Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
10.1109/ACCESS.2020.DOI

This research was funded by the projects AIM, ref. TEC2016-76465-C2-1-R
(AEI/FEDER, UE), ATENTO, ref. 20889/PI/18 (Fundación Séneca, Región de
Murcia), and LIFE (Fondo SUPERA Covid-19 funded by Agencia Estatal Consejo
Superior de Investigaciones Científicas CSIC, Universidades Españolas and
Banco Santander). S. Canovas-Carrasco thanks the Spanish MECD for an FPU (ref.
FPU16/03530) pre-doctoral fellowship. This work was also supported by the
Polish Ministry of Science and Higher Education with the subvention funds of
the Faculty of Computer Science, Electronics and Telecommunications of AGH
University of Science and Technology.

Corresponding author: R. Asorey-Cacheda (e-mail: rafael.asorey@upct.es).

# Understanding the Applicability of Terahertz Flow-guided Nano-Networks for
Medical Applications

SEBASTIAN CANOVAS-CARRASCO1, RAFAEL ASOREY-CACHEDA1, ANTONIO-JAVIER GARCIA-
SANCHEZ1, JOAN GARCIA-HARO1, KRZYSZTOF WOJCIK2, PAWEL KULAKOWSKI3  Department
of Information and Communication Technologies, Universidad Politécnica de
Cartagena, 30202 Spain (e-mail: {sebas.canovas, rafael.asorey,
antoniojavier.garcia<EMAIL_ADDRESS> Jagiellonian University Medical
College, Krakow, Poland (e-mail<EMAIL_ADDRESS> Department of
Telecommunications, AGH University of Science and Technology, 30-059 Krakow,
Poland (e-mail<EMAIL_ADDRESS>
###### Abstract

Terahertz-based nano-networks are emerging as a groundbreaking technology able
to play a decisive role in future medical applications owing to their ability
to precisely quantify figures, such as the viral load in a patient or to
predict sepsis shock or heart attacks before they occur. Due to the extremely
limited size of the devices composing these nano-networks, the use of the
Terahertz (THz) band has emerged as the enabling technology for their
communication. However, the characteristics of the THz band, which strictly
reduce the communication range inside the human body, together with the energy
limitations of nano-nodes make the in-body deployment of nano-nodes a
challenging task. To overcome these problems, we propose a novel in-body flow-
guided nano-network architecture consisting of three different devices: i)
nano-node, ii) nano-router, and iii) bio-sensor. As the performance of this
type of nano-network has not been previously explored, a theoretical framework
capturing all its particularities is derived to properly model its behavior
and evaluate its feasibility in real medical applications. Employing this
analytical model, a thorough sensitivity study of its key parameters is
accomplished. Finally, we analyze the terahertz flow-guided nano-network
design to satisfy the requirements of several medical applications of
interest.

###### Index Terms:

flow-guided nano-networks, body area networks, analytical model, nano-
communications

=-15pt

##  I Introduction

Electromagnetic (EM) nano-communications have been attracting attention in
recent years as a powerful technological advance enabling novel applications
that could revolutionize different areas [1, 2, 3, 4, 5, 6], with particular
relevance to medicine [7, 8, 9, 10, 11, 12, 13]. This emerging field deals
with communication among size-constrained devices (known as nano-devices)
forming nano-networks, and the connection between these nano-networks to the
Internet paving the way for a new paradigm, the Internet of Nano-Things (IoNT)
[1].

As the size of nano-devices can be scaled down to a few cubic micrometers
[12], the integration of a communication module inside each nano-device is a
challenging issue that has been thoroughly addressed in different works in the
literature [14, 15, 16, 17, 18, 19]. All these studies point to the
miniaturization of the antenna embedded into each nano-device, which requires
the use of THz frequencies to communicate, leading to the standardization of
the THz band (0.1-10 THz) as the working frequency spectrum for EM nano-
communications [20]. However, these high EM frequencies entail high path
losses when the signal propagates throughout the medium, greatly reducing the
communication range between nano-devices. This limitation becomes more
restrictive when the physical medium has a high water content, as is the case
of human body tissues (e.g., skin or fat) and fluids (e.g., blood), due to
their high absorption coefficient at THz frequencies [21, 22, 23, 24].

Even though these nano-devices present several technical limitations, they are
able to perform simple tasks in scenarios where larger artificial devices
would be excessively intrusive. This is of special relevance within the human
body, where simple tasks (e.g., monitoring a biomarker and sending warnings)
performed in vivo and in real time would significantly improve the
effectiveness of disease diagnoses. Concretely, we focus on a nano-
communication network composed of micrometer-sized devices (hereafter referred
to as nano-nodes) located inside the human cardiovascular system to assist the
immune system in detecting specific bacteria in blood, severe inflammatory
response that may lead to sepsis, blood flow disorders, and ischemic heart
diseases, as discussed later in this work. This type of network, where nano-
nodes are in constant movement throughout the bloodstream, is known as a flow-
guided nano-network [25].

Flow-guided nano-networks are designed to overcome two important challenges
that must be carefully investigated. First of all, the limited communication
range inside the human body makes direct communication between nano-nodes and
an external device unachievable [22, 24]. Secondly, the amount of energy that
can be stored in a nano-node is scarce due to the constraints derived from
downsizing the battery. To alleviate this restriction and provide a continuous
source of energy, nano-nodes should use a piezoelectric nano-generator to
harvest energy from the environment, as proposed in several works in the
literature [26, 27, 28, 29, 30, 31, 32]. Still, the energy provided by the
piezoelectric generator does not allow nano-nodes to be continuously working,
so they need to alternate between idle (stand-by) and active cycles.

The human vascular system is a closed system, composed of many sub-circuits
(see Fig. 1a), so the nano-devices, when injected into it, travel within the
bloodstream carried by the force of the pumping heart, reaching all the human
tissues in subsequent cycles. Up to now, flow-guided nano-networks aimed at
medical applications have been designed assuming that nano-nodes both sense
the medium (i.e., blood) and transmit the information. However, this approach
is not useful for certain applications where monitoring a critical part of the
body is required. Thus, an additional sensing device positioned at the place
of interest is necessary. With this objective in mind, we have conceived a new
kind of flow-guided nano-network consisting of three different types of
devices (depicted in Fig. 1):

  * •

Nano-node. This is the smallest device, and it continuously flows through the
blood circulatory system, being able to either receive (from a bio-sensor) or
transmit data frames (to a nano-router). It is endowed with a THz
communication module, in line with the proposals in [19, 14], and a
piezoelectric generator to recharge the battery periodically. Due to its
extremely limited resources, a nano-node cannot always be operative, so it can
only perform one task (transmission or reception), and only when its battery
level is high enough.

  * •

Nano-router. A larger and less resource-constrained device that acts as a
middle network tier receiving the data carried by nano-nodes and sending them
to a macro-device able to connect the whole nano-network to a healthcare
provider or, ultimately, to the Internet, following the paradigm of the IoNT
[1, 3, 33, 34]. It is located on a superficial vein so that the distance
between the nano-router and the macro-device (e.g., a wearable) is as short as
possible [22].

  * •

Bio-sensor. A medical device located in specific internal parts of the body
sensing a given medical parameter or vital sign of interest. It is equipped
with a communication module able to transmit the acquired information to the
mobile nano-nodes circulating within the bloodstream.

As the characteristics of this type of flow-guided nano-network where nano-
nodes act as information carriers are not similar to other nano-network
topologies, a theoretical framework capturing all their particularities is
necessary to properly model their behavior. For this purpose, in this paper we
propose a general analytical model to mathematically evaluate different
metrics of this novel kind of flow-guided nano-network. From this, we derive
closed-form expressions for effective throughput, quality of delivery, and
average data delay as functions of the different key parameters considered in
the model. We also employ these expressions to analyze the impact of each one
of these parameters on the nano-network performance.

Figure 1: (a) The closed system of human blood circulation pumped by the heart
(H) to the lungs (L) and other body organs; (b) bio-sensors (green circles)
and nano-routers (yellow circles) indicated on the human vascular system; (c)
mobile nano-nodes circulating along with the red blood cells in blood vessels.

To summarize, the main contributions of this work are the following:

  * •

We design and develop an analytical model capturing all the special
characteristics of this novel THz flow-guided nano-network architecture based
on Markov chains. This model jointly considers: (i) the energy balance in
nano-nodes derived from the energy harvesting and consumption processes, and
(ii) the probability of reception and transmission based on realistic values
of the cardiovascular system and the limited communication range of nano-nodes
due to communication in the THz band. We note that this model is general
enough to be employed in different applications in which this nano-network
could be used.

  * •

We provide insightful results about nano-network performance as a function of
different parameters to accurately predict nano-network behavior in diverse
scenarios. We pay special attention to the required number of nano-nodes in
the network for less intrusiveness and an adequate network dimensioning for
the different medical applications under study.

  * •

We validate the proposed model by simulating this type of flow-guided nano-
network under different conditions. The results obtained from the simulations
have been compared with those derived from the analytical expressions, showing
that the mathematical framework accurately models the performance of this
nano-network architecture.

  * •

We analyze the requirements and limitations of potential flow-guided nano-
networks to be used in medical applications of early-stage detection of
bacterial and viral infections, sepsis, heart attacks, and restenosis,
employing realistic and accepted values for all the parameters employed.

The rest of the paper is summarized as follows. In Section II, several
potential medical applications where this kind of nano-network can play a
critical role are identified. Section III rigorously defines the analytical
model, considering all the characteristics of this new type of flow-guided
nano-network. All the results are discussed in Section IV, confirming the
analytical model by means of simulations. The impact on network performance of
the most relevant parameters is also analyzed and discussed. Finally, Section
V concludes the paper.

