# Oncotripsy: Targeting cancer cells selectively via resonant harmonic
excitation

S. Heyden<EMAIL_ADDRESS>M. Ortiz<EMAIL_ADDRESS>Division of
Engineering and Applied Science, California Institute of Technology, Pasadena,
CA 91125, USA

###### Abstract

We investigate a method of selectively targeting cancer cells by means of
ultrasound harmonic excitation at their resonance frequency, which we refer to
as oncotripsy. The geometric model of the cells takes into account the
cytoplasm, nucleus and nucleolus, as well as the plasma membrane and nuclear
envelope. Material properties are varied within a pathophysiologically-
relevant range. A first modal analysis reveals the existence of a spectral gap
between the natural frequencies and, most importantly, resonant growth rates
of healthy and cancerous cells. The results of the modal analysis are verified
by simulating the fully-nonlinear transient response of healthy and cancerous
cells at resonance. The fully nonlinear analysis confirms that cancerous cells
can be selectively taken to lysis by the application of carefully tuned
ultrasound harmonic excitation while simultaneously leaving healthy cells
intact.

###### keywords:

Oncotripsy , modal analysis , resonance , cell necrosis

##  1 Introduction

In this study, we present numerical calculations that suggest that, by
exploiting key differences in mechanical properties between cancerous and
normal cells, oncolysis, or ’bursting’ of cancerous cells, can be induced
selectively by means of carefully tuned ultrasound harmonic excitation while
simultaneously leaving normal cells intact. We refer to this procedure as
oncotripsy. Specifically, by studying the vibrational response of cancerous
and healthy cells, we find that, by carefully choosing the frequency of the
harmonic excitation, lysis of the nucleolus membrane of cancerous cells can be
induced selectively and at no risk to the healthy cells.

Numerous studies suggest that aberrations in both cellular morphology and
material properties of different cell constituents are indications of various
forms of cancerous tissues. For instance, a criterion for malignancy is the
size difference between normal nuclei, with an average diameter of $7$ to $9$
microns, and malignant nuclei, which can reach a diameter of over $50$ microns
(Berman, 2011). Early studies (Guttman and Halpern, 1935) have shown that the
nuclear-nucleolar volume ratios in normal tissues and benign as well as
malignant tumors do not differ quantitatively. Nucleoli volumes of normal
tissues, however, are found to be significantly smaller than the volume of
nucleoli in cancerous tissues (Guttman and Halpern, 1935). Similarly, the
mechanical stiffness of various cell components has been found to vary
significantly in healthy and diseased tissues. In Cross et al. (2007), the
stiffness of live metastatic cancer cells was investigated using atomic force
microscopy, showing that cancer cells are more than $80\%$ softer than healthy
cells. Other cancer types, including lung, breast and pancreas cancer, display
similar stiffness characteristics. Furthermore, using a magnetic tweezer,
Swaminathan et al. (2011) found that cancer cells with the lowest invasion and
migratory potential are five times stiffer than cancer cells with the highest
potential. Likewise, increasing stiffness of the extracellular matrix (ECM)
was reported to promote hepatocellular carcinoma (HCC) cell proliferation,
thus being a strong predictor for HCC development (Schrader et al., 2011).
Moreover, enhanced cell contractility due to increased matrix stiffness
results in an enhanced transformation of mammary epithelial cells as shown in
Paszek et al. (2005). Conversely, a decrease in tissue stiffness has been
found to impede malignant growth in a murine model of breast cancer (Levental
et al., 2009).

Various experimental techniques have been utilized in order to quantitatively
assess the material properties of individual cell constituents in both healthy
and diseased tissues. The inhomogeneity in stiffness of the living cell
nucleus in normal human osteoblasts has been investigated by Konno et al.
(2013) using a non-invasive sensing system. As shown in that study, the
stiffness of the nucleolus is relatively higher compared to that of other
nuclear domains (Konno et al., 2013). Similarly, a difference in mass density
between nucleolus and nucleoplasm in the xenopus oocyte nucleus was determined
by Handwerger et al. (2005) by recourse to refractive indices. The elastic
modulus of both isolated chromosomes and entire nuclei in epithelial cells are
given by Houchmandzadeh et al. (1997) and Caille et al. (2002), respectively.
Specifically, Houchmandzadeh et al. (1997) showed that mitotic chromosomes
behave linear elastically up to 200% extension. Experiments of Dahl et al.
(2004) additionally measured the network elastic modulus of the nuclear
envelope, independently of the nucleoplasm, by means of micropipette
aspiration, suggesting that the nuclear envelope is much stiffer and stronger
than the plasma membranes of cells. In addition, wrinkling phenomena near the
entrance of the micropipette were indicative of the solid-like behavior of the
envelope.

