# Generalized Ellis-Bronikov traversable wormholes in $f(R)$ gravity with
anisotropic dark matter

C. R. Muniz111E-mail<EMAIL_ADDRESS>Universidade Estadual do Ceará,
Faculdade de Educação, Ciências e Letras de Iguatu, 63500-000, Iguatu, CE,
Brazil.  R. V. Maluf222E-mail<EMAIL_ADDRESS>Universidade Federal do
Ceará, Centro de Ciências, Departamento de Física 60000-000, Fortaleza, CE,
Brazil.

###### Abstract

This paper studies generalized Ellis-Bronikov (E-B) traversable wormholes in
$f(R)$ extended gravity. We assume that these wormholes are supported by
anisotropic dark matter (DM) according to the most often used phenomenological
models, namely those of Navarro-Frenk-White (NFW), Thomas-Fermi (T-F), and
Pseudo-isothermal (PI). Initially, we obtain the field equations in a general
scenario of $f(R)$ theories in the metric formalism for the static and
spherically symmetric Morris-Thorne spacetime. Then we particularize to a
$f(R)$ model with power-law, including the Starobinky modified gravity.
Following, we analyze the energy conditions which are not dependent on the DM
models (Null and Weak Energy Conditions – NEC and WEC) and those which are
model-dependent (Strong and Dominant Energy Conditions SEC and DEC). Finally,
we compare some E-B wormhole solutions and the mentioned DM models, discussing
the feasibility of the wormhole-dark matter system in different scenarios.

$f(R)$ modified gravity; Wormholes; Dark matter.

##  I Introduction

The $\Lambda$CDM model of the standard cosmology, supported by observational
data, preconizes that the stuff of our universe consists of about 4$\%$
baryonic matter, 29.6$\%$ dark matter, and 67.4$\%$ dark energy Planck:2018vyg
. Concerning galaxies and their clusters, dark matter strongly contributes to
the formation, evolution, and coalescence of these structures Trujillo-
Gomez:2010jbn  through the only fundamental interaction that it is apparently
capable of experiencing: gravity.

Dark matter halos are present in the vast majority of galaxies, and thus one
has considered the formation of traversable wormholes inside them (see, e.g.,
Sarkar:2019uhk ; Jusufi:2019knb ; Xu:2020wfm ). These hypothetical objects are
predicted by general relativity (GR) and represent a kind of tunnel in the
spacetime that connects two distant regions of the same universe or two
different universes (see Morris:1988cz ; Visser , and references therein).
This subject has been recently investigated in a more fundamental level from
the J. Maldacena works Maldacena:2013xja ; Maldacena:2017axo ;
Maldacena:2020sxe  and also applied in the context of condensed matter systems
Gonzalez:2009je ; Alencar:2021ejd . Usually, it is required some type of
exotic matter sourcing traversable wormholes. However, in scenarios of
modified theories of gravity, such a feature can change with non-exotic matter
working as a source for the wormhole geometry Pavlovic:2014gba ;
Mehdizadeh:2019qvc ; Sahoo:2020sva ; Moti ; Alencar ; Sadeghi:2022sto .

Modified $f(R)$ gravity has also drawn much attention in the last few years.
It is based on a generalization of Einstein’s field equations by replacing the
Ricci scalar curvature, $R$, with a general function, $f(R)$, in the
gravitational action. A relevant feature of $f(R)$ gravity is that,
differently from $\Lambda$CDM standard cosmology based on GR, it is
unnecessary postulating dark energy or introduce any kind of new matter field
to explain the cosmic inflation Kehagias:2013mya ; Ketov ; Aziz:2021evx ;
Sharma:2022tce  and the present phase of the Universe accelerated expansion
Nojiri ; Sotiriou . In fact, the simplest $f(R)$ model, due to A. Starobinsky
Starobinsky1 ; Starobinsky2 , with an additional Ricci scalar quadratic term
in the Einstein-Hilbert action and therefore only one free parameter, leads to
a de Sitter phase for as long as this term dominates, being compatible with
the experimental data collected from the Planck satellite Akrami .

In this paper, we will investigate the possibility of occurring generalized
Ellis-Bronikov (E-B) traversable wormholes in $f(R)$ extended gravity. The
wormholes are admitted to be sourced by anisotropic dark matter compatible
with the most often used phenomenological models, namely those of Navarro-
Frenk-White (NFW), Thomas-Fermi (T-F) and pseudo-isothermal (PI). Thus, we
will obtain the general field equations of any $f(R)$ theory in the metric
formalism, and then we will particularize to a power-law $f(R)$ model,
including the Starobinky modified gravity. Following, we will analyze the
energy conditions associated with the source, which are not dependent on the
DM models (Null and Weak Energy Conditions – NEC and WEC) and those that are
model-dependent (Strong and Dominant Energy Conditions – SEC and DEC).
Finally, we compare some E-B wormhole solutions with the mentioned DM profiles
and the $f(R)$ Starobinsky-like models regarding the energetic feasibility of
wormhole-dark matter systems.

The paper is organized as follows: Section II reviews the general Morris-
Thorne wormhole solution in $f(R)$ gravity. Section III discusses the
generalized E-B wormhole sourced by anisotropic dark matter in the Starobinsky
$f(R)$ model. In section IV, we analyze the energy conditions of the system
wormhole-dark matter, and then finally present the conclusions and close the
paper in section V.

##  II Wormhole solutions in $f(R)$ gravity

Let us start by considering the following action for a $f(R)$ modified
theories of gravity:

$S=\frac{1}{2\kappa}\int d^{4}x\sqrt{-g}\;f(R)+S_{M}(g^{\mu\nu},\psi)\,,$ (1)

where $\kappa=8\pi G$ will be taken equal to the unit $(\kappa=1$) in order to
keep simplicity and $f(R)$ is initially an arbitrary function of the curvature
scalar. Besides this, $S_{M}(g^{\mu\nu},\psi)$ is the matter action with
$S_{M}=\int d^{4}x\sqrt{-g}\;{\cal L}_{m}(g_{\mu\nu},\psi)$, where ${\cal
L}_{m}$ is the matter Lagrangian density, in which the matter is minimally
coupled to the metric $g_{\mu\nu}$ and $\psi$ denotes the matter fields.

Varying the action (1) with respect to the metric, we get the gravitational
field equation:

$FR_{\mu\nu}-\frac{1}{2}f\,g_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}F+g_{\mu\nu}\Box
F=\,T^{(m)}_{\mu\nu}\,,$ (2)

where we define $F\equiv df/dR$ and
$\Box=g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}$. The energy-momentum
tensor associated with the matter content is given by

$T^{(m)}_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\sqrt{-g}{\cal
L}_{m}}{\delta g^{\mu\nu}}.$ (3)

Considering the contraction of Eq. (2), we arrive to the following relation

$FR-2f+3\,\Box F=\,T\,,$ (4)

which shows that the Ricci scalar is a fully dynamical degree of freedom, and
$T=T^{(m)\mu}{}_{\mu}$ is the trace of the energy-momentum tensor.