##  II Applications

While a few years ago nano-devices in human blood were matters of theoretical
concepts and early prototypes [35], substantial progress has been made
recently and numerous fully developed nano-bot systems have been presented. In
[36], an effective application of nano-sponges cleaning the blood of toxins
was reported. Later, the way a swarm of magnetotactic bacteria loaded with
drugs and steered with a magnetic field could assist in tumor therapies was
demonstrated [37]. Also recently, nano-bots neutralizing pathogenic bacteria
and toxins in blood have been presented [38]. In this section, we focus on
some more challenging applications, where nano-devices should not only
effectively perform their functions in the human vascular system, but also
communicate with each other. Thus, examples of such applications related to
bacteria and viral infections, sepsis, heart attacks, and restenosis, are
discussed in detail.

###  II-A Bacterial blood infections

While the presence of numerous bacteria types is quite common in a human body,
their appearance in the vascular system is very dangerous, as they can then
spread and infect multiple human organs. It is an especially serious problem
in the case of hospital patients with many comorbidities, because their immune
systems are already weakened. The ability to rapidly detect bacteria, like
Pseudomonas, Escherichia coli, Acinetobacter, Staphylococcus, or Streptococcus
in cases of patients in a critical condition, might be a matter of life and
death, as the early antibiotic administration is correlated with increased
survival rate. Even a very low number of these bacteria in blood is dangerous.
Thus, we propose a bio-sensor consisting of antibodies for each of these
bacteria types. Antibodies (immunoglobulins) are protein molecules, about
10-30 nanometers in size, which recognize a epitope unique for different
bacteria. Antibodies may signal the presence of the bacteria in many ways, but
here, we focus on the technique described in [39], where the antibody is
attached to an electrode. After detecting the bacteria, the electrode
resistance changes. Thus, assuming a simple micro-device is connected to the
electrode, this device should be able to measure the resistance and start
transmitting a warning immediately after detecting a change. A single sensor,
no larger than a millimeter, can have antibodies detecting numerous important
bacteria. Multiple sensors, such as these should probably be located close to
important organs like lungs, the bladder, and kidneys, in order to identify
the source of the bacteria.

###  II-B Viral load monitoring

The same technique proposed for bacteria detection [39] can be used for viral
load, i.e., density monitoring [40]. Viruses are responsible for infectious
diseases, but they differ from bacteria in a few important aspects. The main
differences include: (a) the size, as viruses are significantly smaller than
bacteria (ca. 10x); (b) different strategy of replication, as viruses use the
host cell machinery to create their new copies. Viral infections can be
divided into two groups. The first group are acute (severe, but short-lived)
infections, like typical flu viruses, and also COVID-19. These viruses may not
be present in the blood at all, thus the discussed monitoring system will not
be able to identify them. The second group, however, are chronic (long-term)
infections, in which case the viruses are detectable in blood. The most
important infection of this group requiring viral load monitoring is the human
immunodeficiency virus (HIV). This measurement is crucial to evaluate of
therapy results, plan further treatment, and, most importantly, evaluate the
risk of HIV transmission. The threshold for HIV viral load is 1500 copies/ml
[41], and this is a cornerstone of the so-called U=U campaign (Undetectable
Equals Untransmitable). Thus, this task will require a new function from the
proposed system: not only the detection of certain particles but also the
ability to quantitatively measure the concentration of studied elements. In
the case of HIV, this is the number of copies per ml.

###  II-C Sepsis

Sepsis is a state of the human immune system when its reaction to infections
may threaten its own healthy tissues and organs, finally causing death. Sepsis
is signaled by a very high density of cytokines in the blood, in particular
interleukin 6 (IL-6) molecules. IL-6 molecules are normally present in blood
of a healthy individual as well, but in the case of sepsis, their density
suddenly rises from about $1.5\cdot 10^{10}$/ml (a healthy person) to
$3.75\cdot 10^{11}$/ml and higher (for sepsis) [42]. They are produced by
living cells and their lifetime may be modeled by an exponential distribution
with an average of 45 minutes [43].

Sepsis is usually detected via blood analysis in a laboratory. However, this
is quite time consuming, as the blood must be taken from the patient,
delivered to the lab and analyzed. Sepsis occurrence is very often detected
too late, as it could be a matter of hours which determine whether a patient
survives or not. The bio-sensor located inside the vascular system could check
the patient’s state constantly. However, as IL-6 density is quite high even in
healthy people, sensors with antibodies cannot be used, as in the previous
case with bacteria. Such a sensor would be immediately blocked with IL-6
cytokines, detecting and signaling their high density all the time. Instead,
we propose using a special small tube, about 0.25 mm long, implanted inside of
a human vein. At one end, the tube has a membrane that is semi-permeable,
i.e., small molecules like IL-6 cytokines can flow through but living cells
cannot (living cells are quite large, about 10-20 micrometers in diameter).
The other walls of the tube are not permeable. So, the IL-6 can pass through
the membrane and propagate inside the tube by diffusion, finally reaching the
opposite end of the tube after about 15 minutes [44]. At this end, we have a
layer of antibodies matching IL-6 cytokines. When an antibody catches an IL-6,
the electrical resistance of the connected electrode changes (as described in
the previous subsection). The size of the semi-permeable membrane and the
length of the tube are carefully chosen so that only a small number of IL-6
cytokines reach the antibodies for a healthy person. However, when sepsis
appears, the number of IL-6 cytokines is much larger, so the nano-node
connected to the electrode receives a much higher signal and should send a
warning only in this case.

With such a tube, the sensor with antibodies detecting IL-6 cytokines may work
for a long time. As most of the IL-6 molecules brake when propagating through
the tube, only a small amount of them reach the opposite end of it. After
matching an IL-6, an antibody is not active during a certain period of time,
but recovers after that and can work again. Moreover, as the membrane is semi-
permeable, not letting living cells inside, the IL-6 cytokines cannot be
produced inside the tube. A single tube in a vein is enough, because with
sepsis, the density of IL-6 cytokines quickly rises in the entire vascular
system. A second similar device could be considered as a back-up.

###  II-D Heart attacks

The discussed nano-network may also be critically helpful in cardiac issues,
e.g., in the case of people at risk of heart attacks (myocardial infarctions).
A marker which helps to detect the danger of a heart attack very early is a so
called Heart-type Fatty Acid Binding protein (H-FABP). If the H-FABP density
exceeds $1.8\cdot 10^{11}$/ml, this is considered to be a serious warning that
may suggest a myocardial infarction [45]. Here, we also propose a bio-sensor
comprised of a layer of specific antibodies attached to an electrode, but the
layer is not in a tube, instead, it is covered with a shutter, opening just
for the time necessary to make a measurement. The shutter is normally closed,
so the H-FABP present in the blood does not block the bio-sensor completely.
The shutter opens periodically, once per e.g., 15 minutes, in order to measure
H-FABP density. If the density is high, a warning signal is transmitted by the
bio-sensor. It may also be activated externally in order to take the H-FABP
measurement on demand. This procedure can be initiated if a person feels sick,
e.g., has some clinical symptoms like chest pain, dyspnea (shortness of
breath), or low tolerance to physical effort. Clinical symptoms correlated
with high H-FABP levels are a clear indication of a myocardial infarction
[45].

###  II-E Restenosis

One more application is related to stenosis, i.e., the process of the
narrowing of blood arteries, usually caused by atherosclerosis. A popular
solution is to put a stent into the narrowed artery to keep it open. However,
stents frequently get blocked after some time, as some thrombus (solidified
blood) may gather on it and, in consequence, the artery becomes blocked again,
which may cause a heart attack [46]. In order to control the state of a stent,
a micro-sensor can be mounted on it. The sensor on the stent would then
periodically transmit signals to mobile nano-nodes flowing through the artery.
When the process of restenosis becomes significant, which usually means the
lumen of an artery is more than 90% closed, i.e., the stent is covered with
1-1.5 millimeters of thrombus [46], the signals from the sensors on the stent
will be much weaker. Thus, the network can detect this situation and dispatch
a warning to the gateway and then to the external medical devices. About 20-30
sensors can be mounted on a single stent, in order to have clear information
about its condition.

##  III Analytical Model

Considering the limitations implicitly derived from employing nano-devices,
the analytical model for the proposed THz flow-guided nano-network is based on
the following assumptions:

  * •

There are $n$ nano-nodes randomly distributed throughout the human vascular
system, $n\geq 1,n\in N$, with a uniform distribution throughout the total
volume ($V_{t}$). Therefore, the nano-node concentration per unit of volume
can be considered as constant in the entire system. In addition, two types of
larger (less resource-constrained) devices are placed in two different parts
of the system: (i) the nano-router, which is in charge of receiving and
gathering all the information transmitted by nano-nodes (encapsulated in data
frames), and (ii) the bio-sensor, which is continuously transmitting the
collected data from the area of interest to the nano-nodes.

  * •

Nano-nodes constantly move through the cardiovascular system, which can be
modeled as a branched closed circuit (i.e., each branch would be equivalent to
a vein or an artery). The nano-router is placed in one of these branches to
receive the data frames previously taken in another branch, where the bio-
sensor is located. Although the time to complete a round depends on the branch
through which a nano-node flows, on average, nano-nodes take $T$ seconds to
complete a round through the circuit.

  * •

As the vascular system is divided into different subcircuits, the blood only
flows through one of them in each round before returning to the lungs. This
implies the inability of a nano-node to reach both bio-sensor and nano-router
in just one round if they are located in separate subcircuits. Therefore, to
prevent the overestimation of the probability of transmitting a previously
received frame, we assume that the full data transmission from the bio-sensor
to the nano-router requires at least 2 rounds, one round to receive the frame
from the bio-sensor and the following one to transmit it to the nano-router.

