# Measuring and simulating the biophysical basis of the acoustic contrast
factor of biological cells

Cooper Lars Harshbarger  Department of Orthopedics, Balgrist University
Hospital, University of Zurich, Zurich, Switzerland  Institute for
Biomechanics, Swiss Federal Institute of Technology Zurich, Zurich,
Switzerland  Institute for Mechanical Systems, Swiss Federal Institute of
Technology Zurich, Zurich, Switzerland  these authors contributed equally to
this work<EMAIL_ADDRESS>Alen Pavlic  Institute for Mechanical Systems,
Swiss Federal Institute of Technology Zurich, Zurich, Switzerland  these
authors contributed equally to this work  Davide Cesare Bernardoni  Department
of Orthopedics, Balgrist University Hospital, University of Zurich, Zurich,
Switzerland  Institute for Biomechanics, Swiss Federal Institute of Technology
Zurich, Zurich, Switzerland  Amelie Viol  Institute for Mechanical Systems,
Swiss Federal Institute of Technology Zurich, Zurich, Switzerland  Jess Gerrit
Snedeker  Department of Orthopedics, Balgrist University Hospital, University
of Zurich, Zurich, Switzerland  Institute for Biomechanics, Swiss Federal
Institute of Technology Zurich, Zurich, Switzerland  Jürg Dual  Unai Silván
Department of Orthopedics, Balgrist University Hospital, University of Zurich,
Zurich, Switzerland  Institute for Biomechanics, Swiss Federal Institute of
Technology Zurich, Zurich, Switzerland  BCMaterials, Basque Center for
Materials, Applications and Nanostructures, Leioa, Spain

###### Abstract

The acoustic contrast factor (ACF) is calculated from the relative density and
compressibility differences between a fluid and an object in the fluid. To
name but one application, this acoustic contrast can be exploited using
acoustophoretic systems to isolate cancer cells from a liquid biopsy, such as
a blood sample. Knowing the ACF of a cancer cell represents a crucial step in
the design of acoustophoretic systems for this purpose, potentially allowing
the isolation of circulating cancer cells without labels or contact. For
biological cells the static compressibility is different from the high
frequency counterpart relevant for the ACF. In this study, we started by
characterizing the ACF of low vs. high metastatic cell lines with known
associated differences in phenotypic static E-modulus. The change in the
static E-modulus, however, was not reflected in a change of the ACF, prompting
a more in depth analysis of the influences on the ACF. We demonstrate that
static E-modulus increased biological cells through formaldehyde fixation have
an increased ACF. Furthermore, the static E-modulus decreased biological cells
treated with actin polymerization inhibitor cytochalasin D have a decreased
ACF. Complementing these mechanical tests, a numerical COMSOL model was
implemented and used to parametrically explore the effects of cell density,
cell density ratios, dynamic compressibility and therefore the dynamic bulk
modulus. Collectively the combined laboratory and numerical experiments reveal
that a change in the static E-modulus alone might, but does not automatically
lead to a change of the dynamic ACF for biological cells. This highlights the
need for a multiparametic view of the biophysical basis of the cellular ACF,
as well as the challenges in harnessing acoustophoretic systems to isolate
circulating cells based on their mechanical properties alone.

###### keywords:

Acoustofluidics, Acoustic contrast factor, dynamic material properties

## Introduction

Figure 1: (a) Image of the vertical setup showing the camera (far left) and
the AF device being imaged (far right). (b) AF device imaged with the ROI
indicated by the blue dashed box. A piezoelectric transducer glued to the side
or bottom of the device establishes an acoustic field within the device. (c)
Frame of a video of a cell and PS particle in suspension within the AF device.
Wherever possible, the cells were pre-aligned in a $\lambda$ mode. (d) The
analysis of the particle motion is performed in multiple steps: The video
recording of every experiment is split into single images and converted to
black and white. The object trajectories are indicated in the same colors as
the PS particle and cell in c and are generated using the Fiji Plugin
TrackMate. At $t_{0}$ and
$$1.42\text{\,}\mathrm{MHz}$\pm$2\text{\,}\mathrm{kHz}$$ the cells and PS
particle are located in pressure nodes of the $\lambda$ mode where $p_{0}=0$,
as both objects have a positive acoustic contrast factor. The frequency is
changed to $$742.1\text{\,}\mathrm{kHz}$\pm$2\text{\,}\mathrm{kHz}$$, which
alters the location of the zero pressure nodes. This causes the PS particles
and biological cells to migrate during $t_{1}$ away from their previous
position to the middle of the channel. At $t_{2}$ the objects are again at the
zero pressure nodes $p_{0}=0$ of the $\lambda/2$ mode, thus having minimized
their acoustic potential. (e, f) The plots show the position of a biological
cell (e) and PS (f) particle during the migration from a high pressure to a
low pressure region of the $\lambda/2$ mode. The fit is with respect to the
pressure within the system using eq. 22. Using a self written Matlab script,
the ACF of the biological cells can be back-calculated.

Carl Jung states "The greater the contrast, the greater the potential."[1].
This is true in the field of acoustofluidics (AF), the frequency dependent
acoustic pressure wave driven manipulation of particles in a fluid. In the
case of AF, the contrast is termed the acoustic contrast factor (ACF) $\Phi$,
and arises from relative density and compressibility differences between a
fluid and an object in the fluid as described by Yosioka and Kawasima[2]. The
greater the ACF, the greater the acoustic potential and the more the object in
the fluid will interact with the pressure wave. Changes to either the object
or the fluid will lead to a change of potential, whereas the acoustic
potential in an AF system is termed the Gor’kov potential[3]. Exploiting the
acoustic contrast between biological cells and a fluid has been successfully
demonstrated multiple times for the manipulation of cancer cells in
suspension[4, 5, 6, 7, 8]. Therefore, AF is a promising diagnostic tool when
considering the spreading of cancer cells through bodily fluids, a process
known as metastasis which is responsible for 90% of cancer related deaths[9].
In order for cancer cells to enter the metastatic state, they undergo a wide
range of changes, including a decrease in their static E-modulus[10], which
leads to a different mechanical phenotype between the cancer cells located
within the primary tumor and the disseminated cancer cells. This decrease in
the static E-modulus can be measured by gold standard measurement techniques
such as magnetic tweezers[11], AFM[12] and micropipette aspiration [13].
Although the presented methods are powerful and established, there are certain
limitations such as that the biological cells are adherent on tissue culture
plastic, that only one mechanical parameter of the cell of interest is
measured and that the measurement is based off of one small region of the
biological cell. An alternative method to measure the static E-modulus, which
relies on the deformation of the entire biological cell and does not rely on
adherent cells, is real-time deformability cytometry (RT-DC)[14]. Although
these methods have led to a vast catalog of static E-moduli for various cancer
cell types in adherent and non-adherent states, the reported material
properties stem from (quasi-)static measurements. Considering emerging
ultrasound frequency technologies to detect and treat cancer, such as
oncotripsy[15, 16], the question needs to be answered if static material
parameters are sufficient or if lookup tables for material properties need to
be extended with values stemming from dynamic measurements. Such dynamic
material properties can be measured with AF, by focusing cells using
ultrasound frequencies, as illustrated in Fig. 1. By focusing the biological
cells in a suspension with polystyrene (PS) particles with a known ACF, the
ACF of the biological cells can be calculated. This method of measuring the
ACF is contactless, label-free and does not decrease cell viability[17].

There is a growing body of literature where the ACF of biological cells is
measured[18, 19, 20, 21] as summarized in table 1 and new ACF values reported
in Fig. 2. The work prior to this study however mostly results in measurements
of the compressibility of biological cells from which the ACF can be
calculated, whereas the density is assumed[18], measured with methods such as
a neutrally buoyant sample [20] or measured by defocusing of the particles
during sedimentation[21]. Further publications resulted in lookup tables for
cell lines with different metastatic potentials[19] and ACF comparisons
between fixed and non-treated biological cells[20].

Beside these few studies, little is known about the correlation between the
known static E-modulus of biological cells and the dynamic ACF. To help
broaden the understanding of the applicability of known material parameters to
ultrasound frequency applications we increased the static E-modulus[22] of the
bone cancer cell Sarcoma Osteogenic (SaOs)-2 using the protein crosslinking
fixative formaldehyde[23] and decreased static E-modulus[24] using the actin
polymerization inhibitor cytochalasin D[25] (CD). After these treatments the
ACF of the biological cells are measured in a Phosphate Buffer Solution (PBS).
Furthermore, the SaOs-2 parental cell line is paired with its highly
metastatic counterpart Lung Metastasis (LM)5, with the known static E-modulus
measured of $0.95\text{\,}\mathrm{kPa}$ and $0.8\text{\,}\mathrm{kPa}$,
respectively[12] when measured with RT-DC. Thus demonstrating the limitations
of trying to predict the ACF when only taking a change in static E-modulus
into account. To further place the experimental results into context, a
numerical model, seen in Fig. 3 is introduced. The parameter space
investigated with the model includes the density, compressibility and static
E-modulus, as seen in Fig. 4. Thus aiming at closing gaps of unknown dynamic
material properties of cells which would be useful for cancer diagnosis and
treatment.

## Results

We used a vertical setup composed of a widefield microscope to image the PS
particles and cell migrate through the fluid, as seen in Fig. 1. The AF device
was placed in a vertical orientation to potentially eliminate rolling along
the glass surface and the 3 inlet configuration can improve the prefocusing
efficiency, if needed. The standing pressure wave was established by a
piezoelectric transducer powered by a sinusoidal signal of a function
generator increased in magnitude by an amplifier. The frequencies used are in
the $$742.1\text{\,}\mathrm{kHz}$\pm$2\text{\,}\mathrm{kHz}$$ range and were
varied for optimal focusing. A summary of the ACFs of previous publications
is, together with the results of this study, summarized in table 1. The ACF
measured in PBS of the low metastatic potential cell line SaOs-2 is taken as
the baseline and is 0.037 ($n=25$). SaOs-2 formaldehyde fixed cells have an
ACF of 0.050 ($n=10$). SaOs-2 cells treated with cytochalasin D have an ACF of
0.022 ($n=12$). The higher metastatic potential cell line LM5 have an ACF of
0.037 ($n=12$). The average cell diameter for each condition was measured and
is reported in table 1.

Figure 2: The ACF measured in PBS of the low metastatic potential cell line
SaOs-2 is taken as the baseline and is 0.037 ($n=25$). SaOs-2 formaldehyde
fixed cells have an ACF of 0.050 ($n=10$). SaOs-2 cells treated with
cytochalasin D have an ACF of 0.022 ($n=12$). The higher metastatic potential
cell line LM5 have an ACF of 0.037 ($n=12$).

