# Numerical simulation of the coalescence-induced polymeric droplet jumping on
superhydrophobic surfaces

Kazem Bazesefidpar<EMAIL_ADDRESS>Luca Brandt  Outi Tammisola  SeRC
(Swedish e-Science Research Centre) and FLOW, Dept. of Engineering Mechanics,
KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden  Dept. of Energy
and Process Engineering, Norwegian University of Science andTechnology (NTNU),
Trondheim, Norway

###### Abstract

Self-propelled jumping of two polymeric droplets on superhydrophobic surfaces
is investigated by three-dimensional direct numerical simulations. Two
identical droplets of a viscoelastic fluid slide, meet and coalesce on a
surface with contact angle 180 degrees. The droplets are modelled by the
Giesekus constitutive equation, introducing both viscoelasticity and a shear-
thinning effects. The Cahn-Hilliard Phase-Field method is used to capture the
droplet interface. The simulations capture the spontaneous coalescence and
jumping of the droplets. The effect of elasticity and shear-thinning on the
coalescence and jumping is investigated at capillary–inertial and viscous
regimes. The results reveal that the elasticity of the droplet changes the
known capillary–inertial velocity scaling of the Newtonian drops at large
Ohnesorge numbers; the resulting viscoelastic droplet jumps from the surface
at larger Ohnesorge numbers than a Newtonian drop, when elasticity gives rise
to visible shape oscillations of the merged droplet. The numerical results
show that polymer chains are stretched during the coalescence and prior to the
departure of two drops, and the resulting elastic stresses at the interface
induce the jumping of the liquid out of the surface. This study shows that
viscoelasticity, typical of many biological and industrial applications,
affects the droplet behaviour on superhydrophobic and self-cleaning surfaces.

###### keywords:

coalescence-induced droplet jumping, Viscoelasticity, jumping velocity,
superhydrophobic surface, Diffuse-interface method

##  1 Introduction

When two droplets coalesce, the total surface area decreases. Hence, surface
energy is released during this process. If the two droplets are far from a
wall, the new bigger drop oscillates symmetrically until the released surface
energy has been dissipated by viscosity. However, when two drops of micro- or
nanometer size coalesce on a superhydrophobic surface, the presence of a
repellent wall breaks the vertical symmetry and the resulting droplet propels
in the direction perpendicular to the wall [2]. Coalescence-induced jumping
has been reported on a variety of natural repellent surfaces such as cicada,
lacewings [30] and gecko skin [29], and can be exploited in a variety of
applications such as anti-icing [36] and self-cleaning surfaces [29, 30], and
to control heat transfer [10]. Several researchers have studied the different
aspects of the coalescence-induced droplet jumping numerically and
experimentally, including the basic mechanism of the two equal-sized drop
self-propelled jumping [19], and the effects of droplet size mismatch [28,
27], droplet initial velocity [14, 18], surface topology [26, 24, 34, 20, 23],
surrounding gas properties [11, 32, 33], and surface wettability [6]. A few
main results are outlined in the following.

When two equal-sized static drops coalesce on a superhydrophobic surface,
their total surface area decreases. This implies that surface energy is
released and converted into viscous dissipation and kinetic energy, in a
proportion determined by the Ohnesorge number, which represents the ratio
between viscous and capillary-inertial forces. At large Ohnesorge numbers,
corresponding to the viscous regime, the kinetic energy is completely absorbed
by viscous forces, preventing the jumping of the merged droplet [19]. Even at
small Ohnesorge numbers, corresponding to the capillary–inertial regime, only
less than 4 % of the released surface energy converts to vertical
translational kinetic energy, which nevertheless causes the jumping of the
merged droplet. The conversion rate of surface energy into kinetic energy
reduces when the droplets are of unequal sizes, due to the strong asymmetric
flow [28, 27]. The merged droplet attains an asymmetric shape and jumps with
an oblique angle when one of the two droplets has an initial velocity;
moreover, the jumping velocity of the merged droplet increases significantly
above a critical initial velocity [18].

In addition, macrostructures on the surface affect the jumping velocity and
energy transfer rates significantly. The jumping velocity and the conversion
efficiency of surface energy to kinetic energy decrease if the lower contour
of the merging drop falls between the gap of two rectangular grooves, whereas
both jumping velocity and energy conversion increase when the liquid bridge
expands on a triangular prism structure [26]. The critical Ohnesorge number
for droplet jumping depends on both the surface wettability and the ambient
fluid properties such as density and viscosity; in particular, a larger
density contrast between the ambient and drops will cause the merged droplet
to jump higher [11].

In very recent experiments, the effect of the drops’ elasticity on the
coalescence process was studied for both freely suspended drops and sessile
drops with radius $O(1)$ micrometer on the hydrophobic surfaces [8]. They
found that elasticity enhances the curvature of connecting bridge between two
merging drops, and polymer stresses remain confined in a small region around
the liquid bridge between the coalescing drops. However, the induced elastic
stresses were found to be insufficient to alter the temporal evolution of the
bridge in the capillary–inertial regime, and hence elasticity did not change
the flow regime.

In the present work, we perform numerical simulations to study the effects of
the non-Newtonian viscoelastic properties of two equal-sized static droplets
on the coalescence-induced droplet jumping at large Ohnesorge numbers. Our
studies extend from the viscous-capillary to the inertial-capillary regime,
and in the former case we do observe prominent changes due to elasticity. We
use the Cahn-Hilliard Phase-Field method for capturing the interface between
the two phases, and the Giesekus constitutive equation to model the
viscoelasticity of the drops. First, the role of elasticity is investigated by
comparing the vertical velocity and different components of energy for a
Newtonian and Oldroyd-B droplet at the same Ohnesorge number based on the same
zero shear viscosity, while the influence of the liquid shear-thinning
rheology is examined by using Giesekus model.

##  2 Governing equations and Numerical methods

The numerical method used in this work has been described in detail in
Bazesefidpar et al. [3], so we only give a brief outline here. We consider two
immiscible fluids with different densities and viscosities. The outer fluid is
Newtonian with viscosity $\mu_{n}$, whereas the droplets consist of a Giesekus
fluid with solvent viscosity $\mu_{s}$, polymeric viscosity $\mu_{p}$, and the
other non-Newtonian rheological properties as below. To distinguish between
the phases, we introduce a phase-field variable, where $\phi=\pm 1$ in the
bulk fluids and $\phi=0$ at the fluid/fluid interface. This problem can be
modelled with the following coupled equations [35, 1]:

$\displaystyle\rho(\frac{\partial{\mathbf{u}}}{\partial{t}}+({\mathbf{u}}\cdot\nabla){\mathbf{u}})+{\mathbf{J}}\cdot\nabla{\mathbf{u}}=-\nabla{p}+\nabla\cdot{\boldsymbol{\tau}}+\nabla\cdot\mu_{s}(\nabla{{\mathbf{u}}}+\nabla{{\mathbf{u}}^{T}})+G\nabla\phi,$
(1) $\displaystyle\nabla\cdot{\mathbf{u}}=0,$ (2)

the Cahn-Hilliard model:

$\displaystyle\frac{\partial{\phi}}{\partial{t}}+\nabla\cdot({{\mathbf{u}}\phi})=\nabla\cdot(M\nabla
G),$ (3) $\displaystyle
G=\lambda(-\nabla^{2}{\phi}+\frac{1}{\eta^{2}}\phi(\phi^{2}-1)),$ (4)

and the Giesekus constitutive model:

$\displaystyle\boldsymbol{\tau}_{p}+\lambda_{H}(\frac{\partial{\boldsymbol{\tau}_{p}}}{\partial{t}}+{\mathbf{u}}\cdot\nabla{\mathbf{\boldsymbol{\tau}_{p}}}-\boldsymbol{\tau}_{p}\nabla{{\mathbf{u}}}-\nabla{{\mathbf{u}}^{T}}\boldsymbol{\tau}_{p})+\frac{\alpha\lambda_{H}}{\mu_{p}}(\boldsymbol{\tau}_{p}\cdot\boldsymbol{\tau}_{p})$
$\displaystyle=$
$\displaystyle\mu_{p}(\nabla{{\mathbf{u}}}+\nabla{{\mathbf{u}}^{T}}),\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
(5)

