## Abstract

Current methods for predicting stress distribution on a cell surface due to optical trapping forces are based on a traditional ray optics scheme for fixed geometries. Cells are typically modeled as solid spheres as this facilitates optical force calculation. Under such applied forces however, real and non-rigid cells can deform, so assumptions inherent in traditional ray optics methods begin to break down. In this work, we implement a dynamic ray tracing technique to calculate the stress distribution on a deformable cell induced by optical trapping. Here, cells are modeled as three-dimensional elastic capsules with a discretized surface with associated hydrodynamic forces calculated using the Immersed Boundary Method. We use this approach to simulate the transient deformation of spherical, ellipsoidal and biconcave capsules due to external optical forces induced by a single diode bar optical trap for a range of optical powers.

©2010 Optical Society of America

## 1. Introduction

A change in cellular mechanical properties can be an indication of infection and other pathologies [1,2]; for example, red blood cells (RBC) infected by malaria parasites exhibit significantly different membrane elasticity compared to normal healthy cells. These parasites express proteins that stiffen the RBC membrane by interacting with both the plasma and the cytoskeleton. Suresh and associates [3] have recently shown a difference in the average elastic modulus of normal RBCs and those infected with malaria at different growth stages where, in some cases, the elastic modulus nearly doubled. Such findings suggest that measuring cellular mechanical properties may provide a direct route to either detecting disease at early stages or assessing disease progression at later phases.

Several experimental techniques have been used to measure cellular viscoelastic
properties including those relying on direct contact, such as micropipette aspiration
[4] and atomic force microscopy [5]. Recently, non-contact methods have also been
developed that probe cell mechanical properties using optical trapping [6] through either drag-based deformation [7] or with attached colloids [8]. In this approach, a laser is focused to a diffraction-limited
spot with a high numerical-aperture objective, allowing micron-sized objects in solution
to be trapped into the region of highest light intensity [9]. With multiple optical traps, biological cells cannot only be manipulated
but also deformed, stretched, folded and even rotated. Bronkhorst *et
al.* deformed RBCs into a parachute shape using three traps, two at the ends
and one in the middle [6]. Henon *et
al.* deformed RBCs using two silica beads that were adhered to opposite sides
of the cell surface [8], trapped in dual optical
traps, and then pulled by two opposite but equal forces, thereby stretching the cell.
This method directs the laser beam on the beads and away from the cell, reducing the
risk of thermal damage; however, manipulating beads attached to a cell leads to stress
concentration in the membrane that can lead to mechanical injury and/or the measurement
of highly localized properties. Addressing these issues, Käs and associates have
recently developed an optical stretcher in which they trap individual cells along the
axis between two counter-propagating diverging beams thus elongating the cell [10,11]. The
use of such opposing bead-free optical stretchers induces smoother and less localized
stresses with no focusing required. More recently, Applegate *et al.*
[12] implemented optical trapping methods
using simplified schemes that take advantage of the nature of micro-fluidic fluid
dynamics and use diode laser bars to manipulate particles in micro-scale geometries.
This approach employs a single linear optical trap instead of a spot and eliminates the
need for expensive associated optics to stretch cells at high-throughput within
microfluidic flows [13].

To experimentally quantify an individual cell’s elasticity one needs to predict
the stress distribution on the cell surface due to the optical trap and measure the
induced deformation. To calculate optical forces Ashkin [14] did the first numerical work and determined the total forces of a
single-beam gradient laser on solid spheres in the ray-optics (RO) regime. Biological
cells, however, are deformable and can change shape under the influence of external
flows or applied forces. Because they are directly coupled, the local force distribution
and total trapping force can change significantly with the deformation of the cell, and
visa versa. To determine the initial stress distribution before cells begin to deform
under the influence of induced optical forces Guck *et al.* used the RO
method for a dual-beam fiber-optic stretcher and used it to determine the stiffness of
RBCs [10,15,16]. The effect of subsequent
deformation on the stress distribution was neglected; however, assuming a rigid
spherical cell morphology and disregarding deformation in the force calculation is
non-physical and may significantly affect cell elasticity predictions.

In this manuscript, we present a numerical method, dynamic ray tracing (DRT) and calculate the cell stress distribution due to light–cell interactions for arbitrary cell shape and simulate the resulting cell deformation in optical traps and stretchers. We do this by combining two numerical methods. The first is a vector-based ray tracer that determines the stress distribution in optical traps on elastic capsules of arbitrary shape and the second employs a fluid-structure hydrodynamic solver to simulate cell deformation due to both external optical forces and cell-fluid interaction. The combined method is used to simulate dynamic cell stretching of different initial shapes, including spheres, oblate spheroids and bi-concave discoids.

