# Gap-Dependent Hydrodynamic Lubrication in Conformal Contacts

James A. Richards<EMAIL_ADDRESS>Edinburgh Complex Fluids
Partnership and School of Physics and Astronomy, The University of Edinburgh,
James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD,
United Kingdom  Patrick B. Warren<EMAIL_ADDRESS>The Hartree
Centre, STFC Daresbury Laboratory, Warrington WA4 4AD, United Kingdom  Wilson
C. K. Poon<EMAIL_ADDRESS>Edinburgh Complex Fluids Partnership and School of
Physics and Astronomy, The University of Edinburgh, James Clerk Maxwell
Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom

###### Abstract

We show that the hydrodynamic lubrication of contacting conformal surfaces
with a typical texture height gives rise to a universal behaviour in the
Stribeck curve in which the friction coefficient shows an anomalous power-law
dependence on the Sommerfeld number, $\mu\sim S^{2/3}$. When the gap height
drops below the ‘texture length scale’, deviations from $S^{2/3}$ occur, which
may resemble the onset of elasto-hydrodynamic and boundary lubrication. Within
this framework, we analyse literature data for oral processing and find
$S^{2/3}$ scaling with deviations consistent with measured lengthscales.

##  I Introduction

Figure 1: Schematic Stribeck curve, friction coefficient ($\mu=F/N$) as a
function of Sommerfeld number ($S=\eta UN/R$, for sliding speed $U$, lubricant
viscosity $\eta$ and radius of curvature of lubrication geometry $R$). Regimes
of $\mu$ with decreasing $S$: hydrodynamic lubrication (HL), elasto-
hydrodynamic lubrication (EHL) at minimum, and constant boundary lubrication
(BL, shading).

The importance of lubricated contacts between sliding surfaces cannot be
overstated [1, 2]. Such contacts are often characterised by a ‘pin-on-disc
test’, which is analysed in terms of a hemisphere of radius $R$ trapping a
lubricant of viscosity $\eta$ sliding on a flat surface. The friction
coefficient $\mu$, the ratio of the imposed normal force $N$ on the pin to the
tangential drag force $F$ on the disc, $\mu=\frac{F}{N}$, is measured as $N$
and the relative sliding speed $U$ vary, giving $\mu$ as a function of the
Sommerfeld number $S=\eta UR/N$ [3, 4, 5], a Stribeck curve.

The typical Stribeck curve shows three regimes, Fig. 1. At high $S$, $N$ is
purely hydrodynamic in origin [6]. In this hydrodynamic lubrication (HL)
regime, $\mu$ decreases as $N$ increases, reaching a minimum before upturning
due to the onset of elastohydrodynamic lubrication (EHL), where hydrodynamic
stresses begin to deform surface asperities, which however are not yet
touching. Touching signals the onset of the boundary lubrication (BL), where
$\mu$ becomes nearly constant as $S\to 0$.

Figure 2: Non-conformal lubrication. (a) Lubrication geometry, sphere or
cylinder with radius $R$, loading conditions with normal ($N$) and tangential
load ($F$) applied to upper surface [dark (grey) shading], with opposing
forces from fluid [light (blue) shading] due to relative sliding speed ($U$)
between upper and lower surfaces. (b) Narrow-gap region with gap $h(r)$ at a
distance $r\ll R$ from point of closest approach with gap $h_{0}$. Curved
surface in narrow-gap region approximated by parabola, $h(r)=h_{0}+r^{2}/2R$.
(c) Schematic of resulting fluid flow. Couette flow [(orange) dashed lines]
with changing gap results in compensating Poiseuille flow to give changing net
fluid flow [(blue) solid lines and arrows]. (d) Resulting Reynolds lubrication
pressure distribution $p(r)$ schematic. Symmetry implies equal and opposite
pressures for $r\to-r$, negative pressure [(red) shading] neglected due to
cavitation, the half-Sommerfeld boundary condition.

Hydrodynamic lubrication originates from volume conservation in the convergent
thin-film flow of an incompressible lubricant [7]. Consider a ball-on-flat
geometry, Fig. 2a-b, with a plane Couette flux $Uh/2$ (volume per unit width,
here and throughout) in the convergent gap due to the ball sliding along $x$
relative to the plane at velocity $-U$. This varies with the gap, Fig. 2c
(dashed red), and does not conserve volume. The imbalance is corrected by a
gap-dependent pressure-driven in-plane Poisueille flux,
$-\frac{h^{3}}{12\eta}\nabla p$, Fig. 2c (black). The associated pressure
field $p$ provides the normal force to keep the surfaces apart.

Mathematically, balancing the gap-dependent Couette ($\mathbf{J}_{c}$) and
Poisseuille ($\mathbf{J}_{p}$) fluxes translates to
$\nabla\cdot(\mathbf{J}_{c}+\mathbf{J}_{p})=0$. Noting that $\mathbf{J}_{c}$
is exclusively in the $x$ direction while $\mathbf{J}_{p}$ can in general lie
in the $(x,y)$ plane, we therefore require

$\frac{\partial}{\partial
x}\Bigl{(}\frac{Uh}{2}\Bigr{)}-\left[\frac{\partial}{\partial
x}\Bigl{(}\frac{h^{3}}{12\eta}\frac{\partial p}{\partial
x}\Bigr{)}+\frac{\partial}{\partial
y}\Bigl{(}\frac{h^{3}}{12\eta}\frac{\partial p}{\partial
y}\Bigr{)}\right]=0\,.$ (1)

In this Reynolds lubrication equation [6], the pressure $p(x,y)$ is
independent of the vertical coordinate because the gap $h(x,y)$ is everywhere
small compared to the flow’s length and width. Over the bearing area, this
pressure supports $N$, while $F$ has Couette and Poiseuille components:

$N=\int\\!\\!\mathrm{d}{x}\,\mathrm{d}{y}\,p\,,\quad
F=\int\\!\\!\mathrm{d}{x}\,\mathrm{d}{y}\left[\frac{\eta
U}{h}+\frac{h}{2}\left(\frac{\partial p}{\partial x}+\frac{\partial
p}{\partial y}\right)\right].$ (2)

Analytic solutions to Eq. (1) exist only for a few cases. For a ball (radius
$R$) on a flat surface with minimum gap $h_{0}\ll R$, Fig. 2a, Kapitza gave
$h_{0}=(72\pi^{2}/25)RS^{2}\approx 28.4RS^{2}$ [8] while Hamrock has
$h_{0}\approx 34.8RS^{2}$ [1], reflecting different approximations. They also
calculated $p(x,y)$. Similar results for other canonical geometries are
presented in textbooks [2, 1, 9, 10]. Curiously, however, there are few
calculations of and experimental data for $\mu(S)$ in the HL regime. So, while
HL is widely thought to be understood, much of the theory remains untested by
direct experiments.

In an accompanying paper [11], we address this knowledge gap for conformal
contacts, i.e. surfaces with commensurate curvature. We outline scaling
arguments predicting an unexpected $\mu\sim S^{2/3}$ behaviour at large $S$ in
the HL regime, and deviations below some critical $S^{\star}$. Experiments
confirm these predictions. Here, we justify our scaling arguments and present
full calculations, paying particular attention to deviations from scaling
around $S^{\star}$, which may resemble the transition to EHL. Below, after
recalling the scaling theory for the ball-on-flat geometry [12], we report
detailed scaling analyses for conformal contacts to reveals the physics
underlying the $S^{2/3}$ scaling and deviations from it at low $S$. Solving
Eq. (1) by quadrature follows to explore these deviations in depth. We end by
applying our results to some recently-published data [13] to demonstrate the
practical utility of our analysis.

##  II Scaling theory of non-conformal contacts

The two surfaces in a non-conformal contact have different curvatures. Kapitza
[8] solved the sphere on flat, Fig. 2a, analytically. The physics emerges from
a scaling argument [12].

First, expand the convergent gap around its narrowest point, $h(r)\approx
h_{0}+r^{2}/2R$, Fig. 2b. In a region of radius $r_{0}\sim\sqrt{h_{0}R}$, the
gap remains narrow, and we expect from Eq. (1) that

$\frac{Uh_{0}}{r_{0}}\sim\frac{1}{r_{0}}\times\frac{h_{0}^{3}}{\eta}\frac{p}{r_{0}}\quad\Rightarrow\quad
p\sim\frac{\eta Ur_{0}}{h_{0}^{2}}\,.$ (3)

This supports load $N\sim pr_{0}^{2}\sim\eta UR^{3/2}/h_{0}^{1/2}$, or
$h_{0}/R\sim(\eta UR/N)^{2}\sim S^{2}$, confirming Kapitza [8]. Symmetry
predicts equal but opposite pressure in the diverging gap. If this negative
pressure is significantly below atmospheric and the lubricant cavitates, then
the ‘half Sommerfeld boundary condition’ of $p=0$ applies, giving a net normal
force.

The narrow-gap shear rate $\dot{\gamma}\sim U/h_{0}$ gives a shear stress
$\sigma\sim\eta U/h_{0}$ and lateral force $F\sim\sigma r_{0}^{2}\sim\eta UR$,
so that

$\mu=\frac{F}{N}\sim\frac{\eta UR}{N}\sim S\,.$ (4)

Such linear scaling is often assumed in sketches of the Stribeck curve in the
HL regime (e.g., Fig. 2 in Ref. 14).

For a long cylinder, $x_{0}\sim\sqrt{h_{0}R}$ in the narrow gap region. If its
length $L$ satisfies $L\gg x_{0}$, then lubricant flow is exclusively along
the direction of motion rather than through side leakage. The pressure has the
same form as the ball-on-flat case, but now acts over area $\sim x_{0}L$, not
$r_{0}^{2}$, which support a load $N\sim x_{0}Lp_{0}\sim x_{0}^{2}\eta
UL/h_{0}^{2}\sim\eta UL(R/h_{0})$, with associated drag $F\sim\sigma
x_{0}L\sim\eta UL\sqrt{R/h_{0}}$. Now we predict

$S\equiv\frac{\eta
UL}{N}\sim\frac{h_{0}}{R}\,,\quad\mu\sim\sqrt{\frac{h_{0}}{R}}\quad\Rightarrow\quad\mu\sim
S^{1/2}\,.$ (5)

Apparently, the HL of non-conformal contacts gives $\mu\sim S^{\alpha}$ with
$\alpha$ set by the spatial dimension. However, few published Stribeck curves
show either of these scalings.

##  III Scaling theory of conformal contacts

Figure 3: (a) A textured surface of back-to-back wedge bearings. (b) Wedge
profile specified by length $D$ and ‘step height’ $d$ as decrease in gap
($h_{0}+d$ to $h_{0}$) from $x=0$ to $x=D$. Dimensionless gap profile
$\xi=z+1-u$ for scaled co-ordinates, $z=h_{0}/d$ and $u=x/D$, Eq. (A1). (c)
Inlet–half-cylinder profile. Cylinder approximated by parabolic form from
$u=\chi$ to 1, Eq. (A5), inlet region, $h_{0}+d$ at $u=0$ to $\chi$ (or
$x=\chi D$). Increasing $\chi$, and larger inlet, shown from light to dark
shading and with fine to bold lines. (d) ‘Rayleigh’ step gap profile with
equivalent bearing dimensions, Eq. (A7).

The ball or cylinder on plane is characterized by a single length scale, viz.,
the sphere or cylinder radius, $R$. A contact between two equal-curvature
surfaces apparently has no intrinsic length scale. However, the surfaces in a
real conformal contact will show a degree of mesoscale non-flatness with
characteristic length scale $d\gg$ typical asperities. They are therefore
‘textured surfaces’ [15] in which the gap height varies from some minimum
$h_{0}$ to $h_{0}+d$ over the area of contact, Fig. 3a. We restrict ourselves
to surface undulations along the sliding direction of the conformal contact.
Assuming the half-Sommerfeld boundary condition, we consider only the
converging gap regions, each of which is then a slider bearing of length $D\gg
d$ and width $L\gg d$.

For the three slider bearings in Fig. 3, a wedge, a ‘Rayleigh step’, and a
half-cylinder inlet, there is a characteristic ‘step height’, $d$. Two limits
exist, Fig. 4, the ‘long bearing’, $L\gg D$ (Fig. 4a), or the the ‘short
bearing’, $L\ll D$ (Fig. 4b), where volume conservation is ensured by side-
leakage [16]. In each case, there is a large-gap regime ($h_{0}\gg d$) and a
small-gap regime ($h_{0}\ll d$), with cross-over at $h_{0}/d\approx 1$.

Figure 4: Fluid flow in slider bearings. (a) Long-bearing limit with length
much less than width ($D\ll L$) resulting in compensating Poiseuille flow
[dark(blue) arrows] in same $x$ direction as Couette shear flow [light
(orange) arrows)]. (b) Corresponding short-bearing limit with length much
greater than width ($D\gg L$) leading to “side-leakage” in $y$ direction
perpendicular to Couette flow.