$\displaystyle\boldsymbol{\tau}=\frac{(1+\phi)}{2}\boldsymbol{\tau}_{p},$ (6)

In the above equations, $\mathbf{u}(\mathbf{x},t)$ is the velocity vector,
$p(\mathbf{x},t)$ is the pressure, and $\boldsymbol{\tau}(\mathbf{x},t)$ is
the extra stress due to the polymers, equal to $\boldsymbol{\tau}_{p}$ inside
the droplet and $0$ outside (see eq. 6). In the Cahn-Hilliard equation, $G$ is
the chemical potential, $M$ is the mobility parameter, and $\eta$ is the
capillary width of the interface. In the Giesekus model,
$\boldsymbol{\tau}_{p}$ is the polymer stress, $\lambda_{H}$ is the polymer
relaxation time, $\alpha$ is the Giesekus mobility parameter, and the
polymeric retardation time can be related to the polymeric relaxation time by
$\lambda_{r}=\frac{\mu_{s}}{\mu_{s}+\mu_{p}}\lambda_{H}$. In Eq. (4),
$\lambda$ is the mixing energy density, and it is related to the surface
tension in the sharp-interface limit [35] by:

$\displaystyle\sigma=\frac{2\sqrt{2}}{3}\frac{\lambda}{\eta}$ (7)

Fluid $1$ indicates the droplet phase and fluid $2$ represents the surrounding
fluid (air). The density $\rho$ and the dynamic viscosity $\mu$ fields are
expressed using the phase-field variable as:

$\displaystyle\rho=\frac{(1+\phi)}{2}{\rho_{1}}+\frac{(1-\phi)}{2}{\rho_{2}},$
(8)
$\displaystyle\mu=\frac{(1+\phi)}{2}{\mu_{s_{1}}}+\frac{(1-\phi)}{2}{\mu_{s_{2}}},$
(9)

The total viscosity of the non-Newtonian phase is $\mu_{t}=\mu_{s}+\mu_{p}$.
The density satisfies the following relation[1]

$\displaystyle\frac{\partial{\rho}}{\partial{t}}+\nabla\cdot{\rho\mathbf{u}}=-\nabla\cdot\mathbf{J},$
(10)

where $\mathbf{J}=-\frac{(\rho_{1}-\rho_{2})}{2}M\mathbf{\nabla}{G}$. Boundary
conditions imposed on the substrate are, following Jacqmin [16], Qian et al.
[25], the no-slip boundary condition for the velocities:

$\displaystyle\boldsymbol{u}=\mathbf{0},$ (11)

and the static contact angle $\theta_{s}$ for the phase-field variable:

$\displaystyle{\mathbf{n}}\cdot\nabla{\phi}+\frac{1}{\lambda}f_{w}^{\prime}(\phi)=0,$
(12) $\displaystyle
f_{w}(\phi)=\sigma\cos(\theta_{s})\frac{\phi(\phi^{2}-3)}{4}+\frac{(\sigma_{w_{1}}+\sigma_{w_{2}})}{2},$
(13)

where $\mathbf{n}$ is the outward pointing normal vector to the boundary, and
$f_{w}(\phi)$ is a function describing the fluid–solid interfacial tension.
A second-order accurate scheme is employed for the temporal discretization of
Eq. (3) and (1) while a semi-implicit splitting scheme is used to treat the
linear parts implicitly and the non-linear parts explicitly [9]. To avoid the
High-Weissenberg number problem (HWNP), the log-conformation reformulation
(LCR) of equation Eq. (5) [12, 13] is used and advanced in time by a second-
order total variation diminishing (TVD) Runge-Kutta method [15]. Finally, we
use second-order central differences to approximate spatial derivatives,
except for the advection terms in Eq. (3) and (5), where the fifth-order
WENO-Z is used to improve stability and accuracy [4]

Figure 1: Sketch of the chosen computational domain $\Omega_{1}$. Figure 2:
Average velocity integrated over the droplet volume in time, for the chosen
domain (stars) and a bigger domain $\Omega_{2}$ (solid line), to show
independence of the domain size. The parameters are $Oh=0.0076$ and $De=10$.
Figure 3: The evaluation of average velocity of the merged viscoelastic
droplet for different $Cn$ numbers.

##  3 Physical model and computational domain

We consider two equal-sized initially static viscoelastic drops touching a
homogeneous surface (Fig. 1) with a static contact angle of $180^{o}$. When
two adjacent droplets coalesce on a superhydrophobic surface, the formed
liquid bridge impinges on the substrate, and the merged droplet may jump above
the substrate. Here, gravity is neglected, because the droplet radius is
assumed to be much smaller than the capillary length and therefore capillary
forces are expected to dominate.
The capillary-inertial velocity is chosen as the velocity scale [2]
$u_{ci}=\sqrt{\sigma/\left(\rho_{1}r_{0}\right)}$, and the droplets initial
radius as the length scale. This gives rise to seven nondimensional numbers.
Firstly, the Ohnesorge number $Oh=\left(\mu_{1}/\sqrt{\rho_{1}\sigma
r_{0}}\right)$ representing the relative importance of viscous to capillary-
inertial forces; the Weissenberg number
$Wi=\left(\lambda_{H}u_{ci}/r_{0}\right)$ representing the ratio between
elastic and viscous forces; the Peclet number
$Pe=\left(2\sqrt{2}u_{ci}r_{0}\eta\right)/\left(3M\sigma\right)$ representing
the ratio between the advection and diffusion in the Cahn-Hilliard equation.
Furthermore, the Cahn number $Cn=\left(\eta/r_{0}\right)$ is the ratio between
the interface width and the characteristic length scale;
$\beta=\mu_{s}/\left(\mu_{s}+\mu_{p}\right)$ is the ratio between the
polymeric viscosity and total viscosity;
$k_{\mu}=\left(\mu_{2}/\mu_{1}\right)$ is the ratio between the ambient and
droplet viscosities; $k_{\rho}=\left(\rho_{2}/\rho_{1}\right)$ between the
ambient and droplet densities. The different components of the energy are
scaled by $\sigma r_{0}^{2}$. In what follows, all quantities will be
nondimensional unless indicated otherwise.

To quantify the role of the fluid elasticity on the droplet jumping, we will
measure the mass-averaged velocity of the droplet, defined as:

$\displaystyle\overline{v}=\frac{\int\limits_{\Omega}\frac{1}{2}(1+\phi)v_{z}\
d\Omega}{\int\limits_{\Omega}\ \frac{1}{2}(1+\phi)d\Omega}$ (14)

where $\Omega$ is the computational domain (see Fig. 1), and $z$ the direction
perpendicular to the solid substrate. We also analyze the different components
of the energy during the coalescence and jumping. The total energy $E_{T}$ of
an Oldroyd-B fluid is the sum of the surface energy $E_{s}$, kinetic energy
$E_{k}$, and elastic energy $E_{e}$, defined in phase-field framework as [5,
22]:

$\displaystyle\begin{aligned}
E_{s}&=\int\limits_{\Omega}\frac{3Cn}{4\sqrt{2}}\left[{\mid\nabla{\phi}\mid}^{2}+\frac{1}{2Cn^{2}}{(\phi^{2}-1)}^{2}\right]\
d\Omega,\\\
E_{k}&=\int\limits_{\Omega}\frac{1}{2}\left(1+\phi\right)\mathbf{u}\cdot\mathbf{u}\
d\Omega,\\\
E_{e}&=\int\limits_{\Omega}\frac{Oh}{2}\frac{(1-\beta)}{Wi}\operatorname{tr}(\boldsymbol{c}-\ln{\boldsymbol{c}}-\boldsymbol{I})\
d\Omega,\\\ E_{T}&=E_{s}+E_{k}+E_{e},\\\ \end{aligned}$ (15)

where the relationship between polymer stress $\mathbf{\boldsymbol{\tau}_{p}}$
and the conformation tensor $\mathbf{\boldsymbol{c}}$ is

$\displaystyle\mathbf{\boldsymbol{\tau}_{p}}=\frac{(1-\beta)}{Wi}\left(\mathbf{\boldsymbol{c}}-\mathbf{\boldsymbol{I}}\right),$
(16)

The part of the kinetic energy associated to the vertical velocity component
is most relevant here as it can be associated with the jumping motion, while
the rest is related to interface oscillatory motions [19]. We will therefore
consider a translational kinetic energy, defined as,

$\displaystyle
E_{k,tr}=\int\limits_{\Omega}\frac{1}{2}(1+\phi)\overline{v}^{2}\ d\Omega$
(17)

where $\overline{v}$ is the droplet mass-averaged velocity in z-direction, and
$E_{k,os}=E_{k}-E_{k,tr}$ is the part of the kinetic energy associated with
the oscillatory motion.