The base numerical model of a SaOs-2 cell is constructed as described in the
Methods, and is composed of a cytoplasm-nucleoplasm-nucleolus structure as
shown in Fig. 3. The cell and all the layers are assumed to be spherical. The
radius for the SaOs-2 cell is taken from the measurements reported in table 1
($8.80\text{\,}\mathrm{\SIUnitSymbolMicro m}$). The outer radius of the
nucleoplasm ($6.66\text{\,}\mathrm{\SIUnitSymbolMicro m}$) is computed from
previously reported[12] nucleus-cell volume ratio of $0.434$, while the radius
of the nucleolus ($2.14\text{\,}\mathrm{\SIUnitSymbolMicro m}$) follows from
the nucleolus-nucleus volume ratio of $1/30$ reported by Guttman and
Halpern[26]. The mass density in each layer is computed according to the mass
density ratios of the three characteristic cellular layers, as reported by Kim
and Guck[27], while imposing an assumed apparent density of the whole cell of
$1060\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$. The static E-modulus of
each cellular layer is assigned based on the ratios used in the
literature[15], such that the apparent static E-modulus computed through the
rule of mixture results in $E=$1\text{\,}\mathrm{kPa}$$, as reported from RT-
DC measurements[12]. The initial bulk modulus of individual cellular layer is
assigned based on the reported 2% lower speed of sound in nucleus compared to
the cytoplasm[28].

Figure 3: Numerical model of a eukaryotic cell. The ACF and its dependencies
are extracted from the numerical model. (a) The structure of a typical
eukaryotic cell. (b) Simplification of the representation of the cell into
three layers, namely, cytoplasm, nucleoplasm, and nucleolus. (c) Geometry of
an axisymmetric finite-element method model of a eukaryotic cell, surrounded
by fluid and perfectly-matched layer (PML).

In the first step shown in Fig. 4(a), the SaOs-2 model, with the density and
static E-Modulus taken from literature, is calibrated by varying the bulk
modulus $K$ of all the layers through a common multiplier to match the
experimentally measured ACF of $0.037$. The bulk modulus of the calibrated
SaOs-2 model therefore follows as $2.49\text{\,}\mathrm{GPa}$ for the
cytoplasm and $2.24\text{\,}\mathrm{GPa}$ for the nucleus (nucleoplasm and
nucleolus combined). Figure 4(b) shows how the bulk modulus in the range of
interest connects to the Poisson ratio for the relatively low static E-modulus
of $1\text{\,}\mathrm{kPa}$, reported for SaOs-2 cells[12].

Figure 4: (a) The ACF of a SaOs-2 cell depending on the bulk modulus
multiplier (bulk.multiplier) common to the three layers of the SaOs-2 model
with a constant static E-modulus. To calibrate the model, the bulk multiplier
is tuned to the value that yields the ACF of $0.037$ measured experimentally.
(b) The Poisson’s ratio corresponding to the range of bulk modulus relevant
for our study, computed from eq. 16 with a constant static E-modulus; the bulk
modulus of water is shown for reference. (c) The calibrated layer-based
material properties from the SaOs-2 model were used as a basis for the LM5
model, changing only the volume of the cell and the nucleus-cell volume ratio
according to experimental observations[12]. The base LM5 model underestimates
the ACF at $0.0309$, compared to the experimentally measured value of $0.037$.
The common bulk modulus multiplier and density multiplier are varied in (c) to
indicate the combination changes in the density and bulk modulus that provide
the experimentally measured ACF (denoted with the red line). (d) Varying the
E-modulus, while keeping the bulk modulus and the density constant, does not
influence the ACF.

The base numerical model of a LM5 cell is then formed by keeping all the
material properties of individual cell layers from the calibrated SaOs-2 cell
model, but changing the volume of the cell to the volume measured in
experiments. In addition, the nucleus-cell volume ratio was changed from
$0.434$ to $0.569$, according to previously reported[12] values. These changes
lead to a decrease in the ACF for the base LM5 cell to $0.031$ whereas the
value measured in experiments is $\sim$27\text{\,}\mathrm{\char 37\relax}$$
higher at $0.037$. The decrease in the ACF is due to the increased nucleus-
cell volume ratio that decreases the apparent density and the apparent bulk
modulus of the cell. In Fig. 4(c) we explore how a change in the density and
bulk modulus across the three layers of a cell through common multipliers
could cause the increase in the ACF that would match the experimentally
measured ACF of $0.037$ (red line). In Fig. 4(d), we demonstrate that changing
the E-modulus across the cell layers does not directly affect the ACF as long
as the bulk modulus and the density are kept at a fixed value.

## Discussion

Our results reveal that formaldehyde-fixed cells, presumed to have an
increased static E-modulus, have an increased ACF compared to untreated cells.
Furthermore biological cells treated with the actin depolymerizing drug
cytochalasin D (CD), known to reduce the static E-modulus, have a decreased
ACF. This simultaneous change of static E-modulus and ACF is particularly
interesting when considering that a decrease in the static E-modulus
correlates to an increase of the metastatic potential of cancer cells[29],
especially as the static E-modulus of an object can be linked to its
compressibility (eq. 15 and 16) and thus to the ACF of the object, as
demonstrated by eq. 19. This dependence could potentially be used to predict
the increased ACF of the static E-modulus increased biological cells and the
decreased ACF of static E-modulus decreased biological cells as presented in
this study. To test this predictability a hypothesis can be formulated, that
the more malignant cells have a decreased ACF and thus interact less with the
acoustic field. This hypothesis can be tested with the parental cancer cell
line with low metastatic potential SaOs-2 with a static E-modulus of
$0.95\text{\,}\mathrm{kPa}$ and the high metastatic potential cell line LM5
with a static E-modulus of $0.8\text{\,}\mathrm{kPa}$, both static E-moduli
were measured with a RT-DC system[12]. Contrary to the hypothesis, we found
that the difference between the ACFs of the two cell lines with different
metastatic potential is only in the range of a few percent, even though the
sample size is limited. Nevertheless, this opens up the question if the static
E-modulus change is the causation for a change of the ACF or if the change is
merely correlated. The broader implication is that forming a hypothesis about
dynamic material properties based on known static material properties can lead
to imprecise answers and the current literature reflects this uncertainty and
caution must be exercised when making these assumptions.

Cushing et al.[20] found that fixing the biological cells with formaldehyde
leads to an increase of the compresibility together with a decrease of the
density. This leads to an overall decrease of the ACF, which is contrary to
the increase of the ACF found in our study. Important to note is that in
Cushing et al. the measurement of the compressibility was done by speed of
sound measurements on a population of biological cells in suspension and not
on individual biological cells as was used during this study, which could
account for some discrepancies. Furthermore, although the static E-modulus of
the cell should have been increased by the fixation, the authors do not try to
link the static E-modulus and the dynamically measured compressibility.

Wang et al.[19] demonstrated another method to measure the density and
compressibility of biological cells. Although no statistical analysis was
performed, they demonstrated that the compressibility increases with the
metastatic potential. The change in the density however did not follow a trend
when compared to their metastatic potential. The authors do not specify in
which fluid the measurement was performed, therefore it is not possible to
precisely calculate the ACF. Assuming PBS as the fluid in which the
experiments were conducted, we calculate the ACF of the low and high
metastatic cell lines. For these assumed values of the ACF, we could not find
a correlation between the metastatic potential and the ACF, in part stemming
from the simultaneous change of the compressibility and density.

Summa summarum, there are many gaps in the literature when considering the
dynamic properties of biological cells in acoustic fields. Slightly different
methods lead to different trends concerning the ACF, and no correlation is
attempted to be made between known static material properties and the
dynamically measurement data. This is unfortunate as most literature on cell
mechanics reports the static E-modulus, which is the standard output for gold
standard methods such as the quasi-static AFM.