  * •

Due to the motion of nano-nodes within the bloodstream, the integrated nano-
generator periodically charges the battery every $1/f$ seconds, with $f$ as
the frequency of charge. Thus, nano-nodes alternate idle and active cycles.
Due to battery and memory constraints, we consider the worst-case scenario,
that is, that nano-nodes can only transmit or receive one data frame per
active cycle. During idle cycles, nano-nodes just keep the frame stored in
memory.

Figure 2: Coverage volume model.

  * •

As the communication distance of the nano-node/nano-router link is greatly
limited by the high path loss of the THz band experienced in blood[21, 24], we
define $V_{cv}$ as the volume in which nano-nodes are able to send a frame to
the nano-router, that is, the coverage volume. This volume is defined by two
parameters: (i) the communication range ($r$) and (ii) the diameter of the
vein on which the nano-router is placed ($D$). Assuming an isotropic THz
antenna is used [47], the communication range of nano-nodes can be considered
constant across the three dimensions of space. We remark that the model can be
adapted to any type of transceiver just by tuning $V_{cv}$ as needed. Thus,
generally speaking, the volume in which nano-nodes could communicate with the
nano-router would be a sphere with a radius $r$ and the nano-router at its
center. However, all the nano-nodes are confined within a vein, so the real
volume $V_{cv}$ in which nano-nodes can indeed carry out a successful
transmission is determined by the intersection between the already mentioned
sphere and a cylinder with diameter $D$, as shown in Fig. 2. The expression
required to analytically calculate $V_{cv}$ using Cartesian coordinates is
defined by:

$V_{cv}=\int_{\zeta_{min}(x,y)}^{\zeta_{max}(x,y)}\int_{\gamma_{min}(x)}^{\gamma_{max}(x)}\int_{\chi_{min}}^{\chi_{max}}dxdydz,$
(1)

where:

$\chi_{min}=\begin{cases}-\frac{D}{2}&\text{if $r^{2}\geq\frac{D^{2}}{2}$}\\\
-r\sqrt{1-\frac{r^{2}}{D^{2}}}&\text{if $r^{2}<\frac{D^{2}}{2}$}\end{cases}$
(2) $\chi_{max}=\begin{cases}\frac{D}{2}&\text{if
$r^{2}\geq\frac{D^{2}}{2}$}\\\ r\sqrt{1-\frac{r^{2}}{D^{2}}}&\text{if
$r^{2}<\frac{D^{2}}{2}$}\end{cases}$ (3)
$\\!\\!\gamma_{min}(x)\\!=\\!\max\\!\left\\{\frac{-D\\!-\\!\sqrt{D^{2}\\!-\\!4x^{2}}}{2},\\!-\sqrt{r^{2}-x^{2}}\right\\}$
(4) $\gamma_{max}(x)=\frac{-D+\sqrt{D^{2}-4x^{2}}}{2}$ (5)
$\zeta_{min}(x,y)=-\sqrt{r^{2}-x^{2}-y^{2}}$ (6)
$\zeta_{max}(x,y)=\sqrt{r^{2}-x^{2}-y^{2}}$ (7)

  * •

Depending on the part of the circulatory system through which nano-nodes flow,
their velocity is variable. To model this velocity variation, we consider $v$
as the velocity when passing through $V_{cv}$. This velocity influences the
time nano-nodes are within the coverage volume, thus affecting the probability
of successfully transmitting a frame when crossing $V_{cv}$. In contrast, the
velocity variation outside this zone does not impact on the metrics considered
in this model, since it is characterized as an average round time ($T$), a
known value studied in the literature (60 seconds) [48]. Using this approach,
we are able to develop, a priori, a model for any circular fluid system just
by knowing the average round time, thus not needing the instantaneous velocity
of nano-nodes when flowing through every part of the circuit, which can be
really challenging in complex scenarios (e.g., the human cardiovascular
system).

  * •

Without limiting the generality, we assume that nano-nodes cannot perform more
than one transmission when crossing $V_{cv}$ since the time between
transmissions is longer than the time to cross $V_{cv}$. As shown in Fig. 2,
this assumption can be expressed mathematically as $2r/v\ll 1/f$. Therefore,
nano-nodes can only perform one transmission per round.

  * •

A successful transmission can only be achieved when a nano-node starts and
ends the transmission of a data frame within $V_{cv}$. Thus, with $t_{f}$ as
the time to transmit a complete frame, we define the transmission volume
($V_{tx}$) as:

$V_{tx}=\int_{\zeta_{min}(x,y)}^{\zeta_{max,tx}(x,y)}\\!\\!\\!\\!\int_{\gamma_{min,tx}(x)}^{\gamma_{max}(x)}\\!\int_{\chi_{min,tx}}^{\chi_{max,tx}}dxdydz,$
(8)

where $\chi_{min,tx}$, $\chi_{max,tx}$, $\gamma_{min,tx}$, and
$\zeta_{max,tx}$ correspond to expressions (9) to (12).

$\chi_{min,tx}=\begin{cases}-\frac{D}{2}&\text{if
$r^{2}\geq\frac{D^{2}}{2}+\left(\frac{vt_{f}}{2}\right)^{2}$}\\\
-\sqrt{\left(r^{2}-\left(\frac{vt_{f}}{2}\right)^{2}\right)\left(1-\frac{1}{D^{2}}\left(r^{2}-\left(\frac{vt_{f}}{2}\right)^{2}\right)\right)}&\text{if
$r^{2}<\frac{D^{2}}{2}+\left(\frac{vt_{f}}{2}\right)^{2}$}\end{cases}$ (9)

$\chi_{max,tx}=\begin{cases}\frac{D}{2}&\text{if
$r^{2}\geq\frac{D^{2}}{2}+\left(\frac{vt_{f}}{2}\right)^{2}$}\\\
\sqrt{\left(r^{2}-\left(\frac{vt_{f}}{2}\right)^{2}\right)\left(1-\frac{1}{D^{2}}\left(r^{2}-\left(\frac{vt_{f}}{2}\right)^{2}\right)\right)}&\text{if
$r^{2}<\frac{D^{2}}{2}+\left(\frac{vt_{f}}{2}\right)^{2}$}\end{cases}$ (10)

$\gamma_{min,tx}(x)=\max\left\\{\frac{-D-\sqrt{D^{2}-4x^{2}}}{2},-\sqrt{r^{2}-\left(\frac{vt_{f}}{2}\right)^{2}-x^{2}}\right\\}$
(11)

$\zeta_{max,tx}(x,y)=\sqrt{r^{2}-x^{2}-y^{2}}-vt_{f}$ (12)

  * •

The probability of a nano-node being in the zone where a frame can be
completely transmitted ($p_{tx}$) is modeled as the ratio between the
transmission volume ($V_{tx}$) and the total fluid volume ($V_{t}$), that is:

$p_{tx}=\frac{V_{tx}}{V_{t}}$ (13)

  * •

Even though the TS-OOK modulation presents a really low probability of
interference between pulses (due to the extremely short pulse duration), it is
required perfect multiplexing at reception to distinguish pulses from
different transmitters and successfully demodulate more than one frame
simultaneously. Thus, in order to keep the probability of receiving a flawed
frame as low as possible and not to overestimate the performance of the nano-
network, we assume that a collision occurs when the nano-router simultaneously
receives two transmissions from two or more different nano-nodes. Hence, if a
nano-node is performing a transmission within $V_{tx}$, a second nano-node can
collide in two different cases: (i) it starts the transmission outside
$V_{cv}$ but enters while the transmission is still active and (ii) it starts
a transmission in any part of $V_{cv}$. So, there is a collision volume
($V_{cx}$) larger than $V_{cv}$ defined by:

$V_{cx}=\int_{\zeta_{min,cx}(x,y)}^{\zeta_{max}(x,y)}\int_{\gamma_{min}(x)}^{\gamma_{max}(x)}\int_{\chi_{min}}^{\chi_{max}}dxdydz,$
(14)

where:

$\zeta_{min,cx}(x,y)=-\sqrt{r^{2}-x^{2}-y^{2}}-vt_{f}$ (15)

  * •

Similarly to the definition of $p_{tx}$, the probability of a nano-node being
in the collision zone ($p_{cx}$) is modeled as:

$p_{cx}=\frac{V_{cx}}{V_{t}}$ (16)

  * •

As the bio-sensor is a larger device strategically located within a vein or an
artery, covering its whole section, we consider that the probability of a
nano-node passing through it in each round relies on the fraction of the total
blood flow ($\eta$) that circulates through the vein/artery in which the bio-
sensor is placed. In addition, the energy required by a nano-node to listen to
the medium and receive a bit is much lower than to transmit it (approx. 10
times lower [26]) and could even be passively detected without a
preamplification stage [25]. As the presence of a bio-sensor is detected only
once per round, the required energy for reception has a negligible impact on
the total energy consumption of a nano-node. Therefore, we assume that nano-
nodes are able to receive a frame if passing through the bio-sensor vein.
Consequently, the probability of a nano-node receiving a frame in each round
is: $p_{rx}=\eta$.

For the sake of illustration, Fig. III shows a nano-node flowing through the
bloodstream communicating with a nano-router attached to a vein. As the
transmission of a frame is performed every $1/f$, many other nano-nodes remain
discharged when passing through $V_{cv}$ and are not able to successfully send
data to the nano-router. Likewise, other nano-nodes are transmitting outside
the coverage zone, so they do not cause any collision.

Based on these assumptions and considering that the frame is stored during two
rounds (the minimum storage time to carry out a transmission from the bio-
sensor to the nano-router), the throughput achieved by the proposed nano-
network can be modeled by this expression:

$\begin{split}Th&=nfp_{tx}p_{rx}\left(1-p_{cx}p_{rx}\right)^{n-1}=\\\
&=nf\frac{V_{tx}}{V_{t}}\eta\left(1-\frac{V_{cx}}{V_{t}}\eta\right)^{n-1}\end{split}$
(17)

This metric measures the frames per second that can successfully reach the
nano-router. In order to achieve a correct transmission, nano-nodes must
receive an updated frame, store it in their memories, and then transmit it to
the nano-router without collisions. As can be seen, this model is generic
enough to be employed in different applications, where any of the considered
parameters may vary its value.