## 2. Numerical method

We implement an advanced RO technique that determines the variable stress distribution on cells in optical traps and uses this to calculate cell transient deformation. DRT takes into account the change in the cell shape due to cell deformability and cell-fluid interaction as opposed to the traditional ray optics method (TRO) that assumes fixed spherical geometries. Cell-fluid interaction is simulated using a hydrodynamic solver based on the immersed boundary method (IBM) which has been used extensively to simulate fluid-structure interaction in biological systems under external flows [17–19]. In this method the fluid governing equations are solved where the viscous stresses are determined and used to find the membrane deformation. Elastic forces are then determined due to the membrane deformability and converted to forces acting on the fluid. These forces are applied to the fluid as body forces. The details of the numerical implementation and validation of the IBM can be found elsewhere [18]. Here, we modify the approach to simulate cell deformation induced by external optical forces found using a ray-tracer that employs the governing physical model [10] on any discretized cell shape. In our modified IBM code, we incorporate these external optical forces by adding them to the elastic forces and applying them to the fluid.

In the physical model we employ, the cell membrane is assumed to be an infinitesimally
thin hyperelastic neo-Hookean material with negligible bending resistance. This model,
characterized solely by the membrane stiffness *Eh,* has been commonly
used for capsule deformation studies due to its simplicity [18,20,21]. Cells are assumed immersed in an incompressible Newtonian fluid
with the same density *ρ* and viscosity
*μ* as the cytoplasmic fluid but with a different index of
refraction. Optical forces on individual elements of the membrane are combined with the
membrane elastic forces due to cell deformation into the fluid and added to the source
term in the Navier-Stokes equations.

## 3. Traditional ray optics

Lasers emit rays of light that propagate in straight lines in a medium of constant
refractive index until they hit an interface of different index where they change path
and speed by refraction and/or reflection. Such a change in light path requires a
transfer of the momentum to the interface due to conservation. When the interface is an
object, its surface absorbs this momentum and exerts a force proportional to the laser
power by Newton's second law. To calculate the optical force distribution, a RO method
has been frequently used [10,12,15] where
the incoming light is decomposed into individual rays, each with specific intensity and
direction, an approach valid when the object is much larger than the wavelength of the
light [22]. For spherical objects the condition
is given by 2πa/λ>>1 where *a* is the radius of the
sphere and λ the laser wavelength. For cells of typical radius greater than 3
µm and laser wavelengths less than 1 µm this condition is satisfied
[10]. Optical forces are calculated
via

*n*is the index of refraction of the buffer medium,

_{m}*c*the speed of light in vacuum,

*P*the laser power, and

*Q*is the trapping efficiency, a dimensionless factor representing the amount of momentum transferred.

*Q*is independent of the power applied and depends only on object geometry and the reflectance of the medium. Calculation of

*Q*to date has been for spherical rigid cells [14,23] and shapes [24,25]. In what follows we briefly explain the TRO method and extend this to our implementation of a dynamic ray tracer on cells of arbitrary morphology.

The TRO method makes use of spherical cell symmetry to simplify the trapping efficiency
calculation. *Q* is calculated across the front and back sphere surfaces
separately as a function of refractive index and incident laser beam profile, an
approach valid because refracted angles can be determined at both front and back
interfaces from Snell’s law:

*α*and

*β*and are the incident and refracted ray angles (Fig. 1 ) and ${n}_{m},{n}_{p}$are the indices of refraction of the two media.

*Q*at any point on the surface can then be expressed as a function of the reflectance

*R*, the original incident angle

*α*, and the refracted angle

*β*. For simplicity

*Q*is typically decomposed into two components (parallel to and perpendicular to the laser axis) as:

*n =*${n}_{p}$/${n}_{m}$. In this, the incident light source is considered as an infinite number of rays parallel to the vertical axis, an approximation that works well for large particles. Internal and external reflections within the cell are neglected as their effects rapidly diminish when the reflectance

*R<0.005*for all rays. The magnitude at any point is found via $Q=\sqrt{{({Q}^{\perp})}^{2}+{({Q}^{\parallel})}^{2}}\text{}$and the total trapping efficiency, and corresponding total force, found through surface integration. It is important to note that these equations for the trapping efficiency

*Q*are valid only on spherical cells and thus, in using TRO, the corresponding optical forces are calculated once initially and kept constant throughout the simulations.