| Material of interest | $d[$\text{\,}\mathrm{\SIUnitSymbolMicro m}$]$ | $\rho[$\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$]$ | $E[$\text{\,}\mathrm{kPa}$]$ | $G[$\text{\,}\mathrm{kPa}$]$ | $\kappa[$\text{\,}{\mathrm{TPa}}^{-1}$]$ |  $\Phi$ / ACF $[1]$
---|---|---|---|---|---|---|---
a | Water | - | $0.9966$ | - | - | $444.8$ | -
b | NIH/3T3 | - | $1.079$ | $3-5$ | $1.67$ | $378$ | $0.083$
| MCF-7 | - | $1.068$ | $0.31-0.6$ | $0.103$ | $422$ | $0.047$
| HEPG2 | - | $1.087$ | $0.191-0.941$ | $0.0637$ | $428$ | $0.047$
| HT-29 | - | $1.077$ | $4.09$ | $1.36$ | $404$ | $0.060$
| MCF-12A | - | $1.068$ | - | - | $377$ | $0.083$
| RBC | - | $1.099$ | - | - | $331$ | $0.117$
| Polystyrene | - | $1.050$ | - | - | $216$ | $0.175$
c | HNC Tu686 | - | $1.025$ | - | - | $405$ | $0.032$
| HNC 686LN | - | $1.060$ | - | - | $428$ | $0.025$
| HNC M4e | - | $1.080$ | - | - | $432$ | $0.029$
| HNC 37B | - | $1.045$ | - | - | $440$ | $0.012$
d | RBC | - | $1.101$ | - | - | $334$ | $0.109$
| RBCfx | - | $1.091$ | - | - | $356$ | $0.090$
| WBC | - | $1.054$ | - | - | $393$ | $0.051$
| WBCfx | - | $1.045$ | - | - | $400$ | $0.042$
| DU-145 | - | $1.062$ | - | - | $384$ | $0.059$
| DU-145fx | - | $1.036$ | - | - | $404$ | $0.036$
| MCF-7 | - | $1.055$ | - | - | $373$ | $0.066$
| MCF-7fx | - | $1.035$ | - | - | $395$ | $0.043$
| LU-HNSCC-25 | - | $1.061$ | - | - | $377$ | $0.065$
| LU-HNSCC-25fx | - | $1.040$ | - | - | $404$ | $0.038$
| Polystyrene | $5$ | $1.058$ | - | - | $273$ | $0.143$
| Polystyrene | $7$ | $1.059$ | - | - | $276$ | $0.141$
| Melamine | $10$ | $1.500$ | - | - | $124$ | $0.363$
| PMMA | $3$ | $1.184$ | - | - | $173$ | $0.255$
e | MESC2.10 | $13.2$ | - | - | - | - | $0.05$
| MESC2.10-diff4d | $12.2$ | - | - | - | - | $0.08$
f | Perfluoropentane core | $4.2$ | 1.619 | - | - | $670$ | $(-0.014)-(-0.029)$
| & cellulose nanofiber shell |  |  |  |  |  |
g | SaOs-2 | 17.6 | - | 0.95 | - | - | 0.037
| SaOs-2fx | 15.0 | - | - | - | - | 0.050
| SaOs-2 Cyto. D | 18.9 | - | - | - | - | 0.022
| LM5 | 15.5 | - | 0.80 | - | - | 0.037
| Polystyrene | 10.2 | 1.050 | - | - | $250$ | 0.167
h | Cytoplasm | $17.6$ | $1.099$ | $0.31$ | - | $402$ | -
| Nucleoplasm | $13.3$ | $0.996$ | $1.78$ | - | $446$ | -
| Nucleolus | $4.3$ | $1.744$ | $5.36$ | - | $446$ | -
Table 1: Material properties of eukaryotic cells and some reference inorganic
materials. The measurement uncertainty of individual results is omitted for
the sake of clarity. Size $d$ generally represents the diameter of a cell,
particle, or a part of a cell. $\Phi$ is the mathematical symbol for the
acoustic contrast factor (ACF).
a Values for water from Karlsen and Bruus[30] used in numerical models, with
the dynamic viscosity of $0.8538\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$ and
the bulk viscosity of $2.4\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$.
b NIH/3T3 - fibroblast cell; MCF-7 - breast cancer cell; HEPG2 - liver cancer
cell; HT-29 - colon cancer cell; MCF-12A - breast cell; RBC - red blood cell;
Hartono et al.[18]; the compressibility $\kappa=1/K$ is measured via acoustic
manipulation at $3.75\text{\,}\mathrm{MHz}$, the other properties are
referenced within [18]; density of MCF-12A is assumed.
c HNC Tu686, 686LN, M4e, 37B - head and neck cancer cells whereas M4e & 37B
have a higher metastatic potential than Tu686 & 686LN; density and
compressibility from Wang et al.[19]; acoustic manipulation (data assessed
from the graphs) at $1.91\text{\,}\mathrm{MHz}$. Wang et al. do not report the
ACF nor the fluid in which the cells were focused. If the fluid is assumed to
be PBS, these are the according ACFs.
d ‘fx’ denotes a fixed sample. RBC - red blood cell; WBC - white blood cell;
DU-145 - prostate cancer cell; MCF-7 - breast cancer cell; LU-HNSCC-25 - head
and neck squamous cancer cell; PMMA - polymethylmethacrylat; Cushing et
al.[20]; through speed of sound measurements of neutrally buoyant samples at
$3\text{\,}\mathrm{MHz}$; $\Phi$ in PBS
($c=$1508.2\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ and
$\rho=$1004.00\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$$).
e MESC2.10 - human embryonic ventral mesencephalic cell; MESC2.10-diff4d -
human embryonic ventral mesencephalic cell differentiated in a special medium
for 4 days; Augustsson et al.[21]; the acoustic contrast $\Phi$ is measured
via acoustic manipulation at $1.97\text{\,}\mathrm{MHz}$; we multiplied the
average $\Phi$ from [21] by a factor of $3$ to match our definition of $\Phi$.
f Loskutova et al.[31] demonstrated a negative ACF, whereas the ACF is
dependent on the pressure amplitude.
g SaOs-2 & LM5 - human bone cancer cell line with a lower respectively higher
metastatic potential and ACF data generated during this study. Static
E-modulus measured using an RT-DC system and taken from Holenstein et al.[12].
The cell diameter was measured either by a cell counter or by manually
determining the size from the videos used to compute the ACF. Cells were
focused in the $$742.1\text{\,}\mathrm{kHz}$\pm$2\text{\,}\mathrm{kHz}$$
range.
h Data from the calibrated numerical model of the SaOs-2 cell presented in
this study.

One major limiting factor to date hindering the direct comparison is that most
publications deal with measuring only one material property such as the static
E-modulus or the dynamic compressibility. This one parametric view is not
sufficient for even the simplest cell mechanics models such as the elasticity
theory which requires at least two parameters to describe cell mechanics[32],
as seen in eq. 16. This can then necessitate estimates of further material
properties such as the Poisson’s ration $\nu$, and whichever ratio is chosen
has orders of magnitude impact on the calculated static bulk or E-modulus.
This can quickly be seen from the common assumption that cells are
incompressible, meaning that $\nu=0.5$ and would therefore imply an infinitely
large bulk modulus as seen from eq. 16 and Fig. 4(b). In order to try to shed
some light on the usefulness of static mechanical properties reported for
various cell lines, we provide a computational model for a cell in an acoustic
field. This model is insofar interesting as it is a multiparametric model
where the mechanical properties, such as the E-modulus and the bulk modulus,
can be varied independently. The computational model shows on the one hand
that the E-modulus can be varied without changing the ACF as seen in Fig.
4(c). This supports previous findings and the results from our experiments,
that a change in E-modulus might be correlated with a change in the ACF but
does not necessarily need to be the causation. Therefore the hypothesis that
the static E-modulus alone can be used to predict the dynamic material
properties must be rejected. To further demonstrate this possible
independence, the computational model shows that there is a parameter space
where a variation of the apparent bulk modulus and density can result in the
same ACF as seen in Fig. 4(b). This finding is further supported by the
findings of Wang et al. where a change in metastatic potential altered both
the density and dynamic compressibility, thus defying a trend between
metastatic potential and the ACF. There is another method to measure dynamic
properties of a cell (e.g. bulk modulus) - scanning acoustic microscopy[32],
which confirms our predictions of the Poisson’s ratio, and confirms that the
E-modulus can vary independently of the bulk modulus. This means that most
known material properties have an undefined influence on dynamic measurements
and prior knowledge cannot be exploited, such as static E-modulus measurements
from quasi-static methods such as AFM or RT-DC. This is problematic for
applications such as the acoustic focusing and sorting of cells in liquid
biopsies, potential cancer treatment methods such as oncotripsy, ultrasound
neuromodulation[33, 34] and sonogenetics[35]. This indicates an unmet need for
dynamic material properties which in parts could be potentially alleviated by
AF.

## Methods

### Theoretical Background: Acoustic scattering and streaming

The presented work deals with objects in fluids. In order to predict the
motion of the objects, the fundamental equations of the motion of the fluids
need to be introduced. The motion of a viscous fluid is governed by the
compressible Navier-Stokes equations

$\rho\left[\frac{\partial\underline{v}}{\partial
t}+(\underline{v}\cdot\bm{{\nabla}})\underline{v}\right]=-\bm{\nabla}p+\eta\nabla^{2}\underline{v}+\left(\eta_{\mathrm{B}}+\frac{\eta}{3}\right)\bm{\nabla}\left(\bm{\nabla}\bm{\cdot}\underline{v}\right),$
(1)

and the continuity equation

$\frac{\partial\rho}{\partial t}=-\rho\bm{\nabla}\bm{\cdot}\underline{v},$ (2)

with the velocity $\underline{v}$, pressure $p$, the dynamic viscosity $\eta$
and the bulk viscosity $\eta_{\mathrm{B}}$. As the fluid is assumed to be
barotropic, the density $\rho$ is assumed to be a function of pressure $p$
only $\rho=\rho(p)$.

The equations are linearized using the perturbation approach [36].
Accordingly, the physical fields are expanded in a series,
$\square=\square_{0}+\square_{1}+\square_{2}+\dots$, where $\square$
represents the field, while the subscript denotes the respective order.

#### First-order (acoustic) problem

For a quiescent fluid at the zeroth order ($\underline{v}_{0}=\underline{0}$),
the substitution of the perturbed fields into the governing equations yields
the following set of linear first-order equations,

$\displaystyle\rho_{0}\frac{\partial\underline{v}_{1}}{\partial t}$
$\displaystyle=-\bm{\nabla}p_{1}+\eta\nabla^{2}\underline{v}_{1}+\left(\eta_{\mathrm{B}}+\frac{\eta}{3}\right)\bm{\nabla}\left(\bm{\nabla}\bm{\cdot}\underline{v}_{1}\right),$
(3) $\displaystyle\frac{\partial\rho_{1}}{\partial t}$
$\displaystyle=-\rho_{0}\bm{\nabla}\bm{\cdot}\underline{v}_{1},$ (4)

with the equilibrium density $\rho_{0}$. The equation of state,

$\rho_{1}=\frac{1}{c_{0}^{2}}p_{1},$ (5)

is connecting the first-order density with the first-order pressure through
the speed of sound in the fluid $c_{0}$. The first-order fields are assumed to
have a harmonic time-dependency with the factor $\mathrm{e}^{\mathrm{i}\omega
t}$, with the angular frequency $\omega=2\pi f$.

The acoustic fields, comprised of the velocity $\underline{v}_{1}$ and
pressure $p_{1}$, are assumed to be the sums of background fields (bg) and
scattered fields (sc), namely
$(\,)_{1}=(\,)_{1\mathrm{bg}}+(\,)_{1\mathrm{sc}}$. We assume a one-
dimensional plane standing wave along the $z$-direction of the cylindrical
coordinate system. The background velocity field is set to

$\underline{v}_{1\mathrm{bg}}=\mathrm{Re}\left[\frac{\varphi_{{\rm
NewA}}}{2}\mathrm{i}k\left(\mathrm{e}^{\mathrm{i}kz}-\mathrm{e}^{-\mathrm{i}kz}\right)\mathrm{e}^{\mathrm{i}\omega
t}\right]\underline{e}_{z},$ (6)

with the corresponding velocity potential amplitude

$\varphi_{{\rm NewA}}=-\frac{p_{{\rm
NewA}}}{\mathrm{i}\omega\rho_{0}+\left(\eta_{\mathrm{B}}+\frac{4}{3}\eta\right)k^{2}},$
(7)

with pressure amplitude $p_{{\rm NewA}}$, and the wavenumber

$k=\frac{\omega}{c_{0}}-\alpha\mathrm{i},$ (8)

with the attenuation coefficient for viscous fluids [37]

$\alpha=\frac{\omega^{2}}{2c_{0}^{3}\rho_{0}}\left(\eta_{\mathrm{B}}+\frac{4}{3}\eta\right).$
(9)

#### Second-order (streaming) problem

Applying the perturbation theory up to second order to the governing
equations, together with taking the time average
$\langle\square\rangle=\frac{1}{T}\int_{T}\square\mathrm{d}t$ over an
oscillation period $T=1/f$, results in the equations of acoustic streaming
[38],