[t!](topskip=0pt, botskip=0pt, midskip=0pt)figure_2D.png Nano-node
communicating with the nano-router attached to a vein without collisions.

TABLE I: Abbreviations employed throughout Section III. Parameter | Description
---|---
$n$ | Number of nano-nodes
$T$ | Time to complete a round through the system
$V_{t}$ | Total fluid volume
$V_{cv}$ | Coverage volume
$V_{tx}$ | Transmission volume
$V_{cx}$ | Collision volume
$D$ | Diameter of the vein where the nano-router is placed
$r$ | Communication range of nano-devices
$v$ | Nano-node velocity when passing through $V_{cv}$
$t_{f}$ | Time to transmit a complete frame
$1/f$ | Time between active cycles
$\eta$ | Fraction of fluid circulating through the bio-sensor
$p_{tx}$ | Probability of a nano-node being in the transmission volume
$p_{cx}$ | Probability of a nano-node being in the collision volume
$p_{rx}$ | Probability of receiving a frame per round
$p_{s}$ | Probability of successfully transmitting a frame per active cycle
$p_{s,rnd}$ | Probability of successfully transmitting a frame per round
$k$ | Number of rounds that the frame is stored in nano-nodes memory

###  III-A Frame storage

Once the throughput considering that nano-nodes store a frame for the minimum
required time to complete a transmission (i.e., two rounds) is obtained, we
extend the analytical model by analyzing the scenario in which nano-nodes are
able to keep the frame in memory during an arbitrary $k$ number of rounds. To
achieve this, we start by deriving the probability of having a frame stored
($p_{frame}$) as a function of the number of rounds $k$ that nano-nodes keep
the frame, with $k\geq 2$. Likewise, we get the expression of not having a
frame stored ($p_{empty}$). To obtain these probabilities we employ the Markov
chain shown in Fig. 3, where the state $0$ is associated with not having a
frame in memory, and states from $1$ to $k$ stand for having a frame stored in
round numbers $1$ to $k$, respectively. The transition matrix $T_{1}$
associated to this Markov chain is defined as:

$T_{1}=\begin{pmatrix}1-p_{rx}&p_{rx}&0&0&\ldots&0&0\\\ 0&0&1&0&\ldots&0&0\\\
0&0&0&1&\ldots&0&0\\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&0&\ldots&0&1\\\ 1-p_{rx}&p_{rx}&0&0&...&0&0\\\
\end{pmatrix}_{\\!\\!(k+1)\times(k+1)}$ (18)

As the stationary distribution of a Markov chain with a transition matrix $T$
can be obtained by a vector, $\pi$, so that $\pi T=\pi$, we define the vector
$\pi=\\{\pi_{0},\pi_{1},\pi_{2},...,\pi_{k-1},\pi_{k}\\}$ as the stationary
probabilities of being in the states $0$, $1$, $2$, …, $k-1$, $k$,
respectively. Using this property, the series of equations of our Markov chain
is defined as follows:

$\pi_{0}=(1-p_{rx})\pi_{0}+(1-p_{rx})\pi_{k}$ (19)
$\pi_{1}=p_{rx}\pi_{0}+p_{rx}\pi_{k}$ (20) $\pi_{i}=\pi_{i-1},i=2,\ldots,k$
(21) $\sum^{k}_{i=0}\pi_{i}=1$ (22)

Figure 3: Markov chain to derive the probability of having a frame stored.
Probabilities of state changes are shown over each transition.

Eq. 22 is derived from the second condition of the stationary probabilities;
that is, the sum of all the probabilities contained in vector $\pi$ must be 1.

Working with Eqs. 19 to 22, we get the following expressions for the values in
vector $\pi$:

$\pi_{0}=\frac{1-p_{rx}}{1+p_{rx}(k-1)}$ (23)
$\begin{gathered}\pi_{1}=\frac{p_{rx}}{1+p_{rx}(k-1)}\\\
\pi_{1}=\pi_{2}=\ldots=\pi_{k-1}=\pi_{k}\end{gathered}$ (24)

Once the values in vector $\pi$ have been obtained, we can directly calculate
the values of $p_{empty}$ and $p_{frame}$:

$\begin{gathered}p_{empty}=\frac{1-p_{rx}}{1+p_{rx}(k-1)}\\\
p_{frame}=\sum^{k}_{i=1}\pi_{i}=\frac{kp_{rx}}{1+p_{rx}(k-1)}\end{gathered}$
(25)

Upon the expressions of $p_{empty}$ and $p_{frame}$, the raw throughput of the
nano-network as a function of the number of rounds that the frame is stored in
memory can be calculated following the same procedure as in Eq. 17:

$Th_{raw}=nfp_{tx}p_{frame}\left(1-p_{cx}p_{frame}\right)^{n-1}$ (26)

As raw throughput considers all the frames received, some of them could be
repeated if a nano-node sends the frame stored more than once until it
receives a new one. Thus, we also provide an expression for effective
throughput (i.e., not considering repeated frames). To do this, we focus on
the states of the Markov chain in Fig. 3. As we can see, if the received frame
should not be transmitted more than once, there are two valid options: (i) the
frame is transmitted in state $1$ or (ii) the frame is transmitted in any of
the remaining states (from $2$ to $k-1$) but not in any of the preceding ones
(e.g., if the frame is transmitted in state $3$, it cannot have been
transmitted in states $1$ and $2$). Note that it cannot be transmitted in
state $k$, since the nano-node discards the frame to be ready to receive a new
one. Mathematically, considering that only one transmission can be performed
per round, we express the probability of transmitting a frame only once during
$k$ rounds ($p_{eff}$) as:

$\begin{gathered}\begin{split}\\!\\!p_{eff}&=\pi_{1}p_{s}+\pi_{2}p_{s}(1-p_{s})+\pi_{3}p_{s}(1-p_{s})^{2}+\ldots\\\
&+\pi_{k-1}p_{s}(1-p_{s})^{k-2}=\pi_{1}p_{s}\sum^{k-1}_{i=1}(1-p_{s})^{i-1}=\\\
&=\pi_{1}\frac{(1-p_{s})^{k}+p_{s}-1}{p_{s}-1}\end{split}\end{gathered}$ (27)

where:

$p_{s}=p_{tx}(1-p_{frame}p_{cx})^{n-1}$ (28)

Using Eqs. 24 and 27, the effective throughput ($Th_{eff}$) can be directly
obtained as:

$Th_{eff}=nf\frac{\eta\left((1-p_{s})^{k}+p_{s}-1\right)}{\left(1+\eta(k-1)\right)\left(p_{s}-1\right)}$
(29)

###  III-B Quality of Delivery (QoD)

Apart from the throughput, another important metric that can be useful to
predict and measure the performance of a THz flow-guided nano-network is the
probability of capturing and transmitting a frame before a certain number of
rounds. We define this concept as the Quality of Delivery ($QoD$). It will
determine how efficiently the nano-network is able to generate and transmit
useful data. Thus, employing this metric, the number of nano-nodes required in
a nano-network to satisfy a given performance can be more accurately
predicted. To this end, we again rely on Markov chains, specifically the one
shown in Fig. 4. For the design of this Markov chain, the following
assumptions were made:

  * •

State transitions take place in every round, i.e., the state in the Markov
chain does not change until a round is totally completed. To achieve this, we
define the probability of one nano-node successfully transmitting a frame
(without collisions) per round when this frame has been previously received
and stored in memory ($p_{s,rnd}$) as:

$p_{s,rnd}=Tfp_{tx}\left(1-p_{cx}p_{frame}\right)^{n-1}$ (30)

  * •

Each nano-node stores the frame in memory during a generic number of rounds
$k$, following the same approach developed in the previous subsection.

  * •

State $Q$ represents the transmission of the stored frame.

  * •

Nano-nodes cannot receive and transmit in the same round. We assume a worst-
case scenario in which nano-nodes first capture a frame and transmit it, at
the earliest, in the following round. Therefore, if a nano-node stores a frame
for $k$ rounds, it has $k-1$ chances (one per round) to transmit the frame.
This is the reason why state $k$ is not linked to state $Q$, as the round
$k+1$ is devoted to receiving a new frame and not to transmitting the former
one.

The transition matrix $T_{2}$ of this Markov chain is:

$\small\\!T_{2}\\!=\\!\begin{pmatrix}1-p_{rx}&p_{rx}&0&\ldots&0&0&0\\\\[1.0pt]
0&0&1-p_{s,rnd}&\ldots&0&0&p_{s,rnd}\\\\[1.0pt]
0&0&0&\ldots&0&0&p_{s,rnd}\\\\[1.0pt]
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\\[1.0pt]
0&0&0&\ldots&0&1-p_{s,rnd}&p_{s,rnd}\\\\[1.0pt]
1-p_{rx}&p_{rx}&0&\ldots&0&0&0\\\\[1.0pt] 0&0&0&\ldots&0&0&1\\\\[1.0pt]
\end{pmatrix}$ (31)

with a size of $(k+2)\times(k+2)$.