## 4. Dynamic ray tracing

In simulating the shape evolution of a deformable particle under optical forces, the
membrane steady-state geometry is not known *a priori*. Thus, the
traditional ray tracing approach is not suitable and another technique is required to
calculate the optical forces on evolving complex geometries. We show here how DRT can be
used to calculate the optical force distribution on the cell in both its initial
stress-free geometry as well as the transient morphologies as the cell deforms in
response to combined hydrodynamic and optical forces. In DRT a finite number of rays are
issued from a light source with a given intensity and known direction and travel
linearly from the source until they intersect with a surface (defined as an interface
between two media with different refractive indices). To implement this, we divide the
surface into triangular elements with the same surface elements used for both elastic
and optical stress calculations. From Eqs.
(3) and (4), *Q*
can be determined from incident, refracted and reflected ray angles with the laser axis
angle at any point of ray-surface intersection. Thus the elemental optical forces at the
first refraction can be expressed as:

*i*,

*r*and

*t*are the angles formed between the incident, reflected, and transmitted ray with the laser axis (Fig. 2 ). The angle between the incident ray and the surface normal

*α*remains the same in the calculation of reflectance

*R*. Note here that at any surface

*Q*is multiplied by a factor of

*1-R*to account for energy loss from previous refractions and thus the general form of

*Q*from a medium

*x*to another medium

*y*can be written as:

Calculation of the stress distribution therefore requires knowledge of the position where an individual ray hits the surface and the direction of the incoming ray. The first step in implementing a dynamic ray tracer is a ray (semi-line)-triangle intersection algorithm, for which various algorithms techniques are available in the literature [26]. We employ a barycentric approach that determines the intersection of the ray with a plane containing the triangle and then checks whether the point of intersection is inside the triangle [27]. This approach is straightforward and easy to implement but can be computationally consuming; however, optimizing this algorithm does not require much modification [28].

A ray **I** is modeled as a vector with a given origin **e** and
direction **d, I = e +**
*s*
**d** where *s* is the distance along the ray measured from the
origin in the positive direction (*s > 0*, semi-line). We assume rays
are initially parallel to the laser axis, an assumption that works well for non- or
weakly-diverging laser diodes (Fig. 2); however,
any initial ray direction can be simulated by choosing appropriate ray parameters. We
then discretize the surface of the cell into a finite number of non-overlapping 2D
triangular elements. In this, each single ray emanating from the source intersects the
object at a maximum of one triangle and may not intersect the surface at all. For a
triangle with three vertices *A, B* and *C* (Fig. 2), each point on a triangle can be defined
using its barycentric coordinates: $P\text{}=\text{}wA+uB+vC.$ For *P* to be inside/on the triangle the sum of the
barycentric coordinates must equal one; therefore, *P* can be represented
as *P = (1-u-v)A + uB + vC = A + u(B-A) + v(C-A)* with$u>0,v>0,u+v<1$. A given ray **I** then intersects the triangle at
*P* when *P =*
*e**+ s*
** d** for a given

*s*. Combining both conditions we have$P=e+sd=a+u(b-a)+v(c-a).$ For numerical implementation this equation is solved for

*s, u*and

*v*to find the intersection point and the distance of intersection [26].

Though a ray can hit a 3D shape at multiple locations, we are first interested in the
initial contact, as this is the point at which the ray will change direction. To
determine all points of intersection, the process is repeated for all triangles to
determine a value of *s* for each element. Smaller values of
*s* indicate closer objects to the light source; if two objects are
both intersected, the one with the smaller value of *s* is recorded as
the nearest point of intersection *P*. This procedure can be
computationally expensive depending on the number of triangular elements and number of
rays. Advanced techniques for saving computational time that avoid calculating
ray-triangle intersections over elements in the shadow of elements on the
“front-side” of the surface are available in the literature [29].

Once the point of intersection between an incoming ray and a triangular element is
known, one can determine the direction of the reflecting and refracting rays (Fig. 2). Using the normal at a triangle denoted by
**N**, the incident angle can be calculated as $\mathrm{cos}\alpha =I\cdot N/(\left|I\right|\left|N\right|)$. Using Snell’s law, Eq.
(2), one can then calculate the refracted angle*β*and
the direction vector of both reflected and refracted rays via:

Here, rays change both direction and intensity and continue through the cell until they
hit another surface. For refractive index differences typical of cells in water, only
refracted rays are traced and reflected rays can be neglected because of their low
intensity. A refracted ray is now considered a new ray with a new origin (the current
point of intersection *P*) and new intensity. Therefore **I**
*=*
**T** and the previous procedure repeated to find a new contact location. A
spherical cell typically has only two intersection locations, one at the front and one
at the back surface; however, this approach works for any cell geometry where a ray
might encounter multiple external or internal surfaces. For example, in a biconcave
capsule, some rays encounter up to four hits at different surfaces. It is important to
note that when a ray changes direction it is moving from one medium to another and
therefore the indices of refraction are swapped after each hit as the direction of the
force changes.