The numerical setup is as follows. The nondimensional domain size is chosen of
size $\Omega=[0,10]\times[0,10]\times[0,8]$. Two adjacent droplets with
initial radius $1$ are placed above the x–y plane at z=0, see the Fig. 1. We
impose no-slip and no-penetration conditions on the two boundaries in the
z-direction, with static contact angles $\theta_{s}=180^{o}$ at the bottom
wall and $\frac{\partial\phi}{\partial n}=0$ at the top boundary.

Periodic boundary conditions are applied for all variables in the x- and
y-directions. We use 8 grid points across the nominal interface in order to
resolve the sharp gradients, and set the Peclet number to $Pe=\frac{6}{Cn}$,
according to the guidelines in Magaletti et al. [21], Xu et al. [31] to
approach the sharp-interface limit.

To ensure that the chosen grid and domain size are sufficient, we performed
the following numerical tests. Firstly, we examined the effect of the
computational domain on the velocity of the merged droplet by performing an
additional simulation on a larger domain
$\Omega_{2}=[0,15]\times[0,15]\times[0,9]$; the results obtained on
$\Omega_{1}$ match those obtained on the larger domain $\Omega_{2}$, as shown
in Fig. 2. We also tested the grid dependency of the results by comparing the
averaged velocity of the merged viscoelastic droplet for four different
resolutions corresponding to different values of the Cahn number. For this
test, the following values of the dimensionless numbers introduced above were
used:

$\displaystyle Oh$ $\displaystyle=0.0076,$ $\displaystyle Wi$
$\displaystyle=10,$ $\displaystyle\beta$ $\displaystyle=0.1,$ $\displaystyle
k_{\mu}$ $\displaystyle=0.017,$ $\displaystyle k_{\rho}$
$\displaystyle=0.00119.$

Fig. 3 shows that the averaged velocity of the merged drop with $Cn=0.025$ is
almost the same as that obtained with the finer grid $Cn=0.02$. Higher values
of $Cn$ displays non-negligible differences when the simulation time exceeds
2. Thus, we choose $Cn=0.025$, corresponding to a grid with $N_{x}\times
N_{y}\times N_{z}=760\times 760\times 608$ grid points. This satisfactory
convergence is achieved adopting the scaling between the Peclet number and Cn
number suggested by [21, 31].

Table 1: Experimental fluid properties by Yan et al. [33] at $25^{o}C$ $r_{0}(\mu m)$ | $\rho_{1}(\frac{kg}{m^{3}})$ | $\mu_{1}(Pas)$ | $\sigma(\frac{N}{m})$ | $k_{\rho}$ | $k_{\mu}$ | $\theta_{a}^{app}$ | $\theta_{r}^{app}$ | $\Delta\theta^{app}$
---|---|---|---|---|---|---|---|---
290 | 998.2 | 0.001 | 0.072 | $\frac{1}{839}$ | $\frac{1}{58.8}$ | $170.3^{o}$ | $167.7^{o}$ | $2.6^{o}$

(a)

(b)

Figure 4: The coalescence and jumping of two Newtonian droplets on a
superhydrophobic surface at $Oh=0.0076$: (a) Numerical results from present
work ($yz$ view) (b) Experimental data of Yan et al. [33].

(a)

(b)

(c)

Figure 5: Time-dependent quantities of the Newtonian merged drop on a
superhydrophobic surface at $Oh=0.0076$: (a) The average velocity, (b) Total
and surface energies of the merged drop (c) Kinetic and translational kinetic
energies of the merged drop.

##  4 Results

For the results presented here, the density ratio and viscosity ratio are kept
constant ro $k_{\rho}=\frac{1}{839}$ and $k_{\mu}=\frac{1}{58.8}$., following
the values from the experiments in Yan et al. [33].

###  4.1 Newtonian droplets - comparison with experiments

The solver has been validated against several Newtonian and viscoelastic two-
phase flow benchmarks in 2D and 3D [3]. Here, we compare the spontaneous
coalescence and jumping motion of a Newtonian drop on a superhydrophobic
surface with the experimental data of Yan et al. [33]. We choose the same
physical parameters as in the experiment, see Table 1. The influence of the
solid-liquid adhesion on the self-propelled jumping is negligible when the
contact angle hysteresis ($\Delta\theta^{app}$) is less than $10^{o}$ [32, 7],
so we ignore the contact angle hysteresis and impose a static contact angle
$\theta_{s}=180^{o}$ on the bottom wall. Fig. 4 presents the experimental data
[33] for the coalescence of two Newtonian drops on a superhydrophobic surface
and the corresponding numerical results; the visualization shows that the
numerical simulation is able to capture the coalescence and jumping process
accurately in time.

The jumping velocity of the merged water (Newtonian) drop on a
superhydrophobic surface is constant in the capillary-inertial region (i.e.,
$Oh\lesssim 0.1$); [2] reported ${{v}_{j}}\approx 0.2$ for the water drop at
$19^{o}C$ on a textured superhydrophobic surface. Later, [33] reduced the
level of undesired external disturbances and measured a velocity
${{v}_{j}}\approx 0.26$ for self-propelled jumping of water drops upon
coalescence on a superhydrophobic surface. Fig. 5(a) shows the averaged
velocity of the merged droplet in our simulation. The jumping velocity is
measured from the time the bottom of the merged drop leaves the surface. Liu
et al. [19] suggested that a sensible time for extracting the jumping velocity
is the first pseudo-equilibrium, which corresponds to the time when the axial
lengths of the merged droplet in x and y-directions become equal. These axial
lengths are measured with respect to an axis attached to the center of mass of
the merged drop, and density ratio in their numerical simulation was
$k_{\rho}=0.02$. Following this criterion, the jumping velocity of the
Newtonian drop at $Oh=0.0076$ is ${{v}_{j}}\approx 0.21$ in our simulations.
It should be noted that the criterion of Liu et al. [19] may work less well at
our high density ratios, since the merged droplet’s average velocity decreases
very rapidly after its maximum, and this may lead us to underestimate the
jumping velocity. The dimensionless time corresponding to the first pseudo-
equilibrium is $t\approx 3.32$ (dimensional time $t^{*}\approx 1.92ms$) with
the droplet leaving the surface at $t\approx 2.71$ ($t^{*}\approx 1.57ms$) in
both simulation and experiment. The averaged velocity at $t=2.71$ is
${\overline{v}}\approx 0.23$, and the maximum averaged velocity is
${\overline{v}}_{max}\approx 0.26$, see Fig. 5 (a).

Fig. 5 (b) presents the time-evolution of the total and surface energy of the
merging and jumping droplet. The total energy of the droplet decreases over
time due to the viscous dissipation, and this decrease is larger in the
merging process and prior to jumping, because there are highly localized
velocity gradients around the liquid bridge during its impingement on the
substrate. Fig. 5 (c) depicts the total and translational kinetic energy of
the merged droplet, confirming that a small fraction of the released surface
energy is converted into transitional kinetic associated to the jumping
motion.

