ 0.1 and W0 = 0 elsewhere. If
50
the initial condition for the probability density distribution was to decrease the probability density values for the larger time values will decrease making the area underneath the
probability density curve equal to one as it should be. Since the Peclet number is so large
the initial condition persists over the time scale presented. It was observed that as parameters such as the velocity, bead size, and hydraulic diameter changed the probability density
plots changed. The probability density distribution increases as the velocity decrease, the
particle size decrease to nano-sized particles, or the hydraulic diameter is decreased. This
observation implies that this method of calculating probability might be applicable on a
different model where the parameters mentioned above are considered. To find out under
which model this method for calculating probability of binding might be best utilized, a
study of different Peclet numbers was conducted and the results are discussed below.
In Figure 4.4 a-h the Peclet number was increased by a factor 10. This was done to find
out within what Peclet number range will the binding equation mentioned above work best.
From the
The probability density distribution at different dimensionless times was calculated for
Pe numbers ranging from 10−2 to 105 . This was conducted to see at which Pe numbers
is the probability density the probability of binding equation suitable to use. The Peclet
number range for this probability method to best be utilized is between 10 - 100. Between
this range the model is not solely controlled by either diffusion or the initial velocity of
the particle. When the Peclet number is small diffusion controls the model, and when
the Peclet number is large convection controls the motion of a particle and the diffusion
coefficient has little influence on the system as in our experimental case. With a Peclet
number between 10 - 100 the particle positions are dependent on both the diffusion and
velocity and the plots can be used to calculate the binding probability.
51
Figure 4.4: These plots show the dimensionless probability density, W ∗ , versus the dimensionless position, x∗ , at different dimensionless time, t∗ , values for different Peclet
numbers. a) Pe = 10−2 , b) Pe = 10−1 , c) Pe = 1, d) Pe = 10, e) Pe = 102 , f) Pe = 103 , g) Pe
= 104 , h) Pe = 105
52
4.4
Summary
The probability of binding equation was introduced as the integral of the probability
density. Values from the deterministic equation for cell motion were non-dimensionalized
and used a parameters in this equation. The probability of binding is lower as the Peclet
number is decreased; this means that the diffusion has a significant effect, or the velocity is
significantly slowed. There might be some application under these conditions.
53
Chapter 5: Magnetic Phenomena
5.1
Introduction
In the ACBA system magnets play an important part in the total functionality of the
system. The labelling and trapping of the magnetic particles in the system allow for encapsulation and analysis in later stages of the device. But, in order for this to occur it is
important to understand how and why the particles are being trapped within the system.
Several papers explain the behavior of magnetic particles in a microfluidic system and
the way they react in the presence of a magnetic field, but to my knowledge no one has
looked at the behavior of the magnetic particle once it is attached to the cell in this case and
how does this additional object effect the capture efficiency of the magnetic particle in the
system.
In this chapter the different roles magnetic particles play in microfluidic systems will
be addressed. A study will also be conducted to analyze how the diameter of the cell in the
drag calculations effect the capture efficiency of the magnetic particle in the system under
different magnetic strengths and volumetric flow rates.
5.2
Magnetostatics
In the ACBA, the magnetically labeled cell will interact with the externally imposed
magnetic field due to the magnetized disk. The situation is depicted on Figure 5.1. In using
54
Figure 5.1: Dipole diagram of magnetic disk in microfluidic system.
the magnetic field to move the magnetized cell, we seek to determine the cell trajectory
and velocity. In order to determine the trajectory and hence the speed of the cell we must
determine the force on the cell due to the magnetic field. We assume that the external
magnetic field is governed by the laws of magnetostatics. In this section we discuss the
basics of magnetostatics, derive the force exerted by a suparamagnet and then calculate the
magnetic field. Magnetic fields are generally viewed as originating as a result of orbital
rotation of electrons (Kirby, 2010).
5.2.1
Definition of the Magnetic Field
Recall that the Electric Field at any point is defined as the force acting on a single
charge at that point, or
E=
N
F
q
=
0
2
q
4πe r C
(5.1)
where r is the distance from the point charge. This equation is known as Coulomb’s Law.
A magnetic field arises as a result of charges moving in a domain. In analogy with the
55
electric field, the magnitude of the magnetic induction field is given by
FM
Vq
B∝
(5.2)
and B is called the magnetic induction. The direction of B is such that the force is given
by
Fm = q(V × B)
and is called the Lorentz force. Note that a force is exerted on a given charge only if the
magnetic induction is in a direction perpendicular to the direction of motion of the charge
and that V • FM = 0 indicating that the magnetic force does no work. Thus the general
formula for the magnitude of the magnetic induction field is
B=
The unit of the magnetic induction is
Fm
V qsinθ
W eber
m2
where W eber =
sometimes called the magnetic flux density and the unit
W eber
m2
N msec
.
Coul
The quantity B is
= T esla.