The trace equation (4) can be used to simplify the field equations and then
can be kept as a constraint equation. Thus, substituting the trace equation
into Eq. (2) and re-organizing the terms, we end up with the following
gravitational field equation

$G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}R\,g_{\mu\nu}=T^{{\rm
eff}}_{\mu\nu}\,,$ (5)

where the effective stress-energy tensor is given by $T^{{\rm
eff}}_{\mu\nu}=T^{(c)}_{\mu\nu}+\tilde{T}^{(m)}_{\mu\nu}$. The term
$\tilde{T}^{(m)}_{\mu\nu}$ corresponds to

$\tilde{T}^{(m)}_{\mu\nu}=T^{(m)}_{\mu\nu}/F\,,$ (6)

and the curvature stress-energy tensor, $T^{(c)}_{\mu\nu}$, is defined as

$\displaystyle
T^{(c)}_{\mu\nu}=\frac{1}{F}\left[\nabla_{\mu}\nabla_{\nu}F-\frac{1}{4}g_{\mu\nu}\left(RF+\Box
F+T\right)\right]\,.$ (7)

It is also interesting to consider the conservation law for the above
curvature stress-energy tensor. Taking into account the Bianchi identities,
$\nabla^{\mu}G_{\mu\nu}=0$, and the diffeomorphism invariance of the matter
part of the action, which yields $\nabla^{\mu}T^{(m)}_{\mu\nu}=0$, we verify
that the effective Einstein field equation provides the following conservation
law

$\nabla^{\mu}T^{(c)}_{\mu\nu}=\frac{1}{F^{2}}T^{(m)}_{\mu\nu}\nabla^{\mu}F\,.$
(8)

Once the scenario for the $f(R)$ theories of gravity has been presented, we
will study the wormhole solutions in this framework. The following line
element describes the geometry of a static and spherically symmetric
traversable wormholes Morris:1988cz :

$ds^{2}=-e^{2\Phi(r)}dt^{2}+\frac{dr^{2}}{1-b(r)/r}+r^{2}\,(d\theta^{2}+\sin^{2}{\theta}\,d\phi^{2})\,,$
(9)

where $\Phi(r)$ and $b(r)$ are arbitrary functions of the radial coordinate,
$r$, denoted as the redshift function, and the shape function, respectively.
The radial coordinate $r$ decreases from infinity to a minimum value $r_{0}$,
the radius of the throat, where $b(r_{0})=r_{0}$. A fundamental property of a
wormhole is that a flaring out condition of the throat, given by
$(b-b^{\prime}r)/b^{2}>0$ Morris:1988cz , and at the throat
$b(r_{0})=r=r_{0}$, such that the condition $b^{\prime}(r_{0})<1$ is imposed
to have wormhole solutions. It is precisely these restrictions that impose the
NEC violation in classical general relativity. Another condition that needs to
be satisfied is $1-b(r)/r>0$. For the wormhole to be traversable, one must
demand that there are no horizons present, which are identified as the
surfaces with $e^{2\Phi}\rightarrow 0$, so that $\Phi(r)$ must be finite
everywhere. In the analysis outlined below, we consider that the redshift
function is constant, $\Phi^{\prime}=0$, which simplifies the calculations
considerably, and provide interesting exact wormhole solutions.

Relative to the matter content of the wormhole, we impose that the energy-
momentum tensor that threads the wormhole satisfies the energy conditions,
which will be discussed in section IV, and is given by the following
anisotropic distribution of matter

$T^{(m)}_{\mu\nu}=(\rho+p_{t})U_{\mu}\,U_{\nu}+p_{t}\,g_{\mu\nu}+(p_{r}-p_{t})\chi_{\mu}\chi_{\nu}\,,$
(10)

where $U^{\mu}$ is the four-velocity, $\chi^{\mu}$ is the unit spacelike
vector in the radial direction, i.e.,
$\chi^{\mu}=\sqrt{1-b(r)/r}\,\delta^{\mu}{}_{r}$. $\rho(r)$ is the energy
density, $p_{r}(r)$ is the radial pressure measured in the direction of
$\chi^{\mu}$, and $p_{t}(r)$ is the transverse pressure measured in the
orthogonal direction to $\chi^{\mu}$. Taking into account the above
considerations, the stress-energy tensor is given by the following profile:
$T^{(m)\mu}{}_{\nu}={\rm diag}[-\rho(r),p_{r}(r),p_{t}(r),p_{t}(r)]$.

Taking into account the above considerations, the effective field equation (5)
provides the following equations for the wormhole geometry

$\displaystyle-\frac{b^{\prime}}{r^{2}}+\frac{\rho}{F}+\frac{H}{F}$
$\displaystyle=$ $\displaystyle 0,$ (11)
$\displaystyle-\frac{b}{r^{3}}-\frac{p_{r}}{F}+\frac{1}{F}\left[H+F^{\prime\prime}\left(\frac{b}{r}-1\right)+F^{\prime}\left(\frac{b^{\prime}}{2r}-\frac{b}{2r^{2}}\right)\right]$
$\displaystyle=$ $\displaystyle 0,$ (12)
$\displaystyle-\frac{b^{\prime}}{2r^{2}}+\frac{b}{2r^{3}}-\frac{p_{t}}{F}+\frac{1}{F}\left[H+F^{\prime}\left(\frac{b}{r^{2}}-\frac{1}{r}\right)\right]$
$\displaystyle=$ $\displaystyle 0.$ (13)

where the prime denotes a derivative with respect to the radial coordinate,
$r$. The term $H=H(r)$ is defined as

$H(r)=\frac{1}{4}\left(FR+\Box F+T\right)\,,$ (14)

and the Ricci curvature scalar is $R=2b^{\prime}(r)/r^{2}$. Thus, the term
$\Box F$ is encoded by the following expression

$\Box
F=F^{\prime\prime}\left(1-\frac{b}{r}\right)-F^{\prime}\left(\frac{b^{\prime}}{2r}+\frac{3b}{2r^{2}}-\frac{2}{r}\right).$
(15)

So we can explicitly write $H(r)$ as

$H=-\frac{\rho}{4}+\frac{p_{r}}{4}+\frac{p_{t}}{2}+\frac{Fb^{\prime}}{2r^{2}}-F^{\prime}\left(\frac{b^{\prime}}{8r}+\frac{3b}{8r^{2}}-\frac{1}{2r}\right)-F^{\prime\prime}\left(\frac{b}{4r}-\frac{1}{4}\right).$
(16)

Note that the field equations (11)-(13), despite their complexities, form an
algebraic system of equations for the components of the energy-momentum
tensor. From the field equations (11)-(13), and with the help of (16), we can
express the radial and transverse pressures in the following way

$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle-\rho+F\left(\frac{b^{\prime}}{r^{2}}-\frac{b}{r^{3}}\right)+F^{\prime}\left(\frac{b^{\prime}}{2r}-\frac{b}{2r^{2}}\right)+F^{\prime\prime}\left(\frac{b}{r}-1\right),$
(17) $\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle-\rho+F\left(\frac{b^{\prime}}{2r^{2}}+\frac{b}{2r^{3}}\right)+F^{\prime}\left(\frac{b}{r^{2}}-\frac{1}{r}\right).$
(18)

It is worth mentioning that the solutions obtained for $p_{r}$ and $p_{t}$
depend on the energy density $\rho$ and on the explicit forms of the functions
$F(r)$ and $b(r)$. In this work, we are interested in studying the behavior of
these pressures and evaluating the resulting energy conditions in a dark
matter context considering the generalized Ellis-Bronikov model for the form
of the wormhole function $b(r)$. We will also assume the Starobinsky-like
model for the $f(R)$ function, motivated by its phenomenological feasibility
in the cosmological context. Such models will be better exemplified in what
follows.

##  III Generalized Ellis-Bronikov wormhole sourced by anisotropic dark matter
in the Starobinsky-like model

The generalised Ellis–Bronnikov model of wormholes was discussed in Kar:1995jz
; DuttaRoy:2019hij ; Sharma:2021kqb  as a two parameter ($n$ and $r_{0}$)
family of simple Lorentzian wormholes, where $n$ is a free even-integer
exponent and $r_{0}$ is the throat radius. This wormhole presents a shape
function given by

$b(r)=r-r^{(3-2n)}(r^{n}-r_{0}^{n})^{(2-\frac{2}{n})}.$ (19)

The exponent $n=2$ furnishes the original and simplest solution for that
wormhole, in which $b(r)=r_{0}^{2}/r$.

The $f(R)$ model that we will henceforth employ is $f(R)=R+\alpha R^{m}$,
which has a cosmological motivation, namely the pure Starobinsky one ($m=2$).
It was shown that models with other values for $m$ (specifically, $1<m\leq 2$)
can in addition cure the formation of curvature singularities in the
gravitational collapse Bamba .