Full calculations, Sections IV and V, predict the Stribeck curves shown in
Fig. 5. To highlight the physics, we first give scaling analyses for the
universal power-law behaviour at high $S$ and for the details-dependent limit
$S\to 0$.

###  III.1 Large-gap limit, $h_{0}\gg d$

For a long slider bearing in this limit, $\partial/\partial y=0$ and the gap
narrows over $D$ from $h_{0}+d$ to $h_{0}$, and Eq. (1) leads us to expect

$\frac{Ud}{D}\;\sim\;\Bigl{(}\,\frac{1}{D}\times\frac{h_{0}^{3}}{\eta}\frac{p}{D}\,\Bigr{)}\;+\;\Bigl{(}\,\frac{1}{L}\times\frac{h_{0}^{3}}{\eta}\frac{p}{L}\,\Bigr{)}\,.$
(6)

With $D\ll L$, we drop the second term on the RHS: the Poiseuille flow is
predominantly along $x$, Fig. 4a. So,

$p\sim\frac{\eta UDd}{h_{0}^{3}}\,.$ (7)

This and shear stress $\sigma\sim\eta U/h_{0}$ [17] act on area $LD$ to give

$\begin{split}&N\sim\frac{\eta UdLD^{2}}{h_{0}^{3}}\,,\quad F\sim\frac{\eta
U\\!LD}{h_{0}}\\\\[3.0pt] &{}\qquad\Rightarrow\quad
S\sim\frac{d^{2}}{D^{2}}\Bigl{(}\frac{h_{0}}{d}\Bigr{)}^{3}\,,\quad\mu\sim\frac{d}{D}\Bigl{(}\frac{h_{0}}{d}\Bigr{)}^{2}\\\\[6.0pt]
&{}\hskip
100.00015pt\Rightarrow\quad\mu\sim\Bigl{(}\frac{D}{d}\Bigr{)}^{1/3}S^{2/3}\,.\end{split}$
(8)

The ‘anomalous exponent’ of $2/3$ arises from $r_{0}=\sqrt{h_{0}R}$ in Eq.
(3), because a sphere is characterised by a single length scale $R$, while the
$d$ and $D$ in Eq. (7) are constants unrelated to $h_{0}$, i.e. a slider
bearing is specified by two length scales.

In the short-bearing limit ($L\ll D$), we drop the first term on the RHS of
Eq. (6): volume conservation is now entirely managed by side-leakage [16],
Fig. 4b, and we have

$p\sim\frac{\eta UdL^{2}}{Dh_{0}^{3}}\,.$ (9)

This and shear stress $\sigma\sim\eta U/h_{0}$ act on area $LD$ to give

$\begin{split}&N\sim\eta UdL^{3}/h_{0}^{3}\,,\quad F\sim\frac{\eta
U\\!LD}{h_{0}}\\\\[3.0pt] &{}\qquad\Rightarrow\quad
S\sim\frac{d^{2}}{L^{2}}\Bigl{(}\frac{h_{0}}{d}\Bigr{)}^{3}\,,\quad\mu\sim\frac{Dd}{L^{2}}\Bigl{(}\frac{h_{0}}{d}\Bigr{)}^{2}\\\\[6.0pt]
&{}\hskip
100.00015pt\Rightarrow\quad\mu\sim\Bigl{(}\frac{D^{3}}{L^{2}d}\Bigr{)}^{1/3}S^{2/3}.\end{split}$
(10)

The anomalous scaling therefore remains unchanged in the short-bearing limit,
which only introduces new pre-factors.

###  III.2 Small-gap limit, $h_{0}\lesssim d$

Now, the gap profile matters. Figure 3 (a)-(c) show increasing abruptness of
the transition from $h(x=0)=h_{0}+d$ to $h(D)=h_{0}$, with $\chi$ controlling
the inlet length in (b) and (c).

####  III.2.1 Wedge

Figure 5: Stribeck curves, $\mu(S)$, for various gap profiles. (a) Long-
bearing limit, friction coefficient normalised by geometric pre-factors
[$\mu/(d/D)$] as a function of Sommerfeld number scaled by geometric factor
[$S/(d^{2}/D^{2})$] from Eqs. (27) and (28). High-$S$ scaling of $\mu\sim
S^{2/3}$ shown by dotted line. Solid lines: black, wedge profile slider
bearing, Eq. (11); dark (orange), inlet–half-cylinder, Eq. (A5); and, light
(blue), Rayleigh step profile Eq. (21). For step and inlet–half cylinder inlet
length $\chi=0.5$. (b) Running exponent $\alpha$ as a function of reduced gap,
$h_{0}/d$. Lines as in (a). (c) Short-bearing limit for same profiles [dashed
lines, see legend in (a)] from Eqs. (32) and (33). Modified geometric pre-
factors, $\mu/(dD/L^{2})$ and $S/(d^{2}/L^{2})$; $\chi=0.4$. (d) Respective
running exponent.

As $h_{0}\to 0$ in a long wedge with profile

$h=h_{0}+d\left(1-\frac{x}{D}\right)\,,$ (11)

the contact becomes effectively non-conformal with a small-gap region of
extent $x_{0}\sim h_{0}D/d$. From Eq. (1) we expect

$\frac{1}{x_{0}}\times
Uh_{0}\sim\frac{1}{x_{0}}\frac{h_{0}^{3}}{\eta}\frac{p}{x_{0}}\quad\Rightarrow\quad
p\sim\frac{\eta Ux_{0}}{h_{0}^{2}}\,.$ (12)

This pressure and shear stress $\eta U/h_{0}$ acting over area $x_{0}L$ gives

$N\sim\frac{\eta ULD^{2}}{d^{2}}\,,\quad F\sim\frac{\eta
UDL}{d}\quad\Rightarrow\quad\mu\sim\frac{d}{D}\,,$ (13)

i.e. a constant, Fig. 5a (black lines), with the running power law of the
Stribeck curve (i.e. the slope in a double-log plot),
$\alpha=\mathrm{d}{\ln}\mu/\mathrm{d}{\ln}S\to 0$, Fig. 5b. As $h_{0}\to 0$,
our problem becomes equivalent to ‘Taylor scraping flow’ [18] with vanishing
wedge angle, where there are logarithmic divergences in the forces; these are
missed in our scaling analysis, but will emerge in the full calculations in §
IV.2.

For a short bearing, we again consider an effective non-conformal contact with
$x_{0}\sim h_{0}D/d$. Equation (1) now gives

$\frac{Uh_{0}}{x_{0}}\;\sim\;\Bigl{(}\,\frac{1}{x_{0}}\times\frac{h_{0}^{3}}{\eta}\frac{p}{x_{0}}\,\Bigr{)}\;+\;\Bigl{(}\,\frac{1}{L}\times\frac{h_{0}^{3}}{\eta}\frac{p}{L}\,\Bigr{)}\,.$
(14)

If we are only just into the small-gap limit ($h_{0}/d\lesssim 1$), then for a
short bearing ($L\ll D$), $x_{0}/L=(h_{0}/d)(D/L)\gg 1$, and we can neglect
the first term on the RHS of Eq. (14) and obtain

$p\sim\frac{\eta UL^{2}}{x_{0}h_{0}^{2}}\,.$ (15)

This pressure and shear stress $\eta U/h_{0}$ acts on area $x_{0}L$ to give

$\begin{split}&N\sim p\times x_{0}L\sim\frac{\eta UL^{3}}{h_{0}^{2}}\,,\quad
F\sim x_{0}L\times\frac{\eta U}{h_{0}}\sim\frac{\eta ULD}{d}\\\\[3.0pt]
&{}\qquad\Rightarrow\quad S=\frac{\eta
UL}{N}\sim\left(\frac{h_{0}}{L}\right)^{2},\quad\mu\sim\frac{Dh_{0}^{2}}{dL^{2}}\sim\left(\frac{D}{d}\right)S\,,\end{split}$
(16)

which is a linear scaling, Fig. 5c and d (black dashed line).

Deep into the small gap regime, $h_{0}/d\lll 1$, and
$x_{0}/L=(h_{0}/d)(D/L)\ll 1$ even for a short bearing ($D/L\gg 1$).
Discarding the second term on the RHS of Eq. (14), we find

$p\sim\frac{\eta Ux_{0}}{h_{0}^{2}}\quad\Rightarrow\quad N\sim p\times
x_{0}L\sim\eta UL\left(\frac{D}{d}\right)^{2}.$ (17)

With $F$ the same as before, we find

$\mu\sim\frac{d}{D}\,,$ (18)

as in the case of the long bearing in the same small-gap limit.

####  III.2.2 Inlet–half-cylinder

Consistent with the assumptions underpinning Eq. (1), we describe the gap in
the inlet–half-cylinder by a parabola, Fig. 3b,

$h=\Bigl{\\{}\begin{array}[]{lll}h_{0}+d&&0\leq x/D<\chi\,,\\\
h_{0}+d(1-x/D)^{2}/(1-\chi)^{2}&&\chi\leq x/D\leq 1\,,\end{array}$ (19)

with $0\leq\chi\leq 1$. The low-$S$ limit for the long bearing is simple. For
a small gap, the flat inlet region becomes inconsequential and we have a
cylinder-on-flat geometry with $\mu\sim S^{1/2}$, Fig. 5a and b [dark
(orange)].

In the small-gap limit for a short bearing, leakage occurs sideways, so that
we need only consider a small region near the bottom of the half cylinder
($x=D$ in our coordinates) where the gap remains of order $h_{0}$. Equation
(19) shows that this region has dimension $x_{0}=(1-\chi)D\sqrt{h_{0}/d}$. The
rest of the scaling analysis proceeds along the same lines as that for the
wedge in the thin-film limit, only with a different expression for $x_{0}$.

Just into the thin-film regime, $x_{0}/L\gg 1$, we now find, Fig. 5b and d
[dashed dark (orange)],

$\mu\sim\frac{h_{0}}{L^{2}}(1-\chi)D\sqrt{\frac{h_{0}}{d}}\sim\left[\frac{(1-\chi)D}{(Ld)^{1/2}}\right]S^{3/4}.$
(20)

Deep into the thin-film regime, $x_{0}/L\ll 1$, and again following the
analysis for the wedge with a new expression for $x_{0}$ in Eq. (17) we find
$\mu\sim S^{1/2}$, with geometric pre-factors cancelling and the expected
recovery of a long cylinder expression.

####  III.2.3 Rayleigh step

The gap in the Rayleigh step is piecewise continuous,

$h=\Bigl{\\{}\begin{array}[]{lll}h_{0}+d&&0\leq x/D<\chi\,,\\\ h_{0}&&\chi\leq
x/D\leq 1\,.\end{array}$ (21)

For a long-bearing with this profile in the small-gap limit ($h_{0}\ll d$),
the Couette flux in the thin gap region is negligible, and the compensating
Poiseuile flux exists almost entirely to balance the Couette flux in the inlet
of length $\chi D$ and height $d+h_{0}\approx d$. From Eq. (1), we have

$\frac{Ud}{\chi D}\sim\frac{1}{\chi
D}\times\frac{d^{3}}{12\eta}\times\frac{p}{\chi D}\quad\Rightarrow\quad
p\sim\frac{\eta U\chi D}{d^{2}}\,.$ (22)

This acts over the whole length of the bearing (even though the flow in the
narrow gap is negligible) to give the normal force, while the tangential load
is dominated by the narrow gap:

$N\sim\frac{\eta U\\!\chi D^{2}L}{d^{2}}\,,\quad F\sim\frac{\eta
U}{h_{0}}(1-\chi)DL\,,$ (23)

i.e., $N$ is a constant, while $F$ diverges as the gap narrows. So, we expect
that as $S$ decreases, $\mu\sim F/N$ must reach a minimum and then upturns and
diverges as $S=\eta UL/N$ approaches some constant value
$\sim\chi^{-1}(d/D)^{2}$, Fig. 5a and b [light (blue)].

In the short Rayleigh step, sideways Poisueille leakage occurs over the
infinitesimal width of the step itself, so that the conditions necessary for
the validity of the Reynolds lubrication formalism in Eq. (1) no longer apply.

###  III.3 Summary of scaling analysis

Our scaling analysis gives is a universal ‘anomalous’ regime of $\mu\sim
S^{2/3}$ at large $S$ for all three geometries in Fig. 3. When $h_{0}\to d$,
deviations from this scaling occurs. The details are geometry dependent, with
the $S\to 0$ limit derivable by scaling analysis except in the pathological
case of the short Rayleigh step. Solving Eq. (1) by quadrature in Sections IV
and V will verify these scaling conclusions and provide closed forms for
plotting the behaviours shown in Fig. 5 and Table 1 (and Fig. 3 of Ref. 11).
Further, we will show that for a broad class of profiles, with small inlets,
the deviation from $S^{2/3}$ scaling is upwards as $S$ decreases, which mimics
entry into the BL regime even though lubrication remains hydrodynamic in
origin. Understanding such deviation then allows, in § VI, a general way of
analysing experimental data to extract a surface texturing length scale from
the Stribeck curve.