###  4.2 Viscoelastic droplets - elasticity effect

Most of the existing studies are restricted to experiments with water droplets
and numerical simulation of Newtonian drops; [33] newly investigated the
effect of the liquid internal hydrodynamics by conducting experiment for the
self-propelled jumping upon coalescence on a superhydrophobic surface with
ethanol-water and ethylene-glycol solutions. Their experiment shows that the
properties of the droplet affect the coalescence and jumping process
significantly. A very recent experimental study, however, addressed the effect
of the drops’ elasticity on the coalescence process [8], and their findings
will be referred to later in this section. In the following, we investigate
numerically how the droplet elasticity (Weissenberg number) influences the
jumping process for different values of the Ohnesorge number. One way to vary
the Ohnesorge number in experiments is to keep the physical properties
constant and vary its radius, and we adopt this approach in our simulation. It
should be noted that other parameters change also with the droplet radius; the
parameters for each case are summarized in Appendix A.

(a)

(b)

(c)

Figure 6: Quantities of the merged Newtonian and Oldroyd-B drops at $Wi=10$
and $\beta=0.1$. N and V refer to Newtonian and Oldroyd-B drops respectively.
(a) The averaged velocity (b) Total and surface energies of the merged drop at
$Oh=0.0076$ (c) Kinetic and transitional kinetic energies of the merged drop
at $Oh=0.0076$.

####  4.2.1 Small Ohnesorge numbers

To isolate the effect of the droplet elasticity on the coalescence and jumping
on a superhydrophobic surface in the inertial-capillary region, $Oh\lesssim
0.1$, we set $\alpha=0$ in the following simulations. This choice implies that
shear-thinning is eliminated, and the Giesekus model reduces to the Oldroyd-B
model. The average velocities of the merged Newtonian and viscoelastic
droplets are compared at two Ohnesorge numbers, $Oh=0.0076$ and $Oh=0.0373$.
For Newtonian droplets, these values represent the capillary-inertial regime.
The two dimensionless numbers defining the Oldroyd-B model are kept constant,
$Wi=10$ and $\beta=0.1$, while the others vary since we are changing the
droplet radius, see Appendix A.

Fig. 6 (a) shows the time evolution of the average velocity of the merged
droplet for the Newtonian and viscoelastic cases, for both Ohnesorge numbers.
The first observation is that both droplets jump from the surface at these low
Ohnesorge numbers. Let us now consider the blue lines, corresponding to the
smallest Ohnesorge number: Newtonian (solid line) and viscoelastic (dashed
line). We observe that the elasticity of the drop has a negligible effect on
the average velocity prior to and during jumping. However, there is a small
qualitative difference after jumping, where the averaged velocity remains
approximately constant for the viscoelastic droplet, while the velocity of the
Newtonian droplet decreases in time. For the larger Ohnesorge number (green
lines), we observe that the droplet elasticity increases the maximum averaged
velocity. The total energy is dissipated more rapidly in the Newtonian droplet
after departure, while there is less dissipation in the merged viscoelastic
droplet, see Fig. 6 (b). The droplet kinetic energy is presented in Fig. 6
(c): the viscoelastic droplet has more kinetic energy so that it undergoes
larger shape oscillations than the Newtonian droplet. The oscillations of the
viscoelastic droplet are due to its elasticity and independent of the surface
tension, see [17]. The viscoelastic droplet oscillates even at the large
Ohnesorge numbers corresponding to a highly viscous drop [17].An extensional
flow occurs when the two viscoelastic drops are coalescing, so polymer chains
stretch and store elastic energy during the coalescence process, see Fig. 6
(c).

Summarizing, the average and jumping velocity are not considerably affected by
the elasticity of the drops in the inertial-capillary regime. This result is
in line with the experiments of Dekker et al. [8], where elasticity did not
considerably influence the coalescence process in the inertial-capillary
regime. However, quantitatively we found that the oscillations are promoted by
elasticity, and that the average velocity decays less rapidly after the
droplet departure from the superhydrophobic surface.

(a)

(b)

(c)

Figure 7: Quantities of the merged Newtonian and Oldroyd-B drops at $Wi=10$
and $\beta=0.1$. N and V refer to Newtonian and Oldroyd-B drops respectively.
(a) The averaged velocity (b) Total and surface energies of the merged drop at
$Oh=0.118$ (c) Kinetic and transitional kinetic energies of the merged drop at
$Oh=0.118$.

(a)

(b)

Figure 8: The coalescence and jumping of Newtonian and Oldroyd-B drops on a
superhydrophobic surface at $Oh=0.118$, $Wi=10$ and $\beta=0.1$ (a) Newtonian
drops (b) Oldroyd-B drops. The trace of the dimensionless polymeric stresses
are visualized on the surface of the polymeric drop.

####  4.2.2 Large Ohnesorge numbers

For Newtonian drops, viscous forces become the dominant at larger Ohnesorge
numbers ($Oh\gtrsim 0.1$), known as the viscous regime: both jumping and
averaged velocities decrease rapidly with increasing Ohnesorge number due to
the strong viscous dissipation. Let us now examine whether and how this
behaviour changes for viscoelastic droplets at $Oh\gtrsim 0.1$.

The results from the simulations are reported in Fig. 7. First, we note that
the merged viscoelastic droplet gains much larger average velocity than the
Newtonian one during the coalescence process, as seen by comparing the solid
(Newtonian) and dashed (viscoelastic) lines of the same color in panel (a).
Panel (b) of the same figure shows that the released energy is soon damped at
$Oh=0.118$ ( ${t}^{*}\approx 4$), and the total energy (blue solid line)
reaches its equilibrium value $E_{T}^{*}=2^{\frac{2}{3}}4\pi$ consisting of
surface energy only (green solid line). The Newtonian drop resulting from the
coalescence reaches a spherical shape corresponding to its equilibrium, and
stays on the surface without jumping, as shown in Fig. 8.

Let us now consider the polymeric drop (dashed blue and green lines). This
merged polymeric drop has energy available to oscillate and move upwards until
$t=4.0$. The kinetic energies of the Newtonian (solid) and Oldroyd-B droplets
(dashed) are depicted in Fig. 7 (c), and both total kinetic and translational
kinetic energies of the polymeric drop are larger than for the Newtonian drop.
The Oldroyd-B drop also has the additional elastic energy due to the presence
of the polymer molecules, see the black dashed line. These are stretched
during the coalescence and prior to jumping, so that extra elastic energy is
stored and available for the polymeric drop.

A visualisation of the coalescence and jumping of both droplets at $Oh=0.118$
is provided in Fig. 8. The polymeric drops merge faster than the Newtonian
drops, and the the bridge formed due to the coalescence reaches the substrate
sooner. The interface (given by $\phi=0$) of the Oldroyd-B drops is coloured
using the trace of the conformation tensor which indicates the intensity of
polymer stretching. As reported in experiments of [8], the polymeric stresses
are seen to be very concentrated around the merging interface and the
capillary bridge. At the present flow regime, however, we observe significant
changes due to elasticity in both the liquid bridge formation and in the
merging and coalescence process. The merged polymeric drop undergoes a large
deformation in all three directions and jumps out of the surface; this
oscillatory motion and jumping is characteristic of the inertial-capillary
region and is maintained at high $Oh$ when elasticity is present. Indeed, the
Newtonian drop goes rapidly towards its equilibrium condition without
noticeable oscillations, and remains on the surface, as expected in the
viscous region.

To gain a better understanding on the effect of elasticity on the self-
propelled jumping, the flow field is visualized on the central $XZ$ and $YZ$
planes inside the merging drops, see Fig. 9 (a). As concern the $YZ$ plane,
shown in the top row, we depict the interface, identified by the $\phi=0$
value of the order parameter (red contour), velocity field (arrows), and trace
of conformation tensor (colormap) for merging viscoelastic drop at three
times, together with the the interface of merging Newtonian drop (white
contour). This illustrates how the shape of Newtonian and viscoelastic
droplets differ, and confirms the localisation of polymeric stresses at the
merging cross-section. Moreover, the bottom row of the same figure displays
contoues of the trace of the conformation tensor for the viscoelastic drops on
the $XZ$ plane at the same dimensionless times as in Fig. 9 (b). The data
indicate that polymers are most elongated prior to jumping, near the bottom
wall.