Considering that $R=2b^{\prime}(r)/r^{2}$ and $b(r)$ is given by (19), we have

$F(r)=1+2^{m-1}\alpha
m\left[\frac{1}{r^{2}}+\frac{1}{r^{2n}}(2n-3)\left(r^{n}-r_{0}^{n}\right)^{2-\frac{2}{n}}+\frac{2}{r^{n}}(1-n)\left(r^{n}-r_{0}^{n}\right)^{1-\frac{2}{n}}\right]^{m-1}.$
(20)

Finally, let us define the dark matter content that we will use in our
analysis. As in Jusufi:2019knb ; Xu:2020wfm , we will adopt simple but well-
motivated dark matter density profiles with respect to the $\Lambda$CDM
scenario. Compilations of observations from galaxies and their agglomerates,
supported by numerical simulations Dubinski:1991bm ; Navarro:1995iw ;
Navarro:1996gj , suggest that the dark matter halo can be described by the
Navarro-Frenk-White (NFW) density profile, whose analytical expression is
defined as

$\rho=\rho_{NFW}=\frac{\rho_{s}}{\frac{r}{R_{s}}\left(1+\frac{r}{R_{s}}\right)^{2}},$
(21)

where $\rho_{s}$ is the density within the central region dominated by dark
matter and $R_{s}$ is its characteristic radius. The NFW density profile
accounts for a large family of models for dark matter from which the collision
effects between particles are very weak.

The second case analyzed is based on the Bose-Einstein Condensation dark
matter model and can be described by the Thomas-Fermi (TF) profile, whose
application is more relevant in small distances of galaxies. In fact, the
interactions of dark matter particles can no longer be neglected in the inner
regions of galaxies, so it ceases to be cold. The TF profile can be
represented in the form Boehmer:2007um

$\rho=\rho_{TF}=\rho_{s}\frac{\sin(kr)}{kr},$ (22)

with $k=\pi/R_{s}$. The last case to be considered is the Pseudo isothermal
(PI) profile, related to a class of dark matter models present in theories of
modified gravity known as MOND (modified Newtonian dynamics) Begeman:1991iy .
The PI profile is given by

$\rho=\rho_{PI}=\frac{\rho_{s}}{1+\left(\frac{r}{R_{s}}\right)^{2}}.$ (23)

In the context of wormhole studies, these profiles of dark matter density are
examined in Jusufi:2019knb ; Xu:2020wfm  and are depicted here in Fig. 1 as
functions of the radial coordinate, $r$. The NFW model profile seems to grow
indefinitely near the origin $r=0$. However, this feature proceeds from the
relatively low resolution of the used numerical simulations (order of 1 kpc).
Despite this, the radial coordinate is always greater than (or equals to) the
wormhole throat radius, $r_{0}$, and therefore the origin is out of our
analysis. Also, Fig. 1 reveals that close to the wormhole throat, the dark
matter density is positive for all cases and very similar in the TF and PI
models, while the NFW profile is higher than the others.

Figure 1: The behavior of dark matter density $\rho$ as a function of radial
distance $r$ for the three analyzed profiles. The parameter settings are:
$\rho_{s}=0.05$ and $R_{s}=10$ in Planck units.

##  IV Energy conditions

In order to investigate the energy conditions associated with the anisotropic
dark matter as a source for the wormhole, we replace Eq. (20) and its
derivatives, as well as the dark matter density profiles into Eqs. (17) and
(18). Thus, we find the expressions for the radial and lateral pressures,
respectively. As these are quite involved, we will discuss the energy
conditions graphically. Before this, we must notice that, by Eqs. (17)-(18),
the Null Energy Conditions (NEC) and the Weak Energy Conditions (WEC), for
which $\rho+p_{i}\geq 0$ (also $\rho\geq 0$ for the latter), are independent
of the employed dark matter model. On the other hand, the more restrictive of
the energy conditions (Strong Energy Conditions-SEC, $\rho+p_{i}\geq 0$,
$\rho+\sum_{i}p_{i}\geq 0$) is model-dependent.

Figure 2: Radial dependence of the energy density and pressure combinations
for $n=2$ (top) and $n=4$ (bottom) Ellis-Bronikov wormhole solutions, in the
pure Starobinsky $f(R)$ model ($m=2$) and for the three dark matter profiles,
considering $\alpha=0$ (left panel) and $\alpha=0.5$ (right panel),
$\rho_{s}=0.05$, $R_{s}=10$, and $r_{0}=1$, in Planck units.

In Fig. 2, we depict the energy density and pressure combinations as functions
of the radial coordinate in order to examine NEC, WEC, and SEC, considering
that $\rho>0$. Notice that for $n=2$ (the simplest Ellis-Bronikov solution)
and $\alpha=0$ (i.e., in general relativity), none of these conditions are
obeyed in all space ($r\geq r_{0}$), or they are partially satisfied if we
observe that $\rho+p_{t}=0$. For $n\geq 4$, they are all met exactly at the
wormhole throat. However, as we can see through the general expressions valid
at this point, namely

$\left.(\rho+p_{r}+2p_{t})\right|_{r\rightarrow
r_{0}}=\left\\{\begin{array}[]{cc}2^{m}m(3-2m)\alpha\left(-\frac{1}{r_{0}^{2}}\right)^{m}-\frac{2}{r_{0}^{2}}\left(1+\frac{\rho_{s}r_{0}R_{s}^{3}}{\left(r_{0}+R_{s}\right){}^{2}}\right),&\
\ n=2;\\\
2^{m}m(-5+6m)\alpha\left(\frac{1}{r_{0}^{2}}\right)^{m}+\frac{2}{r_{0}^{2}}\left(1-\frac{\rho_{s}r_{0}R_{s}^{3}}{\left(r_{0}+R_{s}\right){}^{2}}\right),&\
\ n=4;\\\
2^{m}m\alpha\left(\frac{1}{r_{0}^{2}}\right)^{m}+\frac{2}{r_{0}^{2}}\left(1-\frac{\rho_{s}r_{0}R_{s}^{3}}{\left(r_{0}+R_{s}\right){}^{2}}\right),&\
\ n\geq 6,\end{array}\right.$ (24)

for $\alpha=0$, a greater concentration of dark matter may make it hard to
fulfill SEC, since the total sum of the pressures with the density becomes
negative when $\rho_{s}>(r_{0}+R_{s})^{2}/(r_{0}R_{s}^{3})$.

Figure 3: The dominant energy condition (DEC): Radial dependence of the energy
density and pressure $p_{r}$ and $p_{t}$ for $n=2$ (top) and $n=4$ (bottom)
Ellis-Bronikov wormhole solutions, in the simplified Starobinsky $f(R)$ model
($m=2$) and NFW model for dark matter, considering $\alpha=0$ (left panel) and
$\alpha=0.5$ (right panel), $\rho_{s}=0.05$, $R_{s}=10$, and $r_{0}=1$, in
Planck units. Figure 4: Parameter space where the regions with values of both
the throat radius and dark matter characteristic density are highlighted,
which allow SEC to be fulfilled at the wormhole throat, for some values of
$m$, with $n=4$, $\alpha=0.8$, $R_{s}=9$, in NFW DM model (Planck units are
used).

Moving away from general relativity (i.e., $\alpha\neq 0$; this constant must
be positive in order to support cosmological observations Kehagias:2013mya ),
in considering $n=2$, NEC, WEC and SEC are not met nearby and at the wormhole
throat; however, SEC is partially obeyed in a limited region of the space
sufficiently far from it. For $n=4$, those conditions are all satisfied nearby
the throat, provided the content of dark matter is not much high. On the other
hand, according to the second one of Eqs. (24), a greater coupling constant
$\alpha$ can compensate for the growth of the dark matter and thus SEC to
continue being satisfied at the throat. This conclusion is also valid for the
solutions in which $n\geq 6$. Away from the throat, the present energy
conditions are partially fulfilled for all the cases.