##  IV Exact lubrication calculations and limits

###  IV.1 General lubrication calculations

We start by deriving a general expression for the Reynolds lubrication
pressure, $p(x,y)$ from Eq. (1) in the long- and short-bearing limits, Fig. 4a
and b respectively. For completeness, we include some material that is
available in textbooks [1].

####  IV.1.1 Long slider bearings ($D\gg L$)

With negligible edge effects and $p=p(x)$, $\partial_{y}\rightarrow 0$, and we
can integrate Eq. (1) with respect to $x$,

$\frac{\partial p}{\partial x}=6\eta
U\Bigl{(}\frac{1}{h^{2}}-\frac{h_{m}}{h^{3}}\Bigr{)}\,,$ (24)

where $h_{m}$ is a constant. Define a dimensionaless gap $\xi=h/d$, where $d$
is some fiducial length such as a ‘step’ height, or the difference in height
across the width of the bearing [$d=h(0)-h(D)$]. In terms of a dimensionless
minimum gap $z=h_{0}/d$, the different bearing profiles are distinguished by
$\delta(u)=\xi(u)-z$, where we have used the dimensionless length $u=x/D$ so
that $0\leq u\leq 1$ spans the length of the bearing. Hereafter, the length-
wise average $\langle\cdots\rangle$ is given by

$\langle{\cdots}\rangle=\frac{1}{D}\int_{0}^{D}\\!\\!\mathrm{d}{x}\,(\cdots)=\int_{0}^{1}\\!\\!\mathrm{d}{u}\,(\cdots)\,.$
(25)

Integrating Eq. (24) gives the pressure drop across the bearing

$\Delta p=6\eta
UD(\langle{\xi^{-2}}\rangle-\xi_{m}\langle{\xi^{-3}}\rangle)/d^{2}\,.$ (26)

Assuming equal inlet and outlet pressures, $\Delta p=0$ and
$\xi_{m}=\langle{\xi^{-2}}\rangle/\langle{\xi^{-3}}\rangle$. Integrating $p$
to obtain the normal and friction forces, Eq. (2), the transverse ($y$)
integral trivially gives a factor of the width $L$. We deal with the $x$
integral in the normal force by integrating Eq. (24) by parts,
$\int\\!\mathrm{d}{x}\,p=-\int\\!\mathrm{d}{x}\,x(\partial p/\partial x)$,
again assuming $\Delta p=0$. After some algebra, we find

$\displaystyle N=\frac{6\eta
U\\!LD^{2}}{d^{2}}\,\frac{\langle{u\xi^{-3}}\rangle\langle{\xi^{-2}}\rangle-\langle{u\xi^{-2}}\rangle\langle{\xi^{-3}}\rangle}{\langle{\xi^{-3}}\rangle}=\frac{\eta
U\\!LD^{2}}{d^{2}}g_{\delta}(z)\,,$ (27) $\displaystyle F=\frac{\eta
U\\!LD}{d}\,\frac{4\langle{\xi^{-1}}\rangle\langle{\xi^{-3}}\rangle-3\langle{\xi^{-2}}\rangle^{2}}{\langle{\xi^{-3}}\rangle}=\frac{\eta
U\\!LD}{d}f_{\delta}(z)\,,$ (28)

where the subscript in the dimensionless normal load, $g(z)$, and drag,
$f(z)$, points to the dependence on the gap profile $\delta(u)=\xi(u)-z$.
These expressions imply $\mu=F/N\sim d/D$ and $S=\eta UL/N\sim d^{2}/D^{2}$,
reproducing the geometric factors in Eq. (8) established by scaling analysis.

The Stribeck curve, $\mu(S)$, then follows parametrically as the gap $z$ (and
so $\langle{\xi^{-n}}\rangle$) varies as a control parameter. Note that the
higher-order averages required in Eqs. (27) and (28) can be computed from
$\langle{\xi^{-1}}\rangle$ and $\langle{u\,\xi^{-1}}\rangle$ from recurrence:

$\langle{\xi^{-n-1}}\rangle=-\frac{1}{n}\frac{\mathrm{d}{\langle{\xi^{-n}}\rangle}}{\mathrm{d}{z}}\,,\quad\langle{u\,\xi^{-n-1}}\rangle=-\frac{1}{n}\frac{\mathrm{d}{\langle{u\,\xi^{-n}}\rangle}}{\mathrm{d}{z}}\,,$
(29)

which follow by differentiating Eq. (25) and using $\xi=z+\delta$.

####  IV.1.2 Short slider bearings ($D\ll L$)

In this limit, we neglect $\partial p/\partial x$ in Eq. (1) and, recalling
that $h$ is $y$-independent, the equation simplifies to

$h^{3}\frac{\partial^{2}p}{\partial y^{2}}=6\eta
U\frac{\mathrm{d}{h}}{\mathrm{d}{x}}\,.$ (30)

Fixing $p(y=0)=p(y=L)=0$, i.e. taking $p$ as the excess pressure, this
integrates to

$p=-\frac{3\eta U}{h^{3}}\frac{\mathrm{d}{h}}{\mathrm{d}{x}}\times y(L-y)\,,$
(31)

giving the normal load

$\begin{split}N=3\eta
U\,\frac{L^{3}}{6}\int_{0}^{1}\\!\frac{\mathrm{d}{u}}{\xi^{3}}\,\frac{\mathrm{d}{\xi}}{\mathrm{d}{u}}=\frac{\eta
U\\!L^{3}}{d^{2}}\,\frac{(2z+1)}{4z^{2}(z+1)^{2}}=\frac{\eta
U\\!L^{3}}{d^{2}}g(z)\,.\end{split}$ (32)

Now, $g(z)$ does not depend on the bearing profile, $\delta$, but only the
dimensionless minimum gap, $z$. The drag also simplifies due to the absence of
Poiseuille flow along the sliding direction,

$F=L\int_{0}^{D}\\!\\!dx\,\frac{\eta U}{h}=\frac{\eta
U\\!LD}{d}\,f_{\delta}(z)\,,$ (33)

with $f_{\delta}(z)=\langle{\xi^{-1}}\rangle$. From Eqs. (32) and (33), $S$
and $\mu$ can again be derived with the geometric pre-factors previously
established in the scaling argument, Eq. (10).

####  IV.1.3 Power-law expansion

In the large-gap limit, $z\gg 1$. For the short bearing, expanding Eq. (32)
gives $g(z)\sim z^{-3}$. For the long bearing, we need to expand the
$\xi$-dependent numerators in Eqs. (27) in the small quantity
$\delta(u)/z\sim\mathcal{O}({1/z})$, whereupon we find

$\displaystyle\langle{\xi^{-n}}\rangle$
$\displaystyle=\int_{0}^{1}\\!\frac{\mathrm{d}{u}\,z^{-n}}{[1+\delta(u)/z]^{-n}}\approx
z^{-n}(1-n\langle{\delta}\rangle/z)\,,$ (34)
$\displaystyle\langle{u\xi^{-n}}\rangle$
$\displaystyle=\int_{0}^{1}\\!\frac{\mathrm{d}{u}\,z^{-n}u}{[1+\delta(u)/z]^{-n}}\approx
z^{-n}(1/2-n\langle{u\delta}\rangle/z)\,.$ (35)

The leading order terms cancel, leaving

$g(z)\approx\frac{6z^{-5}(\langle{\delta}\rangle/2z-\langle{u\delta}\rangle/z}{z^{-3}}=\frac{3(\langle{\delta}\rangle-2\langle{u\delta}\rangle)}{z^{3}}\sim\mathcal{O}({\frac{1}{z^{3}}})\,,$
(36)

as in the short-bearing limit. In the final step we have taken the gap profile
averages to be of order the ‘step’ height and not cancel, which is correct for
all profiles apart from the case of $\chi=1$, which violates the lubrication
approximation. With the tangential load $f(z)\sim z$ in all cases (as
$\langle{\xi^{-n}}\rangle\rightarrow z^{-n}$), we recover $S\propto z^{3}$
with $\mu\propto z^{2}\propto S^{2/3}$, as previously found.

In the small-gap limit, $z+1\rightarrow 1$ and $g(z)=1/4z^{2}\propto z^{-2}$
from Eqs. (27) and (32). This gives $S\propto z^{2}$ and $\mu\propto z$, i.e.
$\mu\propto S^{1/2}$ for all bearing profiles and types, contradicting our
scaling predictions. A more rigorous analysis is needed.

###  IV.2 Specific gap profiles

Profile | $g(z)\propto N$ | $f(z)\propto F$ | $S(z)$ | $\mu(z)$ | $\mu(S)$
---|---|---|---|---|---
Wedge: Long | $6\ln z^{-1}$ | $4\ln z^{-1}$ | $\textstyle\frac{1}{6\ln z^{-1}}$ | $\textstyle\frac{2}{3}$ | $\sim S^{0}$
Short | $\textstyle\frac{1}{4z^{2}}$ | $\ln z^{-1}$ | $4z^{2}$ | $4z^{2}\ln z^{-1}$ | $\sim S^{1}$
IHC: Long | $\textstyle\frac{3(1-\chi)^{2}}{z}$ | $\textstyle\frac{\pi(1-\chi)}{z^{1/2}}$ | $\textstyle\frac{z}{3(1-\chi)^{2}}$ | $\textstyle\frac{\pi z^{1/2}}{3(1-\chi)}$ | $\sim S^{1/2}$
Short | $\textstyle\frac{1}{4z^{2}}$ | $\textstyle\frac{\pi(1-\chi)}{2\sqrt{z}}$ | $4z^{2}$ | $2\pi(1-\chi)z^{3/2}$ | $\sim S^{3/4}$
Step: Long | $3\chi$ | $\textstyle\frac{1-\chi}{z}$ | $\textstyle\frac{1}{3\chi}$ | $\textstyle\frac{1-\chi}{3\chi z}$ | $\to\infty$
Table 1: Small-gap scaling behaviour, $z\ll 1$, for varying gap profiles,
$\delta(u)$, in both the long and short bearing limits (first column) for
wedge, inlet–half-cylinder (IHC) and step. Scaling is given for the normalised
normal load ($g$), tangential load ($f$), Sommerfeld number ($S$) and friction
coefficient ($\mu$) in terms of the minimum gap, respectively. The final
column is the resultant low-$z$ scaling of the Stribeck curve, $\mu(S)$.

For each gap profile, full expressions for the load and drag in the long, Eqs.
(27)–(28), and short, Eqs. (32)–(33), bearing limits can be calculated, see
Appendix, giving the Stribeck curves in Fig. 6 and 7 respectively, and small-
gap scaling limits in Table 1. For all step types and most inlet length
length, we indeed find (see Appendix) $\mathcal{O}(1)$ pre-factors. The
physical picture underlying our scaling analysis therefore holds. Such
analysis can go further: it can treat a changing Poiseuille flow direction
with reducing gap and predicts the recovery of the long bearing limit for
short bearings for $h_{0}\lll d$.

However, one discrepancy highlights an important limitation of our narrow-gap
scaling approach, which gives a constant $\mu$ as $h_{0}\rightarrow 0$ because
both $N$ and $F$ individually approach constant values. From quadrature, we
see that $N$ and $F$ both diverge logarithmically, Table 1, but these cancel
to give a constant $\mu$. In considering only a narrow-gap region with $h\sim
h_{0}$, the cumulative effects of the region with a larger gap are neglected,
as the Poiseuille pressure drop and Couette drag fall off with increasing
$h_{0}$. Where the gap increases quadratically this assumption is correct
[12]. However, it is not strictly correct for a linear profile.

