State Estimation of the Time-Varying and Spatially Localized Concentration of Signal Molecules from the Stochastic Adsorption Dynamics on the Carbon Nanotube-Based Sensors and Its Application to Tumor Cell Detection

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This paper addresses a problem of estimating time-varying, local concentrations of signal molecules with a carbon-nanotube (CNT)-based sensor array system, which sends signals triggered by monomolecular adsorption/desorption events of proximate molecules on the surfaces of the sensors. Such sensors work on nano-scale phenomena and show inherently stochastic non-Gaussian behavior, which is best represented by the chemical master equation (CME) describing the time evolution of the probabilities for all the possible number of adsorbed molecules. In the CME, the adsorption rate on each sensor is linearly proportional to the local concentration in the bulk phase. State estimators are proposed for these types of sensors that fully address their stochastic nature. For CNT-based sensors motivated by tumor cell detection, the particle filter, which is nonparametric and can handle non-Gaussian distributions, is compared to a Kalman filter that approximates the underlying distributions by Gaussians. In addition, the second-order generalized pseudo Bayesian estimation (GPB2) algorithm and the Markov chain Monte Carlo (MCMC) algorithm are incorporated into KF and PF respectively, for detecting latent drift in the concentration affected by different states of a cell.

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Citation: Jang H, Lee JH, Braatz RD (2015) State
Estimation of the Time-Varying and Spatially
Localized Concentration of Signal Molecules from the
Stochastic Adsorption Dynamics on the Carbon
Nanotube-Based Sensors and Its Application to
Tumor Cell Detection. PLoS ONE 10(11): e0141930.
doi:10.1371/journal.pone.0141930

Editor: Adam R Hall, Wake Forest University School
of Medicine, UNITED STATES

Received: July 26, 2015

Accepted: October 14, 2015

Published: November 3, 2015

Copyright: © 2015 Jang et al. This is an open
access article distributed under the terms of the
[Creative Commons Attribution License, which permits](http://creativecommons.org/licenses/by/4.0/)
unrestricted use, distribution, and reproduction in any
medium, provided the original author and source are
credited.

Data Availability Statement: All relevant data are
within the paper and its Supporting Information files.

Funding: This work was supported by the Advanced
Biomass R&D Center(ABC) of Global Frontier Project
funded by the Ministry of Science, ICTand Future
Planning (ABC-2011-0031354). The funder had no
role in study design, data collection and analysis,
decision to publish, or preparation of the manuscript.

Competing Interests: The authors have declared
that no competing interests exist.

RESEARCH ARTICLE
# State Estimation of the Time-Varying and Spatially Localized Concentration of Signal Molecules from the Stochastic Adsorption Dynamics on the Carbon Nanotube-Based Sensors and Its Application to Tumor Cell Detection

Hong Jang 1, Jay H. Lee 1 *, Richard D. Braatz 2

1 Department of Chemical and Biomolecular Engineering, Korea Advanced Institute of Science and
Technology, Daejeon, Korea, 2 Department of Chemical Engineering, Massachusetts Institute of
Technology, Cambridge, Massachusetts, United States of America

- jayhlee@kaist.ac.kr

## Abstract

This paper addresses a problem of estimating time-varying, local concentrations of signal

molecules with a carbon-nanotube (CNT)-based sensor array system, which sends signals

triggered by monomolecular adsorption/desorption events of proximate molecules on the

surfaces of the sensors. Such sensors work on nano-scale phenomena and show inher
ently stochastic non-Gaussian behavior, which is best represented by the chemical master

equation (CME) describing the time evolution of the probabilities for all the possible number

of adsorbed molecules. In the CME, the adsorption rate on each sensor is linearly propor
tional to the local concentration in the bulk phase. State estimators are proposed for these

types of sensors that fully address their stochastic nature. For CNT-based sensors moti
vated by tumor cell detection, the particle filter, which is nonparametric and can handle non
Gaussian distributions, is compared to a Kalman filter that approximates the underlying dis
tributions by Gaussians. In addition, the second-order generalized pseudo Bayesian esti
mation (GPB2) algorithm and the Markov chain Monte Carlo (MCMC) algorithm are

incorporated into KF and PF respectively, for detecting latent drift in the concentration

affected by different states of a cell.

Introduction

Recently, several near-infrared (nIR) fluorescent sensors based on single-walled carbon nanotubes (SWNTs) have been developed for detecting biomolecules in the human body [1–9]. In
response to a continuous incident light source, the SWNT-based sensors detect stepwise
changes in emitted light intensity triggered by monomolecular adsorption and desorption (i.e.,

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

adsorption and desorption at a single-molecular level) of a trace of proximate molecules on the surfaces of the sensors. The nIR fluorescence can penetrate more deeply into tissues than visible fluorescence without photobleaching or overlapping with autofluorescence from biological substrates

[10, 11]. Furthermore, compared with small fluorescent probes [12–22], non-diffusive SWNTs
allow for a precise spatial resolution at the micrometer scale. As a result of these advantages,
SWNTs can act as effective sensing platforms for real-time, direct and selective detection in vivo. In
particular, for nitric oxide (NO) and hydrogen peroxide (H2O2), μM level concentration could be
detected successfully by using this sensing platform and resolve several questions about local generation upon growth factor stimulation and the signalling mechanism in a living cell [7, 8].