Kim et al. (2011) estimated the elastic moduli of both cytoplasm and nucleus
of hepatocellular carcinoma cells based on force-displacement curves obtained
from atomic force microscopy. In addition, Zhang et al. (2002) used
micropipette aspiration techniques in order to further elucidate the
viscoelastic behavior of human hepatocytes and hepatocellular carcinoma cells.
Based on their study, Zhang et al. (2002) concluded that a change in the
viscoelastic properties of cancer cells could affect metastasis and tumor cell
invasion. The increased compliance of cancerous and pre-cancerous cells was
also investigated by Fuhrmann et al. (2011), who used atomic force microscopy
to determine the mechanical stiffness of normal, metaplastic and dysplastic
cells, showing a decrease in Young’s modulus from normal to cancerous cells.

The scope of the present work, and the structure of the present paper, are as
follows. We begin by defining the geometric model and summarizing the material
model and material parameters used in finite-element analyses. Subsequently,
the accuracy of the finite-element model is assessed by means of a comparison
between numerical and analytical solutions for the eigenmodes of a spherical
free-standing cell. We then present eigenfrequencies and eigenmodes of a free-
standing ellipsoidal cell, followed by a Bloch wave analysis to model tissue
consisting of a periodic arrangement of cells embedded in an extracellular
matrix. Finally, resonant growth rates are calculated that reveal that
cancerous cells can selectively be targeted by ultrasound harmonic excitation.
The transient response at resonance of healthy and cancerous cells is
presented in the fully nonlinear range by way of verification and extension of
the findings of the harmonic modal analysis. We close with a discussion of
results.

##  2 Finite element analysis

In this section, we investigate the dynamical response of healthy and
cancerous cells under harmonic excitation. We begin by briefly outlining the
underlying geometric and material parameters used in our analysis, followed by
a verification of the finite element model used for modal analysis. We then
calculate the eigenfrequencies and eigenmodes of both free-standing and
periodic distributions of cells. In this latter case, we determine the full
dispersion relation by means of a standard Bloch wave analysis. Finally, we
present resonant growth rates and simulate the transient response of both
cancerous and healthy cells excited at resonance in a fully-nonlinear setting
by means of implicit dynamics calculations.

###  2.1 Geometry and material parameters

The nucleus, the largest cellular organelle, occupies about $10\%$ of the
total cell volume in mammalian cells (Lodish et al., 2004; Alberts et al.,
2002). It contains the nucleolus, which is embedded in the nucleoplasm, a
viscous solid similar in composition to the cytosol surrounding the nucleus
(Clegg, 1984). In this study, the cytosol is modeled in combination with other
organelles contained within the plasma membrane, such as mitochondria and
plastids, which together form the cytoplasm. For simplicity, we idealize the
plasma mebrane, nuclear envelope, cytoplasm, nucleoplasm, and nucleolus as
being of spheroidal shape. We model the plasma membrane, a lipid bilayer
composed of two regular layers of lipid molecules, in combination with the
actin cytoskeleton providing mechanical strength as a membrane with a
thickness of $10$ nm (Hine, 2005). Similarly, we model the nuclear envelope, a
double lipid bilayer membrane, in combination with the nuclear lamin meshwork
lending it structural support as a $20$ nm thick membrane. We define the
cytoplasm, nucleoplasm, and nucleolus as spheres with radii of $5.8\,\mu$m,
$2.7\,\mu$m, and $0.9\,\mu$m and subsequently scale them by a factor of $1.2$
in two dimensions in order to obtain the desired spheroidal shape. We assume
an average nuclear diameter of about $5\,\mu$m, as reported in Cooper (2000).
Diameters for both cytoplasm and nucleolus follow from Lodish et al. (2004)
and Guttman and Halpern (1935), who report nucleus-to-cell and nucleus-to-
nucleolus volume ratios of $0.1$ and $30.0$, respectively. The geometry with
all cell constituents as used in subsequent finite element analyses is
illustrated in Figure 1. In order to further elucidate the effect of an
increasing nucleus-to-cell volume ratio, as observed experimentally (Berman,
2011), we consider a range of geometries with increasing nuclear and nucleolar
volumes. For all of these geometries, we hold fixed the volume of the
cytoplasm. Furthermore, we assume a constant nuclear-to-nucleolar volume ratio
for both healthy and cancerous cells, as observed by Guttman and Halpern
(1935).

Figure 1: Cell geometry with different cell constituents used in finite-
element simulations.