Two other vectors may be defined in terms of the magnetic induction, the magnetic
field intensity H and the magnetization density M which indicates the extent to which the
magnetic medium is polarized. These vectors are defined as
B = µ0 (H + M)
(5.3)
where µ0 is the magnetic permeability of a vacuum, and has the magnitude µ = 4π ×
mkg
10−7 Coul
2 =
H
m
where H stands for Henry. Analogous to the electrical permeability, we
may define the relative permeability as µr =
µm
.
µ0
The magnetic permeability is entirely
analogous to the electrical permittivity and is a transport property. Most materials have a
magnetic permeability close to that of a vacuum. Thus magnetostatics is characterized by
56
the the three vectors B, H, M. Equation (5.3) means that a permanent magnet of magnetization M in an applied magnetic field H produces the magnetic induction field B.
Another property of a magnetic material that measures the extent with which a body
can be magnetized is called the magnetic susceptibility and is defined by
M = χH
(5.4)
and the susceptibility is usually small for non-magnetic materials; on the other hand for
highly magnetized materials such as permalloy, the susceptibility is very large. This means
that
B = µ0 (1 + χ)H = µm H
5.2.2
(5.5)
Maxwell’s Equations
At steady state the magnetic fields and electric field are independent with the magnetic field being proportional to the electric current. However, when the magnetic field is
time-varying, the electric and magnetic fields are coupled and the resulting equations are
Maxwell’s equations.
Maxwell’s equations, which describe the flow of charge, the electric and magnetic fields
in moving media are
∇ · B = 0 N o magnetic sources
∂B
F araday 0 s Law
∇×E=−
∂t
∂D
Ampere0 s Law
∇ × B = µm J +
∂t
∇ • D = ρe Gauss0 s Law
(5.6)
(5.7)
(5.8)
(5.9)
Here, H is the intensity of the magnetic field; E, the electric field; D, the displacement
field; B, the magnetic induction field; J, the current density; and ρe , the free charge density.
57
If the material is an isotropic, permeable dielectric, then
D = e E, B = µm H
(5.10)
∂E
∇ × B = µm J + e
∂t
(5.11)
and Ampere’s Law becomes
where the magnetic permeability µm is related to the electrical permittivity by c = (µm e )−1/2
and c is the speed of light. The magnitude of magnetic induction fields that are measured
can be on the order of 1T esla. Even when the electric field is time varying, the magnetic
field term in Faraday’s Law is negligible; for example; an electric field of 1V over a 10µm
channel will require a very short time scale of about 10−10 seconds for a magnetic field of
B = 1T to balance Faraday’s Law. Similarly, the left side of Ampere’s Law dominates
unless the electric field is very large. Thus in most cases of practical interest, the electric
and magnetic fields can be decoupled. Note that since
∇•B=0
(5.12)
there is no such thing as a magnetic monopole; all magnetic materials are made up of
dipoles, having north and south poles as depicted on Figure 5.3. The magnetic moment of
a material is defined in the next section.
5.2.3
Electric and Magnetic Dipoles
An electric dipole is set up as a combination of a positive and negative charge and the
expression for the potential is a linear superposition of each according to
q
φ=
4πe
58
1
1
−
r1 r2
(5.13)
Figure 5.2: Geometry for calculating the electric potential field due to a dipole at the point
P.
From Figure 5.2 using the Law of Cosines, r1 = r − d2 cosθ and r2 = r + d2 cosθ, so that
φ=
dcosθ
q
4πe r2 − d42 cos2 θ
(5.14)
qdcosθ
4πe r2
(5.15)
Now for d << r
φ=
The quantity p = qd is called the electric dipole moment and the unit of dipole moment is
1Debye = 1D = 3.336 × 10−30 Cm.
There is an entirely analogous magnetic potential for a magnetic dipole whose potential
may be calculated as follows. It is well known that the magnitude of the magnetic force
0
between two point poles of strength p and p in a vacuum is given by
0
pp
Fd =
4πµ0 r2
59
(5.16)
Figure 5.3: Geometry for calculating the magnetic field due to a magnetic dipole at the
point P .
0
and the force acts along the line in the plane of the dipole. The convention is that if p is a
unit pole pointing north, then the magnetic field is defined by
H=
pr
4πmu0 r3
The magnetic induction field is then B = µ0 H =
p
.
4πr2
(5.17)
If the dipole is located at a surface
of area A, then the strength of the dipole is given by p = Aρm where ρm is the surface
density of the magnetic poles.
Now consider the situation depicted on Figure 5.3 similar to the situation depicted on
Figure 5.2. The magnetic field is written as the linear sum of the two magnetic fields as
r1 r2
ρm A
(5.18)
− 3+ 3
H=
4πµ0
r1 r2
Using the law of cosines as with the electrical dipole and the binomial theorem for d << r
we obtain
ρm A
H=
4πµ0
1
3d
1
3d
− d−r
1 − cosθ + − d + r
1 + cosθ
2
2r
2
2r
60
(5.19)
so that
H=
ρm Ad ˆ
−
d
+
3cosθr̂
4πµ0 r3
(5.20)
where dˆ and r̂ are unit vectors. Now cosθ = d̂ • r̂ and ρm = µ0 M so that with V = Ad we
have
H=
MV ˆ
ˆ • r̂r̂
−
d
+
3
d
4πr3
(5.21)
Since the dipole moment m = ρm Ad we can also write the result in terms of the magnetic
moment as
H=
1 m
m • rr − 3 +3
4πµ0
r
r5
(5.22)
It is this dipole distribution that creates a magnetic force that acts on a magnetic particle in
the channel.