Employing this approach, the $QoD$ for one node matches the probability of
entering in the state $Q$ after a certain number of rounds $m$, denoted by
$\pi_{mQ}$. Thus, we define the vector
$\pi_{m}=\\{\pi_{m0},\pi_{m1},\pi_{m2},\ldots,\pi_{mk},\pi_{mQ}\\}$ as the
probabilities of being in states $0$, $1$, $2$, …, $k$, $Q$, respectively,
after $m$ state transitions. In a Markov chain, $\pi_{m}$ can be calculated
as:

$\pi_{m}=\pi_{0}T_{2}^{m}$ (32)

where $\pi_{0}=\\{\pi_{00},\pi_{01},\pi_{02},...,\pi_{0k},\pi_{0Q}\\}$ stands
for the initial distribution probabilities across the state space of the
chain. To get $QoD$ we assume that the nano-network is already working in a
stationary state. So, values in vector $\pi_{0}$ equal those in vector $\pi$,
obtained in the previous section. The remaining value, $\pi_{0Q}$, is equal to
0. Thus, from Eq. 32, we get the expression for every single value of vector
$\pi_{m}$ in terms of $k$ and $m$:

$\pi_{mQ}=p_{s,rnd}\sum^{k-1}_{i=1}(\pi_{(m-1)i})+\pi_{(m-1)Q}$ (33)
$\pi_{m0}=(1-p_{rx})\pi_{(m-1)0}+(1-p_{rx})\pi_{(m-1)k}$ (34)
$\pi_{m1}=p_{rx}\pi_{(m-1)0}+p_{rx}\pi_{(m-1)k}$ (35)
$\pi_{mi}=(1-p_{s,rnd})\pi_{(m-1)(i-1)},i=2,3,\ldots,k$ (36)

Then, once the probability of sending a frame in $m$ rounds for each node has
been obtained, we derive the general expression of $QoD$ as follows:

$QoD=1-(1-\pi_{mQ})^{n}$ (37)

where $n$ represents the number of nodes in the nano-network. We note that,
even though there is not a direct expression for $QoD$, recursive expressions
from Eq. 33 to 36 can be easily implemented in any numerical computing
environment to obtain the $QoD$ of any flow-guided nano-network, as analyzed
in Section IV.

Figure 4: Markov chain to obtain $QoD$. Probabilities of state changes are
shown over each transition.

###  III-C Delay

Finally, we define the delay of a transmitted frame ($\tau$) as the time
elapsed from its reception by a nano-node to the transmission to the nano-
router, with $kT\geq\tau\geq 2T$. To provide a consistent metric to measure
the average information delay of a nano-network, in this subsection, we derive
an analytical expression for the average nano-network delay ($\tau_{av}$).
This indicates how fresh the information that the entire nano-network provides
to an external agent is, which could be critical for applications that require
real-time information streams.

Assuming the same conditions as in the previous section, we define a new
Markov chain with a state space comprising $2k$ states, as shown in Fig. 5,
with the transition matrix $T_{d}$ shown in Eq. 38.

$T_{d}=\left(\setcounter{MaxMatrixCols}{11}\begin{matrix}1-p_{rx}&\>p_{rx}&0&\dotsm&\>0&0&0&0&\dotsm&\>0&0\\\
0&0&1-p_{s,rnd}&\dotsm&\>0&0&p_{s,rnd}&0&\dotsm&\>0&0\\\
0&0&0&\dotsm&\>0&0&0&p_{s,rnd}&\dotsm&\>0&0\\\
\vdots&\vdots&\vdots&\ddots&\>\vdots&\vdots&\vdots&\vdots&\ddots&\>\vdots&\vdots\\\
0&0&0&\dotsm&\>0&\>1-p_{s,rnd}&0&0&\dotsm&\>0&\>p_{s,rnd}\\\
1-p_{rx}&p_{rx}&0&\dotsm&\>0&0&0&0&\dotsm&\>0&0\\\
0&0&0&\dotsm&\>0&0&0&1&\dotsm&\>0&0\\\ 0&0&0&\dotsm&\>0&0&0&0&\dotsm&\>0&0\\\
\vdots&\vdots&\vdots&\ddots&\>\vdots&\vdots&\vdots&\vdots&\ddots&\>\vdots&\vdots\\\
0&0&0&\dotsm&\>0&0&0&0&\dotsm&\>0&1\\\
1-p_{rx}&p_{rx}&0&\dotsm&\>0&0&0&0&\dotsm&\>0&0\\\
\end{matrix}\right)_{2k\times 2k}$ (38)

As in Subsection III-A, we rely on the stationary distribution of a Markov
chain to derive the expression of the average delay of a transmitted frame.
Thus, we define the vector
$\pi^{d}=\\{\pi^{d}_{0},\pi^{d}_{1},\pi^{d}_{2},\ldots,\pi^{d}_{k},\pi^{d}_{Q2},\pi^{d}_{Q3},\ldots,\pi^{d}_{Qk}\\}$,
so that $\pi^{d}T_{d}=\pi^{d}$. The first $k+1$ states, from $0$ to $k$, are
devoted to nano-node states in which the frame is stored but it has not been
transmitted to the nano-router, as in the previous subsection. States from
$Q_{2}$ to $Q_{k}$ represent the fact that the frame has been transmitted in a
specific round, from $2$ to $k$ (we should note that the frame cannot be
transmitted in the same round as that in which it has been received; that is,
the first round). As a transmitted frame should be counted only once (if the
same frame is transmitted twice, the second transmission is discarded), when
one of the $Q$ states is reached, the next state transitions will be in the
remaining $Q$ states until round number $k$. Based on the expression
$\pi^{d}T_{d}=\pi^{d}$, the equations associated with the stationary
probabilities are:

$\pi^{d}_{0}=(1-p_{rx})\pi^{d}_{0}+(1-p_{rx})\pi^{d}_{k}+(1-p_{rx})\pi^{d}_{Qk}$
(39) $\pi^{d}_{1}=p_{rx}\pi^{d}_{0}+p_{rx}\pi^{d}_{k}+p_{rx}\pi^{d}_{Qk}$ (40)
$\pi^{d}_{i}=(1-p_{s,rnd})^{i-1}\pi^{d}_{1},i=2,3,\ldots,k$ (41)
$\pi^{d}_{Qi}=p_{s,rnd}\sum^{i}_{j=2}(1-p_{s,rnd})^{j-2}\pi^{d}_{1},i=2,3,\ldots,k$
(42) $\sum^{k}_{i=0}{\pi^{d}_{i}}+\sum^{k}_{i=2}{\pi^{d}_{Qi}}=1$ (43)

Working with this series of equations, values in vector $\pi^{d}$ are defined
as follows:

$\pi^{d}_{0}=\frac{1-p_{rx}}{1+p_{rx}(k-1)}$ (44)
$\pi^{d}_{i}=\frac{p_{rx}(1-p_{s,rnd})^{i-1}}{1+(k-1)p_{rx}},i=1,2,\ldots,k$
(45)
$\pi^{d}_{Qi}\\!=\\!\frac{p_{rx}((1-p_{s,rnd})^{i}\\!+\\!p_{s,rnd}\\!-\\!1)}{(p_{s,rnd}-1)((k-1)p_{rx}+1)},i\\!=\\!2,3,\ldots,k$
(46)

According to the design of the Markov chain, the general expression of the
average delay for the frames transmitted can be calculated as:

$\tau_{av}=\sum^{k}_{i=2}\frac{\pi^{d}_{Qi}-\pi^{d}_{Q(i-1)}}{\pi^{d}_{Qk}}\cdot
iT$ (47)

where $T$ stands for the time required to complete a round. We note that the
minimum average delay in this expression is $2$$\cdot$$T$, since we assume
that a nano-node takes at least two complete rounds to send a frame.

Figure 5: Markov chain to derive the average delay. Probabilities of state
changes are shown over each transition.

##  IV Results

In this section, we first evaluate network performance (considering the
figures of merit defined in Section III) as a function of different
parameters, so that the influence of each one of them on the performance of a
THz flow-guided nano-network is thoroughly discussed. Then, we study the
feasibility of using flow-guided nano-networks in medical applications in
terms of their network performance. To do this, we apply the analytical model
to the applications described in Section II, employing realistic and accepted
values for all the parameters employed. All the results obtained throughout
this section have also been validated by means of simulations. Simulations
have been implemented in MATLAB with the following scenario variables: number
of nano-nodes, communication range, diameter of vein, battery charging
frequency, nano-node velocities in the flow, time to complete one round
through the system, time to transmit a complete frame, total volume of the
flow, and simulation seed. The nano-network has been simulated considering a
branched, closed-loop medium (modeling the cardiovascular system) where nano-
nodes flow for an equivalent (simulated) time of 1 hour. Nano-nodes flow
through each branch of the circuit, simulating veins, with a certain
probability equal to the actual percentage of blood flowing through that vein,
containing a total volume of 5 liters (total volume of blood in the human
body) [48]. The average time to complete a round through the cardiovascular
system ($T$) is set to 60 seconds [48]. Every point in the simulations has
been repeated ten times with a different seed.

###  IV-A Performance evaluation of a generic flow-guided nano-network

In this subsection, we conduct a sensitivity analysis for the four most
relevant parameters: (i) number of nano-nodes, (ii) diameter of vein, (iii)
communication range, and (iv) number of rounds storing a frame. To this end,
we set the values of the remaining parameters as follows:

  * •

The time to transmit a complete frame ($t_{f}$) is determined by two factors:
(i) the frame length (in bits) and (ii) the symbol rate. Regarding the first
factor, we assume 64 bits, so frames are short enough to be stored and
transmitted by nano-nodes but large enough to include accurate medical
measurement and some fields of an ad-hoc protocol, aimed at preserving the
integrity and robustness of the transmitted data. For the second factor, we
assume a conservative nano-node symbol rate of $10^{6}$ symbols per second, in
accordance with the related literature [47, 28, 49, 50]. Thus, assuming an
On–Off keying modulation [47] where one symbol equals one bit, the bitrate
($R$) is equal to $1$ Mbps and the value of $t_{f}$ is 64 $\mu$s.