(a)

(b)

Figure 9:  Time evolution of viscoelastic drops coalescence on $XZ$ and $YZ$
planes on a superhydrophobic surface at $Oh=0.118$, $Wi=10$ and $\beta=0.1$.
The interface $\phi=0$ of the merging Newtonian drops are depicted at
$Oh=0.118$ on $YZ$ plane for the comparison : (a) $XZ$ and $YZ$ planes used
for the visualization of the flow field (b) The interface of both Newtonian
and viscoelastic drops on $YZ$ plane at three different times $t$ along with
the velocity field and trace of conformation tensor belong to the viscoelastic
drops. The trace of conformation tensor is visualized on $XZ$ for the
viscoelastic drops.

A possible physical explanation for the polymer effect can be as follows. When
the two initially static drops start to merge, the liquid moves driven by the
capillary pressure towards the center of the expanding bridge. Then, due to
the conservation of mass, the liquid is forced to move in the transverse $XZ$
plane, see the velocity field of the merging viscoelastic drops in Fig. 9 (b)
at $t=0.24$. This flow causes the polymer molecules to stretch in the $XZ$
plane and produce extra elastic stresses, which push the liquid bridge
connecting the two polymeric drops to move faster. When the liquid bridge
interacts with the substrate at $t\approx 1$, the liquid is induced to move
upwards due to the impermeability of the surface. This upward flow converges
toward the $XZ$ plane and causes the polymers to stretch mainly in the
vicinity of the substrate, as shown by the trace of conformation tensor at
$t=1.96$ in Fig. 9 (b). The stretched polymers exert extra elastic stresses on
the interface near the substrate, so the merged polymeric drop jumps out of
the surface. The newly-formed larger drop leaves the surface at $t\gtrsim 3$,
when the trace of the conformation tensor decreases. Later, the polymers are
mainly stretched at the bottom of the merged drop, and two small vortices
appear in that region so that the polymer molecules remain stretched. Thus,
these extra polymer stresses at the bottom of the drop push the drop to move
upward. The polymers are also significantly stretched in the $XZ$ plane around
the interface, and these extra polymer stresses push the interface to
oscillate in the x-direction, see Fig. 9 (b).

These results reveal that the elasticity of the drop plays an important role
at large Ohnesorge numbers in the coalescence and jumping processes of two
initially static equal-sized polymer drops on a superhydrophobic surface;
elasticity also affects the merged droplet motion after its departure as
demonstrated by the oscillatory motion in highly-viscous yet viscoelastic
drops.

(a) Figure 10: The effect of drop shear-thinning $\alpha$ on the averaged
velocity at $Oh=0.118$, $Wi=10$, and $\beta=0.1$.

###  4.3 Viscoelastic droplets - effects of polymeric viscosity ratio and
shear-thinning

The effect of the shear thinning on the self-propelled jumping of two equal-
sized polymeric drops has been studied by performing simulations at
$Oh=0.118$, $De=10$, $\beta=0.1$ and varying $\alpha$; the results show that
the effect of shear thinning is minor and negligible for the self-propelled
jumping of two equal-sized polymeric drops even at large Ohnesorge numbers,
see Fig. 10.

In addition, we have investigated the effect of the polymeric viscosity ratio
$\beta$ on the self-propelled jumping at $Oh=0.118$ and $Wi=10$. Fig. 11
depicts the variation of the averaged velocity of the droplet for
$\beta=0.1-0.8$. The Newtonian droplet velocity is also shown for comparison.
Two regimes can be distinguished: as long as $\beta\lesssim 0.5$, we note a
minor influence on the averaged velocity; for $\beta\gtrsim 0.6$, conversely,
the velocity rapidly converges towards the Newtonian one, so that elasticity
effects become negligible. This can be explained by considering the
retardation time, i.e. the relative time it takes for polymer molecules to be
stretched. The retardation time can be related to the polymeric viscosity
ratio by $\lambda_{r}=\beta\lambda_{H}$, while the flow time scale time is
constant in our simulations since the droplet radius $r_{0}$, velocity scale
$u_{ref}=\sqrt{\frac{\sigma}{\rho_{1}r_{0}}}$, and Weissenberg number $Wi$ are
kept constant. Since the retardation time is increasing by increasing $\beta$,
the polymers do not have time to stretch and store elastic energy during the
coalescence and jumping when $\beta\gtrsim 0.6$. Thus, the polymeric drops
behave like Newtonian drops at large polymeric viscosity ratios.

(a) Figure 11: The effect of polymeric viscosity ratio $\beta$ on the averaged
velocity at $Oh=0.118$ and $Wi=10$.

##  5 Conclusions and outlook

In the present study, three-dimensional direct numerical simulations have been
performed to study the self-propelled jumping of two equal-sized polymeric
drops on a superhydrophobic surface with contact angle of $180^{o}$. The
results demonstrate that the viscoelastic properties of the droplets have a
significant impact on the coalescence and jumping.

At small Ohnesorge numbers (inertial-capillary region), the elasticity effect
is weak before the jumping; however, the averaged velocity of the coalesced
drop does not decay as rapidly as for a Newtonian liquid. Drop shape
oscillations are promoted bythe presence of the polymers.

At large Ohnesorge numbers ($Oh\gtrsim 0.1$) however, profound differences
between polymeric and Newtonian drops are observed during the coalescence and
jumping process. The polymeric drops merge faster than the Newtonian drops,
and the merged drop jumps out of the surface in contrast to their Newtonian
counterparts, which remain on the substrate due to the large viscous
dissipation. Our investigation reveals that the polymers are highly stretched
at the cross-section of the merging droplets during coalescence, and these
stretched chains exert extra elastic stresses on the interface of the merging
drops in the vicinity of the wall, hence helping the polymeric drop to jump
from the surface. These results are obtained with a typical value in the
literature for the polymeric viscosity ratio, i.e. $\beta=0.1$; here, we also
observe that the merged viscoelastic drop behaves like a Newtonian drop when
$\beta\gtrsim 0.7$. The larger $\beta$ corresponds to larger retardation times
in our simulation, so that the polymer molecules do not have enough time to
stretch. Finally, the shear-thinning effect is found to be negligible in the
coalescence and jumping process of two equal-sized drops on a superhydrophobic
surface.

Our results indicate that the elasticity of the droplet can change the viscous
cutoff radius (for example $30\mu m$ for water) for the self-propelled jumping
of drops on superhydrophobic surfaces. Thus, it is expected that polymeric
drops jump from a superhydrophobic surface upon their coalescence with radii
below the viscous cutoff radius for Newtonian drops at the same Ohnesorge
number. Moreover, the merged polymeric drop oscillatory motion is promoted by
the elasticity of the drop in both inertial-capillary and viscous-capillary
regimes.

In this study, we have neglected the contact angle hysteresis, assuming it to
be smaller than $10^{o}$; however, superhydrophobic surfaces may have large
contact angle hysteresis, which might play an important role in the case of
polymeric drops. Studying the effect of the contact angle hysteresis is one of
the possible extensions of this work.

##  6 Acknowledgments

This work was funded by European Research Council (ERC) through Starting grant
no. 852529 MUCUS, and by the Swedish Research Council through grant VR
2017-0489. We acknowledge the computing time on the supercomputer Beskow at
the PDC center, KTH provided by SNIC (Swedish National Infrastructure for
Computing),Sweden.

## Appendix A

In this Appendix, we report the values of the non-dimensional numbers used in
the simulations.