Although the three dark matter models under consideration are used in
different contexts, they present the same properties in the wormhole
formation, as evidenced in the previous analysis of the energy conditions,
that is, the influence of different DM density profiles is very similar. As we
can see, some associated behaviors practically coincide. Therefore, we will
focus on the analysis of the NFW profile in what follows.

Regarding the Dominant Energy Conditions (DEC), in which $-\rho\leq
p_{i}\leq\rho$, Fig. (3) depicts $\rho-|p_{i}|$ as functions of the radial
coordinate, $r$, for diverse situations. Considering $\alpha=0$ and $n=2$,
those conditions are partially met with respect to the lateral pressure, in
all space. For $n\geq 4$, DEC is not satisfied anywhere. When $\alpha\neq 0$,
the energy conditions under consideration are partially satisfied in a few
regions of space, for $n\geq 2$.

In Fig. 4 we depict the parameter space where, in the colored region,
$\rho+p_{r}+2p_{t}\geq 0$ is represented at the wormhole throat, for some
values of $m$. Notice that the parameter space is the largest with respect to
the Starobinsky model ($m=2$) for small throat radius and high DM
concentrations. In other words, it is the most energetically favorable.
However, the other power-law $f(R)$ models ($m<2$) are the most favored when
one considers greater throat radius and low DM densities. Therefore, from the
point of view of fulfillment of SEC, the model with the greater deviation from
GR prevails over the other in scenarios where one has very high DM densities
and microscopic wormholes, as expected.

##  V Conclusions

We have studied the generalized Ellis-Bronikov (E-B) traversable wormhole
solutions, which depend on a free parameter besides the throat radius, in
$f(R)$ extended gravity. The shape function associated with the wormhole was
therefore fixed. Because the object is zero-tidal, we can only consider an
anisotropic dark matter as a source for it. We have analyzed three DM
phenomenological models, namely, those of Navarro-Frenk-White (NFW), Thomas-
Fermi (T-F), and Pseudo-isothermal (P-I). Hence, we found the correct field
equations by particularizing the approach to the Starobinky-like $f(R)$ power-
law model.

Following, we have analyzed the energy conditions, focusing on SEC and DEC,
since they depend on the material source profile, differently from that occurs
with NEC and WEC. Dark matter does not obey SEC anywhere in GR (where the
coupling constant $\alpha$ vanishes) for the simplest E-B wormhole ($n=2$).
These energy conditions are precisely met at the wormhole throat in the other
E-B models ($n\geq 4$). Nevertheless, a higher amount of dark matter can
undermine this fulfillment. Concerning DEC, it is not obeyed anywhere in all
situations.

In taking into account the quadratic gravity, for $n=2$ E-B wormhole, SEC is
partially fulfilled in a limited region of space. On the other hand, these
energy conditions are entirely fulfilled nearby and at the wormhole throat for
$n\geq 4$. On the other hand, a higher concentration of dark matter can spoil
such fulfillment, as in RG case. But a greater coupling constant can restore
it at the wormhole throat, as well as a smaller size of this latter. Fig. 4
illustrates such competition between the involved parameters and some power-
law $f(R)$ models, being the pure Starobinsky model ($m=2$) the most
energetically favorable for small throat radii and high concentration of DM
since the associated parameter space is the largest. Otherwise, the power-law
$f(R)$ models with smaller exponents are the most favored. Still, in this
context, we have shown that, unlike GR, DEC is partially satisfied in limited
regions of space for all $n$.

We have thus shown that different models of DM present the same properties in
the wormhole formation. Finally, we can conclude that anisotropic dark matter
can support a class of traversable wormholes as non-exotic matter, at least in
regions nearby and at the throat, in the simplest phenomenological $f(R)$
extended theory of gravity.

###### Acknowledgements.

The authors thank the Conselho Nacional de Desenvolvimento Científico e
Tecnolóogico (CNPq), grants no 308268/2021-6 (CRM) and no 311732/2021-6 (RVM)
for financial support.

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# autohaem: 3D printed devices for automated preparation of blood smears

Samuel McDermott<EMAIL_ADDRESS>Jaehyeon Kim  Department of Physics,
Cambridge University, UK  Aikaterini Anna Leledaki  Duncan Parry  Louis Lee
Alexandre Kabla  Department of Engineering, Cambridge University, UK
Catherine Mkindi  Ifakara Health Institute, Bagamoyo, Tanzania  Richard Bowman
Department of Physics, Bath University, UK  Pietro Cicuta  Department of
Physics, Cambridge University, UK

###### Abstract

The process of making blood smears is common in both research and clinical
settings, for investigating the health of blood cells and the presence of
blood-borne parasites. It is very often carried out manually. We focus here on
smears for malaria diagnosis and research which are frequently analysed by
optical microscopy and require a high quality. Automating the smear
preparation promises to increase throughput and to improve quality and
consistency of the smears. We present here two devices (manual and motorised)
designed to aid in the making of blood smears. These are fully documented,
open-source hardware, and an important principle was to make them easily
fabricated locally anywhere. Designs and assembly instructions are freely
available under an open licence. We also describe an image analysis pipeline
for characterising the quality of smears, and use it to optimise the settings
and tunable parameters in the two devices. The devices are shown to perform as
well as expert human operators, whilst not requiring a trained operator and
offering potential advantages in reproducibility and standardisation across
facilities.

##  I Introduction

Blood smears are used in diagnosis for a variety of hematological disorders
such as anemia. They are also the preferred method of diagnosis of parasitic
infections, such as malaria. Malaria is a widespread parasitic disease,
affecting a large fraction of the world’s population. Our work is motivated by
our experience in both research laboratories and clinical settings in lower-
income countries, where blood smears are made routinely. In large-scale blood
testing centers (e.g. large hospitals in developed countries) this process is
automated and high throughput, but there are relatively few of these
facilities. We focus here on the challenges faced in the much more widespread
situation of smears being performed by hand. Smear preparation is time
consuming, repetitive and labour-intensive. High quality and consistent blood
smears are essential for analysis by optical microscopy of blood cells, and
therefore to achieve accurate diagnoses or characterisation for research.

The current ‘gold standard’ for malaria diagnosis is by optical microscopy
examination of blood smears [1]. A thin film of the patients bloods is fixed
onto a microscope slide and stained. The microscopist looks at the smear,
counting the parasites in various fields of view. This process is highly
sensitive for clinical malaria, allowing differentiation of malaria species
and parasite stages, and can be used to calculate parasite density in the
blood [2].

Unfortunately, there is ample evidence that performance of routine microscopy
in health facilities is poor [3, 4, 5, 6]. Health facilities preparing good
blood films have been shown to be 24 times more likely to have accurate
microscopy diagnosis than poor quality blood films, and well stained blood
smears are 10 times more likely to have accurate microscopy diagnosis compared
to poorly stained blood films [5]. Moreover, competency in preparing the
samples consistently appears to be lower than required [7, 8, 9]. Common
issues with blood smears include greasy and dirty slides, unevenly spread
films (too thick, thin, or streaky), poorly positioned blood films, too much
or too little blood used, and issues with the Giemsa staining protocol, such
as incorrect dye concentration or inconsistent buffer solution.

Our team is actively developing low-cost automation of the imaging and
analysis of blood smears for malaria [10], which tackles one aspect of
robustness and consistency in the diagnostics. It is clear that even the best
image analysis algorithms will struggle with poor quality blood smears. Poor
quality smears cannot even be used for human diagnosis. In the best scenario,
they need to be remade, delaying effective treatment. This is also true in
research settings, where poor quality smears can hamper experiments and affect
repeatability.

This paper describes our work in creating a series of devices which we call
“autohaem”. autohaem devices aim to enable even non-experts to produce
consistent, high quality, thin film blood smears at low cost. We introduce two
devices, mechanical and electronic, both with completely open designs. They
can be manufactured using 3D printers and assembled using common tools,
enabling labs to build their own devices. This open hardware approach is
already starting to revolutionise various aspects of research laboratories
[11, 12, 10]. 3D printers are becoming increasingly common in the lower-income
countries for prototyping and manufacturing. autohaem devices are fully open-
source, with complete printing and assembly instructions [13, 14], assisting
research laboratories who regularly make blood smears and so supporting
research efforts in countries with lower resources.