  * •

As nano-nodes are powered by means of a piezoelectric generator, we assume
that the mechanical strength caused by a heart rate of 60 beats per minute
will allow us to harvest enough energy to activate and transmit or receive a
frame once per second; that is, $1/f$ = 1 s. To justify this claim, we analyze
the energy balance in each nano-node once a frame has been received, as data
transmission is the most energy-consuming task. On the one hand, the energy
expected to be consumed in each cycle is given by:

$E_{node}=L_{f}WE_{p}+\frac{L_{f}P_{bit}}{f}$ (48)

where $W$ is the probability of transmitting a pulse, since the transmission
of a logical ”1” is carried out by sending a pulse, whereas a logical ”0” is
sent as silence. To ensure that the nano-network is working in all cases, we
assume $W$ = 1 to cover the worst-case scenario, that is, all the frames are
composed of ”1”s (the energy consumption is at its maximum). $E_{p}$ stands
for the energy of a pulse, which is considered 0.1 fJ to get a communication
range of 1 mm [25]. This range was calculated considering the particular
characteristics of THz waves propagation inside the human body, specifically
in blood, and a transmission power of 1 mW using the TS-OOK modulation.
$L_{f}$ is the length of the frame (64 bits) and $P_{bit}$ denotes the power
required by the circuitry to keep a bit stored in memory (2.4 fW [51]). On the
other hand, we consider that each nano-node features a nano-generator able to
gather $\Delta Q$ = 6 pC per compress-release cycle (induced by each
heartbeat) with a generated voltage ($V_{g}$) of 0.2 V, as analyzed in the
literature [25, 27, 26], and a nano-capacitor with a capacitance ($C$) of 10
pF –with a maximum energy capacity $\left(E_{max}=\frac{CV_{g}^{2}}{2}\right)$
of 200 fJ–. Using these parameters, the energy harvesting rate ($\lambda_{h}$)
can be calculated by this expression (taken from [25]):

$\lambda_{h}(E_{nc})=\Delta
Qf_{ng}V_{g}\sqrt{\frac{E_{nc}}{E_{max}}}\left(1\\!-\\!\sqrt{\frac{E_{nc}}{E_{max}}}\right)$
(49)

where $E_{nc}$ is the energy already stored in the nano-capacitor and $f_{ng}$
is the frequency of compress-release cycles (1 Hz). Using this expression and
applying the energy consumption of the nano-node with $W$ = 1, the nano-
generator is able to perform a frame communication every second and stabilize
the energy level at around 140 fJ.

  * •

The velocity of nano-nodes when passing through the nano-router coverage zone
($v$) is set in accordance with the velocity of blood flow in a superficial
vein in the hand [52]; that is, 10 cm/s on average [53].

  * •

The percentage of the blood flow ($\eta$) flowing through the bio-sensor is
set to 10%, taking as a reference a renal vein [54] in which a bio-sensor
could be implanted to detect bacteria from kidneys.

  * •

The diameter of the vein where the nano-router is placed ($D$) is set to 6 mm,
provided that it is not the parameter under study. This value is in line with
a generic application in which the nano-router is placed in a superficial vein
in the hand [53].

  * •

When applicable, the communication range ($r$) is established at 1 mm, a
worst-case value for the THz-based communication in blood according to both
the high path-loss in this medium and the low energy employed to generate the
EM pulses of the modulation in use [25].

####  IV-A1 Number of nano-nodes

Using all the parameters above, Fig. 6 represents the throughput and $QoD$ as
a function of the number of nano-nodes in the nano-network for different
values of $k$, both theoretically (solid line) and obtained by simulation
(dots).

As can be seen in Fig. 6a, nano-network throughput increases almost linearly
as the number of nodes grows with all the values of $k$, since the number of
collisions is still negligible. The impact of $k$ on the throughput can also
be observed, as it increases the probability of a nano-node having one frame
stored in memory. However, when the number of nano-nodes is high enough,
collisions become relevant (see Fig. 6b) and the throughput reaches a maximum.
Then, it progressively decreases until it reaches zero due to the increasing
number of collisions. Another notable fact is that when $k$ increases (that
is, the percentage of nano-nodes ready to transmit a frame is higher), the
maximum throughput is greater. This occurs, fundamentally, because in order to
get this value the density of nodes in the circuit is less than with lower
values of $k$, thus reducing the number of collisions for a similar amount of
transmitting nano-nodes. It can also be noted that as $k$ increases, the
enhancement of the throughput lessens, since the percentage of nano-nodes with
a frame stored in memory increases to 100%.

|  |
---|---|---

Figure 6: Nano-network throughput (a), (b) and $QoD$ (c) versus number of
nano-nodes for different values of $k$. Lines show the throughput obtained
from the proposed analytical model and dots from simulations.

The same trend can be observed in the $QoD$ selected metric. Concretely, Fig.
6c shows the $QoD$ of a flow-guided nano-network with $m$ = 10 (that is, the
probability of transmitting a frame before 10 rounds) as a function of the
number of nano-nodes. As $n$ increases, the value of $QoD$ gets closer to one.
Also in this case, the effect of $k$ is noticeable, showing a significantly
higher $QoD$ as $k$ grows but moving toward a maximum value when it becomes
excessively high.

So far, raising the value of $k$ has had a positive impact on nano-network
performance in terms of throughput and $QoD$. However, as shown in Fig. 7(a),
there is a significant drawback to increasing $k$ because this increases the
average delay ($\tau_{av}$) of the transmitted frames. When nano-nodes keep a
frame in memory for a longer time, the availability is higher, entailing a
higher global throughput, but the freshness of these frames is compromised. It
is also worth noting that, in this scenario, the average delay of the
transmitted frames is completely independent of the number of nano-devices in
the network (see Fig. 7(b)). From these results, we can conclude that the time
each nano-node needs, on average, to transmit a frame from its reception does
not change, although the number of nodes that can transmit becomes higher.
Therefore, there is a trade-off between throughput and information freshness
when varying $k$ that should be considered in the design of a flow-guided
nano-network.

|
---|---

(a)

(b)

Figure 7: (a) $QoD$ and average data delay ($\tau_{av}$) versus $k$ with $n$ =
$10^{4}$ and $m$ = 10. (b) $\tau_{av}$ versus number of nano-nodes for
different values of $k$. Lines show the throughput obtained from the proposed
analytical model and dots from simulations.

####  IV-A2 Diameter of vein

Another relevant factor in nano-network performance is the diameter of the
vein in which the nano-router is implanted. Fig. 8 shows the nano-network
throughput as a function of $D$ for two different values of $n$. As can be
noted in Fig 8(a), if $n$ is not excessively high (i.e., collisions are not
relevant yet), the throughput noticeably grows for values of $D$ below 2 mm
and then it stabilizes for larger diameters. This result is derived from the
communication range of nano-devices, set to 1 mm, so the throughput grows as
$V_{cv}$ gets larger. Interestingly, if the nano-network is working in a state
in which collisions are not numerous, we can confirm that placing the nano-
router in thicker veins does not penalize nano-network performance. However,
if $n$ grows and collisions become more frequent, there is an optimal $D$
where the throughput is maximized (see. Fig 8(b)) at around 0.75 mm. This is
due to the fact that transmission and collision zones increase at different
paces as a function of $D$. The collision zone increases faster than the
transmission zone, making the impact of collisions higher for larger
diameters. Therefore, to reach the maximum throughput in a scenario where the
number of collisions is substantial, the value of $D$ should be optimized. The
$QoD$ shows similar behavior for $10^{4}$ nano-nodes, as shown in Fig. 8(c),
with a sharp increase for low diameters and stabilizing when $D$ grows. This
is because the number of collisions for $10^{4}$ nano-nodes is not relevant.
If $n$ increases, the value of $QoD$ tends to be 1, reducing the impact of
$D$, since the number of nano-nodes needed to hinder the value of $QoD$ is too
high (higher than $10^{7}$) to be relevant.

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(a)

(b)

(c)

Figure 8: Nano-network throughput and $QoD$ versus $D$ for different values of
$k$. In (a), $n$ = $10^{4}$. In (b), $n$ = $10^{7}$, In (c), $n$ = $10^{4}$
and $m$ = 10. Lines show the throughput obtained from the proposed analytical
model and dots from simulations.

####  IV-A3 Communication range

Even though the high path loss in watery mediums strictly limits the
communication range of nano-devices, we should also analyze its impact on
nano-network performance and highlight its relevance, even for small
variations. The fact that future research could reveal that there are some
parts of the human body where blood flow presents lower absorption at THz
frequencies (extending the communication range of nano-devices), therefore
making this analysis useful to highlight how performance can be affected.
Thereby, Fig. 9 shows the throughput as a function of the communication range
($r$), varying it from 0 to 5 mm (values in line with the ranges expected for
nano-networks [1, 24]), with $10^{4}$ nano-nodes. As can be noted, the impact
of $r$ on nano-network performance is the most significant among all the
parameters analyzed. For example, the act of increasing $r$ from 1 to 2 mm
implies an improvement of approximately 8 times in throughput for $D$ = 5 mm
and $k$ = 10. The larger $D$ is, the more noticeable the enhancement will be,
since the volume of the vein within the coverage range is greater. If progress
in nano-communications enabled nano-devices to increase $r$ up to 5 mm inside
the human body, nano-network throughput would increase by a factor of
approximately 70, drastically boosting the potential of flow-guided nano-
networks.

Figure 9: Nano-network throughput versus communication range ($r$) with $n$ =
$10^{4}$ and $k$ = 10. Lines show the throughput obtained from the proposed
analytical model and dots from simulations.

###  IV-B Application

In this subsection, we analyze potential flow-guided nano-networks based on
the applications described in Section II. For all the applications, the nano-
router would be located on the cephalic vein, a superficial vein in the hand
that would allow communication between the nano-router and an external device,
as studied in [24]. On average, the diameter of this vein is about 6 mm with a
flow velocity of 10.9 cm/s, in accordance with the study in [53]. Regarding
the communication range of nano-nodes, we consider a worst-case scenario in
which the communication range is limited to 1 mm. All the parameters
considered in this case of study are specified in Table II.