5.2.4
The Force Induced by a Magnetic Material on a Magnetic Particle
The induced motion of a magnetized particle due to a magnetic force called the Kelvin
force, is termed magnetophoresis (Kirby, 2010). Consider a dipole oriented as on Figure
5.1 (Rosensweig, 1985). The force experienced by this volume element is thus
−Hρm A + (H + dH)ρm A = dHρm A
(5.23)
The dH on the right side is the directional derivative in the direction denoted by d
dH = d • ∇H =
d
(M • ∇)H
M
(5.24)
Since ρm = µ0 M and V = Ad the force per volume is given by
Fd = µ0 (M • ∇H)
61
(5.25)
To find the total force, substitute equation (5.4) into equation (5.25) and multiply by the
volume
Fd = µ0 V χ(H • ∇H)
(5.26)
There has been some controversy in the literature about the nature of the magnetic field
in equation (5.26). Recall that the electrical body force is given by
FE = ρe E
(5.27)
where ρe is the volume charge density and E is the electric field, both quantities taken in
the fluid. On the other hand, Petit et al. (2011) and Bakuzis et al. (2005) argue that the H
in equation (5.26) should be given by that force associated with the magnetic dipole given
by equation (5.22), and this approach is used here.
5.3
Magnetically Induced Cell Transport
The goal of the ACBA device is to manipulate an object within a fluid flow through
the introduction of magnetic particles. Magnetic particles are normally superparamagnetic,
lacking magnetic memory, and can be manipulated only in the presence of a magnetic field.
Outside of that field they behave as normal particles in a fluid flow (Pamme, 2006). Most
magnetic particles used in microfluidic systems are a few nm to several µm in size. This is
advantageous because biomolecules such as antibodies, antigen, DNA, and mRNA can be
attached to the particles to be used in different applications. In the ACBA system magnetic
particles called Dynabeads are 2.8 µm in diameter and have antibodies attached to them
(Henighan et al., 2010). These antibodies are what causes the magnetic particle to attach
to the antigens on the identified circulating tumour cells.
Magnetic particles are used first to label then trap, transport, or separate biological material (Pamme, 2006). Permanent magnets or electromagnets either microfabricated inside
62
of the micofluidic device or placed outside of the microfluidic device are used to control
the movement of the magnetic particles in the system. Permanent magnets produce their
own magnetic field and have the ability to attract a magnetic particle without the use of an
electrical current. An electromagnet, on the other hand, uses electrical currents to produce
a magnetic field thus attracting the magnetic particle.
The ACBA system uses external electromagnets to control the magnetic fields of an
array of permalloy disk 5 µm in diameter and 40 nm in depth located on the internal
silicone surface of the microfluidic device (Henighan et al., 2010). The orientation of the
electromagnets located on the outside of the ACBA system control the magnetic fields of
the permalloy disk. Electromagnets control the magnetic fields in the x, y, and z directions,
Hx , Hy , Hz , on the permalloy disk creating a three dimensional magnetic field as seen
in Figure 5.4. By controlling the maxima locations of the magnetic fields on the disk the
magnetic particle can be moved from disk to disk. This method of trapping and transporting
magnetic particles is called “magnetic tweezers”.
A detailed analysis of the magnitude of the forces acting on a particle in the ACBA
system was conducted in Peng (2011). It was shown that when analyzing a particle in a
micrfluidic system in the presence of body forces the primary forces acting on the particle
are Stokes drag force, electrostatic force, and magnetic forces. The secondary forces of
the random force, the EDL force, van der Waals force, and the force due to gravity were
negligible. Without the presence of an electric field the electostatic force is not present in
this system. The forces remaining are the magnetic force and the Stokes drag force. The
Stokes drag force accounts for the force the fluid exerts on the particle as it moves through
the fluid. This force for a spherical object is calculated by
FS = 6πµap u
63
(5.28)
Figure 5.4: Electromagnets and solenoid control magnetic fields on magnetic disk. Picture
from (Henighan et al., 2010) from NSEC Faculty Dr. Sooryakumar’s group here at OSU,
who investigates the use of “magnetic tweezers” in microfluidic systems.
Figure 5.5: Particle moving along magnetic disk array in a process known as “magnetic
tweezers”. Picture from (Yellen et al., 2007) at Duke University.
64
In the equation above µ is the viscosity of the fluid, u is the relative velocity of the cell,
u = uf − uc where f and c stand for the fluid and cell respectively.