Figure 6: Load and drag dependence with gap height for long-bearing conformal
‘steps’. Dimensionless load, $g=N/(\eta ULD^{2}/d^{2})$, and drag, $f=F/(\eta
ULD/d)$, as a function of reduced gap, $z=h_{0}/d$. (a) Wedge profile load
(solid line) with large gap limit applying to all profiles, $N\propto z^{-3}$
(dashed line), and profile-specific low-$z$ limit, $N\propto\ln(z^{-1})$
(dotted line). (b) corresponding drag with high-$z$ limit, $F\propto z^{-2}$
(dashed line), and low-$z$ limit, $F\propto\ln(z^{-1})$ (dotted line). (c)
Load for inlet–half-cylinder with increasing inlet length, $\chi$ (dark to
light), see legend in (d) for values. (d) Corresponding drag. (e) Load for
Rayleigh step profile with increasing $\chi$, see (f) for values, and low-$z$
limit $\propto z^{-1}$. (f) Corresponding drag. Figure 7: Short bearing load
and drag gap dependence with changing step profile. (a) Dimensionless load,
$g(z)=N/(\eta UL^{3}/d^{2})$, as a function of reduced gap, $z=h_{0}/d$, for
all short-bearing profiles. Scaling in large-gap limit $z\gg 1$, $g\propto
z^{-3}$ (dashed line), and scaling in limit of $z\ll 1$, $g\propto z^{-2}$
(dotted line). (b) Dimensionless drag, $f(z)=F/(\eta ULD/d)$, for wedge gap
profile, Eq. (A1), bold (black) line. Scaling for $z\gg 0$, $f\propto z^{-1}$
applies to all profiles (dashed lines); scaling for $z\ll 1$, $f\propto\ln
z^{-1}$ (dotted line). Inlet–half-cylinder profile, Eq. (A5), with increasing
$\chi$, dark (blue) to light (gold) fine lines. Scaling for $z\ll 1$,
$f\propto z^{-1/2}$. See Table 1 for resulting scaling of Stribeck curve
$\mu(S)$ for $z\to 0$.

##  V Transition from large-gap scaling

To probe how deviations from $\mu\sim S^{2/3}$ scaling first emerges as
$h_{0}\to d$, which will be important for analysing experiments, consider the
‘running exponent’ of the Stribeck curve,

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{\mathrm{d}{\,\ln\mu}}{\mathrm{d}{\,\ln
S}}=1-\frac{\mathrm{d}{\,\ln F}}{\mathrm{d}{\,\ln N}}=1-\frac{\mathrm{d}{\,\ln
f}}{\mathrm{d}{\,\ln g}}$ (37) $\displaystyle=$ $\displaystyle
1-\frac{\mathrm{d}{\,\ln f}}{\mathrm{d}{z}}\left(\frac{\mathrm{d}{\,\ln
g}}{\mathrm{d}{z}}\right)^{-1}\\!\\!=1-\frac{1}{f}\frac{\mathrm{d}{f}}{\mathrm{d}{z}}\left(\frac{1}{g}\frac{\mathrm{d}{g}}{\mathrm{d}{z}}\right)^{-1}\\!\\!.$

###  V.1 Short-bearing limit

Given $g(z)$ for all profiles, Eq. (32), and the recurrence relation for the
derivative of $f(z)$, Eq. (29), we find

${\alpha}=1-\frac{z(z+1)(2z+1)}{2(3z^{2}+3z+1)}\times\frac{\langle{\xi^{-2}}\rangle}{\langle{\xi^{-1}}\rangle}\,.$
(38)

The moments of $\xi$ can be recast explicitly in terms of the gap profile,
$\xi(u)=z+\delta(u)$. In the asymptotic limit of large $z$,

$\Bigr{\langle}{\frac{1}{\xi^{n}}}\Bigl{\rangle}=\frac{1}{z^{n}}\Bigl{[}1-\frac{n\langle{\delta}\rangle}{z}+\mathcal{O}\Bigl{(}\frac{1}{z^{2}}\Bigr{)}\Bigr{]}\,.$
(39)

Then, within the short bearing approximation,

$\alpha=\frac{2}{3}+\frac{\langle{\delta}\rangle-1/2}{3z}+\mathcal{O}\Bigl{(}\frac{1}{z^{2}}\Bigr{)}\,.$
(40)

The sign of initial deviations from $\alpha=2/3$ depends on the average
profile height $\langle{\delta}\rangle$. The short inlet–half-cylinder is
‘blunt’, $\langle{\delta}\rangle<1/2$, only when $\chi$ is small enough, Fig.
8a [light (gold) to dark (blue) for decreasing $\chi$], in which case $\alpha$
initially drops below 2/3 and the Stribeck curve bends upwards from the region
of universal $S^{2/3}$ scaling, mimicking the onset of EHL. The onset of
deviations from $\alpha=2/3$ for the short wedge, Fig. 8a (black dashed), is
at significantly smaller $z$ than for the inlet–half-cylinder because
$\langle{\delta}\rangle=1/2$, so that deviations, which are always positive,
scale as $z^{-2}$.

Figure 8: Running exponent $\alpha=\mathrm{d}{\ln}\mu/\mathrm{d}{\ln}S$ vs gap
around $z\approx 1$. $\alpha<2/3$ corresponding to upwards deviations in
Stribeck curve. (a) Deviation from large-gap scaling (dotted line) for short-
bearing limit. Dashed lines: bold (black), wedge; fine, inlet–half-cylinder
with increasing inlet, $\chi$, dark (purple) to light (yellow), see legend for
values. (b) Long-bearing limit wedge. (c) Long inlet–half-cylinder with
increasing inlet length [dark (purple) to light (yellow)], see legend for
values. (d) Long Rayleigh step with increasing inlet length dark (purple) to
light (gold), see legend for values. Rayleigh step; and, dark (orange),
inlet–half-cylinder. Calculated from Eq. (37) with $f$ and $g$ given in
Appendix.

###  V.2 Long-bearing limit

In the long-bearing limit, we find a more complex expression,

$\alpha=\frac{2}{3}+\frac{2(\langle{u}\rangle\langle{\delta^{2}}\rangle-\langle{u\delta}\rangle\langle{\delta}\rangle)+\langle{u\delta}\rangle^{2}-\langle{u}\rangle\langle{\delta^{2}}\rangle}{3d(\langle{u}\rangle\langle{\delta}\rangle-\langle{u\delta}\rangle)z}{}+\mathcal{O}\Bigl{(}\frac{1}{z^{2}}\Bigr{)}\,.$
(41)

To highlight the structure of the answer we have retained
$\langle{u}\rangle=1/2$, rather than substitute its value. The sign of the
initial deviation from $\alpha=2/3$ is therefore controlled by the numerator
of the second term in this expansion [19]. For the wedge, this reduces to

$\alpha=\frac{2}{3}-\frac{17}{90z^{2}}+\mathcal{O}\Bigr{(}\frac{1}{z^{3}}\Bigl{)}\,.$
(42)

So, $\alpha$ drops below 2/3, Fig. 8b, leading to an upwards deviation of the
Stribeck curve from $S^{2/3}$ scaling. Again, this effect is second order in
$1/z$, as in the short-bearing case.

For the inlet–half-cylinder,

$\alpha=\frac{2}{3}-\frac{2(1+5\chi-30\chi^{2})}{45(1+3\chi)}\frac{1}{z}+\mathcal{O}\Bigl{(}\frac{1}{z^{2}}\Bigr{)}\,.$
(43)

For sufficiently short inlet lengths, i.e. $\chi<(5+\sqrt{145})/60\approx
0.284$, in this narrow-gap limit, $\alpha$ is lowered, Fig. 8c (dark lines),
and the Stribeck curve deviates upwards from the large-gap scaling with
decreasing $S$.

Finally, for the Rayleigh step, Fig. 8d, we find

$\alpha=\frac{2}{3}-\frac{1-2\chi}{3z}+\mathcal{O}\Bigr{(}\frac{1}{z^{2}}\Bigl{)}\,,$
(44)

so that in a similar manner, a suitably ‘blunt’ step, i.e.
$\langle{\delta}\rangle=\chi<1/2$, will give $\alpha<2/3$.

We therefore arrive at a conclusion for all slider bearing types and profiles:
large-gap scaling is $\mu\sim S^{2}/3$ and upwards deviation with decreasing
$S$ as the gap approaches the step height requires a small enough
$\langle{\delta}\rangle$, i.e. a small enough inlet.

##  VI Interpretation of experimental data

We have shown that for a broad class of surface profiles and bearing
dimensions, upwards deviation from $\mu\sim S^{2/3}$ scaling in the HL regime
starts when the gap approaches the step height, i.e. $h_{0}/d\sim 1$. This
also apply to various textured surfaces, Fig. 3(a), so that $d$ can be
estimated from measured Stribeck curves. We end by giving a worked example of
this procedure.

Soltanahmadi et al. [13] obtained the Stribeck curves for a range of
geometries lubricated by molten chocolate. We show their data for a bio-
mimetic tongue against a steel surface, Fig. 9. The
$$2\text{\,}\mathrm{cm}$\times$2\text{\,}\mathrm{cm}$$ ‘tongue’ surface
consists of $80\times$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ high
hemispherical ‘fungiform’ papillae and
$800\times$250\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ high cylindrical
‘filiform’ papillae, each of which is textured on the
$10\text{\,}\mathrm{\SIUnitSymbolMicro
m}100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ scale (see Fig. 3a in Ref. 20).
Their material parameters and applied normal load of $1\text{\,}\mathrm{N}$
implies that the fungiform papillae will compress
$\approx$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and bring the more
numerous filiform papillae into proximity with the lower surface at a gap
height comparable to the papillae texturing.

In the large gap HL regime, where all papillae have comparable gap heights and
the more numerous papilla form dominates, the data presented in Fig. 5B(2) of
Ref. 13 can be replotted as the Stribeck curve for a single filiform papilla
at a normal load of $1/800\,$\mathrm{N}$$, Fig. 9. In the high-$S$ regime, the
scaling law slope is less than unity (fine dashed line), but is consistent
with $\mu\sim S^{2/3}$ (dashed line); fitting the data at $S>10^{-3}$ gives
$\alpha=0.6(1)$. This regime can therefore be interpreted in terms of our
analysis of the HL of a step slider bearing.

Figure 9: Stribeck curve for molten chocolate lubricating a bio-mimetic tongue
sliding on steel. Re-plotted from Ref. 13 as $\mu$ against $S=\eta UD/N$ using
$N=$1.25\text{\,}\mathrm{mN}$$ (see text). Lines: bold dotted, $S^{2/3}$
scaling; fine dashed, $\mu\sim S$. Inset schematic: relevant length scales,
$D\sim$350\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ from papillae diameter, and
$d\sim$30\text{\,}\mathrm{\SIUnitSymbolMicro
m}60\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ from papillae roughness.

The properties of the effective step bearing can be estimated from the point
($\mu^{\star}\simeq 0.17,~{}S^{\star}\simeq 6\times 10^{-3}$) at which the
Stribeck curve deviates from $S^{2/3}$ scaling, Fig. 9 (black star). For a
long bearing, Eq. (8) shows that when $h_{0}=d$, we have

$S^{\star}\sim\frac{d^{2}}{D^{2}}\,,\quad\mu^{\star}\sim\frac{d}{D}\,\quad\Rightarrow\quad\sqrt{S^{\star}}\sim\mu^{\star}\,.$
(45)

With an observed $S^{\star}\simeq 0.006$, our scaling analysis of the HL
regime of a long slider bearing therefore predicts
$\mu^{\star}\sim\sqrt{S^{\star}}=0.08$, which is order-of-magnitude consistent
with the observed $\mu^{\star}\simeq 0.17$ [21] Taking
$D=$350\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ (the diameter of the
cylindrical papillae) then gives an estimate for $d$, either from
$D\sqrt{S^{\star}}\simeq$30\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ or from
$D\mu^{\star}\simeq$60\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, consistent
with the actual papillae roughness on the
$30\text{\,}\mathrm{\SIUnitSymbolMicro m}60\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ scale [20].

Interestingly, Soltanahmadi et al. [13] estimated the gap height in the
vicinity of $(S^{\star},\mu^{\star})$ to be
$h_{0}\gtrsim$undef\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ [see their Fig.
5B(1)] using a phenomenological expression [22] modified from an analytical
form derived from an analysis of EHL [23], which is significantly smaller than
the gap size we have estimated based on our HL analysis. However, the actual
measured gap size is $\simeq$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ (see
Ref. 13 Supporting Information Fig. S8B), which is well within range of our
estimate based on the point of deviation from $S^{2/3}$ scaling being where
$h_{0}\simeq d$, where $d\simeq$50\text{\,}\mathrm{\SIUnitSymbolMicro
m}100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ is the scale of surface
texturing.

##  VII Conclusions

We have shown that the HL of non-conformal and conformal contacts differ
fundamentally. For former, the problem has a single length scale, the minimum
gap, giving ‘trivial’ scaling in the Stribeck curve, e.g. $\mu\sim S$ for
ball-on-flat, independent of the absolute gap. In contrast, the HL of
conformal contacts is non-trivial due to the presence of two independent
length scales, the minimum gap and a surface texturing or ‘step’ height,
giving anomalous $\mu\sim S^{2/3}$ large-gap scaling. As the gap decreases
below the step height, deviation from $S^{2/3}$ occurs. ‘Blunt’ surface
profiles, where the gap remains small over more of the bearing area, show an
‘upwards’ deviation as $S$ drops, reminiscent of entry into the EHL regime. We
used our approach to deduce the length scale of surface texturing from the
data in a recent triborheological study.