## 5. Dynamic ray tracer validation

The DRT can be used to find both the initial stress distribution on the cell surface due to light as well as the transient stress distribution upon deforming. Here, we first calculate the initial stress distribution on spherical capsules to validate our method against the TRO method.

#### 5.1 Trapping efficiency calculation

We first calculate the trapping efficiency *Q* on a sphere using the
two methods. The trapping efficiency *Q* is independent of the light
source and is only function of the cell shape and the index of refraction of the cell
and the medium *n _{p} =* 1.37 and

*n*1.335 [15]. Here we use Eqs. (3) and (4) for the parallel rays case where

_{m}=*Q*can be found as a function of the polar angle on both the front and back surface as shown in Fig. 3 . The 3D DRT is based on shape discretization and we employ a mesh of 20480 triangles. For comparison, we plot only the calculated

*Q*at the first and second quadrant at a plane parallel to the laser axis passing through the center of the cell representing the front and back surfaces. In these calculations we see an average error of 1.43% across the surface while the maximum error calculated across this plane is 10% at the back surface at the peak value, attributed to the convergence of rays on the back surface. The error on the front surface is relatively small because the gradients in the incoming beam are small and each element is intersected by one ray. Transmitted rays however converge on the backside, causing multiple intersections on some elements and no intersections on other elements, increasing optical force gradients and leading to larger localized error.

#### 5.2 Net optical forces

After determining the local *Q* distribution, one can now calculate
induced optical forces by defining the cell size and light source. A spherical cell
representing a swollen erythrocyte has a radius *a* of 3.3 μm.
We simulate a linear diode bar light source that is 200 μm long and 1
μm wide [13] (Fig. 2). The laser wavelength is 833 nm leading to
2πa/λ≈25>>1, satisfying the RO condition. Using the TRO
for a fixed power of 12 mW/μm, a net force of −7.77 pN on the front
surface in the laser axis direction is applied, while 8.23 pN is applied on the back
surface, leading to 0.46 pN of net optical force. The magnitude of the induced
optical forces compares well with values seen in other work on swollen RBCs [10]. Using the DRT, −7.83 pN is applied
on the front surface and 8.50 pN force on the back surface with 0.67 pN of net
optical force, with a calculated error of only 0.75% on the front surface and 3.3% on
the back surface.

## 6. Transient cell deformation

After determining the initial stress distribution, we employ the IBM to deform the cell where the surface optical forces are added as body forces to the surrounding fluid. We then allow the cell to deform until steady state, where the elastic and applied optical forces are equal. Steady-state deformation can then be characterized using the Taylor deformation parameter

where*L*and

*B*are the lengths of the major and minor axes of elongated cells in a specified plane. Here we define

*DF*in the

*x-z*plane as the rays are assumed coming parallel to the z axis and the perpendicular component of the optical forces is in the

*x*direction.

*DF*describes the geometrical deformation from perfect spheres (

*DF = 0*) to highly elongated morphologies.

Using a membrane stiffness of *Eh* = 0.1 dyn/cm, an unstressed cell is
initially placed in a fluid of density *ρ* = 1 g/cm^{3}
and dynamic viscosity *µ* = 0.8 cP. The fluid domain is a cube
with a side 8x the cell radius *a* with periodic boundary conditions. The
grid used in the simulations has 64^{3} nodes with a uniform grid spacing
*h = a/8*. The hydrodynamic intrinsic time scale for cell deformation
is *t _{c}* =

*μa/Eh*[30] or

*t*~10

_{c}^{−3}s for our simulation parameters. With this, we use a time step of 10

^{−5}s to ensure resolution and numerical stability. In Fig. 4 we show the force at both the front and back surface along with the total net force on the cell function of time. From this we see that both the front and back forces decrease as the cell deforms; however, the net force increases by 6%, using the 12 mW/μm laser power to a constant value when the cell reaches a steady-state shape. In Fig. 5 we show that the net translation force is in the z-direction, pushing the cell away from the light source and deforming the cell as expected for non-converging traps. We note that the initial forces at time

*t* = 0*are the forces that remain constant using TRO on a sphere.