5.4
Characterization of the Magnetic Field
The magnetic field in the microchannel produced by the permalloy disks are characterized directly using the Landau-Lifshitz-Gilbert equation in micromagnetic simulation
software (Neudecker et al., 2006; Liu et al., 2007; Ha et al., 2003). Sooryakumar’s research group uses this equation incorporated in a micromagnetic software called OOMMF
to characterize their magnetic fields for the magnetic tweezers (Henighan et al., 2010).
The magnetic fields in Sooryakumar’s group papers (Henighan et al., 2010; Chen et al.,
2013) have four major inputs that effect the magnetic field generated by the micromagnetic
software they include the magnetic strength in all three directions and the rotating frequency of the oscillating field. Other inputs include the material and magnetic properties
of the disk and the microbead.
5.4.1
Trapping Efficiency of a Magnetic Bead
In this section the binding efficiency of a magnetic particle in a constant magnetic field
in a rectangular channel undergoing Poiseuille flow will be explained. From the previous
section the force balance for the system assuming the acceleration is negligible is
FS = −FM
where we assume one-dimensional fully developed flow; on the other hand, the magnetic force is two-dimensional with components in both the x− and y− directions. In
the x−direction,
6πµap (uf − up ) = −FM x
65
Solving for the velocity of the bead, ub is given by,
ub = uf +
FM x
6πµab
(5.29)
Using a process similar to that found in (Haverkort et al., 2009) the fluid velocity for this
system is characterized by a Poiseuille flow where the velocity is only the x direction, uf
can be described as
uf = umax
y2
1− 2
h
(5.30)
In this equation umax = Q( 4W3 h ), where Q is the volumetric flow rate and W is the width
of the channel. Substituting equation(5.30) into equation(5.29) results in
ub = umax
y2
1− 2
h
+
FM
6πµab
(5.31)
The velocity of the particle can be broken down into its different coordinates so that a
relationship between the motion of the particle in the x and y directions
y2
dx
FM x
= umax 1 − 2 +
dt
h
6πµab
dy
Fmy
=
dt
6πµab
By dividing
dx
dt
by
dy
dt
(5.32)
(5.33)
an equation is obtained that describes the change in position of the
particle in the x direction with that of the y direction.
dx
umax 6πµab
=
dy
FM y
y2
FM x
1− 2 +
h
FM y
(5.34)
Actually the above equation is already in dimensionless form with both x and y assumed
to be scaled on the channel height. Thus we write
dx
1
=
1 − y2 + β
dy
α
66
(5.35)
with
α=
FM y
FS
β=
FM x
FM y
Integrating between the limits x to x0 and y to y0 , the following equation is obtained to
describe the particle motion
y03 y 3
−
+ β(y − y0 )
α(x − x0 ) = y − y0 +
3
3
(5.36)
Equation (5.36) has a four-parameter family of solutions depinding on the values of (x0 , y0 , α, beta).
We consider the computational domain to be −l ≤ x ≤ l and −1 ≤ y ≤ 1, where l =
l∗
.
h
For example we can start with l = 4. In this formulation we consider the magnetic disk to
be placed at y = −1 and −γ ≤ x ≤ γ. We look for curves for which the trajectories of the
particles end up at a location on the disk. Actually the minimum distance that any particle
can attain is y = −1 +
ap
.
h
In the actual case we know that the magnetic forces will vary with (x, y); however, here
we assume that both forces are constant so that α and β are constant.
This same method described above to calculate the capture efficiency of a just a magnetic bead will be used in the next section. However, instead of using ab , the bead radius,
for the radius in the Stokes drag equation ac will be used to account for the cell radius. The
reason for this change will be explained in the next section.
5.4.2
Trapping Efficiency of a Magnetic Bead Bound to a Cell
The capture efficiency of a magnetic bead in a microfluidic system using different fluid
transport methods is documented well in literature The capture efficiency of a magnetic
bead in a microfluidic system using different fluid transport methods is documented well
in literature (Furlani et al., 2007; Sinha et al., 2007; Haverkort et al., 2009). These papers
however, fail to assess the capture efficiency of a magnetic bead attached to a larger object
67
in a fluidic system. In the case of the ACBA system the magnetic bead is attached to
the circulating tumor cell. In this section the capture efficiency of magnetic bead will be
assessed with the magnetic properties of the bead and the drag properties of the circulating
tumor cell. Previous authors have either calculated the magnetic and drag force on the
magnetic bead or treated the cell as a magnetic particle, but they have not combined the
two for a analysis of a magnetic bead attached to a larger particle.
Here the characteristics from the magnetic particle and the circulating tumor cell will
be used to predict magnetic particle behavior under the influence of a magnetic field. The
analysis for the magnetic force on the object will come from the parameters of the magnetic
bead while the force due to drag will be calculated using parameters of the cell.
The equations and method used to calculate the efficiency equation in the previous
section will be the same in this situation. The main difference between this case and the
previous case is that the particle used in the drag equation is the cell instead of the particle.
This means that the Stokes drag force in α and β now becomes 6umax πµac . The equations
(5.36) stays the same.
