Measurement of viscoelastic properties of treated and untreated cancer cells using passive microrheology

Experiments have shown that there is an increase in ultrasound backscatter from cells during cell death. Since cell scattering depends on the mechanical property variations, one step towards a better understanding of the phenomenon involves measuring the cells' viscoelastic properties. Two promising techniques used for such studies are particle tracking microrheology (1P) and two-point microrheology (2P). The main aim of this work is to develop and test the ability of both to measure changes in viscous and elastic moduli of breast cancer cells during chemotherapeutic treatments. First, the viscosities of glycerol-water mixtures measured using microrheology were found to be within 5% of rheometer values. The viscous and elastic moduli of 4% and 6% poly(ethylene oxide) solutions were successfully measured at 30°C and 37°C. For MCF-7 cells, a 10-fold increase in the elastic modulus was observed using 1P, without a corresponding increase in the viscous modulus. Thus, it was shown that MCF-7 cells undergo an increase in stiffness during apoptosis.

Experiments have shown that there is an increase in ultrasound backscatter from cells during cell death. Since cell scattering depends on the mechanical property variations, one step towards a better understanding of that phenomenon involves measuring the cells' viscoelastic properties. Two promising techniques used for such studies are particle tracking microrheology (1P) and two-point microrheology (2P). The main aim of this work is to develop and test the ability of both to measure changes in viscous and elastic moduli of breast cancer cells during chemotherapeutic treatments. First, the viscosities of glycerol-water mixtures measured using microrheology were found to be within 5% of rheometer values. The viscous and elastic moduli of 4% and 6% poly(ethylene oxide) solutions were successfully measured at 30°C and 37°C. For MCF-7 cells, a 10-fold increase in the elastic modulus was observed using 1P, without a corresponding increase in the viscous modulus. Thus, it was shown that MCF-7 cells undergo an increase in stiffness during apoptosis. The cell is very heterogeneous over many lengthscales. Adapted from [14] . . .  It has been shown that high frequency ultrasound  has the ability to detect changes in dying cells. For example, the ultrasound backscatter was observed to increase by 6-13 dB as cells started undergoing apopsosis [10,11,23,24].  Adapted from Czarnota et al. [10].
If this change in the backscatter intensity could be fully characterized, it could potentially be used as a novel way of monitoring cancer treatments. However, this process is still not well understood. The cell is a complex structure, and it is not known exactly which part of the cell is mostly responsible for this change in ultrasound scattering. Theoretical models developed by Falou et al. [15] have aimed to develop finite element models of the backscattering from cells in order to understand the underlying physics. However, one of the inputs required for these models is the mechanical properties of both viable and non-viable cells. Therefore, developing a technique to measure properties of the whole cell, as well as different subcellular components, is crucial.

Shear-mode ultrasound Imaging
In conventional longitudinal-wave ultrasound imaging, very high frequencies are used (usually in MHz region) since higher frequencies yield better resolution. However, at those frequencies, the waves are also highly attenuating in tissue. The shear waves generated by ultrasound pulses have a much lower frequency (kHz range). Such shear waves also produce a much better soft tissue contrast when compared to conventional longitudinal waves. Whereas the bulk modulus of soft tissue spans only a small range, the range for the shear modulus is much bigger [34]. The main idea behind this technique is that in solids, both longitudinal and shear waves can travel.
So, an ultrasound beam can be focused at different depths and locations within the tissue and the shear-wave generated by the radiation force of the beam can be detected [34]. The speed of the shear wave depends on the mechanical properties of the material. Hence, a mechanical map of the material at tissue level can be obtained. Recently, ultrasound imaging using shear-waves has been proposed as an alternative method of obtaining the changes in mechanical properties that occur in tissue due to pathology [1,34] for example, images taken before and after treatment will be different. Thus, a novel imaging modality could be developed using shear-mode ultrasound imaging.

Cell Structure and Mechanics
The cell is one of the basic components of complex organisms. There are two main types of animal cells: eukaryotic (found in organisms such as mammals and prokaryotic (found in lower organisms such as bacteria). Eukaryotic cells have complex and dynamic structures, and their properties vary over different length and time scales. For example, the mechanical properties are altered in response to different kinds of chemical and physical stimuli [14]. A schematic of a typical eulkaryotic cell is shown in figure 1.2. The cell structure is highly heterogeneous.
Different locations within the cell also have different properties due to the presence of a variety of sub-cellular structures Therefore, techniques with high spatial and temporal resolution are preferred when studying cellular mechanics. The cell is very heterogeneous over many lengthscales. Adapted from [14] .
The cell's mechanical properties are primarily determined by its actin filament cytoskeleton [44]. Thus, one model that has been used to study cell mechanics is by reconstituting polymer networks (e.g of actin filaments) since there is more control of the properties of these networks [18,45]. From experiments done in vivo, on cancer cells [22,39,28] it has been determined that cells are viscoelastic (for a comprehensive review, see Kaszra et al. [21]). In addition, there have been several mathematical models developed of the cell. Since the cell is very heterogeneous, modelling each and every component is extremely challenging. Several methods have been used. These include modelling the cell as a continuum, using a tensegrity model of the cytoskeleton or modelling the actin filaments as a foam [14]. In the continuum model, the viscoelastic behavior can be obtained using different arrangements of two elements connected in various ways: a linear spring for the elastic component, and a dashpot for a viscous component. Such models can provide a simple but good first approach in the study of cellular biomechanics. Bausch et al. [3], for example, measured the viscoelastic response of NIH 3T3 fibroblast cells using magnetic bead microrheometry and analyzed the viscoelastic response curves using models consisting of springs and dashpots. The tensegrity model uses actin microfilaments as the component providing tension and microtubules provide the compression component [14,20,37]. According to this theory, the interaction between these two components can be used to predict viscoelastic behavior. Finally, when modelling the actin filaments as an open-celled foam (random, solid matrix with open pores) as described by Gibson and Ashby [17], a unit cell model can be used for the cytoskeleton and the effective shear modulus can be estimated by calculatind the mechanical properties of individual fibers [14]. The cell's properties also change due to physiological processes; a cell in a diseased state will behave differently from a non-diseased state [28]. Thus, it is important to develop a better undestanding of the mechanical behavior of cells under various conditions.