TABLE II: Parameters employed in Section IV-B. Parameter | Value
---|---
$D$ | 6 mm
$v$ | 10.9 cm/s
$r$ | 1 mm
$R$ | 1 Mbps
$L_{f}$ | 64 bits
$C$ | 10 pF
$V_{g}$ | 0.2 V
$\Delta Q$ | 6 pC
$E_{p}$ | 0.1 fJ
$P_{bit}$ | 2.4 fW
$1/f$ | 1 s

For the first application (bacterial blood infections), it is necessary to
locate a bio-sensor in a renal vein ($\eta$ = 0.1 [54]) which is able to send
a warning frame (to the nano-router) before 1 hour (3600 seconds) from the
moment the level of bacteria reaches a given threshold. In order to ensure
this condition with a probability of 99%, three parameters must be taken into
account: $QoD$, $\tau_{av}$ and $k$. On one hand, the number of nano-nodes in
the nano-network should be high enough to ensure a $QoD$ equal to 0.99 for a
certain number of rounds ($m_{target}$), defined as: $m_{target}$ =
$\lfloor\tau_{target}/T\rfloor$, where $\tau_{target}$ is the target deadline
required for the application under study. In this case, $\tau_{target}$ = 3600
s, with the value of $m_{target}$ being equal to 60 rounds. On the other hand,
$QoD$ also depends on the value of $k$, with $m_{target}\geq k\geq 2$,
obtaining higher values as $k$ increases (see Fig. 6c). However, as shown in
Fig. 7(a), when $k$ grows, the freshness of the transmitted frames is
compromised, implying an increase of $\tau_{av}$. Therefore, to calculate the
values of $n$ and $k$ to meet a specific deadline, a trade-off between $QoD$
and $\tau_{av}$ should be considered to achieve a good nano-network
performance with acceptable information freshness. To this end, we define the
metric $\tau_{av}\cdot n_{min}$, with $n_{min}$ as the minimum number of nano-
nodes necessary to obtain a $QoD$ equal to 0.99. This metric provides a
balance between the number of nodes required (that decreases when $k$ gets
higher) and the average frame delay (that increases with $k$). Fig. 10 shows
the normalized value of this metric as a function of $k$. The maximum value of
$\tau_{av}\cdot n_{min}$ indicates the equilibrium point between $n_{min}$ and
$\tau_{av}$, obtaining the optimal value of $k$. When $k$ is excessively low,
$n_{min}$ gets too high, even though $\tau_{av}$ is at a minimum, resulting in
a metric value close to zero. Similarly, when $k$ becomes too high, $n_{min}$
decreases but the value of $\tau_{av}$ becomes too high to meet the
application requirement. Results reveal that a nano-network with 6246 nano-
nodes storing a frame for 11 rounds ($k$ = 11) should be able to send a frame
before 1 hour (i.e., 60 rounds) with a $QoD$ of 0.99, thus being enough to
provide robust and reliable service. The throughput provided by this nano-
network is 0.073 frames/min (i.e., one frame every 13.7 minutes, on average),
with $\tau_{av}$ equal to 6.5 minutes.

Figure 10: $\tau_{av}\cdot n_{min}$ vs $k$, so that the value of $QoD$ is
equal to 0.99 for $m$ = $m_{target}$.

The following three applications (i.e., viral load monitoring, sepsis, and
heart attacks) have similar requirements since the nano-network must be
designed to send a warning before a specific time limit. For the viral load
monitoring application, the bio-sensor should be located in the cephalic vein
($\eta$ = 0.0056 [53]) and the warning must be delivered before 1 day
($\tau_{target}$ = 86400 s). Following the same procedure as in the previous
application, the optimal value of $k$ to minimize the number of nano-nodes is
55, while the nano-network would need 580 nano-nodes to meet the requirements.
Regarding sepsis detection, the bio-sensor is placed in the jugular vein
($\eta$ = 0.14 [55]), the deadline is 1 hour ($\tau_{target}$ = 3600 s), while
for heart attacks the bio-sensor is positioned in the cava vein ($\eta$ = 0.35
[56]), and the warning must be reported before 15 minutes ($\tau_{target}$ =
900 s). In these two cases, the minimum number of nano-nodes in the nano-
network is 5397 ($k$ = 11) and 19294 ($k$ = 6), respectively. As can be
observed, the impact of the target deadline on the size of the nano-network is
much greater than the location of the bio-sensor.

Finally, the last application (restenosis) requires a periodic update of the
state of the bio-sensor, located in a coronary artery ($\eta$ = 0.03 [57]), so
the performance requirement relies on the throughput and the average delay. An
update period of 1 hour should be enough to keep medical staff well informed
about a patient’s condition. In this case, we size the nano-network to get a
throughput of 0.033 frames per minute (2 frames per hour), with $\tau_{av}$ =
30 minutes. Thus, the number nano-nodes required for this application would be
2328. The nano-network features obtained with the proposed analytical model
for each application are shown in Table III.

TABLE III: Nano-network features for each application. Application | $n$ | $k$ | $\tau_{target}$ | Throughput (frames/min) |  $\tau_{av}$ (min)
---|---|---|---|---|---
Bacterial blood infections | 6246 | 11 | 1 hour | 0.073 | 6.5
Viral load monitoring | 580 | 55 | 1 day | 0.003 | 28.5
Sepsis | 5397 | 11 | 1 hour | 0.074 | 6.5
Heart attacks | 19294 | 6 | 15 min | 0.286 | 4
Restenosis | 2328 | 58 | - | 0.033 | 30

##  V Conclusion

The use of the THz band in the field of nano-communications could be a real
breakthrough in applied medicine and revolutionize the way information about
the state of human organisms and particular tissues is collected. In pursuit
of this objective, in this paper we have conceptually presented an in-body
flow-guided nano-network composed of three different types of devices: (i)
bio-sensors measuring medical parameters, (ii) nano-nodes circulating within
the bloodstream and working as data carriers, and (ii) nano-routers performing
as gateways and forwarding vital medical information acquired from the nano-
network to external devices directly connected to medical personnel. As
thoroughly reviewed, this nano-network would be able to play an essential role
in diverse applications, concretely, in the early-stage detection of bacterial
and viral infections, sepsis, heart attacks, and restenosis.

Based on this nano-network architecture, we have derived an analytical model
jointly considering the peculiarities of nano-devices (i.e., energy balance
and communication in the THz band) and the complex branched nature of the
cardiovascular system. This model accurately predicts nano-network
performance, employing different figures of merit; that is, throughput,
Quality of Delivery (QoD), and average transmission delay. In addition, this
model can be used to adequately project the number of nano-nodes required for
each application. Nano-network performance has been analyzed as a function of
different parameters, highlighting the impact of (i) the number of nano-nodes,
(ii) the number of rounds that a frame is stored in nano-nodes memory, (iii)
the diameter of the vein where the nano-router is implanted, and (iv) the
nano-nodes communication range. All the results obtained have also been
validated by means of simulations. Finally, we have analyze nano-network
dimensioning based on the particular requirements of each medical application.
To this end, we have tuned the parameter $k$ to minimize the number of nano-
nodes employed to meet the specifications. Outcomes reveal that a nano-network
employing a reasonable number of nano-nodes would be able to effectively
fulfill medical application specifications, reducing intrusiveness. Keeping in
mind the ultra-small size of these nano-nodes, the number of devices
considered (between 580 and 20000), deployed in-body, should be seen as non-
invasive.

To sum up, this work has shown that THz nano-communications can pave the way
for multiple and promising medical applications, being able to perform useful
tasks without needing a huge number of nano-devices. Besides, the analytical
model proposed here represents a step forward in the design and deployment of
flow-guided nano-networks, as it is generic enough to be directly employed or
easily adapted to many other different applications, widening the scope of THz
flow-guided nano-network applicability.