A pipeline for automated analysis of smear quality is presented and used for
device optimisation. Red Blood Cells (RBCs), at the typical hematocrit for
malaria research, are used as the testing media. This pipeline will also be
suitable for a more systematic analysis of blood smear preparation, for
example to help with training and evaluation of technicians.

Figure 1: There are a large number of variables in the key steps of the manual
smear protocol as recommended by the WHO, represented in these images.
Different users (or even the same person over time) therefore have a high
chance of performing the procedure inconsistently, leading to unusable smears
and or making data not comparable across labs or over time. a) Placement of
drop of blood onto microscope slide. b) Pulling back spreader slider into
blood drop until blood is drawn along its edge. c) Pushing the spreader slide
along the microscope slide, at an angle of
$45\text{\,}\mathrm{\SIUnitSymbolDegree}$, pulling the blood behind. d) The
completed blood smear.

##  II Design parameters

In order to achieve a sustainable, useful device, several options have been
considered, materials and parameters were chosen and optimised. This section
describes the main design process, under the constraints of low cost and local
manufacture.

###  II.1 3D printing

The devices need to be able to be 3D printed: PLA was chosen as the material
because it is a widely available and relatively cheap filament used for fused
filament fabrication 3D printers. It is recyclable and biodegradable (under
certain industrial conditions). Other thermoplastics are likely to perform
broadly the same; stiffer materials will provide an improved rigidity to the
frame, but the slide holder geometry might have to be tweaked. If the devices
become contaminated with blood, PLA can be cleaned using 70% ethanol.

To be able to be printed with the widest range of printers possible, it is
necessary to design the 3D printed parts to meet certain requirements. All
parts were designed to be printed with a layer height of
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$. None of the parts requires
support material, which either requires a different material such as PVA, or
extra manufacturing time to clean the parts after printing. The parts are all
designed to have good adhesion to the print bed, by using curved corners and
orientating the parts such that the larger surfaces are on the print bed.
Therefore, they should not require ‘brims’ in order to adhere to the print
bed, but we recommend that the print bed should be heated. Where screws are
likely to be undone, or are required to be extra strong, nut traps are used to
hold nuts in place. The devices have been printed and tested on Ultimaker (S3,
S5) and Prusa (i3 MK3) printers.

Figure 2: The first device presented here to improve smears, the manually
operated autohaem smear. a) 3D-rendered exploded view of autohaem smear
showing the non-3D printed parts. b) Photo of an autohaem smear showing the
assembled device with the two microscope slides in their positions.

###  II.2 Non-printed parts

Non-printed parts were chosen to ensure that the devices were suitable for
assembly around the world. Common machine fixings such as M3 hex head screws
were used. Other non-printed parts are clearly specified in the assembly
instructions, with examples of where to purchase them from.

Likewise, electronic components were specified to be purchasable from many
large electronics suppliers around the world. They can be soldered using a
simple hand soldering iron, and powered using common power supplies.

Figure 3: The autohaem smear device makes smear production more reproducible,
as characterised in this work. Once the slides and drop of blood have been
loaded, the spreader slide remains at a fixed angle and so the operator only
needs to move the slider handle. The sequence of images shows the key steps in
the operation of the non-motorised device. a) Placement of microscope slide in
slot. b), c) Placement of drop of blood on microscope slide. d) Insertion of
spreader slide into slider slot. e) Pulling back slider. f) Pushing slider
forward. g) Lifting up slider and removing microscope slide with smear. h)
Removing spreader slide. (Photos reproduced from [13].)

###  II.3 Spreader blade

Most commercial automated smearing devices use single use plastic blades as a
spreader. This is not a sustainable solution, generating large amounts of
plastic waste. It is also not suitable for use in devices in lower-income
countries and remote testing facilities, where access to procurement of
proprietary single use parts is not possible. We designed autohaem devices to
function solely with microscope slides, which is more sustainable.

##  III Conventional process of smearing

Blood smears are a common process in working with blood and blood diseases,
and especially in lower-income countries where blood smears are made manually.
The WHO’s basic malaria microscopy learner’s guide [15] describes the
currently accepted way for making manual thin blood smears (key steps are
illustrated in Figure 1):

  1. 1.

Drop the blood on the slide.

  2. 2.

Using another slide as a spreader, touch the drop of blood with its edge. The
blood should be drawn along the length of the edge.

  3. 3.

Push the spreader along the side, at an angle of
$45\text{\,}\mathrm{\SIUnitSymbolDegree}$ whilst remaining in constant contact
with the slide. Our analysis of human experts shows this is done at
approximately $6\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$.

The steps in making blood smears for research are essentially the same,
although in many research applications the blood product might be a refined
fraction of the whole blood, for example only the RBCs are typically used for
culturing blood stage malaria in research. For accurate optical microscopy
diagnosis, which in this application implies counting parasites (and in some
cases identifying their species) the thin film blood smears must have RBCs
that are evenly spread. On a macro level this can be seen as an absence of
linear ridges or ‘hesitation marks’. In terms of RBCs density, a sweet spot
needs to be achieved where cells are not overlapping, but are also not so
sparse that many fields of view are required to observe a large enough cell
number. The typical parasetemia for research cultures is
$5\text{\,}\mathrm{\char 37\relax}$. For quantifying parasitemia for clinical
purposes, the CDC recommends 500 RBCs are counted for high parasetemia
($>10\%$) and 2000 RBCs for low parasetemia ($<1\%$) [16].

Figure 4: The second device presented in this work is motorised and improves
ease of use. a) The 3D rendered exploded view of autohaem smear+ showing the
non-3D printed parts. b) Photo of the assembled autohaem smear+ showing the
two microscope slides in their positions. (Photo reproduced from [14].). c)
Schematics of the electronics modules of autohaem smear+.

##  IV The autohaem smear device

autohaem smear is the simpler and more portable of the two devices developed
here. It is entirely mechanical and requires no electricity, making it
suitable for labs or clinics with an unreliable electricity supply. The device
fixes the angle of the spreader slide, ensures that the motion of the spreader
is restricted to the horizontal axis, and that the spreader is in constant
contact with the slide. The user is able to change the speed of their
smearing, but as this is the only user-changeable parameter, it is much
simpler to make reproducible and consistently high quality smears compared to
a fully manual approach.

###  IV.1 Design

The autohaem smear device is shown in Figure 2. It consists of a 3D printed
main body, which contains a slot to place the horizontal microscope slide. At
one end of the device, a living hinge is used to hold two parallel ground
steel rods. The hinge allows the rods to be lifted to give access to the
microscope slide underneath. The rods sit on two rod rests, with integrated
magnets and held together at the other end by a handle. This ensures that the
rods remain secured in position during smearing. On the rods is the slider.
The slider moves smoothly along the rod with two Oilite bushings and has a
handle for the user to move. The slider contains a slit in which the spreader
slide (a second microscope slide) is inserted. The spreader slide rests on the
horizontal slide with a force due to its weight.

###  IV.2 Operation of autohaem smear

Operation of the autohaem smear device is show in Figure 3. Firstly the user
lifts up the metal rods with the handle. They place the microscope slide down
into the slot and place the drop of blood onto the slide. They then lower the
rods into the lower position where they are held in place by the magnets. A
second microscope slide, acting as the spreader, is inserted into the slot of
the slider until it touches the slide underneath. The user then pulls the
handle back until the edge of the spreader meets the blood and the blood runs
along the edge of the spreader. The user then pushes the slider forward with a
steady motion to produce the smear. The slider can then be lifted, the slide
with the smear removed, and the spreader slide removed through the bottom of
the slider to prevent contamination of residual blood. The edge of the
spreader slide should be cleaned, and then can be used as the next microscope
slide. The slide with the smear can then be fixed, stained and examined,
according to the WHO protocols [15].