5.4.3
Results
As mentioned in the previous section a MATLAB code was created to calculate the
capture area of the magnetic bead using equation (5.36). Any bead located between the
minimum and maximum x positions for a given y value will be captured. The edges of the
magnetic disk or in 1-D, magnetic strip were used to find the initial x position of a bead
that will allow it to intersect the specific γ and y0 values representing the borders of the
disk for different values of α. In the results below the β = −1, γ = 0.33 to represent a 10
µm disk with the x orgin at the center of the disk, and y0 = −1 to represent the bottom of
68
1
Particle position y vs x at α = −0.5
Max. Particle Trajectory @ y=1
Min Particle Trajectory @ y=1
Max. Particle Trajectory @ y=0
Min Particle Trajectory @ y=0
0.8
0.6
0.4
y
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−5
−4
−3
−2
−1
0
1
x
Figure 5.6: The x ranges for a particle’s motion at y = 1 and at y = 0 when y0 = −1,
γ = 0.33, β = −1, and α = −0.5.
the channel. The α and β values are negative to account for the negative magnetic force on
the particle. The results are in Figures 5.6 - 5.9.
The trajectory of a particle is the same for different y values however the x position
range varies depending on the y value. This can be seen in Figure 5.6. As the initial y
position of the magnet bead decreases the x position needed for capture increases. At a
starting position at y = 1 the range of capture for a magnetic bead is between −4.33 and
−5 and for a particle whose initial y position is y = 0 the x range of capture is −2 to
−2.67. The distance between the x values do not change due to the similarity of the curve.
Figures 5.7 through 5.9 show the x positions for bead capture increase as the α value
increases. This makes sense because as the α value increases the magnetic force on the
particle decreases so the particle has to be closer to the magnetic strip to be captured. The
smaller the α values magnetic force has a greater influence on the particle motion than the
69
1
Particle position y vs x at α = −0.1
Max Particle Trajectory
Min Particle Trajectory
0.8
0.6
0.4
y
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−16
−14
−12
−10
−8
−6
−4
−2
0
2
x
Figure 5.7: The x range for a particle’s motion at y = 1, y0 = −1, γ = 0.33, β = −1, and
α = −0.1.
1
Particle position y vs x at α = −0.5
Max Particle Trajectory
Min Particle Trajectory
0.8
0.6
0.4
y
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−5
−4
−3
−2
−1
0
1
x
Figure 5.8: The x range for a particle’s motion at y = 1, y0 = −1, γ = 0.33, β = −1, and
α = −0.5.
70
1
Particle position y vs x at α = −5
Max Particle Trajectory
Min Particle Trajectory
0.8
0.6
0.4
y
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
x
Figure 5.9: The x range for a particle’s motion at y = 1, y0 = −1, γ = 0.33, β = −1, and
α = −5.
stokes drag force thus it has the ability to trap the magnetic bead earlier. A large value
for α could also mean that the Stokes drag force is very small. This is taken into account
when looking at the drag caused by just the magnetic particle versus the drag caused by the
circulating tumor cell that is attached to the bead.
5.5
Summary
In this chapter the equation for magnetic force has been derived using the characteristics
of a dipole magnet. How magnets are used in the ACBA device was described in detail.
Magnetic force along with the Stokes drag force was then used to describe the behavior of
a particle in a microfluidic channel in the presence of a permalloy magnetic disk. This disk
had a constant magnetic field. The results were as expected whereas the capture efficiency
71
of the bead with and without the CTC increased as the volumetric flow rate decreased and
as the magnetic strength increased.
72
Chapter 6: Summary and Future Work
In this thesis the different types of fluid and particle transport in microfluidic systems
have been discussed. These transport processes include pressure driven flow, electrokinetic
flow, and magnetically induced flow. Electrokinetic applications to lab-on-a-chip technology is described and the use of magnetic materials to induce biological cell transport in
microfluidic systems is investigated.
In this thesis, the Langevin and Fokker-Plank stochastic equations are derived. The
Langevin equation describes the random motion of a magnetic bead that contains an antibody that is designed to bind to a cancer cell. The Fokker-Planck equation describes
the transition probability density distribution of the magnetic bead along the length of the
channel. A specific form of the Fokker-Plank equation known as a Wiener Process is then
used as a simplified model to describe the probability density of a simplified 1-D problem
undergoing Brownian motion. Though not used in the ACBA device, electroosmosis and
electrophoresis have many applications in the success of a many lab-on-a-chip devices and
in the future it is possible that these transport phenomena may be employed in the ACBA.
In Chapter 4 the probability density equation derived in Chapter 3 is used along with
the known motion of the larger circulating tumorcell to predict the probability of a bead
and cell binding in the system. In Chapter 5 an analysis of magnetic fields is performed
73
and results for the capture area of the magnetic bead with and without a cell attached by an
array of magnets are producecd.
The conclusions of this thesis are as follows,
1. Stochastic differential equations, more specifically the Langevin equation and the
Fokker-Planck equation are useful in describing the random behavior of small particle in microfluidic systems. The Wiener Process, a form of the Fokker Planck
equation, is an simplified equation used to characterize the behavior of a particle
undergoing Brownian motion.