Table 2: Dimensionless numbers used in the simulations at small Ohnesorge numbers, see section 4.2.1 $Case$ | Droplets | $Oh$ | $Wi$ | $\beta$ | $\alpha$ | $\theta_{s}^{b}$ | $k_{\rho}$ | $k_{\mu}$
---|---|---|---|---|---|---|---|---
1 | Viscoelastic | 0.0076 | 10 | 0.1 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
2 | Newtonian | 0.0076 | 0 | 0 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
3 | Viscoelastic | 0.0373 | 10 | 0.1 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
4 | Newtonian | 0.0373 | 0 | 0 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
Table 3:  Dimensionless numbers used in the simulations at large Ohnesorge numbers, see section 4.2.2 $Case$ | Droplets | $Oh$ | $Wi$ | $\beta$ | $\alpha$ | $\theta_{s}^{b}$ | $k_{\rho}$ | $k_{\mu}$
---|---|---|---|---|---|---|---|---
1 | Viscoelastic | 0.118 | 10 | 0.1 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
2 | Newtonian | 0.118 | 0 | 0 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
3 | Viscoelastic | 0.1668 | 10 | 0.1 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
4 | Newtonian | 0.1668 | 0 | 0 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
5 | Viscoelastic | 0.3 | 10 | 0.1 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
6 | Newtonian | 0.3 | 0 | 0 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
Table 4:  Dimensionless numbers used in the simulations focusing on the role of shear thinning, see section 4.3 $Case$ | Droplets | $Oh$ | $Wi$ | $\beta$ | $\alpha$ | $\theta_{s}^{b}$ | $k_{\rho}$ | $k_{\mu}$
---|---|---|---|---|---|---|---|---
1 | Viscoelastic | 0.118 | 10 | 0.1 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
2 | Viscoelastic | 0.118 | 10 | 0.1 | 0.2 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
3 | Viscoelastic | 0.118 | 10 | 0.1 | 0.5 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
Table 5: Dimensionless numbers used in the simulations focusing on the role of the polymer viscosity ratio, see section 4.3 $Case$ | Droplets | $Oh$ | $Wi$ | $\beta$ | $\alpha$ | $\theta_{s}^{b}$ | $k_{\rho}$ | $k_{\mu}$
---|---|---|---|---|---|---|---|---
1 | Newtonian | 0.118 | 0 | 0 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
2 | Viscoelastic | 0.118 | 10 | 0.1 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
3 | Viscoelastic | 0.118 | 10 | 0.5 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
4 | Viscoelastic | 0.118 | 10 | 0.6 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
5 | Viscoelastic | 0.118 | 10 | 0.7 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$
6 | Viscoelastic | 0.118 | 10 | 0.8 | 0 | $180^{o}$ | $\frac{1}{839}$ | $\frac{1}{58.8}$

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# Spatial Aggregation of Holistically-Nested Networks for Automated Pancreas
Segmentation

Holger R. Roth Imaging Biomarkers and Computer-Aided Diagnosis Laboratory
Radiology and Imaging Sciences
Clinical Center
National Institutes of Health
Bethesda, MD 20892-1182, USA. Le Lu Imaging Biomarkers and Computer-Aided
Diagnosis Laboratory
Radiology and Imaging Sciences
Clinical Center
National Institutes of Health
Bethesda, MD 20892-1182, USA. Amal Farag Imaging Biomarkers and Computer-Aided
Diagnosis Laboratory
Radiology and Imaging Sciences
Clinical Center
National Institutes of Health
Bethesda, MD 20892-1182, USA. Andrew Sohn Imaging Biomarkers and Computer-
Aided Diagnosis Laboratory
Radiology and Imaging Sciences
Clinical Center
National Institutes of Health
Bethesda, MD 20892-1182, USA. Ronald M. Summers Imaging Biomarkers and
Computer-Aided Diagnosis Laboratory
Radiology and Imaging Sciences
Clinical Center
National Institutes of Health
Bethesda, MD 20892-1182, USA.

###### Abstract

Accurate automatic organ segmentation is an important yet challenging problem
for medical image analysis. The pancreas is an abdominal organ with very high
anatomical variability. This inhibits traditional segmentation methods from
achieving high accuracies, especially compared to other organs such as the
liver, heart or kidneys. In this paper, we present a holistic learning
approach that integrates semantic mid-level cues of deeply-learned organ
interior and boundary maps via robust spatial aggregation using random forest.
Our method generates boundary preserving pixel-wise class labels for pancreas
segmentation. Quantitative evaluation is performed on CT scans of 82 patients
in 4-fold cross-validation. We achieve a (mean $\pm$ std. dev.) Dice
Similarity Coefficient of 78.01%$\pm$8.2% in testing which significantly
outperforms the previous state-of-the-art approach of 71.8%$\pm$10.7% under
the same evaluation criterion.

##  1 Introduction

Pancreas segmentation in computed tomography (CT) is challenging for current
computer-aided diagnosis (CAD) systems. While automatic segmentation of
numerous other organs in CT scans such as liver, heart or kidneys achieves
good performance with Dice Similarity Coefficients (DSC) of $>$90% [1, 2, 3],
the pancreas’ variable shape, size and location in the abdomen limits
segmentation accuracy to $<$73% DSC being reported in the literature [3, 4, 5,
6]. Deep convolutional Neural Networks (CNNs) have successfully been applied
to many high-level tasks in medical imaging, such as recognition and object
detection [7]. The main advantage of CNNs comes from the fact that end-to-end
learning of salient feature representations for the task at hand is more
effective than hand-crafted features with heuristically tuned parameters [8].
Similarly, CNNs demonstrate promising performance for pixel-level labeling
problems, e.g., semantic segmentation [6, 9, 10, 11]. Recent work in computer
vision and biomedical imaging including fully convolutional neural networks
(FCN) [10], DeepLab [11] and U-Net [9], have gained significant improvements
in performance over previous methods by applying state-of-the-art CNN based
image classifiers and representation to the semantic segmentation problem in
both domains. Semantic organ segmentation involves assigning a label to each
pixel in the image. On one hand, features for classification of single pixels
(or patches) play a major role, but on the other hand, factors such as edges
(i.e., organ boundaries), appearance consistency and spatial consistency,
could greatly impact the overall system performance [8]. Furthermore, there
are indications of semantic vision tasks requiring hierarchical levels of
visual perception and abstraction [12]. As such, generating rich feature
hierarchies for both the interior and the boundary of the organ could provide
important “mid-level visual cues” for semantic segmentation. Subsequent
spatial aggregation of these mid-level cues then has the prospect of improving
semantic segmentation methods by enhancing the accuracy and consistency of
pixel-level labeling.

##  2 Methods

In this work, we present a holistic semantic segmentation method for organ
segmentation in CT which incorporates deeply learned organ interior and
boundary mid-level cues with subsequent spatial aggregation. This approach to
organ segmentation is completely based on machine-learning principles. No
multi-atlas registration and label fusion methods are employed. Our methods
are evaluated on CT scans of 82 patients in 4-fold cross-validation (instead
of “leave-one-patient-out” evaluation often used in other work [1, 2, 3]).

###  2.1 Candidate Region Generation

As a form of initialization, we employ a previously proposed method based on
random forest (RF) classification [6] to compute a candidate bounding box
regions. We only operate the RF labeling at a low probability threshold of
$>$0.5 which is sufficient to reject the vast amount of non-pancreas from the
CT images. This initial candidate generation is sufficient to extract bounding
box regions that completely surround the pancreases in all used cases with
nearly 100% recall. All candidate regions are computed during the testing
phase of cross-validation (CV) as in [6]. Note that candidate region proposal
is not the focus of this work and assumed to be fixed for the rest of this
study. This part could be replaced by other means of detecting an initial
bounding box for pancreas detection, e.g., by RF regression [13] or sliding-
window CNNs [6].

###  2.2 Semantic Mid-level Segmentation Cues

We show that organ segmentation can benefit from multiple mid-level cues, like
organ interior and boundary predictions. We investigate deep-learning based
approaches to independently learn the pancreas’ interior and boundary mid-
level cues. Combining both cues via learned spatial aggregation can elevate
the overall performance of this semantic segmentation system. Organ boundaries
are a major mid-level cue for defining and delineating the anatomy of
interest. It could prove to be essential for accurate semantic segmentation of
an organ.