$\displaystyle\bm{\nabla}\left\langle
p_{2}\right\rangle-\eta\nabla^{2}\left\langle\underline{v}_{2}\right\rangle-\left(\eta_{\mathrm{B}}+\frac{\eta}{3}\right)\bm{\nabla}\left(\bm{\nabla}\bm{\cdot}\left\langle\underline{v}_{2}\right\rangle\right)$
$\displaystyle=-\rho_{0}\bm{\nabla}\bm{\cdot}\left\langle\underline{v}_{1}\underline{v}_{1}\right\rangle,$
(10)
$\displaystyle\rho_{0}\bm{\nabla}\bm{\cdot}\left\langle\underline{v}_{2}\right\rangle$
$\displaystyle=-\bm{\nabla}\bm{\cdot}\left\langle\rho_{1}\underline{v}_{1}\right\rangle.$
(11)

### Theoretical Background: Acoustic contrast factor

A method to measure the acoustic contrast factor of a biological cell is to
subject the biological cell to a standing pressure wave within a fluid cavity.
The standing pressure wave is generated by exciting a piezoelectric transducer
(PT) glued onto an acoustofluidic device, which transmits the vibration of the
PT into the fluid cavity where resonance is established. These vibrations,
when assuming a spherical particle with a radius much smaller than the
acoustic wavelength in an inviscid fluid, give rise to a potential commonly
defined as the Gor’kov potential $U$[3]

$U=\frac{4\pi}{3}r^{3}\biggl{[}f_{1}(\tilde{\kappa})\frac{1}{2\rho_{0}c_{0}^{2}}\langle
p_{1\mathrm{bg}}^{2}\rangle-
f_{2}(\tilde{\rho})\frac{3}{4}\rho_{0}\langle\underline{v}_{1\mathrm{bg}}^{2}\rangle\biggr{]},$
(12)

where $r$ is the radius of the particle, $\langle p_{1\mathrm{bg}}^{2}\rangle$
the first order time averaged square of the incident acoustic pressure and
$\langle\underline{v}_{1\mathrm{bg}}^{2}\rangle$ the first order time averaged
square of the incident acoustic velocity. Furthermore, the monopole scattering
coefficient $f_{1}(\tilde{\kappa})$ which is related to the relative
compressibility between the particle and the medium and the dipole scattering
coefficient $f_{2}(\tilde{\rho})$ which is related to the relative density
between the particle and the medium

$f_{1}(\tilde{\kappa})=1-\tilde{\kappa};\\\
\tilde{\kappa}=\frac{\kappa_{p}}{\kappa_{0}},$ (13)

$f_{2}(\tilde{\rho})=2\frac{\tilde{\rho}-1}{2\tilde{\rho}+1};\\\
\tilde{\rho}=\frac{\rho_{p}}{\rho_{0}},$ (14)

where $\kappa_{p}$ is the compressibility of a particle, $\kappa_{0}$ is the
compressibility of the fluid and $\rho_{p}$ is the density of the particle.
The compressibility is in this case defined as the the inverse of the bulk
modulus $K$

$\kappa=\frac{1}{K}.$ (15)

The bulk modulus is one parameter needed for the linear elastic description of
cell mechanics and is given by

$K=\frac{E}{3(1-2\nu)}$ (16)

for a given static E-modulus and Poisson’s ratio $\nu$. For fluids, bulk
modulus follows as $\rho_{0}c_{0}^{2}$, and the compressibility as
$\kappa_{0}=1/(\rho_{0}c_{0}^{2})$. The acoustic radiation force
$\underline{F}_{ARF}$ is commonly approximated as the negative gradient of the
Gor’kov potential $U$

$\underline{F}_{\mathrm{ARF}}=-\bm{\nabla}U.$ (17)

Given an inviscid standing pressure wave defined through eq. 6 and assuming
$\eta=\eta_{\mathrm{B}}=0$, $\underline{F}_{ARF}$ can be simplified to a one-
dimensional ARF in the $z$ direction

$F^{1D}_{\mathrm{rad}}=4\pi
r^{3}E_{ac}k_{z}sin\left(2k_{z}\left(z+\frac{w}{2}\right)\right)\Phi(\tilde{\kappa},\tilde{\rho}).$
(18)

$k_{z}=\frac{\omega}{c_{0}}$ is the wavenumber in an inviscid fluid, $w$ is
the width of the channel, $\sin(2k_{z}(z+w/2))=1$ for the maximal force,
$E_{ac}$ is the acoustic energy density, the direction of $z$ is shown in Fig.
3 and $\Phi$ is the acoustic contrast factor

$\Phi(\tilde{\kappa},\tilde{\rho})=\frac{1}{3}f_{1}(\tilde{\kappa})+\frac{1}{2}f_{2}(\tilde{\rho}).$
(19)

which is a dynamic quantity related to the scattering of pressure waves on
objects and is related to the relative density and compressibility between a
fluid and an object in the fluid. A table of relevant $\Phi$ values is given
in table 1, whereas objects with a positive ACF migrate to pressure nodes as
seen in Fig. 1 and objects with a negative ACF migrate to pressure anti-nodes.
The drag force acting on a particle moving through a fluid is given by the
Stokes’ drag[39, 40]

$\underline{F}_{str}=-6\pi\eta r\underline{v}_{p},$ (20)

where $\underline{v}_{p}$ is the velocity of the particle. Neglecting the
particle inertia, the particle velocity $v_{z}$ in the $z$ direction can be
calculated by balancing the ARF with the Stokes’ drag[41]

$v_{z}=\frac{2\Phi}{3\eta}r^{2}k_{z}E_{ac}\sin(2k_{z}(z+\frac{w}{2})).$ (21)

With $v_{z}=\frac{\partial{z_{p}}}{\partial{t}}$ and given a particle at
starting position $z_{0}$ at time $t=0$, the transverse path $z_{p}(z_{0},t)$
can be calculated[41]

$z_{p}(z_{0},t)=\frac{1}{k_{z}}\arctan\left\\{\tan\left[k_{z}(z_{0}+\frac{w}{2})\right]\exp(\frac{4\Phi}{3\eta}(k_{z}r)^{2}E_{ac}t)\right\\}-\frac{w}{2}.$
(22)

The resonance frequency for the $n$-th ultrasonic resonance mode of a one-
dimensional standing wave with hard wall boundary conditions is[42]

$f_{res}^{1D}=\frac{c_{0}n}{2w}=\frac{c_{0}}{\lambda_{n}},$ (23)

where $w$ is the width of the channel and $\lambda_{n}$ is the wavelength. For
$n=1$ or $n=2$, the wavelength is equal to $2w$ or $w$ corresponding to the
$\lambda/2$ mode or $\lambda$ mode, respectively.

### Setup

The setup is analogous to the one used in Harshbarger et al.[43] and is
comprised of a function generator (AFG-2225, GW Instek) which is connected to
a computer running a code which allowed for the one click change of the
frequency. The signal from the function generator is amplified (325LA Linear
Power Amplifier, Electronics & Innovation) and monitored using an oscilloscope
(UTD2025CL, UNI-T). The amplified signal is passed on to the piezoelectric
transducer (PT). The vertical setup seen in Fig. 1(a) is built from parts from
the THORLABS Cerna® series. The video feed was generated with a 10x objective
(M Plan Apo 10x / 0.28, Mitutoyu) and a uEye camera (UI-3160CP Rev. 2.1, iDS,
1920 x 1200 pixels, 60 fps). To positively identify the fluorescent PS in the
solution, a blue LED was used. In addition, there was a backlight, shining
through the backside of the device, which resulted in brightfield image, but
also an increased fluorescent signal of the PS particles. The sample flow was
controlled manually whereas the $1\text{\,}\mathrm{m}\mathrm{l}$ syringe was
placed vertically in a holder located above the device in order to avoid
sedimentation of the PS particles and the cells in the tubing and the device.
This vertical orientation is unlike the horizontal orientation presented
in[18, 19, 20, 21]. This vertical orientation is only advisable if the density
of the objects in the fluid are in a similar range as the density of the
fluid, as is the case for biological cells and PS particles in PBS.

### Glass-silicon-glass device

The glass-silicon-glass device, seen in Fig. 1(b), is produced by bonding a
$500\text{\,}\mu\mathrm{m}$ thick glass wafer to a $200\text{\,}\mu\mathrm{m}$
thick silicon wafer. The fluidic channel is patterned onto the exposed silicon
wafer using photolithography (resist: S1828, Shipley, 4’000 $\mathrm{rpm}$;
developer: AZ351B, Microchemicals). The full thickness of the silicon wafer is
etched away with an inductively coupled plasma (ICP) deep reactive ion etching
(DRIE) machine (Estrellas, Oxford instruments). Following the etching, a
$700\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick glass wafer is anodically
bonded onto the exposed silicon wafer. The wafer is diced into individual
devices with a wafer saw (DAD3221, Disco corporation). Fused silica
capillaries ($164\pm$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ outer diameter,
$100\pm$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ inner diameter, Molex) are
inserted into the inlets and outlets of the device to create a fluidic
connection. The capillaries are fixed with a two-component glue (5 Minute
Epoxy, Devcon). Piezoelectric transducers ($10\text{\,}\mathrm{mm}$ length,
$2\text{\,}\mathrm{mm}$ width, $1\text{\,}\mathrm{mm}$ thickness, Pz26,
Meggitt Ferroperm) are glued on using an electrically conductive Epoxy glue
(H20E, Epoxy Technology). Copper cables ($0.15\text{\,}\mathrm{mm}$ diameter)
are attached to the piezoelectric transducers and the electrical connection
was established with an electrically conductive silver paste. The final
dimensions of the devices is given by: device: 50 mm length, 12 mm width, 1.4
mm thickness; focusing channel: 15 mm length, 1 mm width, 0.2 mm height as
seen in Fig. 1c. The 3 inlets results in the cells already being slightly
prefocused close to the walls before the acoustics is turned on. This makes
the prefocusing process in a $\lambda$ mode efficient, if even needed.

### Polystyrene particles

Green fluorescent polystyrene (PS) particles (microParticles GmbH, Germany)
with diameters of $10.23\pm$0.13\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
2.5 w/v% were used for all experiments.

### Cell culture

Two cell lines were used, the low metastatic potential parental cell line
SaOs-2[44] and the highly metastatic cell line LM5. The creation of the LM5
cell line is detailed in[45]. In brief, Sarcoma Osteogenic (SaOs)-2 cells were
isolated from the primary osteosarcoma of an 11 year old Caucasian girl and
were subsequently injected into the tail vein of a nude mouse. After 6 months
the mouse was sacrificed and the metastatic lesion was removed from the lung
of the mouse. The cells isolated from the lung lesion were again injected into
the tail of a new nude mouse. This process was repeated 4 times resulting in
the lung metastasis 5 (LM5) cell line. The anatomical origin of the primary
tumor from which the SaOs-2 cells were extracted is not known. All cell lines
are kept at the standard 37°C and 5% $CO_{2}$ and 95% air. The cell media used
is DMEM - F12 Ham (D8437, Sigma) supplemented with 10% fetal bovine serum
(10270106, Thermo) and 1% P/S. The cells were passaged when 60% confluency was
reached. The diameter of the cells was measured by the CellDrop BF, DeNovix
after removal from the cell culture flask or directly in the video. The
results section always indicated how the cell size was measured.