2. The dimensional and nondimensional transition probability densities of the Wiener
Process were used to describe an microfluidic system that was not bound on either
side with limits ∞ to −∞ and a microfluidic system that had a no flux boundary
condition. The no flux condition modelled a system where flow only occurred in one
direction.
3. The probability of binding can be calculated by taking the integral of the probability
density from two different position values at a certain time.
4. For the model described in this thesis the binding probability obtained did not characterize the binding behavior of the particles. The ACBA system particles are too
large to observe the stochastic behavior (Pe = 3.6 × 106 ). However the principles of
this model may be used to characterize other models with smaller in value parameters
than the ones used in this model.
5. The ACBA system might be better characterized using a deterministic model to describe magnetic bead behaviour.
74
6. The magnetic strength of a magnet varies according to the type of magnet a system. The magnetic field of a permalloy is actually characterized using the LandauLifshitz-Gilbert equation. To simplify the model the magnetic field of a magnetic
block is used instead of the disk.
7. Magnetic force and particle size can be used to find that theoretical capture area of a
magnetic particles in a microfliuidic device.
8. The attachment of a circulating tumor cell to a magnetic bead affects the capture area
of the bead in the system.
Several areas mentioned in this thesis can be studied in greater detail. In the future,
1. The Ornstein-Uhlenbeck process, a more advance model to describe Brownian motion may be used to better characterize the Brownian motion in the system for smaller
particles.
2. A more advanced model the describe the magnetic particles in the ACBA system
needs to be developed to include their deterministic trajectories.
3. The theoretical capture area of a magnetic particle attached and not attached to a circulating tumor cell can be compared to experimental data and a efficiency calculation
can be derived.
75
Appendix A: MATLAB Calculation of Important Constants
76
clc;clear all
%Stokes Einstein Equation (Diffusion Coefficient)
%D=(kb*T)/(6*pi*u*r)
%Variables
Name
Unit
%kb
Boltzmann constant
J/K
%T
Temperature
K
%u
Viscosity
Pa*s
%r
Particle Radius
m
%*************************************************
%
kb=1.3806488e-23;
T=284.15;
u=0.001;
r1=12.5e-6; %Radius of Cell
r2=1.4e-6; %Radius of Antibody
%Calculate the Diffusion Coefficient
D1=(kb*T)/(6*pi*u*r1)
D2=(kb*T)/(6*pi*u*r2)
%*************************************************
%%
%Peclet Number
%Pe=LU/D
%Variables
Name
Unit
%L
Characteristic Length
m
%U
Velocity
m/s
%D
Diffusion Coefficient
mˆ2/s
%*************************************************
L=2e-6; %height of the channel
U=2.0833e-4;
%Calculate the Peclet Number
Pe1=L*U/D1
Pe2=L*U/D2
%*************************************************
%%
%Diffusion Time Over a Certain Distance
%t=xˆ2/(2D)
%Variables
Name
Unit
%x
Distance of Diffusion
m
%*************************************************
x=25e-6; %Diffuse half the width of the channel
%Calculate the Diffusion Time
t1=xˆ2/(D1*2)
t2=xˆ2/(D2*2)
%*************************************************
%%
%Concentration and Molar Concentration of Antibodies
and Antigen
%C=n/V
%C_m=C/NA
%Variables
Name
Unit
%n
#of objects in Solution
%V
Volume of Solution
L
%NA
Avogadro’s number
molˆ-1
%*************************************************
n1=15;
%Number of Cells in Solution
n2=10000; %Number of Antibody in Solution
V=7.5e-3;
NA=6.0221415e23;
%Calculate Concentrations
C1=n1/V;
C_m1=C1/NA
77
C2=n2/V
C_m2=C2/NA