Mechanical Properties Measurement
There have been several techniques developed with the aim of assessing the mechanical behavior of cells under a variety of conditions such as in response to chemical or physical stimuli, metastasis or migration. For example, there have been several attempts made to measure the response of cells to an applied force. Some of the techniques developed include atomic force microscopy (AFM), optical tweezers, magnetic beads and micropipette aspiration [14]. In AFM, a probe attached to a cantilever is used. When that probe touches the surface of a cell, it is deflected. In this way, very high resolution images of live cells can be obtained (probe deflections of up to 1 nm can be measured). The mechanical properties of cells can also be obtained using optical tweezers. For this method, a dielectric bead that has been injected into the cell is trapped in a potential well using two laser beams. From the motion of those beads, properties such as elasticity can be obtained. Magnetic bead rheometry is similar to the optical trap technique. However, instead of using lasers to trap the paramagnetic beads, a magnetic field is applied and this results in the motion of the beads. Finally, there is the method of micropipette aspiration. In this technique, a very thin glass pipette is used to generate a negative perssure after it has been brought in contact with the cell. By analyzing the deformation of the cell caused by the micropipette, one can extract the mechanical properties. For a more thorough description of the techniques mentioned above, see Ethier et al. [14]. Even though there are a host of techniques available to researchers, they all have limitations. For example, in AFM, only one frequency can be probed at a time. When the beads are moved by a magnetic field or using optical means, an active force is exerted on the cell, and this might have an effect on the cell mechanics.
One of the more recently developed techniques used to measure the viscoelstic properties of a wide variety of soft materials is microrheology.There are two main categories in which In this method, laser beams are used to move particles and the response is measured. This technique can be used to measure the rheological response over a broad range of frequencies (theoretically can go up to MHz). Also, the response using a single probe particle at a time can be measured. This allows for very localized measurements [40]. However, by actively manipulating the system one risks causing damage. Also, the active methods usually require comparatively more sophisticated equipment. On the other hand, passive microrheology relies only on the Brownian motion of particles caused by material's inherent thermal energy. This method has the advantage of being minimally invasive, usually only requiring a way of getting the probes into the material. Also, passive microrheology requires less sophisticated and cheaper equipment. In passive microrheology, no external forces are exerted on the system as the motion of particles is due to the inherent thermal energy of the system. In cells, any active force can modify the cell's properties due to cellular reorganization as a response to this force. Hence, passive microrheology possesses an advantage over active methods. Passive methods have been used by several groups in the study of cell properties. Denis Wirtz [44] has written a review of the various cell lines analyzed with particle tracking microrheology, a passive technique. In addition, there have been spatial maps of Swiss 3T3 fibroblast cells developed by Tseng et al. [39] showing that this technique can be used to measure the properties in different locations within the cell. They found that there is a lot more heterogeneity in the cytoskeleton, compared to reconstituted actin filament networks. Moreover, they observed that the cell behaved more like an elastic solid at high frequencies (63 rad/s), but was more like a viscous fluid at low frequencies (1 rad/s). There has also been work done on cancer cells. Li et al. [28] compared the viscoelastic properties of malignant (MCF-7) and benign (MCF-10A) breast cancer cells and showed that the mean squared displacements (MSD) in the malignant cell lines were consistently higher.
Another technique related to particle tracking microrheology, but developed more recently, is two-point microrheology. This technique, developed by Crocker et al. [8], was shown to more acurately measure the viscoelastic properties of complex materials. Instead of extracting the viscoelastic properties using data collected from single particle motion, this method aims to extract the same properties from the cross-correlated motion of particle-pairs. John Crocker's group demonstrated that the viscous and elastic moduli obtained using two-point microrheology were in better agreement with rheometer measurements for 0.25% guar solutions compared to one-point particle tracking microrheology [8]. Moreover, in TC7 cells, it was shown that the two-point MSD is consistently higher than the one-point MSD [7]. One-point and two-point particle tracking microrheology are two methods that could possibly be used in the analysis of a cell's mechanical behavior in response to treatment.

Motivation and Hypothesis
In this thesis, I aim to continue the work started by Ahmed El-Kaffas in our laboratory [13] and use both of the abovementioned passive microrheology techniques to measure the mechanical properties of treated and untreated cancer cells. I hypothesize that both one-point particle tracking microrheology and two-point microrheology can be used to measure the mechanical properties (viscous and elastic moduli) of inhomogeneous viscoelastic materials, breast-cancer cells (MCF-7) and the changes that occur during cell death. Preliminary work has shown that one point particle tracking microrheology can accurately measure the viscous moduli of glycerol/water mixtures as well as detect changes in the viscoelastic nature of guar/water mixtures [13]. In addition, Ahmed El-Kaffas showed that the viscous and elastic moduli of PC3 cells increase as a function of time after treatment using one-point microrheology, with the increase being more apparent in the elastic modulus [13]. However, the two-point microrheology method, which is more accurate in the measurement of the mechanical properties of specimens with elasticity, was not successfully developed.
The first step was to verify the accuracy and precision of both techniques for a purely viscous and a viscoelastic solution. This was done using different concentations of each solution to measure the range over which the techniques can be used. Moreover, since the cells had to be kept at 37°C, the measurements were also done at different temperatures. To do so, experiments were done with glycerol/water and poly(ethylene oxide)/water mixtures. Glycerol, being a Newtonian, purely viscous solution was ideal for the initial measurements. Poly(ethylene oxide) solutions are viscoelastic, inhomogeneous and non-Newtonian, which is why they were chosen to be used as a viscoelastic phantom. After the initial validation stages, both techniques were used to measure the MSD of fluorescent particles embedded within MCF-7 cells treated with a chemotherapeutic and to extract the viscoelastic properties at different frequencies. This thesis hence provided new insight (both qualitative and quantitative) into the changes in the mechanical properties that occur in breast cancer cells during cell death. The results from this work will also aid in obtaining more information on the mechanical properties of tumors at the cellular level, as well as help in developing more accurate models of the ultrasound backscatter from cancer cells undergoing cell death.
Chapter 2 Theory

Brownian Motion
The theory of Brownian motion forms the basis for both passive microrheology techniques described in this chapter. It is a mathematical model put forward by Albert Einstein, among others, to describe the random walk/diffusion of particles embedded in a fluid medium due to the bombardment of these particles by the molecules in solution. By characterizing this random walk, one can extract the mechanical properties of the medium (such as the viscous and elastic moduli). In this chapter, first, the theory of Brownian motion will be described and then the relationship between the material's viscosity and the particles' motion will be derived, following Einstein's formulation. These principles will then be extended to more complex, viscoelastic materials. Finally, there will be a discussion on the various ways in which particles can be embedded within cells.