Large-gap HL is of limited application in engineering applications, which are
typically designed to function near the minimum of the Stribeck curve.
However, this regime becomes important whenever soft matter is involved in
lubrication, e.g., in the oral processing of food [13], the application of
skin cream [24], and the extrusion of ceramic ‘green bodies’ [25] (a ceramic
‘green body’ is the formed object before firing). Future work may therefore
fruitfully extend our analysis to non-Newtonian fluids with, e.g., rate-
dependent rheology and normal stress differences.

## Appendix: gap profile calculations

###  .1 Wedge

In dimensionless form the gap, Eq. (11), of the wedge is

$\xi=z+1-u\,.$ (A1)

The averages required for Eqs. (27) and (28) then evaluate to

$\begin{split}&\textstyle\langle{\xi^{-1}}\rangle=\ln\frac{z+1}{z}\,,\quad\langle{\xi^{-2}}\rangle=\frac{1}{z(1+z)}\,,\quad\langle{\xi^{-3}}\rangle=\frac{1+2z}{2z^{2}(1+z)^{2}}\,,\\\\[3.0pt]
&\textstyle\langle{u\,\xi^{-2}}\rangle=\frac{1}{z}-\ln\frac{z+1}{z}\,,\quad\langle{u\,\xi^{-3}}\rangle=\frac{1}{2z^{2}(1+z)}\,.\end{split}$
(A2)

In the long-bearing limit, the dimensionless functions are

$f=4\ln\frac{1+z}{z}-\frac{6}{1+2z}\,,\quad
g=6\ln\frac{1+z}{z}-\frac{12}{1+2z}\,.$ (A3)

In the limit of $z\to 0$, i.e. with $z+1\to 1$,

$\lim_{z\to 0}\\{f,g\\}=\\{6\ln z^{-1},4\ln z^{-1}\\}\;\Rightarrow\;\lim_{z\to
0}\mu=\frac{2}{3}\,\frac{d}{D}\,.$ (A4)

In the short bearing limit, $f=\langle{\xi^{-1}}\rangle\rightarrow\ln z^{-1}$
follows trivially alongside $g(z)\rightarrow 1/4z^{2}$, as derived in §
IV.1.3.

###  .2 Inlet–half-cylinder

Recasting the profile, Eq. (19), in dimensionless terms,

$\xi=\Bigl{\\{}\begin{array}[]{lll}z+1&&0\leq u\leq\chi\,,\\\
z+(1-u)^{2}/(1-\chi)^{2}&&\chi\leq u\leq 1\,.\end{array}$ (A5)

The averages then evaluate to

$\begin{split}\langle{\xi^{-1}}\rangle&=\textstyle\frac{\chi}{1+z}+\frac{1-\chi}{\sqrt{z}}\tan^{-1}\\!\frac{1}{\sqrt{z}}\,,\\\
\langle{\xi^{-2}}\rangle&=\textstyle\frac{\chi}{(1+z)^{2}}+\frac{1-\chi}{2z}\Bigl{(}\frac{1}{1+z}+\frac{1}{\sqrt{z}}\tan^{-1}\\!\frac{1}{\sqrt{z}}\Bigr{)}\,,\\\
\langle{\xi^{-3}}\rangle&=\textstyle\frac{\chi}{(1+z)^{3}}+\frac{1-\chi}{8z^{2}}\Bigl{(}\frac{3+5z}{(1+z)^{2}}+\frac{3}{\sqrt{z}}\tan^{-1}\\!\frac{1}{\sqrt{z}}\Bigr{)}\,,\\\
\langle{u\,\xi^{-2}}\rangle&=\textstyle\frac{\chi^{2}}{2(1+z)^{2}}+\frac{1-\chi}{2z}\Bigl{(}\frac{\chi}{1+z}+\frac{1}{\sqrt{z}}\tan^{-1}\\!\frac{1}{\sqrt{z}}\Bigr{)}\,,\\\
\langle{u\,\xi^{-3}}\rangle&=\textstyle\frac{\chi^{2}}{2(1+z)^{3}}+\frac{1-\chi}{8z^{2}}\Bigl{(}\frac{1+z+2\chi(1+2z)}{(1+z)^{2}}+\frac{3}{\sqrt{z}}\tan^{-1}\\!\frac{1}{\sqrt{z}}\Bigr{)}\,.\end{split}$
(A6)

These expressions used in Eqs. (27) and (28) (long bearing) or Eqs. (32) and
(33) (short bearing) parametrically gives Fig. 5 [dark (orange) lines]. We do
not show the lengthy expressions for $f(z)$ and $g(z)$ in this case.

For long bearings, a more rigorous analysis of the limits of Eq. (A6) with
$\tan^{-1}(z^{-1/2})/\sqrt{z}\to\pi/2\sqrt{z}-1$ gives $g(z)\to
3(1-\chi)^{2}/z$, Fig. 6c, and $f\to\pi(1-\chi)/\sqrt{z}$, Fig. 6d. The
$1-\chi$ originates from a shorter inlet making a larger cylinder,
$R=D^{2}(1-\chi)^{2}/2d$. The resultant $S$ and $\mu$ are given in Table 1.

For short bearings, taking the limits of Eq. (A6) gives $f(z)\to
2\pi(1-\chi)z^{3/2}$, Fig. 7b. With the $g(z)$ for all short bearings, we
recover $\mu\propto S^{3/4}$ scaling with included $\mathcal{O}({1})$
numerical prefactor for $\chi\lesssim 0.9$, see Table 1.

###  .3 Step

In non-dimensional form, Eq. (21) becomes

$\xi=\Bigl{\\{}\begin{array}[]{lll}z+1&&0\leq u\leq\chi\,,\\\ z&&\chi\leq
u\leq 1\,.\end{array}$ (A7)

The gap averages to calculate the $N$ and $F$, and hence $\mu(S)$, for this
problem are then, rather trivially,

$\textstyle\langle{\xi^{-n}}\rangle=\frac{\chi}{(1+z)^{n}}+\frac{1-\chi}{z^{n}}\,,\quad\langle{u\,\xi^{-n}}\rangle=\frac{\chi^{2}}{2(1+z)^{n}}+\frac{1-\chi^{2}}{2z^{n}}\,.$
(A8)

For the Rayleigh step in the long-bearing limit the dimensionless functions,
Fig. 6f (dark to light with increasing $\chi$), are

$\begin{split}\textstyle f=\frac{3\chi(1-\chi)}{\chi
z^{3}+(1-\chi)(1+z)^{3}}+\frac{\chi}{1+z}+\frac{1-\chi}{z},\,\textstyle
g=\frac{3\chi(1-\chi)}{\chi z^{3}+(1-\chi)(1+z)^{3}}\,.\end{split}$ (A9)

In the large-gap regime while $S^{2/3}$ scaling is recovered for all $\chi$,
the inlet length does control the magnitude high-$S$ limit. Using the profile,
Eq. (A8), to evaluate $g(z)$, Eq. (36), the thick film scaling limit $g\to
3\chi(1-\chi)/z^{3}$, with $f\to 1/z$ trivially. The resultant Stribeck curve
is then,

$\mu(S)=\frac{d}{D}\times\frac{S^{2/3}}{[{3\chi(1-\chi)}]^{1/3}}\,.$ (A10)

Friction is minimised in the large gap limit when the denominator is
maximised, i.e. $\chi=0.5$, and $\mu$ diverges as $\chi\to 0$ or $1$ [26]
Pertinent to our scaling analysis, the $\chi$-dependent factor is again of
$\mathcal{O}({1})$ for $\chi\approx 0.01$ to 0.99.

As $z\rightarrow 0$, $g\to 3\chi$ for long bearings, Fig. 6e, as the first two
powers cancel, reducing the apparent $z^{-2}$ divergence, § IV.1.3. Meanwhile,
the drag force [Eq. (28)] $f\to(1-\chi)/z$ as only the second terms $\propto
z^{-n}$ in Eq. (A8) diverge. The small-$z$ limits are summarised in Table 1.
As detailed in § III, while the equations remain analytic for the short
bearing the lubrication approximation breaks down.

## References

  * Hamrock  _et al._ [2004] B. J. Hamrock, S. R. Schmid, and B. O. Jacobsen,  _Fundamentals of Fluid Film Lubrication_ (Marcel Dekker, New York, NY, 2004).
  * Hirani [2016] H. Hirani,  _Fundamentals of Engineering Tribology with Applications_ (CUP, Cambridge, UK, 2016); G. W. Stachowiak and A. W. Batchelor,  _Engineering Tribology_ (Elsevier, Oxford, UK, 2014) .
  * Gümbel [1914] L. Gümbel, Das Problem der Lagerreibung, Mbl. Berl. Bezirksver. dtsch. Ing. 5, 109 (1914).
  * Hersey [1914] M. D. Hersey, The laws of lubrication of horizontal journal bearings, J. Wash. Acad. Sci. 4, 542 (1914).
  * Stribeck [1902a] R. Stribeck, Die wesentlichen Eigenschaften der Gleit- und Rollenlager, Z. Ver. dtsch. Ing. 46, 1341; 1432; 1463 (1902a).
  * Reynolds [1886] O. Reynolds, On the theory of lubrication and its application to Mr. Beauchamp Tower’s experiments, including an experimental determination of the viscosity of olive oil, Phil. Trans. Royal Soc. Lond. 177, 157 (1886).
  * Lord Rayleigh [1918] Lord Rayleigh, Notes on the theory of lubrication, Phil. Mag. 35, 1 (1918).
  * Kapitza [1955] P. L. Kapitza, Hydrodynamic theory of lubrication during rolling, Zh. Tekh. Fiz. 25, 747 (1955).
  * Faber [1995] T. E. Faber,  _Fluid Dynamics for Physicists_ (CUP, Cambridge, UK, 1995).
  * Guyon  _et al._ [2015] E. Guyon, J.-P. Hulin, L. Petit, and C. D. Mitescu,  _Physical Hydrodynamics_ (OUP, Oxford, UK, 2015).
  * PRL [2023] Manuscript jointly submitted (2023).
  * Warren [2016] P. B. Warren, Sliding friction in the hydrodynamic lubrication regime for a power-law fluid, J. Phys.: Condens. Mat. 29, 064005 (2016).
  * Soltanahmadi  _et al._ [2023] S. Soltanahmadi, M. Bryant, and A. Sarkar, Insights into the multiscale lubrication mechanism of edible phase change materials, ACS Appl. Mater. Interfaces 15, 3699 (2023).
  * Woydt and Wäsche [2010] M. Woydt and R. Wäsche, The history of the Stribeck curve and ball bearing steels: The role of Adolf Martens, Wear 268, 1542 (2010).
  * Gropper  _et al._ [2016] D. Gropper, L. Wang, and T. J. Harvey, Hydrodynamic lubrication of textured surfaces: A review of modeling techniques and key findings, Tribol. Int. 94, 509 (2016).
  * DuBois and Ocvirk [1953] G. B. DuBois and F. W. Ocvirk,  _Analytical derivation and experimental evaluation of short-bearing approximation for full journal bearings_ (US Govt. Printing Office, Washington, DC, 1953).
  * Note [1] Here we neglect the Poiseuille component to the drag as it acts in both directions, Fig. 2c.
  * Batchelor [2000] G. K. Batchelor,  _An introduction to fluid dynamics_ (CUP, Cambridge, 2000).
  * Note [2] Note that the denominator never vanishes, since $\langle{u}\rangle\langle{\delta}\rangle-\langle{u\delta}\rangle=\frac{1}{2}\langle{(1-2u)\delta}\rangle=-\frac{1}{2}\langle{u(1-u)\delta^{\prime}}\rangle$, where the final step involves an integration by parts. Under the assumption that the gap is converging, $\delta^{\prime}\leq 0$, the final expression must be positive.
  * Andablo-Reyes  _et al._ [2020] E. Andablo-Reyes, M. Bryant, A. Neville, P. Hyde, R. Sarkar, M. Francis, and A. Sarkar, 3D biomimetic tongue-emulating surfaces for tribological applications, ACS Appl. Mater. Interfaces 12, 49371 (2020).
  * Note [3] Taking the bearing to be a short bearing the deviation point, $S^{\star}\sim(d/L)^{2}$ and $\mu^{\star}\sim dD/L^{2}$, can also be used to estimate a step height, $d=$12\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, alongside an aspect ratio, $L=0.5D$. The $d$ estimate remains consistent with our order of magnitude analysis and the aspect ratio suggests lubrication is intermediate between the long and short-bearing limits.
  * Myant  _et al._ [2010] C. Myant, M. Fowell, H. A. Spikes, and J. R. Stokes, An investigation of lubricant film thickness in sliding compliant contacts, Tribol. Trans. 53, 684 (2010).
  * de Vicente  _et al._ [2005] J. de Vicente, J. R. Stokes, and H. A. Spikes, The frictional properties of Newtonian fluids in rolling–sliding soft-EHL contact, Tribol. Lett. 20, 273 (2005).
  * Lee  _et al._ [2022] S. Lee, S. R. Kim, H.-J. Lee, B. S. Kim, H. Oh, J. B. Lee, K. Park, Y. J. Yi, C. H. Park, and J. D. Park, Predictive model for the spreadability of cosmetic formulations based on large amplitude oscillatory shear (LAOS) and machine learning, Phys. Fluids 34, 103109 (2022).
  * Jiang  _et al._ [2009] G. Jiang, J. Yang, and J. Gao, Effect of starch on extrusion behaviour of ceramic pastes, Mater. Res. Innov. 13, 119 (2009) .
  * Note [4] In both limits a note of caution should be struck, as the Reynolds lubrication approximation will no longer hold.