Cell-to-cell differences and experimental uncertainties notwithstanding, the
preponderance of the observational evidence suggests that the cytoplasm,
nucleus and nucleolus are ordered in the sense of increasing stiffness.
Neglecting viscous effects, we model the elasticity of the different cell
constituents by means of the Mooney-Rivlin-type strain energy density of the
form

$W(\boldsymbol{F})=\frac{1}{2}\left[\mu_{1}\left(\frac{I_{1}}{J^{2/3}}-3\right)+\mu_{2}\left(\frac{I_{2}}{J^{4/3}}\right)+\kappa\left(J-1\right)^{2}\right],$
(1)

where $\boldsymbol{F}$ denotes the deformation gradient,
$J=\det(\boldsymbol{F})$ is the Jacobian of the deformation, and $\mu_{1}$,
$\mu_{2}$ and $\kappa$ are material parameters. For both cytoplasm and nucleus
in cancerous cells, material parameters corresponding to the data reported by
Kim et al. (2011) are chosen and summarized in Table 1. We additionally infer
the elastic moduli of the nucleolus from Konno et al. (2013) based on a
comparison of the relative stiffnesses of the nucleoli and other nuclear
domains. For membrane elements of the plasma membrane and nuclear envelope, we
choose material parameters corresponding to the cytoplasm and nucleoplasm,
respectively. Furthermore, we infer matrix parameters from the shear moduli
reported by Schrader et al. (2011) for normal and fibrotic livers. For all
parameters, we resort to small-strain elastic moduli conversions, with a
Poisson’s ratio of $0.49$ to simulate a nearly incompressible material, in
order to match experimental values with constitutive parameters. We vary the
stiffness of both cellular components and extra-cellular matrix (ECM) within a
pathophysiologically-relevant range in order to investigate the effect of cell
softening and ECM stiffening on eigenfrequencies. Finally, we assume both
cytoplasm and nucleoplasm to have a mass density of $1$ g/cm3, a value
reported by Moran et al. (2010) as an average cell density, and we set the
density of the nucleolus to $2$ g/cm3 (Birnie, 1976).

|  $\kappa$ [kPa] |  $\mu_{1}$ [kPa] |  $\mu_{2}$ [kPa]
---|---|---|---
Plasma membrane | $39.7333$ | $0.41$ | $0.422$
Cytoplasm | $39.7333$ | $0.41$ | $0.422$
Nuclear envelope | $239.989$ | $2.41$ | $2.422$
Nucleoplasm | $239.989$ | $2.41$ | $2.422$
Nucleolus | $719.967$ | $7.23$ | $7.266$
ECM | $248.333$ | $5.0$ | $5.0$
Table 1: Set of constitutive parameters (bulk modulus $\kappa$ and shear
moduli $\mu_{1}$ and $\mu_{2}$) used in the eigenfrequency analyses.

###  2.2 Verification against analytical solutions

Based on the elastic model described in the foregoing, the remainder of the
paper is concerned with the computation of the normal modes of vibration of
healthy and cancerous cells using finite elements. To this end, we begin by
assessing the accuracy of the finite element model used in subsequent
calculations by means of comparisons to exact solutions. We consider a single
spherical free-standing cell and compare numerically computed eigenmodes with
the analytical solution of Kochmann and Drugan (2012) for a free-standing
elastic sphere with an elastic spherical inclusion.

In the harmonic range, the finite-element discretization of the model leads to
the standard symmetric linear eigenvalue problem

$\left(\boldsymbol{K}-\omega^{2}\boldsymbol{M}\right)\hat{\boldsymbol{U}}=\boldsymbol{0},$
(2)

where $\boldsymbol{K}$ and $\boldsymbol{M}$ are the stiffness and mass
matrices, respectively, $\omega$ is an eigenfrequency of the system and
$\hat{\boldsymbol{U}}$ is the corresponding eigenvector, subject to the
normalization condition

$\hat{\boldsymbol{U}}^{T}\boldsymbol{M}\hat{\boldsymbol{U}}=1.$ (3)

For the spherical geometry under consideration, the modal analysis has been
carried out analytically in closed form by Kochmann and Drugan (2012). For the
homogeneous sphere, they find that the natural frequencies $\omega_{i}$ follow
as the roots of function

$f(y)=\tanh{y}-\frac{y}{1-ky^{2}},$ (4)

with

$y=\sqrt{\frac{\rho\omega^{2}b^{2}}{\lambda+2\mu}}\quad\text{and}\quad
k=\frac{\lambda+2\mu}{4\mu},$ (5)

where $b$ is the outer radius and $\lambda$ and $\mu$ and the Lamé constants.
Furthermore, Kochmann and Drugan (2012) report analytical solutions for an
isotropic linear-elastic spherical inclusion of radius $a$, moduli
$\lambda_{1}$ and $\mu_{1}$, within a concentric isotropic linear-elastic
coating of uniform thickness of outer radius $b$, moduli $\lambda_{2}$ and
$\mu_{2}$. In this case, the eigenfrequencies $\omega_{i}$ follow from the
characteristic equation $\det{\boldsymbol{A}}=0$, where