Underlying Principles of Brownian Motion
The name "Brownian" motion arises from fact that botanist Robert Brown who is traditionally regarded as being one of the first to observe such a phenomenon. Using an inverted microscope, he observed the motion of clarika pollen grains in water. However, he did not adhere to the contemporary view that this irregular motion was due to the pollen grains being "alive". To prove his point, he stored the pollen in alcohol for 11 months. He then showed that the random motion was still observed. He observed similar behavior with other particles such as powdered metals, rocks, and carbon particles [4]. Even though he was unable to explain the theory behind this random motion, he is still acknowledged as the person who discovered "Brownian" motion.
It was not until the kinetic theory of gases was formulated that further development was made in the field of Brownian motion. In particular, the theory that the motion of particles could be due to the collision between the particles and molecules in the fluid was not proven.
Ignace Carbonelle, Joseph Delsaulx and William Ramsey speculated, for example, that if small particles are bombarded from all sides, an imbalance would be caused and the particles would move in a certain direction. They noted that such motion was unpredicable, and hence a mathematical formulation would be impossible. However, there are several characteristics of Brownian motion that were noted [19]: • The particles can move both in straight lines and rotate. Both of these types of motion are irregular.
• The Brownian motion of the particles are independent of one another, provided there are no collisions between these particles.
• If the particles are smaller in the medium, the motion is more vigorous.
• The motion does not depend on the chemical properties of the particles • In less dense fluids, the motion is more vigorous.
• At higher temperatures, the particles move more vigorously.
• If effects such as evaporation are removed, the particles keep on moving.
One of the first people who attempted to describe Brownian motion mathematically was Thorvald N. Thiele in 1880 [26] in a paper on the principle of least squares. Louis Bachelier also proposed a theory in his PhD thesis in 1900. He based his theory on the stochastic behaviour of stock markets [2]. However, it was Marian Smoluchowski [36] and Albert Einstein [12] who independently formulated a solution to the Brownian motion problem. Smoluchowski published a one dimensional model that described the Brownian motion of a particle. There were, however, several assumptions made in this model. The mass, M, of the test particles were assumed to be much greater than the mass, m, of the bombarding molecules. He also assumed that the particle motion was only in one dimension, and that the probability for the test particle to be bombarded from any side was equal. Moreover, it was also assumed that after every collision, the test particle's velocity changed by an equal amount. These assumptions over-simplified the problem and as a result, his theory could only describe Brownan motion in a qualitative manner. For example, in a more realistic Brownian motion problem, it cannot be assumed that there are an equal number of collisions from each side of the test particle as it is moving. Also, the change in velocity of the particle after each collision won't be the same; it will follow a distribution.
Albert Einstein derived a more complete mathematical description of this phenomenon. He started by assuming that colloidal particles could be considered as large molecules [19] and that atoms did exist. Based on these assumptions, he tried to predict the behavior of other particles surrounded by atoms. By concentrating on the average behavior of a large number of particles, he essentially proved that Brownian motion was a stochastic process characterized by a diffusion constant dependent on parameters such as Avogadro's number N, the absolute temperature, the test particle's size and the solution's viscosity. Indeed, his theory was not dissimilar to Bachelier's theory on the price fluctuation in the stock stock market [2]. This was all theoretical work, though. The experimental confirmation of Albert Einstein's theory, and hence proof that atoms did exist was brought by Jean Perrin at the Sorbonne in Paris [19,33], work for which he received the Nobel prize in 1926.

Mathematical Formulation of Brownian Motion and the Stokes-Einstein Equation
As mentioned previously, Einstein's investigations on the osmotic pressure exerted by colloidal particles led him to an equation relating the diffusion coefficient to the fluid's viscosity, the test particle's size, the absolute temperature, and Avogadro's number [19]. The main assumtion he made was that the colloidal particles were much bigger than the particles in the solvent. Then, he went on to find the relationship between the diffusion of the particles and their displacement under the random walk. The following derivation has been adapted from Truskey et al. [38] and follows the same approach used by Albert Einstein in a one dimensional case. Consider the situation depicted in figure 2.1 in which N colloidal particles are suspended in a solvent in a region of length, L. Assume that the concentration of the particles is C(x).
If the solution is in thermodynamic equilibrium, C(x) is usually uniform, with an exception being when the particles are being acted upon by an external force, F x (x). In that case, the diffusion flux is balanced by a negative flux, -v x C(x), caused by convection. This can be . This is a closed system, where no particles are allowed to escape. The length of the container is L. expressed using the following equation : where v x is the average velocity of all colloid particles due to thermal motion of the solvent particles and D is the diffusion coefficient. Now, if the Reynold's number is low (laminar flow regime), the velocity, v x , is directly proportional to F x . Thus, where β is a constant known as the frictional drag coefficient.
If equation 2.2 is substituted into equation 2.1, we obtain the following:

BROWNIAN MOTION CHAPTER 2. THEORY
A concept that will be used for the next few steps is that of "virtual displacement", δu(x).
In essence, it is an arbitrary displacement that preserves the equilibrium state of the system and goes to zero at the boundaries. Since this is a closed system which is in equilibrium, the virtual change, δ, in the Gibbs free energy, G, has to be zero. In other words, where H is the enthalpy, T is the absolute temperature and S is the entropy.
Moreover, since the system is closed and there is no expansion, the change in enthalpy, δH, is equal to the change in the particles' internal energy, i.e, The virtual change in entropy, δS, can be calculated using the Bolzmann equation. Generally, if the volume of a particle system increases from V 1 to V 2 , the net change in entropy is given by: If we assume that ∂δu ∂x 1, we can substitute equation 2.7 into equation 2.6 to obtain: If the above equation is integrated, the virtual change in entropy can be obtained: Note that to obtain equation 2.9, one has to make use of the boundary condition that the virtual displacement goes to zero at x = 0 and x = L.
Now that expressions for δG, δH and δS have been obtained, equations 2.4, 2.5 and 2.9 can be combined: Since the virtual displacement δu is arbitrary, the integral is only equal to zero if the integrand is zero. Therefore, Comparing equation 2.3 and equation 2.11, it can be concluded that the diffusion coefficient, D, is For spherical particles of radius, a, in a solvent of viscosity η, the frictional drag coefficient Therefore, the Stokes-Einstein equation can be rewritten as If it is assumed that the average displacement of a particle is ∆r(τ) given within a certain lag time or time interval, τ, the diffusion coefficient in n dimensions, D n , is obtained from the following equation: Therefore, if the motion of particles embedded within a solution is monitored, and the tracks analyzed, the viscosity, η, of that solution can be obtained:

Viscosity, Elasticity and Viscoelasticity
The viscosity of a material is a measure of that material's resistance to flow as a fluid in response to either a shear stress (force per unit area applied parallel to a material's surface) or tensile stress (force per unit area applied perpendicular to the material's surface). In everyday terms, viscosity can be regarded as a measure of a fluid's "thickness". For example, water is "thin", implying it has a low vicosity. Honey, on the other hand, is "thick", meaning it has a high viscosity. The elasticity of a material is a measure of the propensity of a solid to return back to its original shape after deformation due to an applied stress. In a purely viscous fluid such as glycerol, energy is always dissipated due to viscous flow, and the material never regains its shape. On the other hand, for a purely elastic material, energy is stored in the material and when the deformation-causing stress is removed, the material regains its original shape. One example is a spring. The energy is stored when it is stretched, and it reverts back to its original shape when the external force is removed (assuming the elastic limit is not exceeded, and the object does not undergo permanent plastic deformation). A viscoelastic material is one that exhibits both viscous and elastic behaviour. The strain for a purely viscous material is linear with time. For a purely elastic material, it is instantaneous. However, for a viscoelastic material, there is a time-dependence on the strain (which might be nonlinear) [31]. There are several properties associated with viscoelastic materials: 1. The stress-strain curve exhibits hysteresis (see figure 2.2). So, when a load is applied and then removed, there is energy lost (for example, as heat).

VISCOSITY, ELASTICITY AND VISCOELASTICITY CHAPTER 2. THEORY
2. When a constant strain is applied, the material tries to relieve the stress. This is known as stress relaxation.
3. When a constant stress is applied, the material tends to deform slowly (increasing strain).
This is called the creep.
The creep and stress relaxation are two very important terms needed to understand viscoelastic substances. The creep, J(t), is given by the ratio of the instantaneous strain in response to the applied stress whereas the stress relaxation, G(t), is given by the ratio of the instantaneous stress to the applied strain.
It should be noted that both the viscous and elastic moduli have a frequency dependence, as shown in equation 2.19. Thus a material might behave more like an elastic solid at high frequencies, but behaves like a viscous liquid at low frequencies. One example used by Wirtz et al. [44] is S illyPutty . When left on its own for long timescales (i.e, sampled at very low frequency), it will flatten out, thus behaving more like a liquid. However, Silly Putty can also bounce and regain its shape when thrown against a wall, behaving more like an elastic solid during the short timescale impact (high frequency).
A rheometer is used to measure the viscous and elastic properties of different materials as a function, for example, of frequency and temperature. It typically works by applying a sinusoidally varying strain, (t), of a given amplitude, 0 , and frequency, ω, and then measuring the resulting stress in the material. This stress, σ(t), can be broken down into a sine and a cosine component.
From equation 2.20 above, one can extract the elastic modulus from the sine (in-phase) component with G (ω) = σ 0 and the viscous modulus from the cosine (out of phase) component with

Microrheology
Rheometers have been used to probe the properties of various viscoelastic materials and cells are known to have a behavior that is dependent on the rate at which the deformation is applied. However, it is not possible to probe the full behaviour of cells using a conventional, macroscopic rheometer. To compensate for the inadequacy of using a rheometer to probe a cell's viscoelastic behavior over a wide range of frequencies, the technique of microrheology was developed. Microrheology refers to a family of techniques developed to study the mechanical properties of soft materials (such as polymer solutions and cells) at very small length-scales (sub-micron). For rheometer measurements, typically a few milliliters of sample are required.
As a result, the study of rare samples cannot be done. In microrheology, only a few microliters are required. In addition, a rheometer can only measure the average properties of materials whereas for microrheology, the local viscoelastic response can be measured. This can be particularly advantageous for inhomogeneous materials such as cells. There are two passive microrheology techniques that will be used in this thesis, namely one-point particle tracking microrheology and two-point microrheolgy.

One Point Particle Tracking Microrheology (1P)
In one point particle tracking or 1P microrheology, the thermally-induced Brownian motion of thousands of micron size particles is tracked using video microscopy. The motion of such particles can be determined to sub-pixel accuracy using specialized software written by Crocker et al. [6]. Then, by calculating the mean-squared displacement (MSD) and using this in the Stokes-Einstein equation, the viscoelastic parameters can be obtained.
In theory, the frequency range measureable by 1P microrheology exceeds that of a rheometer. It has been shown that this upper limit is about 100 kHz for 1P as a result of inertial effects [16,27]. In practice, however, this upper limit depends on the equipment available. When using a CCD camera coupled with a microscope, for example, frequencies of only up to 100 Hz can be achieved, depending on the desired spatial and temporal resolution. This limit de-pends on the maximum amount of time each particle remains in the field of view, which varies depending on the different microenvironments. Another advantage of 1P is that it can be used to probe spatially localized behavior by only tracking specific particles. This is especially help- where F u ∆r 2 (t) is the unilateral fourier transform. The unilateral fourier transform has a range of zero to infinity. This range cannot be obtained using particle tracking microrheology since this technique has lower and upper limits as explained previously. There are several numerical methods that have been used to estimate the fourier transform from the finite range obtained in experiments. The one used in this thesis was developed by Thomas Mason [30].
In this method, the complex shear modulus is estimated by measuring the log-derivative of the From the slope, the magnitude of the complex shear modulus can be calculated.
where Γ is the gamma function. After computing the complex shear modulus at each timepoint and MSD, the viscous and elastic moduli can be obtained using:

Two-point Particle Tracking Microrheology (2P)
One point particle tracking microrheology has been shown to work very well for homogeneous systems such as glycerol. However, its ability to determine the mechanical properties of more complex fluids that are heterogeneous has been questioned. This is particularly the case when the probes are much smaller than the length scale of heterogeneity. To visualize the problem with 1P, consider the situation depicted in figure 2.3. This might for example be a polymer solution. One of the probes might be present in an area with a lot of polymer strands, while another probe might be in a region with no polymer chains. The MSD of those two probes are going to differ by a significant amount and will scale differently with lag time. As a result, the viscoelastic moduli calculated from those two probes will be different. caused by the motion of one particle will affect the motion of another particle and that this effect will vary depending on the distance, d, between each of them. As a result, the crosscorrelation of particle pairs can be used to obtain a measure of the whole material's strain field [40]. In a homogeneous solution, the two-point cross correlation function, D rr is proportional to each particle's MSD and decays as a d , where a is the particle radius. However, this is not the case in heterogeneous system, where the a d dependance only occurs over a small range. Therefore, when doing data analysis for 2P, one has to choose a range over which the proportionality relation stated above holds when computing the two-point equivalent of the MSD.
The computational power required for 2P data analysis is greater than for 1P since crosscorrelation between all possible particle pairs over all lag times has to be calculated. In addition, the number of tracers required is larger. In our lab, it has been estimated that about 3000 distinct tracks are required to obtain statistically significant values for the MSD, which is in agreement with the arguments presented by Crocker et al. [8] in his discussion on the statistical requirements for computing the two-point MSD.
The first step in calculating the 2P MSD is to obtain the vector displacements of all the particles at all the specified lag times, τ, for all time points, t: where r i α (t) is the displacement for particle i at time t using a coordinate , α, from a given coordinate system. In this thesis, the polar coordinate system is used. So, when computing the 2P MSD in the radial direction, α = d, which is the distance between particle pairs as shown in figure 2.4. Then, the ensemble-averaged cross correlation of all possible pairs of particle displacements, can be calculated using the tensor product between particles i and j that are a distance, d, apart for a lag time of τ. This is then averaged over all possible particle pairs (i, j) i j and times, t, as shown in the equation below: As mentioned previously, 2P microrheology has been shown to measure the bulk viscoelastic properties more accurately than 1P for inhomogeneous systems such as cells. Moreover, it has been shown that this method is not as sensitive to the interactions between tracer particles and the materials. However, it also has some disadvantages. One of them is related to the number of particles required to obtain statistically significant results. Whereas for 1P only about 100 particles are needed (for homogeneous materials), for 2P up to 3000 different tracks are required. Crocker et al. [7] have also reported similar results. In addition, 2P is very sensitive to noise caused by vibrations and sample drift because particle motion due to these are highly correlated. This is particularly relevant to cells where active processes within the cell can cause super-diffusive motion of the probes. Hence, bead selection is a very important step in the data analysis.

Methods for Embedding Beads into Cells
For both 1P and 2P, fluorescent particles embedded within the sample have to be tracked. It is important that those particles undergo only Brownian motion and are not affected by other processes within the cell. Some have tracked endogenous particles such as lipid granules or mitochondria present in the cell [25,46]. An advantage of this technique is that no foreign materials are introduced into the cell. However, the particles are more susceptible to motion caused by intracellular processes.
Valentine et al. [41] showed that carboxyl-modified particles injected directly into the cell using a Xenoworks Micromanipulator and microinjection system (Sutter Instrument, California: http://www.sutter.com) do not interact significantly with materials in the cytoplasm (e.g. proteins) and their motion is mostly Brownian. One drawback of this technique, though, is that the cells can only be outside of the incubator for a maximum of 30 min to reduce stress due to changes in the environment and minimize contamination. As a result, only a limited number of cells can be injected, which means that there are not enough beads to properly perform 2P analysis.
It has also been shown that polystyrene microspheres can enter the cell via endocytosis [22,42]. For this technique, no injection needles are used, so the cell is not damaged. Moreover, there are about 10 times more beads per cell compared to the microinjection technique in our experience. Hence more tracks are obtained. However, some of those particles can be trapped in vesicles that are bound to the cell membrane, which will affect their motion [42]. In addition, some of those particles can be actively transported within the cell. Exogenous particles such as fluorescent beads can also disrupt the cell structure as they can, for example, bind to the cytosleketon [42] whereas endogenous particles do not.
In the end, a compromise was required between having enough particles for tracking, reducing cell-microsphere interaction and not damaging the cell. With all of the above taken into consideration, it was decided that for any subsequent experiments, bead uptake by endocytosis would be used. the only difference was that care was taken not to track particles close to the cell membrane or particles that exhibited directed motion (an indication that these were being actively transported).

Poly(ethylene oxide) (PEO)
Poly(ethylene oxide) of molecular weight 900 kD was purchased in powder form from Acros Organics (New Jersey, USA). Two different w/w solutions (4 % and 6 %) were made by dissolving the powder in distilled water as outlined by Williams et al [43]. In addition, both solutions were centrifuged at 2000 rpm for 10 min to remove any air bubbles and other residues (such as undisolved powder). Then, 500 µl of the 0.20 µm diameter fluorescent beads were added and the solutions were again gently stirred overnight.