###  IV.3 Angle of smear

The angle that the spreader slide makes with the microscope slide is
constrained by the autohaem smear. By designing modifications of the slider
component, it is therefore possible to adjust this angle to investigate how it
effects the quality of the smear, in order to determine the optimal angle.

##  V The autohaem smear+ device

The second device presented here is autohaem smear+. This is an electro-
mechanical device, suitable for laboratories or hospitals with a high
throughput of blood smears. One user could prepare the next blood sample while
the device is smearing, or could operate 2 devices simultaneously, dropping
the blood on one, while the other one is smearing. The device is able to
control the angle and speed of the spreader slide, whilst keeping it level and
in constant contact with the microscope slide. It is possible to program the
speed of the smear (e.g. to optimise for the average environmental conditions
of particular labs, which might affect the smear), and this speed should then
be set constant for every smear, regardless of user.

###  V.1 Hardware design

The device, as shown in Figure 4, consists of a 3D printed main body. This has
a slot in which holds the microscope slide in position, and houses the
electronics. The slider is also 3D printed, and like in the autohaem smear
holds the slide at a fixed angle. It is attached to a lead screw nut to drive
its motion on one end. On the other end is an inverted hook with a magnet on
top. A ground steel metal rod is held above the slide, and is used to guide
the slider. The rod prevents the slider from rotating counter-clockwise, and
the magnet prevents the slider from rotating clockwise. There is also housing
for the Nema 17 motor and the limit switch.

###  V.2 Electronics and Arduino code design

The slider is driven by an Iverntech Nema 17 stepper motor with an integrated
100 mm Tr8x8 lead screw. This was chosen because it is a common component in
the z-axis drive of 3D printers, and so are readily available. As it is a
stepper motor, its position is set precisely, enabling the slider to move at
precise positions and speeds. It is driven by a Polulu DRV8825 stepper motor
driver. This driver was chosen as it meets the high (1.5 A) current rating of
the motor without additional cooling. It also has current control, so that a
maximum current output can be set, allowing higher voltages and therefore
higher step rates. The driver is plugged into a stepper motor driver expansion
board. These generic boards (also usually used for 3D printers and CNC
machines) provide convenient, solder-free pinouts to the driver, along with a
motor connector, a decoupling capacitor, and dipswitches for microstepping.

The motor has a step size specification of
$1.8\text{\,}\mathrm{\SIUnitSymbolDegree}$ and the lead screw pitch of 8 mm,
meaning that 200 steps of the motor will move the slider 8 mm. This was micro-
stepped down to 1/2 step, to increase smoothness, whilst being able to
maintain a suitable speed (the Arduino Uno can support approximately 4000
steps per second). This results in the motor moving the slider 8 mm in 400
half-steps, at a maximum speed of
$8\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$. A 12 V, 2 A DC power supply
is used to power the motor, connected through an external DC socket onto the
motor controller expansion board.

The device is controlled using an Arduino Uno, chosen for its availability and
the ease of using it to control external electronic components. The motor
driver expansion board is connected to the Arduino through the ‘step’,
‘direction’ and ‘enable’ pins. The Arduino is also connected to a limit
switch, which defines the position of the slider at start up and an
illuminated LED button, which is pressed to start the smear, and illuminates
whilst the motor is moving. The Arduino code makes use of the ‘AccelStepper’
library [17], which provides functions for moving the motor gracefully though
acceleration and deceleration, as well as providing convenient functions for
positioning and moving the motor.

Figure 5: Operation of autohaem smear+ only requires the push of a button,
once the slides and the blood drop have been loaded. The device performs very
reproducibly and autonomously, so higher overall throughput can be achieved by
preparing the blood while the device operates or by having multiple devices
running simultaneously. The sequence of images shows the key steps in the
operation of the electro-mechanical device. a) Plugging in device, device
performs calibration. b) Rotating slider and placing microscope slide in slot.
c), d) Placing drop of blood on microscope slide. e) Rotating slider into
position. f) Inserting spreader microscope slide. g) Pressing button to start
smear. h) Device is producing the smear. i) Removing spreader slide. (Photos
reproduced from [14].)

###  V.3 Operation of autohaem smear+

As shown in Figure 5, the user starts the operation of the device by plugging
it in. On startup, the device performs an initial calibration: The slider
moves backward until it makes contact with the limit switch, then moves
forward to its idle position. The user then rotates the slider around the axis
of the lead screw, in order to place the microscope slide in the slot. They
place their drop of blood in the centre of the two circle cutouts. The user
rotates the slider back into position and inserts the spreader microscope
slide into the slot in the top. Upon pressing the button, the device starts to
create the smear. The LED illuminates to show the device is in operation and
the slider starts to move forwards. The slider then slows down as it enters
the region where the blood drop is. The slider microscope slide moves through
the drop of blood, so that the blood is drawn along the edge of the spreader
slide. The slider then moves forward at a user defined speed until it reaches
its idle position, where it stops. The user rotates the slider and removes the
spreader slide from underneath, preventing contamination with any residual
blood. The spreader slide can then be cleaned and re-used as the microscope or
spreader slider. The microscope slide is then removed. When microscope slides
are in short supply, they can be cleaned and reused after examination [15].

###  V.4 Angle and speed of smear

Two parameters can be controlled when using autohaem smear+: the angle that
the spreader slide makes with the microscope slide, and the speed of the
slider as it creates the smear. These two parameters can be adjusted to find
the optimal angle and speed of the device.

##  VI Availability and Documentation

Both autohaem devices have been designed to be open source and accessible from
their inception. The 3D models have been designed in OpenSCAD, with version
control using git and hosted on GitLab [18, 19]. Code reuse between the
different designs was achieved using submodules [20]. The OpenSCAD files are
compiled using GitLab’s CI/CD tools, enabling continuous development and
compilation of the STL parts. The Arduino code is also hosted on Gitlab [21].

Good documentation is key to enabling others to replicate designs or
collaborate on future improvements. The autohaem device assembly instructions
[13, 14] are written to be compiled with GitBuilding [22], allowing clear
instructions that are easy to version control. They are built using GitLabs
CI/CD tools and hosted on GitLab pages, meaning there are constantly up to
date instructions. They consist of a Bill of Materials, step-by-step
instructions with photos for reference, electronics diagrams [23], and a guide
for loading the software onto the Arduino.

##  VII Characterisation of smear quality and Optimisation of devices

In order to test the performance of the devices, they were tested using human
blood. Currently the assessment of blood smear quality is done using
subjective judgement: a person is judged on their ability to make a good smear
when their assessor judges that it looks correct. This is clearly a process
that does not easily lead to general quality standards. Here we present a
methodical, analytical way of judging blood smears, allowing much finer
gradation of success.

###  VII.1 Blood smearing

####  VII.1.1 Blood preparation

$200\text{\,}\mathrm{\SIUnitSymbolMicro L}$ of $3\text{\,}\mathrm{\char
37\relax}$ hematocrit human RBCs in RPMI were spun down and
$150\text{\,}\mathrm{\SIUnitSymbolMicro L}$ supernatant was removed. The blood
cells were resuspended and $3\text{\,}\mathrm{\SIUnitSymbolMicro L}$ was
pipetted for each blood smear.

####  VII.1.2 Manual creation of smears

Four researchers who regularly make blood smears (experts) each created five
blood smears using their usual manual technique.

####  VII.1.3 autohaem smear blood smears

autohaem smear was used by a non-expert to make five blood smears for each of
these angles:
[$30\text{\,}\mathrm{\SIUnitSymbolDegree}$,$40\text{\,}\mathrm{\SIUnitSymbolDegree}$,$50\text{\,}\mathrm{\SIUnitSymbolDegree}$,$60\text{\,}\mathrm{\SIUnitSymbolDegree}$,$70\text{\,}\mathrm{\SIUnitSymbolDegree}$].
The speed of the smear was made as consistent as possible (approximately
$5\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$).