### Trajectory measurement protocol

The cells were removed from a T75 culture flask using Trypsin and centrifuged
for $5\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ at $300\text{\,}\mathrm{g}$.
After spinning down the cells, the excess culture medium was removed. If
needed, the cells were treated by either:

  * •

fixing the biological cells with $1\text{\,}\mathrm{mL}$ of a 4% formaldehyde
solution (ROTI Histofix, CARL ROTH) for 30 minutes at room temperature,
whereas formaldehyde functions as a protein crosslinker. Afterwards, the cells
were washed twice with PBS to remove the fixative. Or

  * •

with $300\text{\,}\mathrm{\SIUnitSymbolMicro L}$ cytochalasin D, which is an
inhibitor of actin filament polymerization, whereas the actin filaments play a
fundamental role in controlling cell shape and mechanical properties[24]. A
cytochalasin D (Sigma-Aldrich) stock solution of
$5\text{\,}\mathrm{mg}\text{\,}{\mathrm{mL}}^{-1}$ was diluted down to
$333\text{\,}\mathrm{nmol}\text{\,}{\mathrm{mL}}^{-1}$. The cells were
incubated for 30 minutes at room temperature

or were left untreated. $25\text{\,}\mathrm{\SIUnitSymbolMicro L}$ of the
$10.23\pm$0.13\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ PS particle solution
were added to the non-treated or treated cells. PBS was added to bring the
final volume to $1\text{\,}\mathrm{mL}$. The PS particles and biological
solution was loaded into a $1\text{\,}\mathrm{mL}$ syringe, and sequentially
flown into a pre-characterised acoustofluidic device. The flow was turned off
during the recording of the video sequence. Once a video sequence was
recorded, a fresh suspension of PS particles and biological cells were flown
into the device. The default frequency continuously applied was
$f=$1.42\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$\pm$2\text{\,}\mathrm{kHz}$$
as this corresponds to a $\lambda$ mode, i.e. two pressure nodes in the
device. This allows for a prealignment of the PS particles and biological
cells, and reduces the measurement error due to a minimized variability of the
starting position. The video recording was started, and the frequency was
changed from a $\lambda$ to a $\lambda/2$ mode found at
$f=$742.1\text{\,}\mathrm{kHz}$\pm$2\text{\,}\mathrm{kHz}$$.

### Image analysis

The generated videos were analysed using Fiji[46]. The videos were converted
to an Image Sequence. This allows for the selection of the Image Sequence
starting from the first movement of the PS particles and cells in solution
until the last frame where movement could be seen. The particles are
identified and separated into either PS or cells by using one of two methods:
The machine learning software ilastik[47] can be used. This allows for the
automatic detection and classification of particles of interest, but requires
a training set. For videos with low particle numbers, it is faster to track
the particles by hand by going through the Image Sequence frame by frame and
marking the particle of interest. The particle tracks are calculated using the
Fiji Plugin TrackMate.

### Data analysis

The biological cell and PS particle trajectories from TrackMate are imported
into a custom MATLAB script. Importing the known material parameters, seen in
table 1, of the PS particles and PBS and using the trajectory eq. 22, the
acoustic energy density $E_{ac}$ within the system can be calculated. Using
the extracted $E_{ac}$, the ACF of the biological cells in the fluid
suspension can be calculated.

### Numerical model

In the scope of our study, we developed a finite element method (FEM) model of
an eukaryotic mammalian cell whereas the structure is based on the model of
Heyden and Ortiz[15], consisting of nucleolus, nucleoplasm, and cytoplasm,
which are modeled as elastic solids. A graphical summary of the simplified
model of the cell is given in Figure 3.

The FEM model is built in COMSOL Multiphysics® v. 5.6 [48], following and
building upon the general approach of Baasch et al.[49]. The perturbed
equations 3, 4 and 5 of acoustic scattering and equations 10 and 11 of
acoustic streaming are solved consecutively. The scattering is solved in a
frequency domain study, while the streaming is computed in a stationary study,
using the results of the scattering study as source terms in equations 10 and
11. The solid domains are modeled via the Solid Mechanics interface, where the
appropriate material models are defined. The fluids in the frequency domain
study are modeled with a Thermoviscous Acoustics interface with the adiabatic
formulation, and afterwards, in the stationary study, with a Creeping Flow
interface.

At the first order, we impose the continuity of velocity and stress at the
fluid-solid interface. The fluid is modelled as unbounded, and far away from
the particle the first-order fields converge to the background fields, defined
by eq. 6. At the second order, the no-slip boundary condition is imposed on
the Lagrangian velocity of a fluid at the fluid-solid interface, in order to
compensate for the oscillations of the interface at the first order. The
Lagrangian velocity is defined as the summation of the Eulerian streaming
velocity $\left<\underline{v}_{2}\right>$ and the Stokes drift [50, 51]

$\underline{v}_{\mathrm{SD}}=\left\langle\left(\int{\underline{v}_{1}}\mathrm{d}t\bm{\cdot}\bm{\nabla}\right)\underline{v}_{1}\right\rangle,$
(24)

which consequently translates into the boundary condition

$\left\langle\underline{v}_{2}\right\rangle=-\underline{v}_{\mathrm{SD}}\quad\text{at
the interface}.$ (25)

The streaming due to the attenuation of the background standing wave in the
absence of the particle is negligible [38, 49] and was neglected in the
present study.

The FEM model is axisymmetric to limit the computational effort. In the
frequency domain study, the fluid domain surrounding the cell is surrounded by
a perfectly-matched layer (PML) that absorbs any incoming waves, in order to
avoid any influence of the outer wall on the cell. For the steady study, where
the acoustic microstreaming is computed, the PML is replaced by a no-slip
boundary condition, and the wall effects are avoided by ensuring that the
fluid domain is large enough.

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## Acknowledgements

We would like to thank the Balgrist Foundation and ETH Zurich for their most
generous financial support.

## Author contributions statement

C.H. conceived the experiment, designed the setup and produced the
microfluidic devices; C.H., D.B. and A.V. conducted the experiments; A.P.
created the numerical model; C.H. and A.P. analyzed the results and wrote the
draft of the manuscript; J.S., J.D. and U.S. reviewed the manuscript.

## Competing interests

The authors declare no competing interests.


# On some relations between the Perimeter, the Area and the Visual Angle of a
Convex Set

J.Bruna, J.Cufí and A.Reventós

###### Abstract

We establish some relations between the perimeter, the area and the visual
angle of a planar compact convex set. Our first result states that Crofton’s
formula is the unique universal formula relating the visual angle, length and
area. After that we give a characterization of convex sets of constant width
by means of the behaviour of its isotopic sets at infinity. Also for this
class of convex sets we prove that the existence of an isotopic circle is
enough to ensure that the considered set is a disc.

##  1 Introduction

In this paper we establish some relations between geometric quantities
associated to a planar compact convex set $K$ with boundary of class ${\cal
C}^{2}$. More precisely we consider the perimeter $L$, the area $F$ of $K$ and
the visual angle $w=w(P)$ of $K$ from a point $P\notin K$.

The starting point is the classical Crofton’s formula

$\displaystyle\int_{P\notin K}(w-\sin w)dP=\frac{L^{2}}{2}-\pi F.$ (1.1)

This equality easily follows from standard arguments of Integral Geometry, see
for instance [5]. Another approach to (1.1) is given in [1]. The natural
question arises whether separate formulas for $L^{2},F$ alone exist, or
equivalently, whether replacing $w-\sin w$ by some other function $f(w)$ one
gets a different linear combination of $L^{2},F$. For specific domains this is
indeed possible, for instance when $K$ is a disc one has several formulas of
this type, like

$L^{2}=\int_{P\notin K}\frac{4}{3}\sin^{3}w\,dP,\qquad 4\pi F=\int_{P\notin
K}\frac{\sin^{3}w}{\cos^{2}(w/2)}\,dP,$

see [5], p.59.

Our first result (Theorem 2.1) states, under suitable conditions, that if a
function $f$ satisfies

$\int_{P\notin K}f(\omega(P))dP=aL^{2}+bF$

for each compact convex set $K$, with $a,b\in\mathbb{R}$, two constants that
do not depend on $K$, then

$f(w)=\lambda(w-\sin w),$

for some constant $\lambda\in\mathbb{R}$. A consequence of this result is that
no universal formula exists giving say the area $F$ in terms of the visual
angle alone. We believe, though, that a formula like

$F=\int_{P\notin K}f(\omega(P))g(P)\,dP,$

with $f,g$ universal functions not depending on $K$ might exist.

By the result in [3], $K$ is completely determined by the visual angle $w(P)$
outside a big ball containing $K$. It is then natural to ask how specific
properties of $K$ can be read from this knowledge.

In our next results, we do so in terms of the assymptotic behaviour of $w(P)$
as $P$ goes to infinity. Now, it is easy to see that, denoting by
$w(R,\theta)$ the visual angle of $K$ from the point of polar coordinates
$(R,\theta)$, the quantity $R\,w(R,\theta)$ remains bounded and

$\lim_{R\to\infty}R\,w(R,\theta)=a(\theta),$ (1.2)

where $a(\theta)$ is the width of $K$ in the direction $\theta$, meaning that
the knowledge of the asymptotic behaviour of $w$ at infinity amounts to know
the width function. Whence, assymptotic statements can just involve quantities
that depend only on the width function. For instance, the perimeter of the
convex set, that is related to the width by the formula
$2L=\int_{0}^{2\pi}a(\theta)\,d\theta,$ can be known from the behaviour of the
visual angle at infinity by

$2L=\lim_{R\to\infty}R\int_{0}^{2\pi}w(R,\theta)\,d\theta,$ (1.3)

see Theorem 3.1 and Remark 1.

For central symmetric convex sets the area is determined by the width, and
this enable us to obtain the following formula in the spirit of (1.3),

$\lim_{R\to\infty}\int_{0}^{2\pi}R^{2}[w(R,\theta)^{2}-w_{\theta}(R,\theta)^{2}]\,d\theta=8F,$
(1.4)

where $w_{\theta}$ means partial derivative of $w$ with respect to $\theta$
(see Corollary 3.3 and Remark 1).