MCF-7 Cells
Two weeks prior to any experiment, fresh MCF-7 cells were thawed and transferred to a cul-

Experimental Setup
The experimental setup used for all particle tracking microrheology experiments is shown in figure 3.1. The microscope used was an Olympus IX-71 inverted microscope (Olympus Inc., Markham, ON: www.olympus.com). Three different objectives were available (10X, 40X, and 100X), the latter being used with oil immersion. In light microscopy, the smallest object that can be resolved is on the order of 0.40 µm. Therefore, to allow for the use of smaller polystyrene beads, a fluorescence illumination system (EXFO Inc., Quebec, QC: www.exfo.com) was connected to the apparatus.
Videos were recorded using a Retiga EXi (QImaging Inc., Surrey, BC: www.qimaging.com) CCD camera connected to a PC. The software chosen to record videos was Streampix 3.0 (Norpix Inc., Montreal, QC: www.norpix.com). This software allows for the modification of various parameters that can affect the framerate and contrast. For example, reducing image exposure time can increase the frame-rate, but will reduce contrast between the beads and their background. Moreover, longer exposure time will lead to dynamic errors because the particle moves during the finite frame acquisition time [35]. One of the requirements for particle tracking is that each particle must cover at least 4 pixels (otherwise the position of each particle cannot be determine accurately). Since binning the video pixels increases contrast and frame-rate, but decreases the number of pixels per bead, a compromise was required between having a good contrast and a high framerate. It was decided that a framerate between 26 frames-per-second (fps) was acceptable for the cell experiements (where 0.10 µm diameter beads were used). To achieve that, a 400x400 pixel frame size was used with an exposure time of 10 ms. below relating the viscosity of a glycerol-water mixture, µ, to its concentration and temperature [5]. The experimental results for 1P were compared to the theoretical predictions to validate the experimental procedure for a purely viscous solution. In addition, the viscosity was measured using the rheometer for all five solutions. The experimentally determined viscosity of the 69% and 70% solutions were compared to the theoretical predictions to determine the sensitivity of the microrheology method. The formula used for the theoretical calculations was:

Wh i t e L i g h t S o u r c e Ob j e c t i v e He a t e r S y s t e m P e r s o n a l Co mp u t e r
where • µ w : Viscosity of pure water, • µ g : Viscosity of pure glycerol, • α: Weighing factor dependant on temperature and concentration (0-1).

Poly(ethylene oxide)
Similar volumes of PEO solution, one for each concentration (4% and 6%), were sealed in PC20 chambers using microscope coverslips. Again, for each experiment, the temperature was set to 30°C and the solutions were allowed to settle for 1 hour. For each concentration, twenty videos were captured, at 10 min intervals using the same settings as for glycerol. The rheometer was also used to measure the bulk viscous and elastic moduli of each solution for comparison. Finally, in order to test the accuracy with which 2P microrheology could measure the viscous and elastic moduli at another temperature (that of cells), ten videos were captured of the 4% solution at 37°C.

MCF-7 Cells
The cells plated in the Delta-T dishes mentioned in section 3.2.3 were FBS and insulin starved 24 hour prior to any experiments. Paclitaxel, a chemotherapeutic agent, was introduced at a concentration of 20 ng/ml to two of those culture dishes and two other dishes were left untreated. After placing one dish in the Delta-T dish temperature controller and allowing the sample to settle for 10 min, thirty videos (1000 frames each, different locations) were captured at 0, 12, 24 and 48 hours after exposure to paclitaxel using the 100X magnification objective. Care was taken to set the focal plane about halfway between the surface of the cells and the bottom of the culture dish to avoid boundary effects and microspheres attached to the cell membrane. This procedure was then repeated for each of the culture dishes (including the controls). Usually, for each run of the experiment, sixty "treatment" videos and sixty "control" were obtained for each timepoint. It should be noted that even though the temperature was maintained at 37°C during video capture, the culture dishes were only allowed to be outside of the incubator for a maximum of 30 min to avoid cell contamination. Also, the contrast between the fluorescent particles and background is smaller for cell imaging compared to glycerol or PEO. There are also fewer beads in each field of view for the cells since the cell plating density was about 20% and each cell contained 10 beads on average. Therefore, to minimize loss of contrast due to photobleaching and to allow for recording from a greater number of fields-of view, the cell videos were kept shorter.
Finally, in collaboration with Ahmed El-Kaffas, the above experiment was repeated with 0.19 µm microinjected polystyrene beads to determine whether the results would change if a different embedding method was used (For a detailed explanation of the bead injection procedure see [13]). Five culture dishes had to be used since the number of beads available per dish was much smaller (about 10 per dish). Again, the cells were serum and insulin starved 24 hours prior to treatment, and videos were captured at 0, 12, 24 and 48 hours. However, due to the limited amount of beads present, only 1P data analysis was feasible.