$\begin{split}&A_{11}=\frac{\cos(jx)}{jx}-\frac{\sin(jx)}{j^{2}x^{2}},\\\
&A_{12}=\frac{\sin(jkx)}{j^{2}k^{2}x^{2}}-\frac{\cos(jkx)}{jkx},\\\
&A_{12}=-\frac{\cos(jkx)}{j^{2}k^{2}x^{2}}-\frac{\sin(jkx)}{jkx},\\\
&A_{21}=0,\\\
&A_{21}=4k_{2}^{2}\frac{\cos(jk)}{j^{2}k^{2}}+\left(\frac{1}{jk}-\frac{4k_{2}^{2}}{j^{3}k^{3}}\right)\sin(jk),\\\
&A_{23}=4k_{2}^{2}\frac{\sin(jk)}{j^{2}k^{2}}-\left(\frac{1}{jk}-\frac{4k_{2}^{2}}{j^{3}k^{3}}\right)\cos(jk),\\\
&A_{31}=-k\left(4k_{1}^{2}\frac{\cos(jx)}{j^{2}x^{2}}+\left(\frac{1}{jx}-\frac{4k_{1}^{2}}{j^{3}x^{3}}\right)\sin(jx)\right),\\\
&A_{32}=4k_{2}^{2}\frac{\cos(jkx)}{j^{2}k^{2}x^{2}}+\left(\frac{1}{jkx}-\frac{4k_{2}^{2}}{j^{3}k^{3}x^{3}}\right)\sin(jkx),\\\
&A_{33}=4k_{2}^{2}\frac{\sin(jkx)}{j^{2}k^{2}x^{2}}-\left(\frac{1}{jkx}-\frac{4k_{2}^{2}}{j^{3}k^{3}x^{3}}\right)\cos(jkx),\end{split}$
(6)

with dimensionless quantities

$\displaystyle k_{1}$
$\displaystyle=\sqrt{\frac{\mu^{I}}{\lambda^{I}+2\mu^{I}}},\quad
k_{2}=\sqrt{\frac{\mu^{II}}{\lambda^{II}+2\mu^{II}}},\quad$ $\displaystyle k$
$\displaystyle=\sqrt{\frac{\lambda^{I}+2\mu^{I}}{\lambda^{II}+2\mu^{II}}},\quad
j=\sqrt{\frac{\rho\omega^{2}b^{2}}{\lambda^{I}+2\mu^{I}}}$ (7)

and with $x=a/b$.

Constitutive parameters |  $\kappa_{1}$ [Pa] |  $\mu_{1}$ [Pa] |  $\kappa_{2}$ [Pa] |  $\mu_{2}$ [Pa]
---|---|---|---|---
Solid sphere | $1.0$ | $1.0$ | $1.0$ | $1.0$
Spherical inclusion | $1.0$ | $0.1$ | $1.0$ | $0.1$
Material/ geometric parameters |  $\rho_{1}$ [kg/m3] |  $\rho_{2}$ [kg/m3] |  $r_{inner}$ [m] |  $r_{outer}$ [m]
Solid sphere | $10^{-3}$ | $10^{-3}$ | $3$ | $30$
Spherical inclusion | $10^{-3}$ | $10^{-3}$ | $3$ | $30$
Table 2: Set of geometric-, material- and constitutive parameters used in analytical eigenfrequency calculations and finite-element analysis. |  $\omega_{lowest}$ [rad/s] (I) |  $\omega_{lowest}$ [rad/s] (II)
---|---|---
Analytical solution | 3.72394 | 1.17402
Finite element analysis | 3.71717 | 1.17547
Relative error (%) | 0.181797 | 0.123507
Table 3: Lowest radial eigenfrequency from analytical solution and finite-
element analysis for the cases of a solid sphere (I) and a spherical matrix
with inclusion (II).

Table 3 shows a comparison between analytical and finite-element values of the
fundamental frequency of a solid sphere and a sphere with a high-contrast
spherical inclusion for the particular choice of parameters listed in Table 2.
The finite-element values correspond to a mesh of $\approx$ $15,000$ linear
tetrahedral elements, representative of the meshes used in subsequent
calculations. As may be seen from the table, the finite-element calculations
may be expected to result in frequencies accurate to $\sim 10^{-3}$ relative
error.