####  2.2.1 Holistically-Nested Nets:

In this work, we explicitly learn the pancreas’ interior and boundary image-
labeling models via Holistically-Nested Networks (HNN). Note that this type of
CNN architecture was first proposed by [12] under the name “holistically-
nested edge detection” as a deep learning based general image edge detection
method. We however find that it can be a suitable method for segmenting the
interior of organs as well (see Sec. 3.0.2). HNN tries to address two
important issues: (1) training and prediction on the whole image end-to-end
(holistically) using a per-pixel labeling cost; and (2) incorporating multi-
scale and multi-level learning of deep image features [12] via auxiliary cost
functions at each convolutional layer. HNN computes the image-to-image or
pixel-to-pixel prediction maps (from any input raw image to its annotated
labeling map), building on fully convolutional neural networks [10] and
deeply-supervised nets [14]. The per-pixel labeling cost function [10, 12]
offers the good feasibility that HNN/FCN can be effectively trained using only
several hundred annotated image pairs. This enables the automatic learning of
rich hierarchical feature representations (contexts) that are critical to
resolve spatial ambiguity in the segmentation of organs. The network structure
is initialized based on an ImageNet pre-trained VGGNet model [15]. It has been
shown that fine-tuning CNNs pre-trained on the general image classification
task (ImageNet) is helpful to low-level tasks, e.g., edge detection [12].

Figure 1: Schematics of (a) the holistically-nested nets, in which multiple
side outputs are added, and (b) the HNN-I/B network architecture for both
interior (left images) and boundary (right images) detection pathways. We
highlight the error back-propagation paths to illustrate the deep supervision
performed at each side-output layer after the corresponding convolutional
layer. As the side-outputs become smaller, the receptive field sizes get
larger. This allows HNN to combine multi-scale and multi-level outputs in a
learned weighted fusion layer (Figures adapted from [12] with permission).

####  2.2.2 Network formulation:

Our training data $S^{I/B}=\left\\{(X_{n},Y^{I/B}_{n}),n=1,\dots,N\right\\}$
is composed of cropped axial CT images $X_{n}$ (rescaled to within
$\left[0,\dots,255\right]$ with a soft-tissue window of $[-160,240]$ HU); and
$Y^{I}_{n}\in\left\\{0,1\right\\}$ and $Y^{B}_{n}\in\left\\{0,1\right\\}$
denote the (binary) ground truths of the interior and boundary map of the
pancreas, respectively, for any corresponding $X_{n}$. Each image is
considered holistically and independently as in [12]. The network is able to
learn features from these images alone from which interior (HNN-I) boundary
(HNN-B) predication maps can be produced.

HNN can efficiently generate multi-level image features due to its deep
architecture. Furthermore, multiple stages with different convolutional
strides can capture the inherent scales of (organ edge/interior) labeling
maps. However, due to the difficulty of learning such deep neural networks
with multiple stages from scratch, we use the pre-trained network provided by
[12] and fine-tuned to our specific training data sets $S^{I/B}$ with a
relatively smaller learning rate of $10^{-6}$. We use the HNN network
architecture with 5 stages, including strides of 1, 2, 4, 8 and 16,
respectively, and with different receptive field sizes as suggested by the
authors111https://github.com/s9xie/hed..

In addition to standard CNN layers, a HNN network has $M$ side-output layers
as shown in Fig. 1. These side-output layers are also realized as classifiers
in which the corresponding weights are
$\bm{\mathrm{w}}=(\bm{\mathrm{w}}^{(1)},\dots,\bm{\mathrm{w}}^{(M)})$. For
simplicity, all standard network layer parameters are denoted as
$\bm{\mathrm{W}}$. Hence, the following objective function can be defined222We
follow the notation of [12].:

$\small\mathcal{L}_{\mathrm{side}}(\bm{\mathrm{W}},\bm{\mathrm{w}})=\sum^{M}_{m=1}{\alpha_{m}}{l_{\mathrm{side}}^{(m)}}(\bm{\mathrm{W}},\bm{\mathrm{w}}^{m}).$
(1)

Here, $l_{\mathrm{side}}$ denotes an image-level loss function for side-
outputs, computed over all pixels in a training image pair $X$ and $Y$.
Because of the heavy bias towards non-labeled pixels in the ground truth data,
[12] introduces a strategy to automatically balance the loss between positive
and negative classes via a per-pixel class-balancing weight $\beta$. This
allows to offset the imbalances between edge/interior ($y=1$) and non-
edge/exterior ($y=0$) samples. Specifically, a class-balanced cross-entropy
loss function can be used in Equation 1 with $j$ iterating over the spatial
dimensions of the image:

$\small
l^{(m)}_{\mathrm{side}}(\bm{\mathrm{W}},\bm{\mathrm{w}}^{(m)})=-\beta\sum_{j\in
Y_{+}}\log Pr\left(y_{j}=1|X;\bm{\mathrm{W}},\bm{\mathrm{w}}^{(m)}\right)-\\\
(1-\beta)\sum_{j\in Y_{-}}\log
Pr\left(y_{j}=0|X;\bm{\mathrm{W}},\bm{\mathrm{w}}^{(m)}\right).$ (2)

Here, $\beta$ is simply $|Y_{-}|/|Y|$ and $1-\beta=|Y_{+}|/|Y|$, where
$|Y_{-}|$ and $|Y_{+}|$ denote the ground truth set of negatives and
positives, respectively. The class probability
$Pr(y_{j}=1|X;W,w^{(m)})=\sigma(a_{j}^{(m)})\in[0,1]$ is computed on the
activation value at each pixel $j$ using the sigmoid function $\sigma(.)$.
Now, organ edge/interior map predictions
$\hat{Y}ˆ{(m)}_{\mathrm{side}}=\sigma(\hat{A}ˆ{(m)}_{\mathrm{side}})$ can be
obtained at each side-output layer, where
$\hat{A}ˆ{(m)}_{\mathrm{side}}\equiv\\{a_{j}^{(m)},j=1,\dots,|Y|\\}$ are
activations of the side-output of layer $m$. Finally, a “weighted-fusion”
layer is added to the network that can be simultaneously learned during
training. The loss function at the fusion layer $L_{\mathrm{fuse}}$ is defined
as

$\small\mathcal{L}_{\mathrm{fuse}}(\bm{\mathrm{W}},\bm{\mathrm{w}},\bm{\mathrm{h}})=Dist\left(Y,\hat{Y}_{\mathrm{fuse}}\right),$
(3)

where
$\hat{Y}_{\mathrm{fuse}}\equiv\sigma\left(\sum^{M}_{m=1}h_{m}^{\hat{A}_{\mathrm{side}}}\right)$
with $h=\left(h_{1},\dots,h_{M}\right)$ being the fusion weight. $Dist(.,.)$
is a distance measure between the fused predictions and the ground truth label
map. We use cross-entropy loss for this purpose. Hence, the following
objective function can be minimized via standard stochastic gradient descent
and back propagation:

$\small(\bm{\mathrm{W}},\bm{\mathrm{w}},\bm{\mathrm{h}})^{\star}=\mathrm{argmin}\left(\mathcal{L}_{\mathrm{side}}(\bm{\mathrm{W}},\bm{\mathrm{w}})+\mathcal{L}_{\mathrm{fuse}}(\bm{\mathrm{W}},\bm{\mathrm{w}},\bm{\mathrm{h}})\right)$
(4)

##### Testing phase:

Given image $X$, we obtain both interior (HNN-I) and boundary (HNN-B)
predictions from the models’ side output layers and the weighted-fusion layer
as in [12]:

$\small\left(\hat{Y}^{I/B}_{\mathrm{fuse}},\hat{Y}_{\mathrm{side}}^{I_{1}/B_{1})},\dots,\hat{Y}_{\mathrm{side}}^{I_{M}/B_{M}}\right)=\bm{\mathrm{HNN-I/B}}\left(X,(\bm{\mathrm{W}},\bm{\mathrm{w}},\bm{\mathrm{h}})\right).$
(5)