Data Analysis
After capturing the videos using Streampix, each of them was exported to a different TIFF stack for further analysis using Matlab 2009b. The code used for particle tracking was originally written by Crocker et al. [6] in IDL. This was then transcribed to Matlab by Daniel Blair and Eric Dufresne (http://physics.georgetown.edu/matlab/). The first step in the tracking process is to filter each frame of the video so as to remove noise, correct for image imperfections and hence improve signal-to-noise ratio between the particles and the background (as shown in figure 3.2(a) and (b)). A bandpass filter is used to remove both low spatial frequency and high spacial frequency noise. The lower limit for the bandpass filter is always set to one pixel whereas the upper limit is set to a few times the radius of the particle.
Next, as shown in can be set to 12 pixels, which corresponds to 1 µm. Since the microspheres appear to be twice ther actual size under fluorescence, the 1 µm radius of gyration is equal to twice the apparent diameter. In addition, by using all the information about each bright spot, such as the brightness distribution, the centre of each particle could be determined to sub-pixel accuracy. All of the above information (brightness, size, shape) can be used to distinguish unsuitable particles from suitable ones. The suitable particles tend to be circular, bright, and of a specific size. In contrast, the unsuitable particles and noise tend to be elliptical, less bright, and/or irregularly sized. The usual strategy for feature-finding was to change the input parameters and look at what kind of particles were being accepted or rejected by the algorithm. For cell experiments, more care was taken and each particle track was carefully analyzed. Particles found near cell boundaries were rejected since they were more likely to be bound to the cell membrane.
Oversized particles were also rejected since those were most probably several beads that had formed aggregates. Also, tracks that showed particles moving in a specific direction instead of undergoing Brownian motion were discarded. This was done by comparing the distribution of particle displacements in the x and y planes and check whether it was normal.
Particle tracks were formed by relating the locations of particles in each frame. First, the user had to specify a maximum displacement threshold. This threshold was chosen so that it is smaller than the maximum distance between particles (so as not to link two different particles into one track) but smaller than the expected maximum frame-to-frame displacement of a particle. This threshold, of course, depends on the material being used. For example, it was determined that for the 70% glycerol solution, a 0.20 µm polystyrene microsphere had a maximum displacement of approximately 5 pixels. Thus, if the minimum particle separation is 10 pixels, the threshold above has to be between 5 and 10 pixels. Sometimes, in a very noisy video, spurious particles can get tracked by the algorithm despite all the precautions taken above. These tracks are, however, short and the maximum displacement is small. Therefore, by specifying a minimum track length in the routine, one can eliminate those tracks. The microspheres move in three dimensions, but the tracking is only done in two dimensions. Thus, particles might move in and out of the focal plane. To avoid tracking the same particle twice simply because it went out of the focal plane for a few frames, a "memory" parameter was used.
For example, setting the "memory" to 5 would mean that if a particle is out of the focal plane for a maximum of five consecutive frames, then reappears at aproximately the same location, it counted as a single particle. Otherwise, it counted as the beginning of a new track. An example of tracked particles is shown in figure 3.2 [9]. The routine computes the average distance moved by each tracked particle from frame to frame in the x and y directions. The drift can then be reduced by substracting this average from the actual distance moved.
The MSD can then be calculated from the dedrifted tracks using the mean of all particle displacements from frame to frame for every possible lag time: where • ∆r 2 (τ): MSD at lag time τ, • r(t + τ): position of one particle at time, t + τ, • r(t): position of one part at time, t.
For example, if the 5000 frames long video has a framerate of 50 fps, the minimum lag time is 1 50 s and all frames are sampled to calculate the MSD. On the other hand the maximum lag time is 5000 50 = 100s and to calculate the MSD, only the first and last frames are used. The 1P elastic and viscous moduli can then be directly computed using the code provided by Dr.
Kilfoil using the MSD, bead radius and temperature as input parameters as required by the generalized Stokes-Einstein equation. The computation for the 2P moduli is more involved. There were four main parameters required by the algorithm. First, one had to specify a minimum and maximum distance over which to compute the cross-correlation function. Usually, the minimum distance must be set to a value that is larger than the length-scale of heterogeneity. Lau et al. [25] found that a minimum correlation distance of 2 µm is good. As mentioned previously, the two-point correlation approximation, D rr , is only valid over the range where is inversely proportinal to 1 d . The actual range depends on different factors such as the solution being used, the number of beads present. Thus, the maximum correlation distance has to be chosen bearing this in mind. From experience, it was found that a maximum correlation distance of 10 µm was usually acceptable. Another input parameter needed to calculate the 2P correlation is the number of bins between the minimum and maximum correlation distances in which the data is stored. If a large number of bins is used, more points are obtained. However, the statistical power of each point is smaller. On the other hand, choosing fewer bins gives more statistical power. This also depended on the material being studied and the bead concentration. For our experiments, 25 bins was found through trial and error measurements to give accurate results.
The fourth parameter needed is the maximum lag time over which the microspheres will be correlated. Since the particles drift in and out of the focal plane, at the higher lag times, there might be less particles being tracked. Hence, one has to set the maximum lag time to be less than the time during which a particle remains in focus. This value is usually set to one or two seconds, depending on the number of particles available at the higher lag times. If there are not enough particles available at the higher lag times, the uncertainty in the MSD will be greater, and this will lead to a bigger uncertainty in the viscoelastic moduli. The two-point correlation function can be then converted into a 2P MSD, from which the viscous and elastic moduli can be calculated (similar to 1P). It should be noted that for the 2P analysis, all field of views are processed simultaneously in contrast to 1P where each field of view is analyzed separately.        The viscous and elastic moduli for both PEO concentrations were calculated using 1P and 2P (with the same data set being used for each method). This experiment was repeated three times and the average and standard deviation were calculated. The moduli calculated using 2P are within one standard deviation of the rheometer results whereas for 1P they are significantly

MCF-7
For the MCF-7 experiments cells were prepared and beads were internalized as described in section 3.2.3. The first step was to determine whether either 1P or 2P could be used to measure the viscous and elastic moduli of untreated MCF7 cells and whether the results were reproducible. Figure 4.9 gives a comparison between the moduli derived using the 1P and 2P methods on cells for three separate trials of the experiment (one week apart). Again, the same dataset were used for both 1P and 2P. In each case, 60 fields-of-view were captured.    . This leads to a greater dynamic error as explained by [35]. This source of error might also be relevant to the other concentrations, but the effect is not as pronounced.
Another source of error might be due to the temperature fluctuations. The temperature in the rheometer could be more accurately controlled. It was estimated that in the rheometer, the temperature was 30±0.1°C. Hence, it was to be expected that the percentage deviation would be smaller. However, the temperature for the microrheology experiments was controlled by heating the objective, which in turn heated the PC-20 chamber holding the glycerol sample.
As a result, the temperature uncertainty might have been higher (up to ±1°C Moreover, increasing the frame size from 400x400 to something larger will also increase the number of cells being imaged. However, this will lead to a decrease in framerate. In the end, a compromise must always be found between increasing the frequency and capturing enough beads for analysis.
Before performing any experiment, the cells are serum and insulin starved to stop cell division. However, it was not possible to determine in which part of mitosis each cell was in when treated. It might be possible that cells in different stages of the cell cycle have different mechanical properties. For example, during prophase, the nucleus becomes denser as the chromosomes form. Similar changes might also occur in the cytoplasm. This might explain some of the differences in figure 4.9. It could be that for different experiments, cells in different stages of cell division were used for analysis. Even then, a significant change in the average viscoelastic properties was obsserved when using 1P analysis. Moreover, during apoptosis, the cell membrane permeability increases. Such an increase would result in an influx of water and other molecules into the cytoplasm, which might lead to a decrease in viscosity. However, this was only observed at the lower frequencies in our experiments.