####  VII.1.4 autohaem smear+ blood smears

autohaem smear+ was used by a non-expert to make five blood smears for each of
these angles and speeds:
[$30\text{\,}\mathrm{\SIUnitSymbolDegree}$,$40\text{\,}\mathrm{\SIUnitSymbolDegree}$,$50\text{\,}\mathrm{\SIUnitSymbolDegree}$,$60\text{\,}\mathrm{\SIUnitSymbolDegree}$,$70\text{\,}\mathrm{\SIUnitSymbolDegree}$];
[$3\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$,
$4\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$,
$5\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$,
$6\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$,
$7\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$,
$8\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$].

Figure 6: The smear analysis pipeline can be used to evaluate the performance
of the autohaem devices, but also potentially for teaching and evaluation of
human operators. Once the blood smears have been produced, they are imaged
under a microscope producing several micrographs. These micrographs are imaged
using Cellpose, with the default model and by specifying the average RBC size
in pixels. Cellpose generates a set of masks, one set for each micrograph. The
set contains a number of segmented regions, each one corresponding to an
individual RBC. These masks can be used as an input for CellProfiler, with a
custom CellProfiler pipeline used to extract relevant metadata about the
images and RBCs to be stored in data files for further analysis.

####  VII.1.5 Fixing and staining

All the smears were fixed by covering the smear with methanol for
$30\text{\,}\mathrm{s}$. They were then stained by covering the smear in
$20\text{\,}\mathrm{\char 37\relax}$ Giemsa in distilled water (7.2pH) for 10
minutes before rinsing and drying, similar to the standard WHO protocol [15].

Table 1: Comparison of performance of segmentation tools for RBC smears. 10 fields of view of manually created blood smears were segmented using each tool. The images were also segmented by a human and the number of corrected identified (true positive), unidentified (false negative) and wrongly-identified cells was calculated for each tool’s output. From these results it was determined that Cellpose had the best precision and sensitivity. Segmentation Tool | Precision | Sensitivity
---|---|---
CellProfiler (Thresholding) | 0.934 | 0.844
Weka | 0.988 | 0.917
ImageJ (Hough transform) | 0.970 | 0.948
Cellpose | 0.995 | 0.983

###  VII.2 Imaging

All the blood smears were imaged on a Nikon Ti-E microscope using a
$60\text{\,}\times$, 1.4NA objective lens. The camera was a FLIR Grasshopper 3
colour camera, the images were converted to greyscale for analysis with
CellPose [24]. From a starting position, 20 images were captured moving along
the length of the smear (approximately 4mm). The size of the field of view is
$1920\times 1200$ pixels ($188\text{\,}\mathrm{\SIUnitSymbolMicro m}$ $\times$
$117\text{\,}\mathrm{\SIUnitSymbolMicro m}$).

###  VII.3 Smear analysis pipeline

The smear analysis pipeline uses well documented existing tools to
quantitatively evaluate the quality of thin film blood smears. The pipeline is
illustrated in Figure 6. It is run as a single python notebook file [25].

The images, converted into grayscale, are segmented using Cellpose [24].
Cellpose is a generalist deep learning-based segmentation algorithm for cells
which does not require model retraining or parameter adjustments. Cellpose was
chosen as the segmentation tool for RBCs as it had the best precision and
sensitivity when tested using test smears of RBCs (Table 1). The typical
diameter of the cells (in pixels) is measured and used as a parameter to aid
the software, and Cellpose’s default training model was used. The software
outputs a .png image for each original image, where each ‘object’ or RBC
appears as a different brightness level in the image, creating a series of
masks corresponding to each RBC. Cellpose was used to segment all the images
using its python module [26].

Figure 7: We optimised the two devices to the point that they have a
performance as good as, or better than, expert operators. These graphs show
the comparison of results from smears created by human experts (a mean taken
across the experts), versus autohaem smear, and autohaem smear+. The values
for the mean human expert are the center of the colourmaps, meaning that
regions in blue have a higher value and the regions in red have a lower value.
a) Median number of RBCs within a single field of view. b) Interquartile range
of RBCs within a single field of view. c) Mean index of aggregation per field
of view. d) Mean adjacent neighbours per RBC. e) Mean eccentricity per RBC.

These masks are then processed using CellProfiler [27]. CellProfiler is a
software tool that can create modular image analysis pipelines, with many
useful computational tools. A pipeline [25] was generated using the GUI, and
then run using the command line. This pipeline ingested the masks and
extracted metadata based on their filenames. Any RBCs on the boundary of the
image were removed. Its modules were used to calculate the following
parameters from each image:

  * •

RBC count: The number of RBCs per field of view.

  * •

RBC location: The centre coordinate of each RBC.

  * •

RBC neighbours: How many adjacent neighbours each RBC has. This includes
neighbours which were previously filtered for being on the boundary of the
image.

  * •

RBC eccentricity: How circular each of the RBCs are.

Having processed the images through the smear analysis pipeline, CSV data
files were compiled containing information about the fields of view and the
RBCs contained within. Using these records, the following analyses can be
carried out for each operator, device, angle and speed (these measures were
averaged over the 20 Fields of View per smear, for five smears):

  * •

Median number of RBCs per field of view: To determine the density of the
smear. Too few RBCs in a field of view means that more fields of view are
required to be examined. Too many RBCs in a field of view means that the image
will be crowded and it will be hard to observe.

  * •

Interquartile range of RBCs per field of view: To determine how consistent the
smears are, or whether there are ranges of high and low densities. A lower
interquartile range indicates a more consistent smear.

  * •

Index of aggregation: The Clark and Evans index of aggregation is a standard
measure of clustering of a point pattern [28]. It is the ratio of the observed
mean nearest neighbour distance to that expected for a Poisson distribution of
the same intensity. The nearest neighbour distance for each RBC, $r_{i}$ (that
are not touching the boundary of the field of view, but including distances to
those which are) was calculated using CellProfiler. The mean distance,
$\overline{r_{O}}$, was found for each field of view:

$\overline{r_{O}}=\frac{\sum r_{i}}{n}.$

The expected distance to the nearest neighbour, $\overline{r_{E}}$ was also
calculated:

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\frac{n}{s}$
$\displaystyle\overline{r_{E}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\rho}},$ (1)

where $n$ is the number of cell in the field of view, and $s$ is the size of
the field of view in pixels. The deviation of the observed pattern from the
expected random pattern is measured using the ratio for index of aggregation,
$R$:

$R=\frac{\overline{r_{O}}}{\overline{r_{E}}}.$

If the spatial pattern is random $R=1$, if there is aggregation $R$ tends to
0, and if there is a regular pattern, $R$ tends to 2.15.

  * •

Adjacent neighbours: The mean count of adjacent (touching) neighbours per RBC.
This is a measure of how clumped the RBCs are in the smear, a value of zero
indicates that there are no touching or overlapping RBCs in the field of view.

  * •

RBC eccentricity: The average eccentricity of the RBCs in a field of view.
Defined by CellProfiler [29] as ‘the eccentricity of the ellipse that has the
same second-moments as the region. The eccentricity is the ratio of the
distance between the foci of the ellipse and its major axis length.’ A RBC
with an eccentricity of zero is a perfect circle, and an eccentricity of 1 is
a line. A good blood smear with have RBCs that retain their original circular
shape. If the smear is done poorly, for example by pushing the blood rather
than pulling it, the RBCs will tend to be oval.