Using higher laser diode powers from 12 mW/μm to 72 mW/μm we simulate
the deformation of the cell until it reaches a steady-state shape using TRO and DRT
methods. Using TRO we calculate the optical forces initially at the surface of the
spherical cell and keep these constant throughout the simulation, thus neglecting the
effect of change in shape on the optical force calculation. While using DRT however, the
forces are re-calculated every 10 time steps as they vary with the change of the cell
shape. We record the net optical force and the corresponding cell deformation when the
cell reaches a steady-state shape and plot it for the two methods in Fig. 6
. From this we see the non-linear behavior of the net force with laser power as
*Q* changes with cell deformation. Here, calculated values of
F_{net} compared to those when neglecting shape change differ by 42% even
though the corresponding *DF _{xz}* changes only 3.65% at the higher laser power. Note that, at lower laser
intensities, the two techniques converge as expected.

## 7. Non-Spherical Cell Shapes

To test the approach on more complicated structures, we simulate stretching with different initial geometries and laser powers. The complex quiescent biconcave disk shape of normal RBCs prevents the use of TRO as the symmetry along the y-axis allows light rays to hit the RBC surface at four different positions before exiting as illustrated in Fig. 7 . To describe the cross section of RBCs, Evans and Fung [31] developed:

where *C _{0}* = 0.207161,

*C*= 2.002558 and

_{1}*C*= −1.122762, the radius

_{2}*a*is 4 μm, and the surface area is 140 μm

^{2}. In many experimental studies, RBCs are osmotically swollen and assume a spherical shape. To demonstrate the flexibility of DRT, we model both unswollen (biconcave) and swollen (oblate spheroid) RBCs. For the swollen case, we assume an average radius of 2.8 μm, corresponding to a perfect sphere of the same surface area and radius 3.3 μm, and average deformation parameter

*DF*

_{xz}= 0.126 as seen in experimental work for swollen bovine RBCs [13]. These dimensions lead to a semi major to semi minor axes ratio of 1.28:1.

Using DRT we find the transient stress distribution and simulate swollen and unswollen
RBC deformation until steady state. DRT allows us to determine the exact stress
distribution on the cell surface without approximation as previously studied [16]. Cells are considered initially static, then
exposed to the laser diode bar source, and finally allowed to reach a steady-state shape
where *DF*
_{xz} is recorded. Here, the optical forces applied by the diode laser bar are
the only external forces on the cell and any flow is induced only by the cell
deformability and its translation. Beginning initially with spherical cells for
comparison and using a range of laser powers, 12 mW/μm to 72 mW/μm, we
see the net optical force induced by a 12 mW/μm laser power is 0.57 pN with a
final induced deformation of *DF*
_{xz} = 0.0158. Oblate spheroids on the other hand are subjected to more force
at identical powers. For example a net force of 0.79 pN is calculated when using the
same 12 mW/μm power with a net deformation of *DF*
_{xz} = 0.0177, higher than the spherical case (Fig. 8
).

In this, net deformation was calculated as the final minus the initial deformation (for
spheres the initial *DF* was zero). For the normal RBC case, we observe
that the net optical force is significantly higher for the same 12 mW/μm power
with a 1.29 pN net optical force induced by rays intersecting the surface at four
different positions with stretching forces on both the back and front surface. The net
deformation however is smaller *DF*
_{xz} = 0.0062. Figures 9(a)
and 9(b) (Media
2 and Media
3) shows the steady-state shape of swollen and
unswollen RBCs. Table 1
summarizes the net optical forces on different geometries for the same
power and corresponding net deformation. From these measurements it is clear that the
shape of the cell strongly influences net cell stretching force and deformation.

## 8. Conclusions

We present a dynamic ray tracing method for optical cell manipulation modeling. This method overcomes the limitations of traditional ray optics best suited for light interactions with rigid, spherical cells and allows determination of the transient stress distribution on deformable cells of arbitrary shape subject to cell-fluid interactions. Our simulations indicate that the applied optical forces change substantially with the deformation of the cell, significantly altering prediction of cell forces under conditions of high laser diode powers. We also simulate the deformation of different RBC shapes (spherical, oblate spheroids and bi-concave discoids) to demonstrate the flexibility of the approach for modeling cells of greatly different structure and morphology.

## Acknowledgments

The authors would like to acknowledge financial support provided by the National Institutes of Health Grant No. RO1 AI079347-01.

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