Chapter 5
Conclusion and Future Work

Conclusions
The techniques of one-point particle tracking microrheology (1P) and two-point microrheology (2P) were used in this thesis to measure the changes in mechanical properties that occur when MCF-7 cells undergo cell death. The hypothesis was that both one-point particle tracking microrheology and two-point microrheology could be used to measure the mechanical properties (viscous and elastic moduli) of inhomogeneous viscoelastic materials, breast-cancer cells (MCF-7). The first step was to validate the experimental procedure with a purely viscous (glycerol) and a viscoelastic (PEO) solution. For glycerol, I demonstated that both 1P and 2P can be used to accurately measure the changes in viscous modulus and viscosity that occur with increasing temperature and concentration. Using a PEO solution, I was also able to show that although 1P can be used to determine whether the viscosity and elasticity change when the temperature and/or concentration is increased, it does not do so accurately. To obtain accurate measurements of the viscous and elastic moduli of PEO, the 2P analysis method had to be used. It was estimated that about 3000 distinct particles were required to obtain values for the viscous and elastic moduli in agreement with rheometer measurements.
For the MCF-7 cells, the first objective was to confirm whether either method could be used to detect the viscoelastic nature of the cells on a consistent basis. The experiment was hence carried out three times over the span of three weeks. It was observed that even though the value for the moduli obtained on each of the three days were of the same order of magnitude, and followed the same trend, there were still some differences, especially at the lower frequencies.
The differences were mostly attributed to natural variability within the cell line (for example, due different cell size and densities), uncertainties in the temperature as well as errors in the experimental procedure, as explained in the discussion section of chapter 4. Then the cells were treated with paclitaxel and the viscoelastic moduli calculated using both the 1P and 2P methodologies were calculated. Using 1P, it was observed that at the low frequencies, there is an increase in the elastic modulus, but not in the viscous modulus. These changes were not observed in the control experiments. For 2P, no corresponding trends were observed. Thus the hypothesis that 2P could be used to observe the changes that occur in the cell's mechanical properties as it undergoes apoptosis is inconclusive and further studies are required. The 1P results are also consistent with the results obtained by Pelling et al. [32] using AFM to calculate the moduli. It was shown that more beads are likely required for the 2P method to give more accurate results.

Future Work
Both 1P and 2P microrheology are very useful tools in the study of cell mechanics. In this work, the tracked particles were embedded within the cell by endocytosis. However there are other methods to insert tracers into cells. It would be important to compare the results obtained using various methods of embedding cells, such as microinjection and ballistic injection. The ballistic injection method is particularly interesting in that it has all the advantages of microinjection, without most of its drawbacks. The number of particles available for tracking is much larger. Thus, more accurate 2P results might be obtained. Also, the damage to the cell is less than when using a microinjection needle. It will also be interesting to compare the same results to those obtained when using endogenous particles. By using different types of particles, one can obtain a more complete picture of the cell's properties.
Another important step would be to measure how the properties change depending on location within the cell (such as the nucleus and cytoplasm). For example, by probing different regions in the cytoplasm of Swiss 3T3 fibroblast cells, Tseng et al. [39] showed that there is a high degree of heterogeneity. Right now, all particles are embedded within the cytoplasm.
Using the ballistic injection method, it will be possible to embed particles in all regions of the cell. Then, by restricting the particle tracking to specific locations, a mechanical map at different times after treatment could be obtained for different regions within the cell. It will therefore be important to develop the necessary techniques to make such measurements possible. Developing such a mechanical map will also shed more light on the question of which part of the cell is predominantly responsible for the increase in ultrasound backscatter when apoptotic cells are imaged at high frequencies.
Currently, the range of frequencies which can be probed is limited ( 1-100 rad/s). If we want to apply the results from these experiments to models of ultrasound backscatter from cell, we need to develop technology that allows us to capture videos at much higher framerates (i.e higher frequencies). A simple solution would be to acquire a CCD camera which can capture images at higher-speeds without compromising resolution or contrast. It might also be possible to extrapolate the results from the low frequency measurements to higher frequencies, something which has not been done so far using passive microrheology techniques. There are also some other improvements that can be done to the experimental setup. The temperature distri-bution at the bottom of the culture dish is currently unknown and this leads to uncertainties in the actual temperature at the time of measurements. Using an infrared camera to determine the temperature profile will help reduce such an experimental error. Moreover, it is not known whether the fluorescent light is having any effect on the temperature. Again, an infrared camera will help determine that. Also, one of the requirements for optimal cell culture is to have a 5% CO 2 environment while capturing the videos. This is not possible using the current experimental setup. Creating a chamber around the culture dish on the microscope stage and perfusing that chamber with 5% carbon dioxide might improve the robustness of our experimental results. In our cell experiments, it was also very hard to image late apoptotic cells since these are non-adherent. Some materials such as polyethyleneimine and poly-L-lysine have been used to enhance cell adhesion. It is important to obtain more data on the properties of cells in late apoptosis (or necrosis) since one hypothesis is that the more significant changes in the mechanical properties occur during that stage. Moreover, right now, it is not possible to tell whether the cells are undergoing apoptosis or necrosis. Using markers that tag for specific types of cell death might help us differentiate between the two and further refine our experimental results.
It is also not possible to determine which cells are in the early or late apoptotic stage. At each time-point, an average measurement is taken, under the assumption that the number of dying cells increases with time. To further refine the experimetal procedure, the cells can be tagged with fluorescent markers such as Annexin V and propidium iodide (PI). Annexin V, which fluoresces green, binds to phosphatidylserine, which is translocated to the outer cell membrane during early apoptosis whereas PI stains nectotic cells with red fluorescence. Finally, the focus in our laboratory is on breast cancer. The only cell line studied so far has been the MCF-7 cells. Thus, for completeness, it will be very important to perform similar studies on different cell lines such as AML, ZR-75-1 (breast carcinoma showing oestrogen dependence) or MDA-MB-231 (virulent and rapidly-growing ductal carcinoma) cells.