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|  Sebastian Canovas-Carrasco received the B.S. degree in telecommunication
systems engineering and the M.S. degree in telecommunications engineering from
the Universidad PolitÈcnica de Cartagena, Spain, in 2014 and 2016,
respectively, where he is currently pursuing the Ph.D. degree. His research
interest includes the electromagnetic wireless nanonetworks,
radiocommunications at the THz band, and machine learning.
---|---
|  Rafael Asorey-Cacheda received his M.Sc. degree in Telecommunication
Engineering (major in Telematics and Best Master Thesis Award) and his Ph.D.
(cum laude and Best PhD Thesis Award) in Telecommunication Engineering from
the Universidade de Vigo, Spain, in 2006 and 2009, respectively. He was a
researcher with the Information Technologies Group, University of Vigo, Spain
until 2009. Between 2008 and 2009 he was also R&D Manager at Optare Solutions,
a Spanish telecommunications company. Between 2009 and 2012, he held an
Ángeles Alvariño position, Xunta de Galicia, Spain. Between 2012 and 2018, he
was an associate professor at the Centro Universitario de la Defensa en la
Escuela Naval Militar, Universidade de Vigo. Currently, he is an associate
professor at the Universidad Politécnica de Cartagena. He is author or co-
author of more than 60 journal and conference papers, mainly in the fields of
switching, wireless networking and content distribution. He has been a
visiting scholar at New Mexico State University, USA (2007-2011) and at
Universidad Politécnica de Cartagena, Spain (2011, 2015). His interests
include content distribution, high-performance switching, peer-to-peer
networking, wireless networks, and nano-networks.
---|---
|  Antonio-Javier Garcia-Sanchez received his Industrial Engineering degree
(M.S. degree) in 2000 from the Universidad Politécnica de Cartagena (UPCT),
Spain. Since 2001, he has joined the Department of Information Technologies
and Communications (DTIC), UPCT, obtaining his Ph.D. in 2005. Currently, he is
an Associate Professor at the UPCT. He is (co)author of more than 70
conference and journal papers, thirty of them indexed in the Journal Citation
Report (JCR). He has been the head of several research projects in the field
of communication networks and optimization, and he currently is a reviewer of
several journals listed in the ISI-JCR. He is also inventor/co-inventor of 9
patents or utility models, and he has been a TPC member or Chair in about
thirty International Congresses or Workshops. His main research interests are
in the areas of wireless sensor networks (WSNs), streaming services, smart
grid, IoT, and nano-networks.
---|---
|  Joan Garcia-Haro (M’91) received his M.S. degree and Ph.D in
telecommunication engineering from the Universitat Politècnica de Catalunya,
Barcelona, Spain, in 1989 and 1995, respectively. He is currently a Professor
with the Universidad Politécnica de Cartagena (UPCT). He is author or co-
author of more than 70 journal papers mainly in the fields of switching,
wireless networking and performance evaluation. Prof. Garcia-Haro served as
Editor-in-Chief of the IEEE Global Communications Newsletter, included in the
IEEE Communications Magazine, from April 2002 to December 2004. He has been
Technical Editor of the same magazine from March 2001 to December 2011. He
also received an Honorable Mention for the IEEE Communications Society Best
Tutorial paper Award (1995). He has been a visiting scholar at Queen’s
University at Kingston, Canada (1991-1992) and at Cornell University, Ithaca,
USA (2010- 2011).
---|---
|  Krzysztof Wojcik received his M.Sc. and Ph.D. degrees in biophysics from
the Jagiellonian University in Krakow, Poland 2003 and 2015, respectively, and
an M.D. from the Jagiellonian University Medical College in Krakow, Poland
2007. He was an Assistant at Division of Cell Biophysics Faculty of
Biochemistry, Biophysics and Biotechnology Jagiellonian University
(2007-2014). At present he is an Assistant Professor Department of Allergy,
Autoimmunity and Hypercoagulability in II Chair of Internal Medicine JUMC. His
research interests include systemic vasculitis, confocal microscopy and STED
(Stimulated Emission Depletion) techniques and their applications in
autoantibodies research, as well as the use of fluorescent probes in
nanocommunications.
---|---
|  Pawel Kulakowski received his Ph.D. in telecommunications from the AGH
University of Science and Technology in Krakow, Poland, in 2007, and currently
he is working there as an assistant professor. He spent about 2 years in total
as a post-doc or a visiting professor at Technical University of Cartagena,
University of Girona, University of Castilla-La Mancha and University of
Seville. He was involved in research projects, especially European COST
Actions: COST2100, IC1004 and CA15104 IRACON, focusing on topics of wireless
sensor networks, indoor localization and wireless communications in general.
His current research interests include molecular communications and nano-
networks. He was recognized with several scientific distinctions, including 3
awards for his conference papers and a governmental scholarship for young
outstanding researchers.
---|---


# Generating Diverse Indoor Furniture Arrangements

Ya-Chuan Hsu<EMAIL_ADDRESS>University of Southern CaliforniaLos AngelesUSA
,  Matthew C. Fontaine<EMAIL_ADDRESS>University of Southern CaliforniaLos
AngelesUSA ,  Sam Earle<EMAIL_ADDRESS>New York UniversityNew YorkUSA ,
Maria Edwards<EMAIL_ADDRESS>New York UniversityNew YorkUSA ,  Julian
Togelius<EMAIL_ADDRESS>New York UniversityNew YorkUSA and  Stefanos
Nikolaidis<EMAIL_ADDRESS>University of Southern CaliforniaLos AngelesUSA

(2022)

###### Abstract.

We present a method for generating arrangements of indoor furniture from
human-designed furniture layout data. Our method creates arrangements that
target specified diversity, such as the total price of all furniture in the
room and the number of pieces placed. To generate realistic furniture
arrangement, we train a generative adversarial network (GAN) on human-designed
layouts. To target specific diversity in the arrangements, we optimize the
latent space of the GAN via a quality diversity algorithm to generate a
diverse arrangement collection. Experiments show our approach discovers a set
of arrangements that are similar to human-designed layouts but varies in price
and number of furniture pieces.

Furniture arrangement, Generative adversarial network, Latent space
illumination, Quality-diversity algorithm

††copyright: rightsretained††journalyear: 2022††conference: SIGGRAPH ’22
Posters; August 07-11, 2022; Vancouver, BC, Canada††booktitle: Special
Interest Group on Computer Graphics and Interactive Techniques Conference
Posters (SIGGRAPH ’22 Posters), August 07-11, 2022††doi:
10.1145/3532719.3543244††isbn: 978-1-4503-9361-4/22/08

##  1\. Introduction

Scene synthesis has been heavily studied by the graphics community with
applications in AR/VR gaming and autonomous agent or vehicle testing. Our work
focuses on the problem of arranging indoor furniture, which has applications
in testing robots deployed in homes. Existing methods rely on learning
relative positions of furniture from a database (Wang et al., 2018), grouping
objects by common human activity (Fu et al., 2017), or completing partially
arranged floor plans (Paschalidou et al., 2021).

We propose to generate collections of arrangements that vary according to
desired attributes specified by a designer, rather than generating solutions
with diverse spacial relations. For example, an interior designer may propose
a collection of layouts that are affordable for a variety of budgets. Our
method generates a collection of furniture layouts via a three stage process.
First, we train StyleGAN3 (Karras et al., 2021) on furniture occupancy grids
to match furniture layout patterns of human designers. To recover 3D objects
from the generated occupancy grid, we run a geometric sweep algorithm that
fits bounding rectangles to labelled pixels. Finally, we search the latent
space of StyleGAN3 with a quality diversity algorithm to produce a collection
of arrangements that vary according to price and number of objects in the
scene.

##  2\. Encoding furniture arrangement floor plans

To generate diverse floor plans, we need to train a generative model that is
capable of producing novel arrangements, stylistically matching the layouts
from human designers. As a generative model, we choose StyleGAN3, a GAN
capable of generating photorealistic images. To generate arrangements instead
of images, we train StyleGAN3 to produce occupancy grids that model floor
locations containing specific objects. To treat occupancy grids as images in
the StyleGAN3 training pipeline, we color-code the furniture categories, such
as tables, chairs, storage, and beds, of each object. However, assigning
arbitrary colors makes learning occupancy grids more challenging for
StyleGAN3, since conceptually similar categories may be far apart in color
space. Hence, we learn a 3D embedding of furniture categories that we map to
RGB color values.

To produce a 3D color embedding of each category, we first map each category
name to a 512-dimensional vector embedding via the Universal Sentence Encoder
(Cer et al., 2018). We then reduce the encoding to a 3D embedding using t-SNE
(Van der Maaten and Hinton, 2008). Finally, we scale each channel to the range
$[0,255]$ to match an RGB color. We assign colors to 25 umbrella categories
representing a total of over a thousand categories of furniture.

##  3\. Recovering 3D furniture locations and orientations

StyleGAN3 is capable of generating novel occupancy grids that match the
layouts designed by humans. However, the generated occupancy grids produced
are imperfect. To recover 3D furniture locations and orientations from the
grid, we run a rectangle sweep algorithm to find the best bounding rectangle
for a given color and subregion. The algorithm minimizes the cost (Euclidean
distance in color space) of changing all pixel colors within the rectangle to
the RGB color code of the furniture category (Fig. 1).

##  4\. Generating diverse arrangements

While StyleGAN3 can generate arrangements stylistically similar to our
training data, we wish to generate arrangements at varying price points and
with different quantities of furniture. We can frame this problem as a latent
space illumination (LSI) problem (Fontaine et al., 2021b), where we search the
latent space of StyleGAN3 via quality diversity (QD) optimization (Pugh et
al., 2015). To search latent space via QD, we must specify both objective and
measure functions. For the objective, we minimize the repair cost of
transforming occupancy grids into furniture locations. This objective ensures
final arrangements match the occupancy grid as closely as possible. For
measure functions, we compute the price of all furniture extracted from the
arrangement and the number of furniture included in the final arrangement. We
solve the LSI problem with the state-of-the-art QD algorithm, Covariance
Matrix Adaptation MAP-Elites (CMA-ME) (Fontaine et al., 2020).

##  5\. Experimental results

We verify our approach by generating a collection of diverse arrangements.
Fig. 1 visualizes the archive of occupancy grids generated by StyleGAN3 and an
extracted 3D scene from the occupancy grid. The top-right corner of the
archive contains arrangements with a high number of expensive furniture
pieces. The bottom-left corner is an empty room. Note that there are empty
cells in the archive since it is impossible to have empty rooms with a high
price or a room with several furniture pieces with a low price.

Figure 1. Top: Arrangements in the archive. Bottom: A generated arrangement
(left) stored in the orange cell is repaired (middle) and visualized in 3D
(right).

Figure 2. Arrangement in the pink cell from Fig. 1

We compare the arrangements in the orange and pink cell (see Fig. 2). Even
though the orange cell has more furniture pieces, the pink cell arrangement is
more expensive due to the ‘designer walnut vanity unit’ and the ‘California
king wood bed’.

##  6\. Limitations and future work

An important challenge for our method is extracting 3D furniture locations and
orientations from the occupancy grid generated by StyleGAN3. Due to imperfect
occupancy grids, extracting furniture locations may result in overlapping
furniture. Future repair methods will also aim to address issues such as the
stove created in Fig. 1, a result of a large block of noise. Additionally, we
intend to extend arrangement features to include room accessibility by
guaranteeing all furniture is accessible (Fontaine et al., 2021a; Zhang et
al., 2020).

###### Acknowledgements.

We thank Floorplanner for providing indoor furniture data. We thank Gert-Jan
van der Wel, Valentin Bichkovsky, Vincent van Gemert for assisting with the
data and the Floorplanner API.

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