**footnotetext: These authors contributed equally to this
work.${\dagger}$${\dagger}$footnotetext: Shared senior
authorship.11institutetext: Department of Radiology, Northwestern University,
Chicago, USA
11email: {debesh.jha, nikhil.tomar<EMAIL_ADDRESS>
22institutetext: Indian Institute of Technology Guwahati, Assam, India
22email: {vanshalisharma, mkb<EMAIL_ADDRESS>
33institutetext: Department of Gastroenterology, Jefferson Health NJ, Cherry
Hill, USA
33email<EMAIL_ADDRESS>
44institutetext: SimulaMet, Oslo, Norway
44email: {steven, michael<EMAIL_ADDRESS>Department of
Medicine and Emergencies - Mölndal Sahlgrenska University Hospital, Region
Västra Götaland, Sweden & Department of Molecular and Clinical Medicin,
Sahlgrenska Academy, University of Gothenburg, Sweden
55email<EMAIL_ADDRESS>
# GastroVision: A Multi-class Endoscopy Image Dataset for Computer Aided
Gastrointestinal Disease Detection

Debesh Jha∗ 11 Vanshali Sharma∗ 22 Neethi Dasu 33 Nikhil Kumar Tomar 11 Steven
Hicks 44 M.K. Bhuyan 22 Pradip K. Das 22 Michael A. Riegler 44 Pål Halvorsen
44 Ulas Bagci${\dagger}$ 11 Thomas de Lange${\dagger}$ 55

###### Abstract

Integrating real-time artificial intelligence (AI) systems in clinical
practices faces challenges such as scalability and acceptance. These
challenges include data availability, biased outcomes, data quality, lack of
transparency, and underperformance on unseen datasets from different
distributions. The scarcity of large-scale, precisely labeled, and diverse
datasets are the major challenge for clinical integration. This scarcity is
also due to the legal restrictions and extensive manual efforts required for
accurate annotations from clinicians. To address these challenges, we present
GastroVision, a multi-center open-access gastrointestinal (GI) endoscopy
dataset that includes different anatomical landmarks, pathological
abnormalities, polyp removal cases and normal findings (a total of 27 classes)
from the GI tract. The dataset comprises 8,000 images acquired from Bærum
Hospital in Norway and Karolinska University Hospital in Sweden and was
annotated and verified by experienced GI endoscopists. Furthermore, we
validate the significance of our dataset with extensive benchmarking based on
the popular deep learning based baseline models. We believe our dataset can
facilitate the development of AI-based algorithms for GI disease detection and
classification. Our dataset is available at https://osf.io/84e7f/.

###### Keywords:

Medical image GastroVision Gastrointestinal diseases.

##  1 Introduction

Gastrointestinal (GI) cancers account for 26% of cancer incidences and 35% of
cancer-related deaths worldwide. In 2018, there were approximately 4.8 million
new cases of GI cancer and 3.4 million deaths [7]. The five major types of GI
cancers are colo-rectal (1.93 million cases; third most common cancer),
pancreas (466,003 deaths; lowest survival rate), liver (905,677 cases),
stomach (1.09 million cases), and esophagus (604,100 cases) [13]. These cancer
cases are predicted to increase by 58%, and related deaths could show a 73%
rise by 2040 [7]. Early detection of such cancers and their precursors can
play an important role in improving the outcome and make the treatment less
invasive. Some of the common examinations performed for GI cancer detection
include endoscopy or esophagogastroduodenoscopy (EGD), capsule endoscopy,
colonoscopy, imaging studies (MRI, X-ray, ultrasound, CT scan, or PET scan) or
endoscopic ultrasound (EUS). Endoscopy is widely accepted as the gold standard
for detecting abnormalities of the bowel lumen and mucosa, upper endoscopy for
esophagus, stomach, and the duodenum and colonoscopy for the large bowel and
rectum GI tract for abnormalities, respectively.

The endoscopies are performed by nurses or doctor endoscopists. The assessment
of the endoscopy examinations is operator dependent, and the assessment and
therapeutic decision vary between endoscopists. Consequently, the quality and
accuracy of detection and diagnosis of lesions are attributed to the level of
the operator skills and efforts of the endoscopists. Despite various measures
taken to provide guidance for operators, significant lesion miss rates are
still reported. For example, there is evidence of colon polyp miss rates of up
to 27% due to polyps and operator characteristics [2, 20]. Considering the
shortcomings of the manual review process, various automated systems are
adopted to provide AI-based real-time support to clinicians to reduce lesion
miss rates and misinterpretation of lesions to ultimately increase detection
rates. A microsimulation study reports a 4.8% incremental gain in the
reduction of colorectal cancer incidence when colonoscopy screening was
combined with AI tools [6]. Such findings motivated research work in
healthcare to adopt AI as a potential tool for GI cancer detection. Gastric
cancer, inflammatory bowel disease (IBD), and esophageal neoplasia are some of
the GI tract findings already being investigated using AI techniques [1].
Despite AI being adopted in some hospitals for clinical applications, the
integration of AI into the extensive clinical setting is still limited.
Integrating AI techniques with regular clinical practices is multifactorial
and poses serious concerns regarding its implementation in large-scale
healthcare systems. One of the significant factors is the algorithmic bias,
which worsens when the system learns from the annotations handled by a single,
non-blinded endoscopist who may have personal thresholds to label the
findings.

Moreover, most existing AI models depend on data acquired from a single
center, which makes them less valid when faced with a varied patient
population. This leads to spectrum bias under which AI systems encounter
performance drops due to the significant shift in the original clinical
context and the test sample population. In such cases, unexpected outcomes and
diagnostic accuracy could be obtained using automated tools. Such bias issues
could reach the clinical systems at any point of the process, including data
collection, study design, data entry and pre-processing, algorithm design, and
implementation. The very beginning of the process, i.e., data collection, is
of utmost importance for reproducibility and to perform validations on images
from a diverse population, different centers, and imaging modalities. To
develop scalable healthcare systems, it is vital to consider these challenges
and perform real-time validations. However, the scarcity of comprehensive data
covering a range of real-time imaging scenarios arising during endoscopy or
colonoscopy makes it difficult to develop a robust AI-based model. Although
much progress has been made on automated cancer detection and classification
[16, 19], it is still challenging to adapt such models into real-time clinical
settings as they are tested on small-sized datasets with limited classes.

Some classes in the dataset could be scarce because some conditions or
diseases occur less often. Consequently, such findings are not frequently
captured and remain unexplored despite requiring medical attention. AI-based
detection of these findings, even with a small sample count, can significantly
benefit from techniques like one-shot or few-shot learning. These techniques
allow the AI models to learn patterns and features indicative of the
condition, thus, enabling accurate diagnosis with minimal training data. Apart
from the above-mentioned limitations, many existing datasets are available on
request, and prior consenting is required, which delays the process and does
not guarantee accessibility. Therefore, in this paper, we publish
GastroVision, an open-access multi-class endoscopy image dataset for automated
GI disease detection that does not require prior consenting and can be
downloaded easily with a single click. The data covers a wide range of classes
that can allow initial exploration of many anatomical landmarks and
pathological findings.

The main contributions of this work are summarized below:

  * •

We present an open-access multi-class GI endoscopy dataset, namely,
Gastrovision, containing 8,000 images with 27 classes from two hospitals in
Norway and Sweden. The dataset exhibits a diverse range of classes, including
anatomical landmarks, pathological findings, polyp removal cases and normal or
regular findings. It covers a wide range of clinical scenarios encountered in
endoscopic procedures.

  * •

We evaluated a series of deep learning baseline models on standard evaluation
metrics using our proposed dataset. With this baseline, we invite the research
community to improve our results and develop novel GI endoscopy solutions on
our comprehensive set of GI finding classes. Additionally, we encourage
computer vision and machine learning researchers to validate their methods on
our open-access data for a fair comparison. This can aid in developing state-
of-the-art solutions and computer-aided systems for GI disease detection and
other general machine learning classification tasks.

Table 1: List of the existing datasets within GI endoscopy.

Dataset |  Data type |  Size |  Accessibility
---|---|---|---
Kvasir-SEG [17] |  Polyps |  1,000 images†♣ |  Public
HyperKvasir [11] |  GI findings |  110,079 images & 374 videos |  Public
Kvasir-Capsule [24] |  GI findings⋄ |  4,741,504 images |  Public
Kvasir [22] |  GI findings |  8,000 images |  Public
CVC-ColonDB [10] |  Polyps |  380 images† ‡ |  As per request∙
ETIS-Larib Polyp DB [23] |  Polyps |  196 images† |  Public
EDD2020 [4, 3] |  GI lesions |  386 images†♣ |  Public
CVC-ClinicDB [9] |  Polyps |  612 images† |  Public
CVC-VideoClinicDB [8] |  Polyps |  11,954 images† |  As per request
ASU-Mayo polyp database [25] |  Polyps |  18,781 images† |  As per request∙
KID [18] |  Angiectasia, bleeding, inflammations⋄ |  $>$ 2500 images, 47 videos |  Public∙
PolypGen [5] |  Polyps |  1,537 images†♣ & 2,225 video sequence, 4,275 negative frame |  Public
SUN Database [21] |  Polyps |  158,690 video frames♣ |  As per request
GastroVision (ours) |  GI findings |  8,000 image frames |  Public
†Segmentation ground truth ∙Not available now ‡Contour ⋄Video capsule
endoscopy ♣ Bounding box information

##  2 Related Work

Table 1 shows the list of the existing dataset along with data type, their
size, and accessibility. It can be observed that most of the existing datasets
in the literature are from colonoscopy procedures and consist of polyps still
frames or videos. These are mostly used for segmentation tasks. Most of the
existing datasets are small in size and do not capture some critical
anatomical landmarks or pathological findings. In the earlier GI detection
works, the CVC-ClinicDB [9] and CVC-ColonDB [10] were widely used. CVC-
ClinicDB is developed from 23 colonoscopy video studies acquired with white
light. These videos provide 31 video sequences, each containing one polyp,
which finally generates 612 images of size 576 $\times$ 768\. CVC-ColonDB
consists of 300 different images obtained from 15 random cases. Similarly,
ETIS-Larib Polyp DB [23] is a colonoscopy dataset consisting of 196 polyp
frames and their corresponding segmentation masks. Recently, Kvasir-SEG [17]
dataset has been introduced that comprises of 1,000 colonoscopy images with
segmentation ground truth and bounding box coordinate details. This dataset
offers a diverse range of polyp frames, including multiple diminutive polyps,
small-sized polyps, regular polyps, sessile or flat polyps collected from
varied cohort populations. The dataset is open-access and is one of the most
commonly used datasets for automatic polyp segmentation.

The ASU-Mayo Clinic Colonoscopy Video (c) database [25] is a copyrighted
dataset and is considered the first largest collection of short and long video
sequences. Its training set is composed of 10 positive shots with polyps
inside and 10 negative shots with no polyps. The associated test set is
provided with 18 different unannotated videos. CVC-VideoClinicDB [8] is
extracted from more than 40 long and short video sequences. Its training set
comprises 18 different sequences with an approximate segmentation ground truth
and Paris classification for each polyp. SUN Colonoscopy Video Database
comprises 49,136 polyp frames and 109,554 non-polyp frames. Unlike the
datasets described above, this dataset includes pathological classification
labels, polyp size, and shape information. It also includes bounding box
coordinate details. The PolypGen [5] dataset is an open-access dataset that
comprises 1,537 polyp images, 2,225 positive video sequences, and 4,275
negative frames. The dataset is collected from six different centers in Europe
and Africa. Altogether the dataset provides 3,762 positive frames and 4,275
negative frames. These still images and video frames are collected from varied
populations, endoscopic systems, and surveillance expert in Norway, France,
United Kingdom, Egypt, and Italy and is one of the comprehensive open-access
datasets for polyp detection and segmentation.