After that we deal with compact convex sets of constant width for which we
obtain two results. To state the first one we note that formula (1.2) says
that the convex set $K$ has constant width $a$ if and only if the visual angle
$w$ behaves like $a/R$ at infinity, which in turn roughly says that the
isotopic sets $C(\alpha)=\\{P:w(P)=\alpha\\}$ behave like the circles of
radius $R=\frac{a}{\alpha}$. Our result provides a precise quantitative
formulation of this fact by establishing that $C(\alpha)$ tends to a circle in
the sense that it tends to satisfy the equality in the isoperimetric
inequality. More precisely, if $L(\alpha),F(\alpha)$ denote respectively the
length and area of $C(\alpha)$, one has

$\lim_{\alpha\to 0}\frac{L(\alpha)^{2}}{4\pi F_{\alpha}}=1,$

if and only if $K$ has constant width, see Theorem 4.1.

The second result deals with isotopic sets that are actually circles, not just
assymtotically. Green’s ([2]) proved that if $K$ has an isotopic circle
$C_{\alpha}$, with $\alpha$ an irrational multiple of $\pi$, or
$\pi-\alpha=(m/n)\pi$, $m$ even, $m$ and $n$ relatively prime, then $K$ is a
disc. In general $K$ can have an isotopic circle without being a disc. Later
on Nitsche [4] proved that if $K$ has two concentric isotopic circles, then
$K$ is a disc. We prove in Theorem 5.1 that if $K$ has constant width and just
one isotopic circle, then it is a disc.

At the end we give the relationship between the area $F$ and the perimeter $L$
of a convex set and the radius of an isotopic circle, see (5.2) and (5.4). As
a consequence we obtain the inequalities

$F\leq\pi R^{2}\sin^{2}(\frac{\alpha}{2}),\quad L\leq 2\pi
R\sin(\frac{\alpha}{2}),$

where $R$ is the radius of the isotopic circle and $\alpha$ the constant
visual angle on this circle.

##  2 On Crofton’s Function

As we have said in the Introduction, the Crofton’s formula (1.1) is proven
with standard arguments of integral geometry, but it is also a direct
consequence of the general formula for integrating functions of the visual
angle given in [1]. Concretely, equation (16) in [1], says that for a
differentiable function $f:[0,\pi]\rightarrow\mathbb{R}$, satisfying
$f(w)=O(w^{3}),$ $w\to 0$, one has

$\displaystyle\int_{P\notin K}f(\omega)dP$ $\displaystyle=$
$\displaystyle-f(\pi)F+\frac{L^{2}}{2\pi}M(f)$ (2.1) $\displaystyle+$
$\displaystyle\pi\sum_{k\geq
2,even}c_{k}^{2}\left(M(f)+2\sum_{j=1,odd}^{k-1}\alpha_{j}\right)$
$\displaystyle+$ $\displaystyle\pi\sum_{k\geq
3,odd}c_{k}^{2}\left(-2\sum_{j=2,even}^{k-1}\alpha_{j}\right),$

with

$\alpha_{j}=\int_{0}^{\pi}f^{\prime}(\omega)j\cos(j\omega)d\omega,\qquad
M(f)=\int_{0}^{\pi}\frac{f^{\prime}(\omega)}{1-\cos\omega}d\omega,$

and $c_{k}^{2}=a_{k}^{2}+b_{k}^{2}$, where $a_{k},b_{k}$ are the Fourier
coeficients of the support function $p(\varphi)$ of the compact convex set
$K$. Notice that up to a constant, $\alpha_{j}$ is the $j$-th Fourier
coefficient of $f^{\prime}$ in the basis $\\{\cos jw\\}$. The above equality
for $f(w)=w-\sin w$ gives immediately Crofton’s formula.

We shall prove now that the function $w-\sin w$ is the only one that can
provide a Crofton’s type formula.

######  Theorem 2.1.

Let $f:[0,\pi]\longrightarrow\mathbb{R}$ be a differentiable function with
$f(w)=O(w^{3}),$ $w\to 0$, such that for every compact convex set $K$,

$\displaystyle\int_{P\notin K}f(\omega(P))dP=aL^{2}+bF,$ (2.2)

being $a,b\in\mathbb{R}$ some constants not depending on $K$, and $w(P)$ the
visual angle of $K$ from the point $P$. Then $f$ is, up to a constant factor
$\lambda$, the Crofton function $f(\omega)=\omega-\sin\omega$. In this case
$a=\lambda/2$, $b=-\pi\lambda$.

###### Proof.

We consider the family of convex sets given by the support functions

$\displaystyle p(\varphi)=1+t\cos(m\varphi),\qquad 0\leq\varphi\leq 2\pi,$
(2.3)

for $m\in\mathbb{N}$. The condition of convexity $p+p^{\prime\prime}>0$ is
satisfied if $0<t<\frac{1}{m^{2}-1}$.

Then, the perimeter $L$ and the area $F$ of these convex sets are

$\displaystyle L=\int_{0}^{2\pi}p\,d\varphi=2\pi,\qquad
F=\frac{1}{2}\int_{0}^{2\pi}(p^{2}-p^{\prime
2})\,d\varphi=\pi-\frac{\pi}{2}(m^{2}-1)t^{2}.$

Now we combine (2.1) and (2.2) to obtain, with $m$ even,

$\displaystyle a4\pi^{2}+b(\pi-\frac{\pi}{2}(m^{2}-1)t^{2})$ $\displaystyle=$
$\displaystyle-f(\pi)(\pi-\frac{\pi}{2}(m^{2}-1)t^{2})+2\pi M(f)$
$\displaystyle+$ $\displaystyle\pi
M(f)t^{2}+2\pi(\alpha_{1}+\alpha_{3}+\dots+\alpha_{m-1})t^{2}.$

Equalizing the coefficients of $t^{2}$ we obtain

$(m^{2}-1)(b+f(\pi))+2M(f)+4(\alpha_{1}+\alpha_{3}+\dots+\alpha_{m-1})=0.$

Changing $m$ by $m+2$ it follows

$((m+2)^{2}-1)(b+f(\pi))+2M(f)+4(\alpha_{1}+\alpha_{3}+\dots+\alpha_{m-1}+\alpha_{m+1})=0.$

Substracting these two last equalities we get for $m\geq 2$, $m$ even,

$b+f(\pi)=-\frac{\alpha_{m+1}}{m+1}=-\int_{0}^{\pi}f^{\prime}(w)\cos((m+1)w)\,dw.$

From Riemann-Lebesgue’s Lemma it follows that $b+f(\pi)=0$, and
$\alpha_{j}=0$, for $j$ odd, $j\geq 3$.

As well, for $m$ odd, we have from (2.1)

$\displaystyle a4\pi^{2}+b(\pi-\frac{\pi}{2}(m^{2}-1)t^{2})$ $\displaystyle=$
$\displaystyle-f(\pi)(\pi-\frac{\pi}{2}(m^{2}-1)t^{2})+2\pi M(f)$
$\displaystyle-$ $\displaystyle
2\pi(\alpha_{2}+\alpha_{4}+\dots+\alpha_{m-1})t^{2}.$

Equalizing the coefficients of $t^{2}$ and using that $b+f(\pi)=0$ we obtain

$\alpha_{2}+\alpha_{4}+\dots+\alpha_{m-1}=0,\quad m\geq 3$

and so $\alpha_{j}=0$, for $j$ even, $j\geq 2$.

Hence,

$f^{\prime}(\omega)=a_{0}+a_{1}\cos(\omega)$

and

$f(\omega)=a_{0}\,\omega+a_{1}\sin(\omega)+c,$

for some constants $a_{0},a_{1},c$.

But since $f(w)=O(w^{3}),$ $w\to 0$, we get $f(0)=f^{\prime}(0)=0$ and so
$c=0,\,a_{0}+a_{1}=0$. This proves the Theorem with $\lambda=a_{0}$.

∎

##  3 Behaviour of the visual angle at infinity

The goal of this section is to obtain information about the convex set
observing it from a point that goes to infinity.

First of all we will see that the perimeter of a convex set can be evaluated
by integrating the visual angle on circles of increasing radius.

The circle $C_{R}$ centered at the origin with radius $R$, can be parametrized
by means of the support function $p(\varphi)$ of $K$ in the following way. To
each value of $\varphi$ one associates the point $P(R,\varphi)$ given by the
intersection of $C_{R}$ with the half-straight line, taken in the direct
sense, of slope $\varphi+\pi/2$, and at a distance $p(\varphi)$ of the origin.

Then we have

######  Theorem 3.1.

Let $K$ be a compact convex set of perimeter $L$ and denote by $w(R,\varphi)$
the visual angle of $K$ from the point $P(R,\varphi)$. Then

$2L=\lim_{R\to\infty}R\int_{0}^{2\pi}w(R,\varphi)\,d\varphi.$

###### Proof.

The visual angle $w=w(R,\varphi)$ verifies the fundamental relation

$\arccos\frac{p(\varphi)}{R}+\arccos\frac{p(\varphi+\pi-w)}{R}=\pi-w(R,\varphi),\quad
0\leq\varphi\leq 2\pi,$

for every $R>0$ such that $C_{R}$ contains $K$, where $p(\varphi)$ is the
support function of $K$.

From this equation it follows

$\displaystyle p^{2}+p_{1}^{2}+2pp_{1}\cos(w)=R^{2}\sin^{2}(w),$ (3.1)

where $p=p(\varphi)$, $p_{1}=p(\varphi+\pi-w)$.

Then from (3.1) we have that $\lim_{R\to\infty}w(R,\varphi)=0$ and

$\displaystyle\lim_{R\to\infty}R\,w(R,\varphi)=\lim_{R\to\infty}R\sin(w(R,\varphi))=a(\varphi),\quad
0\leq\varphi\leq 2\pi,$ (3.2)

where $a(\varphi)=p(\varphi)+p(\varphi+\pi)$ is the width of $K$ in the
direction $\varphi$.

The limit in (3.2) is uniform in $\varphi$. In fact,
$w(R,\varphi)\leq\tilde{w}(R)$ where $\tilde{w}(R)$ is the visual angle of the
smallest circle centered at the origin and containing $K$ from a point at
distance $R$ from the origin. So $w(R,\varphi)$ tends to zero uniformly in
$\varphi$ when $R\to\infty$ and we deduce, from (3.1) and the uniform
continuity of $p(\varphi)$, that the convergence in (3.2) is uniform.

Then the result follows by integration in (3.2). ∎

Motivated by Theorem 3.1 we can ask if there is an analogous result involving
the area of $K$. We can answer this question for central symmetric compact
convex sets. The basic result is

######  Theorem 3.2.

Let $K$ be a compact convex set and let $w=w(R,\varphi)$ be the visual angle
of $K$ at the point $P(R,\varphi)$. Denote by
$a(\varphi)=p(\varphi)+p(\varphi+\pi)$, where $p(\varphi)$ is the support
function of $K$, the width of $K$ in the direction $\varphi$. Then

$\lim_{R\to\infty}\int_{0}^{2\pi}R^{2}(w(R,\varphi)^{2}-w_{\varphi}(R,\varphi)^{2})\,d\varphi=\int_{0}^{2\pi}(a(\varphi)^{2}-a^{\prime}(\varphi)^{2})\,d\varphi,$

where $w_{\varphi}$ denotes the derivative with respect to $\varphi$.