%*************************************************
%%
%Thermal Velocity
%vth=sqrt((kbT)/(m))
%Variables
Name
Unit
%kb
Boltzmann constant
J/K
%T
Temperature
K
%m
mass
kg
%*************************************************
m1=2.08e-13; %Cell mass
m2=1.5e-14; %Antibody mass
%Calculate Thermal Velocity
vth1=sqrt((kb*T)/m1)
vth2=sqrt((kb*T)/m2)
%*************************************************
%%
%Epsilon Squared
%e_2=mU/BL
%Variables
Name
Unit
%kb
Boltzmann constant
J/K
%T
Temperature
K
%m
mass
kg
%*************************************************
B2=(6*pi*u*r2)
LL=2400e-6
%Calculate Epsilon_2
e_22=sqrt((m2*U)/(B2*LL))
78
Appendix B: MATLAB Code for Transition Probability Density and
Probability Density
79
clear all; close all; clc
%Stokes Einstein Equation (Diffusion Coefficient)
%D=(kb*T)/(6*pi*u*r)
%Variables
Name
Unit
%kb
Boltzmann constant
J/K
%T
Temperature
K
%u
Viscosity
Pa*s
%r
Particle Radius
m
%**************************************************************************
kb=1.3806488e-23;
T=284.15;
u=0.001;
r1=12.5e-6; %Radius of Cell
r2=1.4e-6; %Radius of Antibody
%Calculate the Diffusion Coefficient
D1=(kb*T)/(6*pi*u*r1)
D2=(kb*T)/(6*pi*u*r2)
%**************************************************************************
L=2400e-6; %Diffusion length of the channel
U=2.0833e-4;
%Calculate the Peclet Number
Pe1=L*U/D1
Pe2=L*U/D2
%Divide nodes into equal spacing
% dx=lengthx/(n-1);
lengthx=1;
n=161;
dx=lengthx/(n-1);
x=0:dx:lengthx;
m=n;
lengtht=1;
dt=lengtht/(m-1);
for i=1:m
t(i)=(i-1)*dt;
tpr(i)=t(i);
end
for k=1:m
for i=1:n
%Dimensional transition probability density
P(i,k)=1/(2*sqrt(pi*t(k)*D2))*exp(-x(i)ˆ2/(D2*4*t(k)));
%Dimensionaless transition probability density
P2(i,k)=sqrt(Pe2)*(1/(2*sqrt(pi*t(k))))*exp(Pe2*(-x(i)ˆ2/(4*t(k))));
%Call P or P2 depending on which form of the equation is used
P2(i,k)=2*P(i,k);
end
end
% Integrate to get probability for given initial probability
% W(xpr,t)=.5 for abs xpr<1 0 otherwise
lengthint=0.1*lengthx;
nint=n;
dxint=lengthint/(nint-1);
xint=0:dxint:lengthint;
% evaluate transition prob over integration domain
for k=1:m
80
for i=1:n
for j=1:n
probint(j)=1/(2*sqrt(pi*t(k)*D2))*exp(-(x(i)-xint(j))ˆ2/(4*t(k)*D2));
probint2(j)=sqrt(Pe2)*(1/(2*sqrt(pi*t(k))))*(exp(Pe2*(-(x(i)-xint(j))ˆ2/
(4*t(k))))+exp(Pe2*(-(x(i)+xint(j))ˆ2/(4*t(k)))));
end
prob(i,k)=sum(probint)*dxint ...
- 1/2*probint(1)*dxint-1/2*probint(n)*dxint;
prob(i,k) = prob(i,k)/lengthint;
%
prob(i,k)=trapz(xint,probint2);
end
end
%
ppp = zeros(n,m);
for k=2:m
for i=2:n
dummy(i,k)=trapz(x(1:i),prob(1:i,k));
end
end
dummy(end,1)=1;
for i=2:n
for k=2:m
ppp(i,k)=trapz(t(1:k),dummy(i,1:k));
end
end
for k=1:m
dummy_total(k)=trapz(x,ppp(:,k));
end
max(max(ppp))
%Transition Probability Density Plot
figure(1)
set(gca,’FontSize’,15)
surf(t,x,P2);
% h = colorbar;
% set(h, ’ylim’, [0 1.2])
% caxis([0 1.45])
colorbar;
xlabel ’t’;
ylabel ’x’;
%title ’Transtition Probability Density’;
%Probability Density Plot
figure(2)
set(gca,’FontSize’,15)
surf(t,x,prob);
% h = colorbar;
% set(h, ’ylim’, [0 1.2])
% caxis([0 1.45])
colorbar;
xlabel ’t’;
ylabel ’x’;
%title ’Probability Density’;
%Probability Plot
figure(3)
set(gca,’FontSize’,15)
surf(t,x,dummy);
% h = colorbar;
% set(h, ’ylim’, [0 1.2])
81
% caxis([0 1])
colorbar;
xlabel ’t’;
ylabel ’x’;
%title ’Probability’;
axis equal
%Probability desnity(PD) curve for specific time nodes
figure(4)
set(gca,’FontSize’,15)
%PD Curve at node 3
plot(x,prob(:,3),’color’,[1 0 0], ’LineWidth’,2);
hold on
%PD Curve at node 90
plot(x,prob(:,90),’color’,[0 1 0], ’LineWidth’,2);
%PD Curve at node 160
plot(x,prob(:,160),’color’,[0 0 1], ’LineWidth’,2);
hold off
%Legend displays time equivalent to node position
legend(’t*=0.0186’,...