Table 2: The results of the nearest neighbour search to find the optimal parameters. The expert’s mean was modified to generate optimal values which could be used in the nearest neighbour search to find the optimal parameters of the two devices. Smear quality measurements | Experts’ mean | Optimal values |  | autohaem smear
---
($30\text{\,}\mathrm{\SIUnitSymbolDegree}$)
| autohaem smear+
---
($50\text{\,}\mathrm{\SIUnitSymbolDegree}$,
$7\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$)
Median RBC count per field of view | 55.6 | 55.6 | 54.0 | 66.5
Interquartile range of RBC count per FoV | 21.6 | 0 | 35.5 | 23.5
Mean index of aggregation | 1.11 | 2.15 | 1.09 | 1.17
Mean adjacent neighbours per RBC | 0.17 | 0 | 0.331 | 0.197
Mean eccentricity per RBC | 0.498 | 0 | 0.535 | 0.469
Figure 8: Typical Fields of View of blood smears produced in the three
conditions: (a) expert humans, (b) autohaem smear and (c) autohaem smear+, in
each case chosen with RBC densities close to the median density for each
group. autohaem smear is set at the optimum angle of
$30\text{\,}\mathrm{\SIUnitSymbolDegree}$. The density of RBCs is close to
that obtained by the human experts, and remains evenly distributed across the
field of view (f.o.v.). The autohaem smear+ is set at its optimum angle of
$50\text{\,}\mathrm{\SIUnitSymbolDegree}$ and speed of
$7\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$. The RBC density is higher
here than that of the human experts but there is reduced aggregation across
the field of view. This performance means that a non-expert is able to use
autohaem smear and autohaem smear+ to produce a blood smear as good, or
better, than that of a human expert. Images are converted to greyscale because
Cellpose uses greyscale images for segmentation. Scale bars
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

##  VIII Results

The smear analysis was compared across the human experts, autohaem smear, and
autohaem smear+, as shown in Figure 7. Using these values, the optimum
parameters for the devices were found.

###  VIII.1 Device performance

The performance of the devices was compared to the human experts, in Figure 7.
The colourbars are centered on the value of the mean human expert, such that
squares that are blue have a higher value than the experts, and squares that
are red have a lower value.

####  VIII.1.1 Density of smear

For autohaem smear, increasing the angle of the spreader slide results in a
higher density of RBCs (Figure 7 a)). This reaches a maximum at
$60\text{\,}\mathrm{\SIUnitSymbolDegree}$, with a median value of 120 RBCs per
field of view. Beyond $60\text{\,}\mathrm{\SIUnitSymbolDegree}$ the density
decreases. This can also be seen in the plot for autohaem smear+, where
increasing the angle corresponds to an increase in density until a maximum is
reached. A similar effect is seen for increasing speed–higher speeds produce a
higher density, until a maximum is reached. This means the highest density of
RBCs is produced at $60\text{\,}\mathrm{\SIUnitSymbolDegree}$ (the same as
autohaem smear) and $6\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$. The
mean human expert smear density is between the lowest and highest density of
the devices, showing that the devices produce smears in the expected range of
densities.

The reason for the reduction at high angles and speeds is because the spreader
slide tends to stick-slip with these parameters. The resulting smear consists
of sharp ridges. This effect is quantified in the Interquartile Range (IQR) of
the density of RBCs across the fields of view (Figure 7 b)). For autohaem
smear+, the IQR at low speeds and angles are lower, indicating that the Fields
of View have consistent densities of RBCs. At higher speeds and angles, the
IQR increases, showing that the ridges seen in the smears are producing
inconsistent RBC densities across the fields of view. The smears produced by
moderate angles and speeds have an IQR consistent with that of the human
experts.

####  VIII.1.2 Index of aggregation

In Figure 7 c), it can be seen that the smears which have a low speed and
medium angle have a lower index of aggregation, indicating more clumping. At
the lowest angle, $30\text{\,}\mathrm{\SIUnitSymbolDegree}$, the clumping is
low due to the lower density of the smear. At the higher speeds the smear is
more uniform.

####  VIII.1.3 Adjacent neighbours

In Figure 7 d), smears created at low angles with low speeds tend to have RBCs
with fewer adjacent neighbours. Those Fields of View with the highest density
of RBCs intuitively have RBCs with more adjacent neighbours. For the Fields of
View with intermediate densities, there appears to be no relation between the
parameters and the number of adjacent neighbours.

####  VIII.1.4 Eccentricity

In Figure 7 e), there appears to be no correlation between the parameters and
the eccentricity of the RBCs. The slowest and lowest angle RBCs have some of
the highest eccentricities, as the blood starts to get ahead of the spreader
blade and is pushed rather than pulled.

###  VIII.2 Optimal parameters

In order to determine the optimal parameters (angle, speed) for the devices,
nearest neighbour searching was performed. Each of the set of parameters was
given a ‘smear vector’, describing its performance in the five analyses. Two
such smear vector sets were created for these parameters, one for autohaem
smear and one for autohaem smear+.

The mean of human experts was also given a smear vector, with values shown in
Table 2. The measurements were scaled so that they were equally important. The
optimal smear vector was also calculated by matching the median density to
that of an human expert, assigning the optimal index of aggregation to 2.15
(upper limit indicating a regular spatial pattern), and assigning all the
other measurements optimal values of 0.

A nearest neighbour search was performed between the optimal smear vector and
both smear spaces. For autohaem smear, the optimal angle was found to be
$30\text{\,}\mathrm{\SIUnitSymbolDegree}$, with its smear vector shown in
Table 2. For autohaem smear+, the optimal angle and speed were found to be
$50\text{\,}\mathrm{\SIUnitSymbolDegree}$ and
$7\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$, with its smear vector also
shown in Table 2. These values agree with the preconceived notions of how
humans are taught to make manual smears. They are close to the recommended
$45\text{\,}\mathrm{\SIUnitSymbolDegree}$ angle, and correspond to a speed
that is close to those of human experts
($6\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$). Typical images of smears
produced by the expert humans and the two devices at their optimum parameters
are shown in Figure 8.

##  IX Conclusions

In this work we have developed and presented the autohaem range of devices for
automated blood smearing. autohaem smear is a mechanical device and autohaem
smear+ is an electro-mechanical device. The devices are designed to be
sustainable and all the designs and assembly instructions are available under
an open source licence.

An automated smear analysis pipeline was created and used here as a tool for
quantifying the quality of blood smears. We propose that it can have further
applications in assessing human proficiency in making blood smears.

Both devices were operated exploring two tunable parameters. Once optimised,
autohaem smear can fix the angle that a spreader slide makes with the
microscope slide, and autohaem smear+ can also fix the speed that the slider
moves, making smears more consistent. In this work, blood smears were made by
the two devices and by human experts using the manual protocol. The quality of
the smears was quantified and the optimum parameters of angle and speed
determined for the two autohaem devices. The devices were shown to perform on
par with human experts, even when operated by users with no experience of
making blood smears. These devices hence promise to deliver better
standardisation and robustness of quality across labs and over time.

The next stage of this work will be to analyse the performance of these
devices using whole blood. Improvements to the devices will include providing
better protection to perform in non-ideal environments, for example rural
clinics that will be dustier than a laboratory microbiology safety cabinet,
and to map out optimal device settings as a function of different temperature
and humidity. On the hardware side, a better suspension of the spreader slide
would be an improvement and might reduce ridges in the smears.

### Contributions

SM designed the study, was the lead designer and maintainer of the devices,
software, and assembly instructions, prepared the blood samples and reagents,
performed testing of the devices, carried out the data analysis, and drafted
the manuscript; JK contributed to the device designs, software, and assembly
instructions and performed testing of the devices; AAL, DP and LL contributed
to the device designs and data analysis; AK contributed to the device designs
and the design of the study, and coordinated the study; CM contributed to the
design of the study; RB contributed to the device designs and the design of
the study; PC conceived of the study, designed the study, coordinated the
study, and helped draft the manuscript. All authors gave final approval for
publication and agree to be held accountable for the work performed therein.

###### Acknowledgements.

SM, AK, CM, PC were funded by a GCRF QR grant awarded by Cambridge University.
RWB is supported by a Royal Society award URF\R1\180153.

## Data Availability Statement

All materials and data are available at: https://gitlab.com/autohaem.

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