###### Proof.

We begin by proving that

$\displaystyle\lim_{R\to\infty}R\,w_{\varphi}(R,\varphi)=a^{\prime}(\varphi),$
(3.3)

uniformly on $\varphi$. In fact, differentiation of equation (3.1) with
respect to $\varphi$ gives

$Rw_{\varphi}=\frac{R(pp^{\prime}+p_{1}p^{\prime}_{1}+(p^{\prime}p_{1}+pp^{\prime}_{1})\cos(w))}{R^{2}\sin(w)\cos(w)+p_{1}p^{\prime}_{1}+pp^{\prime}_{1}+pp_{1}}.$

Taking limits and according (3.2) we have

$\lim_{R\to\infty}Rw_{\varphi}(R,\varphi)=\lim_{R\to\infty}\frac{Ra(\varphi)a^{\prime}(\varphi)}{Ra(\varphi)+p^{\prime}(\varphi+\pi)a(\varphi)+p(\varphi)p(\varphi+\pi)}=a^{\prime}(\varphi).$

Since the convergence in (3.2) is uniform and the functions $p(\varphi)$ and
$p^{\prime}(\varphi)$ are uniformly continuous, the convergence in (3.3) is
also uniform.

As a consequence

$\lim_{R\to\infty}R^{2}\int_{0}^{2\pi}(w(R,\varphi)^{2}-w_{\varphi}(R,\varphi)^{2})\,d\varphi=\int_{0}^{2\pi}(a(\varphi)^{2}-a^{\prime}(\varphi)^{2})\,d\varphi,$

as we wanted to prove. ∎

We note that the integral in the right-hand side of the above equality is two
times the area of the convex set having $a(\varphi)$ as its support function.

As a consequence of the above Theorem we obtain for the especial case of
central symmetric convex sets the following result.

######  Corollary 3.3.

Let $K$ be a compact convex set symmetric with respect to the origin and of
area $F$. With the notation in Theorem 3.2 one has

$F=\frac{1}{8}\lim_{R\to\infty}\int_{0}^{2\pi}R^{2}(w(R,\varphi)^{2}-w_{\varphi}(R,\varphi)^{2})\,d\varphi.$

###### Proof.

By the central symmetry of $K$ one has $p(\varphi)=p(\varphi+\pi)$,
$0\leq\varphi\leq 2\pi$, and so $a(\varphi)=2p(\varphi)$ and so
$a^{2}-a^{\prime 2}=4(p^{2}-p^{\prime 2})$ that, integrating with respect to
$\varphi$ and according Theorem 3.2, gives the result. ∎

######  Remark 1.

If we denote by $w(R,\theta)$ the visual angle of $K$ from the point of polar
coordinates $(R,\theta)$ we can consider

$\int_{0}^{2\pi}Rw(R,\theta)\,d\theta$

which is in general diferent from

$\int_{0}^{2\pi}Rw(R,\varphi)\,d\varphi.$

The relation between $\theta$ and $\varphi$ is
$\theta=\varphi+\arccos\frac{p(\varphi)}{R}$.

So one has

$\displaystyle\int_{0}^{2\pi}R\omega(R,\theta)\,d\theta$ $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}Rw(R,\varphi)(1-\frac{p^{\prime}(\varphi)}{\sqrt{R^{2}-p(\varphi)^{2}}})d\varphi.$

As a consequence, Theorem 3.1 gives

$2L=\lim_{R\to\infty}\int_{0}^{2\pi}Rw(R,\theta)d\theta,$

as stated in the introduction and analogously Corollary 3.3 gives

$\lim_{R\to\infty}\int_{0}^{2\pi}R^{2}[w(R,\theta)^{2}-w_{\theta}(R,\theta)^{2}]\,d\theta=8F,$

so that in these results we can use both polar coordinates $(R,\theta)$ or the
coordinates $(R,\varphi)$ associated to the convex set.

##  4 A characterization of convex sets of constant width by means of isotopic
sets

Given a compact convex set $K$ we denote by $C_{\alpha}$ the (isotopic) set of
points in the plane from which $K$ is seen with angle $\alpha$. In view of the
isoperimetric inequality we will say that the sets $C_{\alpha}$ tend to a
circle, as $\alpha\to 0$, if

$\lim_{\alpha\to 0}\frac{L(\alpha)^{2}}{4\pi F_{\alpha}}=1,$

where $L(\alpha)$ is the length of $C_{\alpha}$ and $F(\alpha)$ is the area
enclosed by $C_{\alpha}$.

######  Theorem 4.1.

Let $K$ be a compact convex set. Then the isotopic sets $C_{\alpha}$ of $K$
tend, as $\alpha\to 0$, to a circle if and only if $K$ is of constant width.

###### Proof.

It is known, see for instance [1], that the points $(X,Y)$ in $C_{\alpha}$ can
be parametrized by $\varphi$ as follows

$\displaystyle X$ $\displaystyle=$
$\displaystyle-\frac{1}{\sin\alpha}(p\sin(\varphi-\alpha)+p_{1}\sin\varphi)$
$\displaystyle Y$ $\displaystyle=$
$\displaystyle\frac{1}{\sin\alpha}(p\cos(\varphi-\alpha)+p_{1}\cos\varphi)$

with $p=p(\varphi)$ the support function of $K$, and
$p_{1}=p(\varphi+\pi-\alpha)$.

Hence, the length $L(\alpha)$ of $C_{\alpha}$ is given by

$L(\alpha)=\int_{0}^{2\pi}\sqrt{X^{\prime 2}+Y^{\prime 2}}\,d\varphi.$

A direct computation shows that

$\displaystyle
L(\alpha)=\frac{1}{\sin(\alpha)}\int_{0}^{2\pi}\sqrt{\Delta(\varphi,\alpha)}\,d\varphi$
(4.1)

with

$\Delta(\varphi,\alpha)=p^{2}+p_{1}^{2}+p^{\prime 2}+p_{1}^{\prime
2}+2(pp_{1}+p^{\prime}p_{1}^{\prime})\cos(\alpha)+2(pp_{1}^{\prime}-p^{\prime}p_{1})\sin(\alpha).$

So we have

$\displaystyle\lim_{\alpha\rightarrow 0}L(\alpha)\sin(\alpha)$
$\displaystyle=$
$\displaystyle\int_{0}^{2\pi}\sqrt{(p(\varphi)+p(\varphi+\pi))^{2}+(p^{\prime}(\varphi)+p^{\prime}(\varphi+\pi))^{2})}\,d\varphi$
$\displaystyle=$
$\displaystyle\int_{0}^{2\pi}\sqrt{a(\varphi)^{2}+a^{\prime}(\varphi)^{2}}\,d\varphi,$

where $a(\varphi)$ is the width of $K$ in the direction $\varphi$.

On the other hand, the area $F(\alpha)$ enclosed by $C_{\alpha}$ satisfies
(see [1])

$\lim_{\alpha\to 0}(F(\alpha)\sin^{2}\alpha)=\frac{L^{2}}{\pi}+2\pi\sum_{k\geq
2,even}c_{k}^{2}.$

Thus

$\displaystyle\lim_{\alpha\to 0}\frac{L(\alpha)^{2}}{4\pi
F(\alpha)}=\lim_{\alpha\to 0}\frac{L(\alpha)^{2}\sin^{2}(\alpha)}{4\pi
F(\alpha)\sin^{2}(\alpha)}=\frac{[\int_{0}^{2\pi}\sqrt{a(\varphi)^{2}+a^{\prime}(\varphi)^{2}}\,d\varphi]^{2}}{4L^{2}+8\pi^{2}\sum_{k\,even}c_{k}^{2}}.$

If $K$ is a convex set of constant width $a$, $L=\pi a$ and $c_{k}=0$ for $k$
even. Hence

$\displaystyle\lim_{\alpha\to 0}\frac{L(\alpha)^{2}}{4\pi F(\alpha)}=1,$

that proves one of the implications of the theorem.

Before looking at the converse, let us check that the width $a(\varphi)$
satisfies

$2\pi\int_{0}^{2\pi}a(\varphi)^{2}\,d\varphi=4L^{2}+8\pi^{2}\sum_{k\,even}c_{k}^{2}.$

Indeed,

$\displaystyle 2\pi\int_{0}^{2\pi}a(\varphi)^{2}\,d\varphi$ $\displaystyle=$
$\displaystyle
4\pi\int_{0}^{2\pi}p(\varphi)^{2}\,d\varphi+4\pi\int_{0}^{2\pi}p(\varphi)p(\varphi+\pi)\,d\varphi$
$\displaystyle=$ $\displaystyle 4\pi(2\pi
a_{0}^{2}+\pi\sum_{k}c_{k}^{2})+4\pi(2\pi
a_{0}^{2}+\pi\sum_{k}(-1)^{k}c_{k}^{2})$ $\displaystyle=$ $\displaystyle
4L^{2}+8\pi^{2}\sum_{k\,even}c_{k}^{2}.$

Hence

$\lim_{\alpha\to 0}\frac{L(\alpha)^{2}}{4\pi
F(\alpha)}=\frac{[\int_{0}^{2\pi}\sqrt{a^{2}+a^{\prime
2}}\,d\varphi]^{2}}{2\pi\int_{0}^{2\pi}a^{2}\,d\varphi}.$

Now assuming that the above limit is equal to $1$ we have

$\displaystyle\left[\int_{0}^{2\pi}\sqrt{a^{2}+a^{\prime
2}}\,d\varphi\right]^{2}=2\pi\int_{0}^{2\pi}a^{2}\,d\varphi.$ (4.2)

But this equality implies that $a(\varphi)$ is constant. In fact (4.2) says
that equality holds in the isoperimetric inequality applied to the curve given
in polar coordinates by $r=a(\varphi)$.

∎

##  5 Isotopic circles

In this section we consider the particular case in which the isotopic set
$C_{\alpha}$ of a compact convex set $K$ is a circle. We will say that
$C_{\alpha}$ is an isotopic circle of $K$.

It is known that if a compact convex set $K$ has two concentric isotopic
circles, then $K$ is a disc, see [4]. The existence of only one isotopic
circle is not enought to conclude that $K$ is a disc, for instance all the
ellipses have an isotopic circle with $\alpha=\pi/2$, see [2].

In fact we can provide a family of compact convex sets having an isotopic
circle with visual angle $\alpha=\pi/2$ and different from discs or ellipses.
The examples given in [2] do not have this property.