Apart from the lower GI-related datasets, there are a few datasets that
provide combined samples of upper and lower GI findings. For example,
HyperKvasir [11] is a multi-class GI endoscopy dataset that covers 23 classes
of anatomical landmarks. It contains 110,079 images out of which 10,662 are
labeled and 99,417 are unlabeled images. The EDD2020 dataset [4, 3] is a
collection of five classes and 386 still images with detection and
segmentation ground truth. The classes are divided into 160 non-dysplastic
Barrett’s, 88 suspicious precancerous lesions, 74 high-grade dysplasia, 53
cancer, and 127 polyps with overall 503 ground truth annotations. The Kvasir-
Capsule [24] is a video capsule endoscopy dataset comprising 4,741,504 image
frames extracted from 117 videos. From the total frames, 4,694,266 are
unlabeled, and 47,238 frames are annotated with a bounding box for each of the
14 classes. Similarly, KID [18] is a capsule endoscopy dataset with 47 videos
and over 2,500 images. The images are annotated for normal, vascular,
inflammatory, lymphangiectasias, and polypoid lesions.

The literature review shows that most GI-related datasets focus on a single
specific finding, such as colon polyps. Some of the datasets are small in size
and have ignored non-lesion frames, which are essential for developing
algorithms to be integrated into clinical settings. Additionally, many of
these datasets are available on request and require approval from the data
providers resulting in further delays. A few datasets like Kvasir,
HyperKvasir, Kvasir-Capsule and KID provide multiple GI findings. However,
Kvasir-Capsule and KID are video capsule endoscopy datasets. The Kvasir
dataset has only eight classes, whereas Hyperkvasir has 23 classes. In
contrast, our GastroVision dataset has 27 classes and covers more labeled
classes of anatomical landmarks, pathological findings, and normal findings.
Additionally, we establish baseline results on this dataset for GI disease
detection and classification, offering valuable research resources for
advancing GI endoscopy studies.

##  3 GastroVision

Here, we provide detailed information about the dataset, acquisition protocol,
ethical and privacy aspects of data and suggested metrics.

Figure 1: Example images from the gastrointestinal tract showing distinct
findings from the upper and lower GI tract. Figure 2: The figure shows the
number of images per class. Some classes have few samples because of the
rarity of the findings and the technical challenges associated with obtaining
such samples in endoscopic settings. Figure 3: Resolutions of the 8,000 images
of GastroVision.

###  3.1 Dataset details

GastroVision is an open-access dataset that incorporates 8,000 images
pertaining to 27 different labeled classes (Fig. 1). Most images are obtained
through White Light Imaging (WLI), while a few samples are acquired using
Narrow Band Imaging (NBI). These classes are categorized into two broad
categories: Upper GI tract and Lower GI tract. The number of images under each
class is presented in Fig. 2. These classes indicate findings acquired from GI
tract. It can be observed that the sample count is not balanced across
classes, which is generally experienced in the medical image acquisition
process as some findings occur less often. Releasing these classes in the
dataset will allow the researchers to leverage the fast-emerging AI techniques
to develop methods for detecting such rare but clinically significant
anomalies. All the images are stored in JPG format, and the overall size is
around 1.8 GB. The resolution details of the images can be found in Fig. 3.
GastroVision is provided as a structured directory, with each class having a
specific folder. For example, the ‘Accessory tools’ folder contains all images
featuring diagnostic and therapeutic tools.

####  3.1.1 Upper GI Tract:

Upper GI endoscopy examines the esophagus, stomach, and duodenum. The various
classes covered in this GI tract are discussed below as three subcategories:
normal findings, anatomical landmarks, and pathological findings. A detailed
categorization is shown in Fig. 1. The normal stomach serves as a critical
site for initial digestion, while the duodenal bulb, the first part of the
small intestine, is critical for nutrient absorption. Anatomical landmarks are
used as reference points to indicate a specific location and assist in
navigating during endoscopy procedures. The gastroesophageal junction is an
anatomical area where esophagus joins the stomach also alining to the normal
z-line, a transitional point where the esophagus’s squamous epithelium and the
stomach’s columnar mucosa lining join. Pylorus is a sphincter connecting the
stomach and the duodenum, the first part of the small intestine.

Apart from these anatomical landmarks, any pathological conditions may be
encountered during endoscopy. Esophagitis, the most common abnormality, is
characterized by an inflammation of the esophagus. This disease is graded
based on its severity according to the Los Angeles classification. For
example, grade B refers to the condition when the mucosal break is limited to
the mucosal fold and is more than 5 mm long. In grade D, mucosal break affects
75% of the esophageal circumference. Long standing esophagitis may cause
Barett’s esophagus, a condition in which the cells of the esophagus’s lining
start to change, and tissues appear red. This is a precancerous condition.
Other frequent lesions observed are polyps, abnormal tissue growth or ulcers.
Gastric polyps are abnormal growths in the stomach lining. Ulcers are the open
sores in the stomach or duodenum that can lead to discomfort and bleeding.
Esophageal varices result from portal hypertension, causing swollen veins in
the esophagus. Erythema refers to redness, often indicating inflammation and
blood in the lumen denotes bleeding. Accessory tools aid in investigating and
diagnosing upper and lower GI tract conditions for targeted treatment.

####  3.1.2 Lower GI Tract:

The lower GI tract is examined by colonoscopy to investigate any abnormalities
in the colon, the rectum, and the terminal ileum (the last part of the small
bowel). Here, we covered one more subcategory, therapeutic interventions, in
addition to normal findings, anatomical landmarks, and pathological findings.
A detailed class-wise division is shown in Fig. 1.

The normal mucosa and vascular pattern in the large bowel is essential for
absorbing water and electrolytes. The different anatomical landmarks
associated with lower GI include cecum (first part of the large intestine),
visualizing the appendiceal orifice, ileocecal valve (sphincter muscle between
ileum and colon), and the small bowl. During the colonoscopy, these anatomical
landmarks act as reference points to prove complete examination. Retroflexion
in the rectum is performed to visualize a blind zone, using the bending
section of the colonoscope to visualize the distal area of the colon, called
rectroflex-rectum. The terminal ileum, the last part of the small intestine,
aids in nutrient absorption. Colon diverticula, small pouch-like protrusions,
can form along the colon’s weakened wall, often in the sigmoid colon [12].

During the colonoscopy, the endoscopist navigates through these landmarks and
looks for abnormalities such as polyps, angiectasia, and inflammation like
ulcerative colitis. Angiectasia is a common lesion representing abnormal blood
vessels and is responsible for obscure recurrent lower GI bleeding. These can
easily be distinguished from the normal vessels shown in Fig. 1. Colorectal
cancer occurs in the colon or rectum. One of the early signs of this
colorectal cancer can be detected through colon polyps. Mucosal inflammation
in the large bowel may be caused by different factors, such as infections or
chronic inflammatory conditions.

Apart from the aforementioned pathological conditions, several therapeutic
interventions are adopted to treat the detected anomalies effectively. It
frequently involves the removal of the lesion/polyp. The surrounding of such
resected polyps, also called the resection margins or resection sites, are
then considered for biopsies. To enhance lesion demarcation, a solution
containing indigo carmine is injected, making resection easier. The appearance
of blue color underneath the dyed-lifted-polyp provides accurate polyp
margins. After resecting such polyps, the underlying region, known as dyed-
resection-margin, appears blue. These margins are important to examine for any
remaining tissue of the resected polyp.

###  3.2 Dataset acquisition, collection and construction

####  3.2.1 Data acquisition and collection:

The dataset images are acquired from two centers (Department of
Gastroenterology, Bærum Hospital, Vestre Viken Hospital Trust (VV), Norway and
Karolinska University Hospital, Stockholm, Sweden) using standard endoscopy
equipment from Olympus (Olympus Europe, Germany) and Pentax (Pentax Medical
Europe, Germany). A team of expert gastroenterologists, one junior doctor, and
two computational scientists were involved in the labelling of the images and
the related review process. It is worth noting that for dataset collection, we
labeled some of the unlabeled images from the HyperKvasir dataset and included
them in our dataset. Additionally, we labeled the images acquired from the
Karolinska University Hospital to their respective classes for developing a
diverse and multi-center “GastroVision” dataset.

####  3.2.2 Ethical and privacy aspects of the data:

The dataset is constructed while preserving the patients’ anonymity and
privacy. All videos and images from Bærum hospitals were fully anonymized,
following the GDPR requirements for full anonymization. Hence, it is exempted
from patient consent. The files were renamed using randomly generated
filenames. The Norwegian Privacy Data Protection Authority approved this
export of anonymized images for research purposes. As the dataset development
procedure involved no interference with the medical treatment or care of the
patient, it has also been granted an exemption for approval by Regional
Committee for Medical and Health Research Ethics - South East Norway.
Similarly, the data collection process at Karolinska University Hospital,
Sweden, is completely anonymized as per the GDPR requirements.

###  3.3 Suggested metrics

Standard multi-class classification metrics, such as Matthews Correlation
Coefficient (MCC), micro and macro averages of recall/sensitivity, precision,
and F1-score, can be used to validate the performance using our dataset. MCC
provides a balanced measure even in cases with largely varying class sizes. A
macro-average will compute the metric independently for each class and then
take the average, whereas a micro-average will aggregate the contributions of
all classes to compute the metric. Recall presents the ratio of correctly
predicted positive observations to all the original observations in the actual
class. Precision is the ratio of correctly predicted positive observations to
all the positive predicted observations. F1-score integrates both recall and
precision and calculates a weighted average/harmonic mean of these two
metrics.

##  4 Experiments and Results

In this section, we describe the implementation details, technical validation
and the limitation of the dataset.

###  4.1 Implementation Details

All deep learning diagnostic models are trained on NVIDIA TITAN Xp GPU using
PyTorch 1.12.1 framework. A stratified sampling is performed to preserve the
similar distribution of each class during 60:20:20 training, validation, and
testing split formation. The images are resized to $224\times 224$ pixels, and
simple data augmentations, including random rotation and random horizontal
flip, are applied. All models are configured with similar hyperparameters, and
a learning rate of $1e^{-4}$ is initially set with $150$ epochs. An Adam
optimizer is used with the ReduceLROnPlateau scheduler. More description about
the implementation details and dataset can be found on our GitHub page
{\dagger}{\dagger}${\dagger}$https://github.com/DebeshJha/GastroVision.

Table 2: Results for all classification experiments on the Gastrovision
dataset.

| Macro Average | Micro Average |
---|---|---|---
Method | Prec. | Recall | F1 | Prec. | Recall | F1 | MCC
ResNet-50 [14] | 0.4373 | 0.4379 | 0.4330 | 0.6816 | 0.6816 | 0.6816 | 0.6416
Pre-trained ResNet-152 [14] | 0.5258 | 0.4287 | 0.4496 | 0.6879 | 0.6879 | 0.6879 | 0.6478
Pre-trained EfficientNet-B0 [26] | 0.5285 | 0.4326 | 0.4519 | 0.6759 | 0.6759 | 0.6759 | 0.6351
Pre-trained DenseNet-169 [15] | 0.6075 | 0.4603 | 0.4883 | 0.7055 | 0.7055 | 0.7055 | 0.6685
Pre-trained ResNet-50 [14] | 0.6398 | 0.6073 | 0.6176 | 0.8146 | 0.8146 | 0.8146 | 0.7921
Pre-trained DenseNet-121 [15] | 0.7388 | 0.6231 | 0.6504 | 0.8203 | 0.8203 | 0.8203 | 0.7987

| Pre-trained DenseNet-121 |
---|---|---
Class | Precision | Recall | F1-score | Support
Accessory tools | 0.93 | 0.96 | 0.95 | 253
Barrett’s esophagus | 0.55 | 0.32 | 0.4 | 19
Blood in lumen | 0.86 | 0.91 | 0.89 | 34
Cecum | 0.33 | 0.17 | 0.23 | 23
Colon diverticula | 1 | 0.33 | 0.5 | 6
Colon polyps | 0.78 | 0.87 | 0.82 | 163
Colorectal cancer | 0.63 | 0.41 | 0.5 | 29
Duodenal bulb | 0.72 | 0.76 | 0.74 | 41
Dyed-lifted-polyps | 0.86 | 0.86 | 0.86 | 28
Dyed-resection-margins | 0.94 | 0.92 | 0.93 | 49
Esophagitis | 0.5 | 0.23 | 0.31 | 22
Gastric polyps | 0.6 | 0.23 | 0.33 | 13
Gastroesophageal_junction_normal z-line | 0.65 | 0.85 | 0.74 | 66
Ileocecal valve | 0.74 | 0.7 | 0.72 | 40
Mucosal inflammation large bowel | 1 | 0.33 | 0.5 | 6
Normal esophagus | 0.72 | 0.82 | 0.77 | 28
| Normal mucosa and vasular pattern in the
---
large bowel
0.81 | 0.87 | 0.84 | 293
Normal stomach | 0.9 | 0.86 | 0.88 | 194
Pylorus | 0.8 | 0.92 | 0.86 | 78
Resected polyps | 0.33 | 0.11 | 0.17 | 18
Retroflex rectum | 0.75 | 0.43 | 0.55 | 14
Small bowel_terminal ileum | 0.86 | 0.85 | 0.85 | 169
Table 3: Class-wise performance associated with the best outcome obtained
using pre-trained DenseNet-121. Figure 4: Confusion matrix for the best
outcome obtained using pre-trained DenseNet-121. Figure 5: Two-dimensional
t-SNE embedding for GastroVision. The pre-trained DenseNet-121 model, which is
further trained on our training set, is used to extract features. Some sample
images are shown with either a specific or a broader (due to multiple
overlapping classes) categorization.