###  2.3 Learning Organ-specific Segmentation Object Proposals

“Multiscale Combinatorial Grouping” (MCG333https://github.com/jponttuset/mcg.)
[16] is one of the state-of-the-art methods for generating segmentation object
proposals in computer vision. We utilize this approach to generate organ-
specific superpixels based on the learned boundary predication maps HNN-B.
Superpixels are extracted via continuous oriented watershed transform at three
different scales
$(\hat{Y}_{\mathrm{side}}^{B_{2}},\hat{Y}_{\mathrm{side}}^{B_{3}},\hat{Y}_{\mathrm{fuse}}^{B})$
supervisedly learned by HNN-B. This allows the computation of a hierarchy of
superpixel partitions at each scale, and merges superpixels across scales
thereby, efficiently exploring their combinatorial space [16]. This, then,
allows MCG to group the merged superpixels toward object proposals. We find
that the first two levels of object MCG proposals are sufficient to achieve
$\sim 88\%$ DSC (see Table 1 and Fig. 2), with the optimally computed
superpixel labels using their spatial overlapping ratios against the
segmentation ground truth map. All merged superpixels $\mathcal{S}$ from the
first two levels are used for the subsequently proposed spatial aggregation of
HNN-I and HNN-B.

Figure 2: “Multiscale Combinatorial Grouping” (MCG) [16] on three different
scales of learned boundary predication maps from HNN-B:
$\hat{Y}^{B_{2}}_{\mathrm{side}}$, $\hat{Y}^{B_{3}}_{\mathrm{side}}$, and
$\hat{Y}^{B}_{\mathrm{fuse}}$ using the original CT image as input (shown with
ground truth delineation of pancreas). MCG computes superpixels at each scale
and produces a set of merged superpixel-based object proposals. We only
visualize the boundary probabilities $p>10\%$.

###  2.4 Spatial Aggregation with Random Forest

We use the superpixel set $\mathcal{S}$ generated previously to extract
features for spatial aggregation via random forest classification444Using
MATLAB’s TreeBagger() class.. Within any superpixel $s\in\mathcal{S}$ we
compute simple statistics including the 1st-4th order moments and 8
percentiles $[20\%,30\%,\dots,90\%]$ on CT, HNN-I, and HNN-B. Additionally, we
compute the mean $x$, $y$, and $z$ coordinates normalized by the range of the
3D candidate region (Sec. 2.1). This results in 39 features describing each
superpixel and are used to train a random forest classifier on the training
positive or negative superpixels at each round of 4-fold CV. Empirically, we
find 50 trees to be sufficient to model our feature set. A final 3D pancreas
segmentation is simply obtained by stacking each slice prediction back into
the space of the original CT images. No further post-processing is employed
and spatial aggregation of HNN-I and HNN-B maps for superpixel classification
is already of high quality. This complete pancreas segmentation model is
denoted as HNN-I/B-RF or HNN-RF.

##  3 Results & Discussion

####  3.0.1 Data:

Manual tracings of the pancreas for 82 contrast-enhanced abdominal CT volumes
were provided by a publicly available
dataset555http://dx.doi.org/10.7937/K9/TCIA.2016.tNB1kqBU. [6], for the ease
of comparison. Our experiments are conducted on random splits of $\sim$60
patients for training and $\sim$20 for unseen testing in 4-fold cross-
validation. Most previous work [1, 2, 3] use the leave-one-patient-out (LOO)
protocol which is computationally expensive (e.g., $\sim 15$ hours to process
one case using a powerful workstation [1]) and may not scale up efficiently
towards larger patient populations.

####  3.0.2 Evaluation:

Table 1 shows the improvement from HNN-I to using spatial aggregation via HNN-
RF based on thresholded probability maps (calibrated based on the training
data), using DSC and average minimum distance. The average DSC is increased
from 76.99% to 78.01% statistically significantly (p$<$0.001, Wilcoxon signed-
rank test). In contrast, using dense CRF (DCRF) optimization [11] (with HNN-I
as the unary term and the pairwise term depending on the CT values) as a means
of introducing spatial consistency does not improve upon HNN-I noticeably
(avg. DSC of 77.14%, see Table 1). To the best of our knowledge, our result
comprises the highest reported average DSC (in testing folds) under the same
4-fold CV evaluation metric [6]. Strict comparison to previous methods (except
for [6]) is not directly possible due to different datasets utilized. Our
holistic segmentation approach with spatial aggregation advances the current
state-of-the-art quantitative performance to an average DSC of 78.01% in
testing. To the best of our knowledge, this is the highest DSC ever reported
in the literature. Previous state-of-the-art results range from $\sim$68% to
$\sim$73% [3, 4, 5]. In particular, DSC drops from 68% (150 patients) to 58%
(50 patients) under the leave-one-out protocol as reported in [3]. Our methods
also perform with the better statistical stability (i.e., comparing 8.2%
versus 18.6% [1], 15.3% [2] in the standard deviation of DSCs). The minimal
DSC value is 34.11% for HNN-RF, whereas [1, 2, 3, 6] all report patient cases
with DSC $<$10%. A typical patient result achieving a DSC close to the data
set mean is shown in Fig. 3. Furthermore, we apply our trained HNN-I model on
a different CT data set66630 training data sets at
https://www.synapse.org/#!Synapse:syn3193805/wiki/217789. with 30 patients,
and achieve a mean DSC of 62.26% without any re-training on the new data
cases, but if we average the outputs of our 4 HNN-I models from cross-
validation, we achieve 65.66% DSC. This demonstrates that HNN-I may be highly
generalizable in cross-dataset evaluation. Performance on that separated data
will likely improve with further fine-tuning.

Table 1: 4-fold cross-validation: The DSC and average minimum distance (Dist)
performance of our implementation of [6], optimally achievable superpixels,
HNN-I, and HNN-RF spatial aggregation, and DCRF (best performance in bold).

DSC[%] | [6] | Opt. | HNN-I | HNN-RF | DCRF
---|---|---|---|---|---
Mean | 71.42 | 88.08 | 76.99 | 78.01 | 77.14
Std | 10.11 | 2.10 | 9.45 | 8.20 | 10.58
Min | 23.99 | 81.24 | 24.11 | 34.11 | 16.10
Max | 86.29 | 92.00 | 87.78 | 88.65 | 88.30

Dist[mm] | [6] | Opt. | HNN-I | HNN-RF | DCRF
---|---|---|---|---|---
Mean | 1.53 | 0.15 | 0.70 | 0.60 | 0.69
Std | 1.60 | 0.08 | 0.73 | 0.55 | 0.76
Min | 0.20 | 0.08 | 0.17 | 0.15 | 0.15
Max | 10.32 | 0.81 | 5.91 | 4.37 | 5.71

##  4 Conclusion

In this paper, we present a holistic deep CNN approach for pancreas
segmentation in abdominal CT scans, combining interior and boundary mid-level
cues via spatial aggregation. Holistically-Nested Networks (HNN-I) alone
already achieve good performance on the pixel-labeling task for segmentation.
However, we show a significant improvement (p$<$0.001) by incorporating the
organ boundary responses from the HNN-B model. HNN-B can improve supervised
object proposals via superpixels and is beneficial to train HNN-RF that
spatially aggregates information on organ interior, boundary and location. The
highest reported DSCs of 78.01%$\pm$8.2% in testing is achieved, at the
computational cost of 2$\sim$3 minutes, not hours as in [1, 2, 3]. Our deep
learning based organ segmentation approach could be generalizable to other
segmentation problems with large variations and pathologies, e.g., tumors.

Figure 3: Examples of the RF pancreas segmentation (green) using the proposed
approach in testing with the manual ground truth annotation (red). Case with
DSC close to the data set mean and the maximum are shown. The percentange of
total cases that lie above a certain DSC with RF are shown on the right. 80%
of the cases achieve a minimum DSC of 74.13%, and 90% of the cases achieve a
DSC of 69.0% and higher.

##### Acknowledgments

This work was supported by the Intramural Research Program of the NIH Clinical
Center.

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