To construct this family we remark that, from (3.1), it follows that $K$ has
an isotopic circle $C_{\pi/2}$ of radius $R$ if

$p(\varphi)^{2}+p(\varphi+\pi/2)=R^{2}.$

If we write the Fourier series for the function $p(\varphi)^{2}$ as

$p(\varphi)^{2}=\sum_{-\infty}^{\infty}c_{k}e^{ik\varphi},$

it follows

$p(\varphi)^{2}+p(\varphi+\pi/2)^{2}=\sum_{-\infty}^{\infty}c_{n}e^{in\varphi}(1+e^{in\pi/2}),$

so that this quantity is constant if and only if

$c_{k}=0,\quad k\neq 2+4m,\quad m\mbox{ integer.}$

Then, any positive $2\pi$-periodic function $p(\varphi)$ with
$p+p^{\prime\prime}>0$, such that the Fourier series of $p(\varphi)^{2}$ has
only coefficients $c_{k}$ with $k$ congruent to $2$ module $4$, will give rise
to a convex set seen from angle $\pi/2$ from a circle. For instance take

$p(\varphi)=\sqrt{15+9\cos^{2}(\varphi)+4\sin^{2}(\varphi)+\cos(6\varphi)}.$

###  5.1 Convex sets of constant width with an isotopic circle

For compact convex sets of constant width the above quoted result in [4],
about convex sets with two isotopic circles, can be improved. Concretely we
have

######  Theorem 5.1.

If a compact convex set $K$ of constant width has an isotopic circle, then $K$
is a disc.

###### Proof.

Assume that $K$ has an isotopic circle of radius $R$ with visual angle
$\alpha$. Then equation (3.1) applied with angle $\varphi+\pi$ instead of
$\varphi$ gives

$p(\varphi+\pi)^{2}+p(\varphi-\alpha)^{2}+2p(\varphi+\pi)p(\varphi-\alpha)\cos(\alpha)=C,$

with $C$ some constant. By the condition of constant width,
$p(\varphi)+p(\varphi+\pi)=a$, one has

$(a-p(\varphi))^{2}+p(\varphi-\alpha)^{2}+2(a-p(\varphi))p(\varphi-\alpha)\cos(\alpha)=C.$

Changing the constant one can write

$p(\varphi)^{2}-2ap(\varphi)+p(\varphi-\alpha)^{2}+2ap(\varphi-\alpha)\cos(\alpha)-2p(\varphi)p(\varphi-\alpha)\cos(\alpha)=C.$

Replacing $\varphi$ by $\varphi-\alpha$ and taking into account that
$p(\varphi)$ is $2\alpha$-periodic (see [2]) it follows

$p(\varphi-\alpha)^{2}-2ap(\varphi-\alpha)+p(\varphi)^{2}+2ap(\varphi)\cos(\alpha)-2p(\varphi-\alpha)p(\varphi)\cos(\alpha)=C.$

Substracting the last two equalities it follows

$2a\bigg{(}p(\varphi-\alpha)-p(\varphi)+(p(\varphi-\alpha)-p(\varphi))\cos(\alpha)\bigg{)}=0$

and so

$p(\varphi)=p(\varphi-\alpha),$

that is, $p(\varphi)$ is $\alpha$-periodic.

Then equation (3.1) reads

$p(\varphi)^{2}+p(\varphi+\pi)^{2}+2p(\varphi)p(\varphi+\pi)\cos(\alpha)=C,$

that together with $p(\varphi)+p(\varphi+\pi)=a$ gives

$p(\varphi)^{2}+p(\varphi)^{2}-2ap(\varphi)+2ap(\varphi)\cos(\alpha)-2p(\varphi)p(\varphi)\cos(\alpha)=C,$

or

$(2p(\varphi)^{2}-2ap(\varphi))(1-\cos(\alpha))=C.$

In conclusion $p(\varphi)$ is, for $0\leq\varphi\leq 2\pi$, a solution of a
second degree equation $x^{2}+mx+n=0$, with $m,n\in\mathbb{R}$ and hence it is
constant and $K$ is a disc. ∎

In fact this result can be thought as a consequence of Nitsche’s result that
assumes the existence of two isotopic circles, because in the case of constant
width one of the isotopic circles is given at the infinity by Theorem 4.1.

###  5.2 Relationship between the area of a convex set and the radius of an
isotopic circle

We compare the area of the convex set $K$ with the area enclosed by an
isotopic circle of $K$.

######  Theorem 5.2.

Let $K$ be a compact convex set of area $F$ that has an isotopic circle
$C_{\alpha}$ of radius $R$. Then

$F\leq F_{R}\sin^{2}(\frac{\alpha}{2}),$

with $F_{R}=\pi R^{2}$.

###### Proof.

In [1] it is proved the equality

$\displaystyle
F(\alpha)\sin^{2}(\frac{\alpha}{2})=F+\frac{\pi}{4\cos^{2}(\frac{\alpha}{2})}\sum_{k\geq
2}\big{(}2(k^{2}+1)\cos^{2}(\frac{\alpha}{2})+g_{k}(\alpha)\big{)}c_{k}^{2}$
(5.1)

expressing the area $F(\alpha)$ enclosed by the isotopic set $C_{\alpha}$ of a
compact convex set $K$ in terms of the area $F$ of $K$, the Fourier
coefficients $a_{k},b_{k}$ of the support function of $K$
($c_{k}^{2}=a_{k}^{2}+b_{k}^{2}$), and Hurwitz’s functions $g_{k}(\alpha)$
given by

$g_{k}(\alpha)=1+\frac{(-1)^{k}}{2}((k+1)\cos(k-1)\alpha-(k-1)\cos(k+1)\alpha).$

It is known from [2] that when $C_{\alpha}$ is a circle and $K$ is not a disc
then $\alpha=\pi-\frac{m}{n}\pi$ with $(m,n)=1$, and $m$ odd. We will assume
from now on that $K$ is not a disk.

In this case $p(\varphi)$ is $(2\pi/n)$-periodic and so $k=\mu n$,
$\mu\in\mathbb{N}$, and $g_{k}(\alpha)=1+(-1)^{\mu}\cos(\alpha)$.

So, equality (5.1) says

$\displaystyle F_{R}\sin^{2}(\frac{\alpha}{2})=$ (5.2) $\displaystyle
F+\frac{\pi}{4\cos^{2}\frac{\alpha}{2}}\sum_{\mu}(2(\mu^{2}n^{2}-1)\cos^{2}(\frac{\alpha}{2})+1+(-1)^{\mu}\cos(\alpha))c_{n\mu}^{2}$

and since the coefficient of $c_{n\mu}^{2}$ is positive for each $\mu$, we
obtain the desired inequality.

∎

###  5.3 Relationship between the perimeter of a convex set and the radius of
an isotopic circle

Now we compare the perimeter of a convex set with the length of one of its
isotopic circles.

######  Theorem 5.3.

Let $K$ be a compact convex set of perimeter $L$ that has an isotopic circle
$C_{\alpha}$ of radius $R$. Then

$L\leq L_{R}\sin(\frac{\alpha}{2}).$

with $L_{R}=2\pi R$.

###### Proof.

Let us consider the Fourier series of the functions $p=p(\varphi)$ and
$p_{1}=p(\varphi+\pi-\alpha)$ given by

$\displaystyle p$ $\displaystyle=$ $\displaystyle a_{0}+\sum_{k\geq
1}a_{k}\cos(k\varphi)+b_{k}\sin(k\varphi)$ $\displaystyle p_{1}$
$\displaystyle=$ $\displaystyle a_{0}+\sum_{k\geq
1}A_{k}\cos(k\varphi)+B_{k}\sin(k\varphi)$

with

$\displaystyle A_{k}$ $\displaystyle=$
$\displaystyle(-1)^{k+1}(-a_{k}\cos(k\alpha)+b_{k}\sin(k\alpha))$
$\displaystyle B_{k}$ $\displaystyle=$
$\displaystyle(-1)^{k+1}(-a_{k}\sin(k\alpha)-b_{k}\cos(k\alpha)).$

Then

$\int_{0}^{2\pi}pp_{1}\,d\varphi=\int_{0}^{2\pi}\bigg{(}a_{0}^{2}+\sum_{k\geq
1}(a_{k}A_{k}\cos^{2}(k\varphi)+b_{k}B_{k}\sin^{2}(k\varphi))\bigg{)}\,d\varphi,$

and substituting the given values of $A_{k},B_{k}$, it follows

$\displaystyle\int_{0}^{2\pi}pp_{1}\,d\varphi=\frac{L^{2}}{2\pi}-\pi\sum_{k\geq
1}(-1)^{k+1}c_{k}^{2}\cos(k\alpha).$ (5.3)

Assuming that $C_{\alpha}$ is an isotopic circle of radius $R$, integrating
the equality (3.1) on this circle and taking into account (5.3), we obtain

$\displaystyle 2L^{2}(1+\cos(\alpha))+4\pi^{2}\sum_{k\geq
1}(1+(-1)^{k}\cos(\alpha)\cos(k\alpha))c_{k}^{2}=L_{R}^{2}\sin^{2}(\alpha),$

where $L_{R}=2\pi R$.

Considering, as in the previous section, that $\alpha=\pi-\frac{m}{n}\pi$ with
$(m,n)=1$ and $m$ odd, the above equation reads

$\displaystyle
L^{2}+2\pi^{2}\sum_{\mu,even}c_{n\mu}^{2}+2\pi^{2}\tan^{2}(\frac{\alpha}{2})\sum_{\mu,odd}c_{n\mu}^{2}=L_{R}^{2}\sin^{2}(\frac{\alpha}{2}).$
(5.4)

In particular we have the inequality

$L\leq L_{R}\sin(\frac{\alpha}{2}).$

∎

Acknowledgement. The authors are grateful to A. Gasull for various
conversations on the subject that have contributed to the proof of Theorem
4.1. We also thank E. Gallego for his useful comments.

## References

  * [1] J. Cufí, E. Gallego, and A. Reventós, _On the integral formulas of Crofton and Hurwitz relative to the visual angle of a convex set_ , Mathematika 65 (2019), no. 4, 874–896.
  * [2] J.W. Green, _Sets subtending a constant angle on a circle_ , Duke Math. J. 17 (1950), 263–267.
  * [3] Á. Kurusa, _You can recognize the shape of a figure from its shadows!_ , Geom. Dedicata 59 (1996), no. 2, 113–125.
  * [4] J. C. C. Nitsche, _Isoptic characterization of a circle (proof of a conjecture of M. S. Klamkin)_ , Amer. Math. Monthly 97 (1990), no. 1, 45–47.
  * [5] L.A. Santaló, _Integral Geometry and Geometric Probability_ , second ed., Cambridge University Press, Cambridge, 2004.

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193
Bellaterra, Barcelona, Catalonia.

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