###  4.2 Technical Validation

To evaluate the presented data for technical quality and classification tasks,
we performed a series of experiments using some state-of-the-art deep learning
models. The purpose of this preliminary validation is to provide baseline
results that can be referred to for comparison by future researchers. We
carried out multi-class classification using CNN-based models, namely,
ResNet-50 [14], ResNet-152 [14], EfficientNet-B0 [26], DenseNet-121 [15], and
DenseNet-169 [15], considering their competent performance in GI-related
image-based tasks in the literature [27]. Note that we have only included
classes with more than 25 samples in the experiments, which resulted in 22
classes in total. However, we also release the other classes with fewer
samples to welcome new interesting findings in areas similar to one-shot
learning.

The different experiments performed include (a) ResNet-50: The model is
randomly initialized, and an end-to-end training is done, (b) Pre-trained
ResNet-50 and (c) Pre-trained DenseNet-121: The models are initialized with
pre-trained weights, and then all layers are fine-tuned, (d) Pre-trained
ResNet-152, (e) Pre-trained EfficientNet-B0 and (f) Pre-trained DenseNet-169:
The models are initialized with pre-trained weights, and only the updated last
layer is fine-tuned. All the above pre-trained models use ImageNet weights.
The associated results are shown in Table 2. It can be observed that the best
outcome is obtained using the pre-trained DenseNet-121. A class-wise analysis
using the same model is provided in Table 3 and Fig. 4. It shows that while
most classes achieved satisfactory prediction outcomes, a few proved to be
very challenging for the classification model. For a more detailed analysis,
we plotted a two-dimensional t-SNE embedding for GastroVision (Fig. 5). The
classes like Normal stomach, Dyed-resection-margins, which present a clear
distinction in the t-SNE embedding, are less often misclassified. The above
points could be the reasons for the F1-score of 0.88 and 0.93 in the case of
Dyed-resection-margins and Normal stomach classes, respectively. On the other
hand, there are some overlapping classes such as Cecum and Normal mucosa and
vascular pattern in the large bowel or Colorectal cancer and Colon polyps
which do not present clear demarcation with each other and hence, are likely
to be misclassified.

Considering the overall results and many overlapping classes (without distinct
clustering), it can be inferred that classifying GI-related anatomical
landmarks and pathological findings is very challenging. Many abnormalities
are hard to differentiate, and the rarely occurring findings have higher
chances of getting misclassified. This presents the challenge of developing a
robust AI system that could address multiple aspects important for GI image
classification, e.g., many findings are subtle and difficult to be identified,
and some findings are not easily acquired during the endoscopy procedure,
which results in less number of data samples. Such underrepresented classes
need to be explored with some specific algorithms specially designed to
leverage the availability of a few hard-to-find samples. Thus, the potential
of the baseline results and associated issues and challenges motivate the need
to publish this dataset for further investigations.

###  4.3 Limitation of the dataset

Our dataset, GastroVision, is a unique and diverse dataset with the potential
to explore a wide range of anatomical and pathological findings using
automated diagnosis. Although this labeled image data can enable the
researchers to develop methods to detect GI-related abnormalities and other
landmarks, it lacks segmented annotations in the current version, which could
further enhance the treatment experience and surgical procedures. It is
important to note that some classes (for example, colon diverticula, erythema,
cecum, esophagitis, esophageal varices, ulcer and pylorus) have only a few
images. Despite this limitation, our dataset is well suited for one-shot and
few-shot learning approaches to explore some GI-related conditions that have
still not received attention in medical image analysis. In the future, we plan
to extend the dataset by including more classes and a larger number of
samples, along with ground truth for some of the classes that could be used
for segmentation purposes as well as images with higher resolution from the
most recent endoscopy systems.

##  5 Conclusion

In this paper, we presented a new multi-class endoscopy dataset, GastroVision,
for GI anomalies and disease detection. We have made the dataset available for
the research community along with the implementation details of our method.
The labeled image data can allow researchers to formulate methodologies for
classifying different GI findings, such as important pathological lesions,
endoscopic polyp removal cases, and anatomical landmarks found in the GI
tract. We evaluated the dataset using some baseline models and standard multi-
class classification metrics. The results motivate the need to investigate
better specific techniques for GI-related data. Having a diverse set of
categories labeled by expert endoscopists from two different centers,
GastroVision is unique and valuable for computer-aided GI anomaly and disease
detection, patient examinations, and medical training.

## Acknowledgements

D. Jha is supported by the NIH funding: R01-CA246704 and R01-CA240639.V.
Sharma is supported by the INSPIRE fellowship (IF190362), DST, Govt. of India.

## References

  * [1] Abadir, A.P., Ali, M.F., Karnes, W., Samarasena, J.B.: Artificial intelligence in gastrointestinal endoscopy. Clinical endoscopy 53(2), 132–141 (2020)
  * [2] Ahn, S.B., Han, D.S., Bae, J.H., Byun, T.J., Kim, J.P., Eun, C.S.: The miss rate for colorectal adenoma determined by quality-adjusted, back-to-back colonoscopies. Gut and liver 6(1), 64 (2012)
  * [3] Ali, S., Dmitrieva, M., Ghatwary, N., Bano, S., Polat, G., Temizel, A., Krenzer, A., Hekalo, A., Guo, Y.B., Matuszewski, B., Gridach, M., Voiculescu, I., Yoganand, V., Chavan, A., Raj, A., Nguyen, N.T., Tran, D.Q., Huynh, L.D., Boutry, N., Rezvy, S., Chen, H., Choi, Y.H., Subramanian, A., Balasubramanian, V., Gao, X.W., Hu, H., Liao, Y., Stoyanov, D., Daul, C., Realdon, S., Cannizzaro, R., Lamarque, D., Tran-Nguyen, T., Bailey, A., Braden, B., East, J.E., Rittscher, J.: Deep learning for detection and segmentation of artefact and disease instances in gastrointestinal endoscopy. Medical Image Analysis 70, 102002 (2021)
  * [4] Ali, S., Ghatwary, N., Braden, B., Lamarque, D., Bailey, A., Realdon, S., Cannizzaro, R., Rittscher, J., Daul, C., East, J.: Endoscopy disease detection challenge 2020. arXiv preprint arXiv:2003.03376 (2020)
  * [5] Ali, S., Jha, D., Ghatwary, N., Realdon, S., Cannizzaro, R., Salem, O.E., Lamarque, D., Daul, C., Riegler, M.A., Anonsen, K.V., et al.: A multi-centre polyp detection and segmentation dataset for generalisability assessment. Scientific Data 10(1), 75 (2023)
  * [6] Areia, M., Mori, Y., Correale, L., Repici, A., Bretthauer, M., Sharma, P., Taveira, F., Spadaccini, M., Antonelli, G., Ebigbo, A., et al.: Cost-effectiveness of artificial intelligence for screening colonoscopy: a modelling study. The Lancet Digital Health 4(6), e436–e444 (2022)
  * [7] Arnold, M., Abnet, C.C., Neale, R.E., Vignat, J., Giovannucci, E.L., McGlynn, K.A., Bray, F.: Global burden of 5 major types of gastrointestinal cancer. Gastroenterology 159(1), 335–349 (2020)
  * [8] Bernal, J., Aymeric, H.: MICCAI endoscopic vision challenge polyp detection and segmentation (2017)
  * [9] Bernal, J., Sánchez, F.J., Fernández-Esparrach, G., Gil, D., Rodríguez, C., Vilariño, F.: Wm-dova maps for accurate polyp highlighting in colonoscopy: Validation vs. saliency maps from physicians. Computerized Medical Imaging and Graphics 43, 99–111 (2015)
  * [10] Bernal, J., Sánchez, J., Vilarino, F.: Towards automatic polyp detection with a polyp appearance model. Pattern Recognition 45(9), 3166–3182 (2012)
  * [11] Borgli, H., Thambawita, V., Smedsrud, P.H., Hicks, S., Jha, D., Eskeland, S.L., Randel, K.R., Pogorelov, K., Lux, M., Nguyen, D.T.D., et al.: Hyperkvasir, a comprehensive multi-class image and video dataset for gastrointestinal endoscopy. Scientific Data 7(1), 1–14 (2020)
  * [12] Crafa, P., Diaz-Cano, S.J.: Changes in colonic structure and mucosal inflammation. In: Colonic Diverticular Disease, pp. 41–61 (2022)
  * [13] Globocan: Cancer today (2020), https://gco.iarc.fr/today/fact-sheets-cancers
  * [14] He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition (CVPR). pp. 770–778 (2016)
  * [15] Huang, G., Liu, Z., Van Der Maaten, L., Weinberger, K.Q.: Densely connected convolutional networks. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 4700–4708 (2017)
  * [16] Jha, D., Ali, S., Tomar, N.K., Johansen, H.D., Johansen, D., Rittscher, J., Riegler, M.A., Halvorsen, P.: Real-time polyp detection, localization and segmentation in colonoscopy using deep learning. IEEE Access 9, 40496–40510 (2021)
  * [17] Jha, D., Smedsrud, P.H., Riegler, M.A., Halvorsen, P., Lange, T.d., Johansen, D., Johansen, H.D.: Kvasir-SEG: a segmented polyp dataset. In: Proceedings of the International Conference on Multimedia Modeling (MMM). pp. 451–462 (2020)
  * [18] Koulaouzidis, A., Iakovidis, D.K., Yung, D.E., Rondonotti, E., Kopylov, U., Plevris, J.N., Toth, E., Eliakim, A., Johansson, G.W., Marlicz, W., et al.: Kid project: an internet-based digital video atlas of capsule endoscopy for research purposes. Endoscopy international open 5(6), E477 (2017)
  * [19] Li, K., Fathan, M.I., Patel, K., Zhang, T., Zhong, C., Bansal, A., Rastogi, A., Wang, J.S., Wang, G.: Colonoscopy polyp detection and classification: Dataset creation and comparative evaluations. arXiv preprint arXiv:2104.10824 (2021)
  * [20] Mahmud, N., Cohen, J., Tsourides, K., Berzin, T.M.: Computer vision and augmented reality in gastrointestinal endoscopy. Gastroenterology report 3(3), 179–184 (2015)
  * [21] Misawa, M., Kudo, S.e., Mori, Y., Hotta, K., Ohtsuka, K., Matsuda, T., Saito, S., Kudo, T., Baba, T., Ishida, F., et al.: Development of a computer-aided detection system for colonoscopy and a publicly accessible large colonoscopy video database (with video). Gastrointestinal endoscopy 93(4), 960–967 (2021)
  * [22] Pogorelov, K., Randel, K.R., Griwodz, C., Eskeland, S.L., de Lange, T., Johansen, D., Spampinato, C., Dang-Nguyen, D.T., Lux, M., Schmidt, P.T., et al.: Kvasir: A multi-class image dataset for computer aided gastrointestinal disease detection. In: Proceedings of the 8th ACM on Multimedia Systems Conference. pp. 164–169 (2017)
  * [23] Silva, J., Histace, A., Romain, O., Dray, X., Granado, B.: Toward embedded detection of polyps in wce images for early diagnosis of colorectal cancer. International journal of computer assisted radiology and surgery 9(2), 283–293 (2014)
  * [24] Smedsrud, P.H., Thambawita, V., Hicks, S.A., Gjestang, H., Nedrejord, O.O., Næss, E., Borgli, H., Jha, D., Berstad, T.J.D., Eskeland, S.L., et al.: Kvasir-capsule, a video capsule endoscopy dataset. Scientific Data 8(1), 1–10 (2021)
  * [25] Tajbakhsh, N., Gurudu, S.R., Liang, J.: Automated polyp detection in colonoscopy videos using shape and context information. IEEE transactions on medical imaging 35(2), 630–644 (2015)
  * [26] Tan, M., Le, Q.: Efficientnet: Rethinking model scaling for convolutional neural networks. In: Proceedings of the International conference on machine learning. pp. 6105–6114 (2019)
  * [27] Thambawita, V., Jha, D., Riegler, M., Halvorsen, P., Hammer, H.L., Johansen, H.D., Johansen, D.: The medico-task 2018: Disease detection in the gastrointestinal tract using global features and deep learning. In: Proceedings of the MediaEval 2018 Workshop (2018)