###  2.3 Eigenfrequencies and eigenmodes

In order to obtain a first estimate of the spectral gap between cancerous and
healthy cells, we consider the eigenvalue problem of single free-standing
cells. To this end, cytoplasm, nucleoplasm and nucleolus are discretized using
linear tetrahedral elements, while linear triangular membrane elements are
used for the plasma membrane and nuclear envelope. A typical finite element
mesh for a cell geometry with a ratio of $n/c=1$ and a total of $40,349$
elements is shown in Figure 2. For the calculation of eigenfrequencies, meshes
containing $\sim 16,000$ elements are used for geometries ranging from a ratio
of $n/c=1$ to $n/c=2$. In addition to the accuracy assessment of Section 2.2,
we have assessed the convergence of the free-standing cell finite-element
model by considering five different meshes of $2,171$, $3,596$, $8,608$,
$11,121$, and $15,215$ elements. From this analysis, we find that the accuracy
in the lowest eigenfrequency for the finest mesh is of the order of $0.2\%$,
which we deem sufficient for present purposes. All cell constituents are
modeled by means of the hyperelastic Mooney-Rivlin model described in Section
2.1, with the materials constants of Table 1.

Figure 2: Finite-element mesh of the plasma membrane, cytoplasm, nuclear
envelope, nucleoplasm, and nucleolus (for the purpose of visualization, the
extracellular matrix is omitted). Figure 3: Lowest eigenfrequency for varying
cell stiffness and increasing nucleolus/nucleoplasm-to-cytoplasm volume ratios
$n/c$.

Figure 3 shows the calculated lowest eigenfrequency, rigid-body modes
excluded, for different cell geometries and varying material properties. In
the calculations, the nucleolus/nucleoplasm-to-cytoplasm volume ratio is
increased incrementally in the range of $n/c=1.0$ to $n/c=2.0$, resulting in
six different test geometries. Furthermore, material properties are varied
within a pathophysiologically relevant range, whereby a value of $100\%$
cancerous potential corresponds to values presented in Table 1.

Since cancerous cells are more than $80\%$ softer than healthy cells (Cross et
al., 2007), we vary the elastic moduli in Table 1, with full values
representing cancerous cells and increased moduli representing healthy cells.
In addition, decreased elastic moduli of the extracellular matrix (ECM) are
expected in healthy tissues (Schrader et al., 2011). Figure 3 summarizes the
calculated lowest eigenfrequency for different percentages of the parameter
values presented in Table 1. Thus, a cancerous potential of $80\%$ reflects an
increase in elastic moduli of $20\%$ for material parameters of the different
cell constituents with a simultaneous decrease in elastic moduli of the ECM by
$20\%$. The shaded area in Figure 3 illustrates a typical gap in the lowest
natrual frequency for a nucleolus/nucleoplasm-to-cytoplasm volume ratio of
$n/c=1.0$, with $\omega=501,576$ rad/s for cancerous cells and
$\omega=271,764$ rad/s for a reduction in cancerous potential by $80\%$, the
expected value for healthy cells (Cross et al., 2007)). An even higher
spectral gap is recorded by additionally taking the growth in
nucleolus/nucleoplasm-to-cytoplasm volume ratio into account, as
experimentally observed in cancerous cells (Berman, 2011).

A more detailed comparison of the spectra of free-standing healthy and
cancerous cells, corresponding to cancerous potentials of $20\%$ and $100\%$,
respectively, is presented in Table 4, which collects the computed lowest ten
eigenfrequencies for a cell geometry with volume ratio $n/c=1.0$. From this
table we observe that free-standing cancerous cells have a ground
eigenfrequency of the order of $500,000$ rad/s, whereas free-standing healthy
cells have a ground eigenfrequency of the order of $270,000$ rad/s, or a
healthy-to-cancerous spectral gap of the order of $230,000$ rad/s. In
addition, their higher eigenfrequencies overlap with the ground eigenfrequency
of cancerous cells. Therefore, special attention is required to examine
whether or not excitation of cancerous cells might trigger healthy cells to
resonate. Indeed, figures of merit other than natural frequency, including
growth rates of resonant modes and energy absorption, may also be expected to
play an important role in differentiating the response of cancerous and
healthy cells. These additional figures of merit are investigated in Section
2.6.

Figure 4: Eigenmodes corresponding to different resonance frequencies for a
ratio of $n/c=1.0$ and a cancerous potential of $100\%$.

The eigenmodes corresponding to different resonance frequencies for a ratio of
$n/c=1.0$ and a cancerous potential of $100\%$ are shown in Figure 4. It may
be noted from the figure how each mode represents different characteristic
deformation mechanisms of the various cell constituents. Knowledge of the
precise modal shape may therefore help to target lysis of specific cell
components. Thus, shear deformation may be expected to dominate at a frequency
of $558,031$ rad/s, whereas volumetric deformations may be expected to be
dominant at $576,073$ rad/s. These differences in deformation mode open the
way for targeting specific cell constituents for lysis, such as the plasma
membrane at a frequency of $816,846$ rad/s.