’t*=0.559’,’t*=0.994’)
xlabel(’x*’)
ylabel(’W*’)
%title(’Pe=100000’)
grid on
xlim([0 1])
82
Appendix C: MATLAB Code for Magnetic Bead Capture at Different
Alpha and y Positions
83
clc;clear all
dalph=0.1;
%alpha=[0:dalph:2];
beta=-1;
h=15e-6 %height of the channel from the centerline
ab = 1.4e-6;
y = (-h+ab)/h;
ymax = (h-ab)/h;
xo= 0;
x = 5e-6/h;
u = 0.001;
y0=-1;
x0=.33;
al=2;
N=21;
dy=-al/(N-1);
yy=[1:dy:-1];
x02=-x0
alpha=-5;
for i=1:N
x(i)=x0+(yy(i)-y0+y0ˆ3/3-yy(i)ˆ3/3)/alpha+beta*(yy(i)-y0);
end
for i=1:N
x2(i)=x02+(yy(i)-y0+y0ˆ3/3-yy(i)ˆ3/3)/alpha+beta*(yy(i)-y0);
end
alpha2=-0.1;
for i=1:N
x3(i)=x0+(yy(i)-y0+y0ˆ3/3-yy(i)ˆ3/3)/alpha2+beta*(yy(i)-y0);
end
for i=1:N
x4(i)=x02+(yy(i)-y0+y0ˆ3/3-yy(i)ˆ3/3)/alpha2+beta*(yy(i)-y0);
end
% Plot the path
figure(1)
plot(x,yy,x2,yy,’Linewidth’, 3)
x = x(1);
y = 1;
line(’XData’, [x x x(1)], ’YData’, [-1 y y], ’LineWidth’, 3, ...
’LineStyle’, ’-.’, ’Color’, ’k’);
x = x2(1);
y = 1;
line(’XData’, [x x x2(1)], ’YData’, [-1 y y], ’LineWidth’, 3, ...
’LineStyle’, ’-.’, ’Color’, ’k’);
xlabel(’$x$’,’Interpreter’,’LaTex’,’FontSize’,18);
ylabel(’$y$’,’Interpreter’,’LaTex’,’FontSize’,18);
title(’Particle position $y$ vs $x$ at $\alpha = -5$ ’,’Interpreter’,’LaTex’,’FontSize’,18);
legend(’Max Particle Trajectory’,’Min Particel Trajectory’);
figure(2)
plot(x3,yy,x4,yy,’LineWidth’,3)
x = x3(1);
y = 1;
line(’XData’, [x x x3(1)], ’YData’, [-1 y y], ’LineWidth’, 3, ...
’LineStyle’, ’-.’, ’Color’, ’k’);
x = x4(1);
y = 1;
line(’XData’, [x x x4(1)], ’YData’, [-1 y y], ’LineWidth’, 3, ...
’LineStyle’, ’-.’, ’Color’, ’k’);
xlabel(’$x$’,’Interpreter’,’LaTex’,’FontSize’,18);
ylabel(’$y$’,’Interpreter’,’LaTex’,’FontSize’,18);
84
title(’Particle position $y$ vs $x$ at $\alpha = -0.1$ ’,’Interpreter’,’LaTex’,’FontSize’,18);
legend(’Max Particle Trajectory’,’Min Particle Trajectory’);
al=1;
N=21;
dy=-al/(N-1);
y3=[0:dy:-1];
for i=1:N
x5(i)=x0+(y3(i)-y0+y0ˆ3/3-y3(i)ˆ3/3)/alpha2+beta*(y3(i)-y0);
end
for i=1:N
x6(i)=x02+(y3(i)-y0+y0ˆ3/3-y3(i)ˆ3/3)/alpha2+beta*(y3(i)-y0);
end
figure(3)
plot(x3,yy,’g’,x4,yy,’c’,x5,y3,’b:’,x6,y3,’r:’,’LineWidth’, 3)
% x = x5(1);
% y = 0;
% line(’XData’, [x x x5(1)], ’YData’, [-1 y y], ’LineWidth’, 3, ...
%
’LineStyle’, ’-.’, ’Color’, ’k’);
% x = x6(1);
% y = 0;
% line(’XData’, [x x x6(1)], ’YData’, [-1 y y], ’LineWidth’, 3, ...
%
’LineStyle’, ’-.’, ’Color’, ’k’);
xlabel(’$x$’,’Interpreter’,’LaTex’,’FontSize’,18);
ylabel(’$y$’,’Interpreter’,’LaTex’,’FontSize’,18);
title(’Particle position $y$ vs $x$ at $\alpha = -0.5$ ’,’Interpreter’,
’LaTex’,’FontSize’,18);
legend(’Max. Particle Trajectory @ y=1’,’Min Particle Trajectory @ y=1’,
’Max. Particle Trajectory @ y=0’,’Min Particle Trajectory @ y=0’);
x = x3(1);
y = 1;
line(’XData’, [x x x3(1)], ’YData’, [-1 y y], ’LineWidth’, 3, ...
’LineStyle’, ’-.’, ’Color’, ’k’);
x = x4(1);
y = 1;
line(’XData’, [x x x4(1)], ’YData’, [-1 y y], ’LineWidth’, 3, ...
’LineStyle’, ’-.’, ’Color’, ’k’);
x = x5(1);
y = 0;
line(’XData’, [x x x5(1)], ’YData’, [-1 y y], ’LineWidth’, 3, ...
’LineStyle’, ’-.’, ’Color’, ’k’);
x = x6(1);
y = 0;
line(’XData’, [x x x6(1)], ’YData’, [-1 y y], ’LineWidth’, 3, ...
’LineStyle’, ’-.’, ’Color’, ’k’);
85
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