|  $\omega_{1}$ [rad/s] |  $\omega_{2}$ [rad/s] |  $\omega_{3}$ [rad/s] |  $\omega_{4}$ [rad/s] |  $\omega_{5}$ [rad/s]
---|---|---|---|---|---
Cancerous | 501576 | 502250 | 508795 | 532132 | 537569
Healthy | 271764 | 274141 | 364259 | 364482 | 367413
|  $\omega_{6}$ [rad/s] |  $\omega_{7}$ [rad/s] |  $\omega_{8}$ [rad/s] |  $\omega_{9}$ [rad/s] |  $\omega_{10}$ [rad/s]
Cancerous | 538512 | 557291 | 667107 | 678287 | 678771
Healthy | 375570 | 376000 | 380063 | 424226 | 425327
Table 4: Comparison of the lowest ten eigenfrequencies for a cell geometry
with a ratio of $n/c=1.0$ and a cancerous potential of $100\%$ (cancerous)
versus a cancerous potential of $20\%$ (healthy) obtained from a free
vibration analysis.

###  2.4 Bloch wave analysis

The preceding spectral analysis for a free-standing cell can be extended to a
tissue consisting of a periodic arrangement of cells embedded into an
extracellular matrix. In this case, the analysis can be carried out by
recourse to standard Bloch wave theory. Within this framework, the
displacement field is assumed to be of the form

$\boldsymbol{u}(\boldsymbol{x})=\hat{\boldsymbol{u}}(\boldsymbol{x})e^{i\boldsymbol{k}\cdot\boldsymbol{x}},$
(8)

where $\boldsymbol{k}$ is the wave vector of the applied harmonic excitation
and the new unknown displacement field $\hat{\boldsymbol{u}}(\boldsymbol{x})$
is defined within the periodic cell (Bloch, 1929). By periodicity, the values
of wave vector $\boldsymbol{k}$ can be restricted to the Brillouin zone of the
periodic lattice. Substitution of representation (8) into the equations of
motion results in a $\boldsymbol{k}$-dependent eigenvalue problem. The
corresponding eigenfrequencies $\omega_{i}(\boldsymbol{k})$ define the
dispersion relations of the tissue. Details of implementation of Bloch-wave
theory in elasticity and finite-element analysis may be found in Krödel et al.
(2013).

Figure 5: Displacement elastodynamic boundary value problem with applied
harmonic excitation. Figure 6: First irreducible Brillouin zone and chosen
$k$-path.

In the present analysis, we consider a cubic unit cell of size $a=15$ $\mu$m
and the finite element discretization shown in Figure 2. The extracellular
matrix (ECM), not shown in the figure, is also discretized into finite
elements. Figure 6 shows the first irreducible Brillouin zone, which is itself
a cube of size $2\pi/a$. In order to visualize the dispersion relations, we
choose the $k$-path along the edges and specific symmetry lines of the
Brillouin zone also shown in Figure 6. The path allows for the elliptical
shape of the cells, with only one symmetry axis. The computed dispersion
relations for both cancerous and healthy cells then follow as shown in Figures
7 and 8, respectively, for the lowest $50$ eigenfrequencies. Similarly to the
calculations presented in Section 2.3 for free-standing cells, the lowest
eigenfrequencies of the healthy tissue are shifted uniformly towards lower
values with respect to the eigenfrequencies of the cancerous tissue, with
significant spectral gaps of the order of $200,000$ rad/s between the two.

Figure 7: Disperison relations for a cancerous cell, whereby the k-path is
traversed with a refinement of two points between neighboring nodes. Figure 8:
Disperison relations for a healthy cell, whereby the k-path is traversed with
a refinement of two points between neighboring nodes.

We recall that the computed ground eigenfrequency of free-standing cancerous
cells is of the order of $\omega\sim 500,000$ rad/s. In addition, from the
properties of Table 1 we may expect a cancerous-tissue shear sound speed of
the order of
$c=\sqrt{\frac{{\mu}}{{\rho}}}=\sqrt{\frac{3\kappa(1-2\nu)}{2\rho(1+\nu)}}\sim
0.8$ m/s (cytoplasm) to $c\sim 7.2$ m/s (nucleolus). Therefore, at resonance,
the corresponding wave number of the applied harmonic excitation is of the
order of $k\sim\omega/c\sim 725,000$ rad/m (cytoplasm) and $k\sim 69,444$
rad/m (nucleolus) or, correspondingly, a wavelength of the order of
$\lambda\sim 2\pi/k\sim 10^{-5}$ m (cytoplasm) and $\lambda\sim 9\cdot
10^{-5}$ m (nucleolus), which is larger than a typical cell size. It thus
follows that the regime of interest here is the long-wavelength regime,
corresponding to the limit of $k\to 0$ in the preceeding Bloch-wave analysis.
Consequently, in the remainder of the paper we restrict attention to that
limit. The corresponding boundary value problem takes the form sketched in
Figure 5 and consists of the standard displacement elastodynamic boundary
value problem with harmonic displacement boundary conditions applied directly
to the boundary.

$\omega_{n}$ [rad/s] | 501576 | 502250 | 532132 | 537569
---|---|---|---|---
$r_{n,cancerous}\,||\hat{U}_{n}||$ [$\frac{\mu m}{s}$] | $8.862\cdot 10^{6}$ | $9.179\cdot 10^{6}$ | $-3.898\cdot 10^{8}$ | $-2.863\cdot 10^{7}$
$\omega_{n}$ [rad/s] | 496165 | 496165 | 519049 | 545277
$r_{n,healthy}\,||\hat{U}_{n}||$ [$\frac{\mu m}{s}$] | $-3.882\cdot 10^{6}$ | $-3.882\cdot 10^{6}$ | $-2.032\cdot 10^{6}$ | $-0.335\cdot 10^{6}$
Table 5: Comparison of growth rate ratios $r_{n,cancerous}$ and
$r_{n,healthy}$ with $r_{n}=\frac{F_{n}}{2\omega_{n}}$. Shown are values
corresponding to the lowest four cancerous eigenfrequencies
$\omega_{n,cancerous}$ and their related closest healthy eigenfrequencies
$\omega_{n,healthy}$.

Figure 9: Comparison of $u_{n}$ during transient response simulations in the
linearized kinematics framework for a cancerous cell (with a cancerous
potential of $100\%$) excited at one of its resonance frequencies $\omega_{c}$
and a healthy cell (with a cancerous potential of $20\%$) excited at an
eigenfrequency $\omega_{h}$ which is closest to $\omega_{c}$.

###  2.5 Resonant growth rates

The spectral gap, or gap in the lowest eigenfrequencies, between healthy and
cancerous cells and tissues provides a first hint of sharp differences in the
response of healthy and cancerous tissue to harmonic excitation. In
particular, the preceding analysis shows that the fundamental frequencies of
the cancerous tissue may be in close proximity to eigenfrequencies of the
healty tissue, which appears to undermine the objective of selective
excitation of the cancerous tissue. However, a complete picture requires
consideration of the relative energy absorption characteristics and growth
rates of resonant modes. To this end, we consider the modal decomposition of
the displacement field

$\boldsymbol{U}(t)=\sum_{n=1}^{N}u_{n}(t)\hat{\boldsymbol{U}}_{n},$ (9)

where $(\hat{\boldsymbol{U}}_{n})_{n=1}^{N}$ are eigenvectors obeying the
orthogonality and normalization condition (3) and $(u_{n}(t))_{n=1}^{N}$ are
time-dependent modal amplitudes obeying the modal equations of motion

$\ddot{u}_{n}(t)+\omega^{2}_{n}u_{n}(t)=\hat{\boldsymbol{U}}^{T}_{n}\boldsymbol{F}_{ext}(t)=F_{n}(t).$
(10)

In this equation, $\omega_{n}$ is the corresponding eigenfrequency,
$\boldsymbol{F}_{ext}(t)$ is the external force vector and $F_{n}(t)$ is the
corresponding modal force. For a harmonic excitation of frequency
$\omega_{ext}$, eq. (10) further specializes to

$\ddot{u}_{n}(t)+\omega_{n}^{2}u_{n}(t)=F_{n}\cos{\omega_{ext}t},$ (11)

where now $F_{n}$ is a constant modal force amplitude. At resonance,
$\omega_{ext}=\omega_{n}$, the amplitude of the transient solution starting
from quiescent conditions grows linearly in time and the transient solution
follows as

$u_{n}(t)=\frac{F_{n}}{2\omega_{n}}t\sin({\omega_{n}t}).$ (12)

We thus conclude that the growth rate of resonant modes is

$r_{n}=\frac{F_{n}}{2\omega_{n}}.$ (13)

Figure 9 shows the growth properties of $r_{n}$ for two different cases. In
the first case, a cancerous cell is excited at its resonant frequency of
$\omega_{c}=501,576$ rad/s, whereas the healthy cell is excited at its closest
resonance frequency of $\omega_{h}=496,165$ rad/s. In the second case,
eigenfrequencies of $538,512$ rad/s and $545,277$ rad/s are investigated. The
simulations reveal that the growth rate of the resonant response of the
cancerous cells is much faster than that of the healthy cells, which opens a
window for selectively targeting the former.

###  2.6 Transient response at resonance

The preceding analysis has been carried out with a view to understanding the
resonant response of cells and tissues under harmonic excitation in the
harmonic range. In this section, we seek to confirm and extend the conclusions
of the harmonic analysis by carrying out fully nonlinear implicit dynamics
simulations of the transient response of healthy and cancerous cells under
resonant harmonic excitation. In this analysis, a geometry of ratio $n/c=1$ is
considered, Figure 2, together with material parameters of Table 1. We
restrict attention to the long wavelength limit, i. e. to ultrasound radiation
of wavelengths larger than the cell size. In keeping with this limit, we
enforce harmonic displacement boundary conditions directly as shown in Figure
1 in order to mechanically excite the cell. The strength of the harmonic
excitation used in the calculations is $\hat{u}_{0}=0.04$ $\mu$m.

In the simulations, we track the transient amplification of the cell response
up to failure. We assume that failure occurs when the stress in the
cytoskeletal polymer network, which constitutes the structural support for
cell membranes, reaches a threshold strength value. Lieleg et al. (2009) found
that the macroscopic network strength can be traced to the microscopic
interaction potential of cross-linking molecules and other cytoskeletal
components such as actin filaments. Here, we assume a rupture strength of the
order of $30$ Pa based on strength values of a single actin/cross-linking
protein bond reported in Lieleg et al. (2009).

Figure 10 shows the fully-nonlinear transient response of healthy and
cancerous cells at the resonant frequency of the latter. It can be seen from
the figure that stresses in both the plasma membrane and nuclear envelope of
the cancerous cell grow at a much faster rate than in healthy cells. For the
harmonic excitation under consideration, the strength of the nuclear envelope
of the cancerous cell reaches the rupture strength at time $t_{\rm
lysis}\approx 71\mu$s, while, at the same time, the level of stress in the
healthy cells is much lower. Figure 11 furthermore illustrates the kinetic and
potential energy of the nuclear envelope during excitation at resonance of
both healthy and cancerous cells.

From transient response simulations, the energy that needs to be supplied
until the point of rupture is reached is

$\begin{split}E_{\rm lysis}&=\int_{t=0}^{t_{\rm
lysis}}\int_{\partial\Omega}\boldsymbol{t}\cdot\dot{\boldsymbol{u}}\,dS\,dt\\\
&\approx\sum_{i=0}^{n_{\rm
lysis}}\sum_{j\in\partial\Omega}\frac{1}{2}\big{(}\boldsymbol{F}_{j}(t_{i+1})+\boldsymbol{F}_{j}(t_{i})\big{)}\cdot\big{(}\boldsymbol{u}_{j}(t_{i+1})-\boldsymbol{u}_{j}(t_{i})\big{)},\end{split}$
(14)

where $\boldsymbol{t}$ is the applied traction on the boundary
$\partial\Omega$, $\boldsymbol{u}$ is the displacement vector,
$\boldsymbol{F}_{j}(t_{i})$ is the force acting on surface node $j$ at time
$t_{i}$, and $\boldsymbol{u}_{j}(t_{i})$ is the corresponding displacement
vector. For a cell geometry with a ratio of $n/c=1.0$ and a cancerous
potential of $100\%$, calculations give a value of $228$ pJ for the energy per
cell required for lysis. Assuming an average cell size of 20 $\mathrm{\mu}$m,
a time to lysis of 70 $\mathrm{\mu}$s and a tumor of 1 cm in size, this energy
requirement translates into a power density requirement in the range of 0.8
W/cm2.

Figure 10: Transient response of a cancerous cell (top) and a healthy cell
(bottom) at resonance for a ratio of $n/c=1.0$ and a cancerous potential of
$100\%$. Shown is the Frobenius norm of first Piola-Kirchhoff stress tensor.

Figure 11: Kinetic energy (left) and potential energy (right) of the nuclear
envelope during excitation at resonance of both healthy and cancerous cells.

##  3 Discussion and outlook

In this study, we have presented numerical calculations that suggest that
spectral gaps between hepatocellular carcinoma and healthy cells can be
exploited to selectively bring the cancerous cells to lysis through the
application of carefully tuned ultrasound harmonic excitation, while keeping
healthy cells intact. We refer to this procedure as oncotripsy. A normal mode
analysis in the harmonic range reveals the existence of a healthy-to-cancerous
spectral gap in ground frequency of the order of $230,000$ rad/s, or $36.6$
kHz. Further analysis of the growth rates of the transient response of the
cells to harmonic excitation reveals that lysis of cancerous cells can be
achieved without damage to healthy cells. These findings point to oncotripsy
as a novel opportunity for cancer treatment via the application of carefully
tuned ultrasound pulses in the frequency range of 80 kHz, duration in the
range of 70 $\mathrm{\mu}$s and power density in the range of 0.8 W/cm2. This
type of ultrasound actuation can be readily delivered, e. g., by means of
commercial low-frequency and low-intensity ultrasonic transducers. Evidently,
the present numerical calculations serve only as preliminary evidence of the
viability of oncotripsy, and further extensive laboratory studies would be
required in order to confirm and refine the present findings and definitively
establish the viability of the procedure.

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