This sensor technology presents some challenges as well as opportunities. A sensor array
system where multiple CNT-based sensors distributed on a small area potentially can be used
to follow time-varying, local concentrations of target molecules in vivo and in real time with
precise spatial resolution. In turn, precise spatiotemporal control of these molecules may
become feasible with the advent of appropriate actuators. Challenges in the experimental side
include selective sensor design for target molecules in a desired detection range and actuator
design for the spatiotemporal control at micro-scale. On the system’s side, an immediate challenge is the development of an on-line state estimation method that can effectively extract concentration information from the stochastic adsorption data.

Some methods have been proposed for quantifying local concentrations of signal molecules
near CNT-based sensors [23–25]. The estimation task is made challenging by the fact that the
adsorption/desorption event is highly stochastic given a small number of molecules involved at
the nanoscale sensor’s surface. Conventional methods like least squares are limited in terms of
performance for such problems. For a more accurate estimation, chemical master equation
(CME) describing the evolution of the probability distribution among all possible adsorption
states (i.e., the number of adsorbed molecules on the sensor) has been used in the estimation
formulation. Based on the exact solution of the CME, maximum likelihood estimation (MLE)
has been proposed [23–25]. However, the previous works assumed a constant concentration
and performed the estimation with a batch set of data, which is not realistic for a sensor system
working in a real-time environment in which concentrations show dynamic, time-varying
behavior. What is needed is a full state estimation method that can fully and recursively utilize
the information coming from the sensors to follow the local concentration in real time.

Bayesian methods have been a popular choice for state estimation of stochastic systems
owing to its flexible, convenient formulation and theoretical rigor. For Gaussian systems, only
the first two moments of the probability density function (PDF) have to be followed and the
Kalman filter (KF) provides a simple solution to the problem. However, data from the CNTbased sensor system shows highly non-Gaussian characteristics that follow convolved binomial
distributions [24]. For highly non-Gaussian systems, a class of sequential Monte Carlo methods known as particle filters (PFs) can be attractive as a nonparametric method that can handle
any distribution shape [26]. The PF methods represent the required posterior PDF as a set of
random samples and associated weights.

This article mainly proposes an effective recursive state estimator for estimating time-varying, local concentrations of signal molecules using the stochastic adsorption and desorption
time-profiles onto the surface of the CNT-based sensors. By tracking the concentration of the
signal molecules with the help of a rigorously formulated stochastic state estimator, we can
gain further insights into their roles in biological systems or the effects of other species on
them. The stochastic nature of the adsorption and desorption at the molecular level brings in
the chemical master equation (CME) at the sensor level and makes the problem a challenging
one that cannot be easily handled by the conventional state estimation techniques. Hence, the
state estimation problem studied in this article has not been addressed before in the literature.

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

To test the feasibility and potentials of the proposed method, we test it in the context of a
sensing problem, which is admittedly simplistic and artificial but still is inspired by the real biological problem. Given the known parameters in the model, performances of the KF and PF
methods are examined in terms of both accuracy of estimated local concentration of the signal
molecule and computational cost. The nano-sensors have previously been used for detecting
and measuring signal molecules in human body, to follow the concentrations of signal molecules like nitric oxide (NO) and hydrogen peroxide (H2O2), which are consistently generated
from enzymes in vascular endothelial cells to regulate various physiological and pathological
processes [6,23–25]. Their concentration levels are known to be affected significantly by cell
states, the switching behavior of which is simplistically represented by a hidden Markov model
in our case study. To solve the simulated estimation problem, KF and PF are designed with the
second-order generalized pseudo-Bayesian estimation (GPB2) algorithm and the Markov
chain Monte Carlo (MCMC) algorithm respectively, for the Markov jump system with nanosensors. Their performances are compared for the case of a single sensor as well as of multiple
sensors.

Methods

Single-molecule Sensor System

Carbon nanotube-based sensor. The basic mechanism of SWNT-based sensors is optical
detection of discretized light intensity changes induced by adsorption and desorption of target
molecules on the sensor’s surface at nano-scale. To enhance the sensitivity and selectivity for
target molecules, usually present at the micromolar (μM) concentration level, the SWNT surface is functionalized by wrapping the nanotube with various polymers such as collagen [7] or
certain DNA sequences [8] (Fig 1). The variation in the SWNT wrapping controls the adsorption rates of different analytes present. For example, collagen-SWNTs have shown different,
selective time-profiles of adsorption and desorption events for H2O2, H +, and Fe(CN)63− in dif
ferent concentration ranges [6]. Importantly, all time-profile data had reversible features,
which indicate adsorption and desorption rates of similar magnitudes.

The maximum number of adsorbed molecules is experimentally found to be around 10 [8],
and this number is consistent with the maximum number of excitons (an excition is an electron
and positive hole pair, which remain near each other due to electrostatic Coulomb force and is
free to move through a semiconducting material) diffusion-limited segments on the SWNT [1]
for which an average length is about 1~2 μm. So several SWNT-based sensors can be placed in
a small area less than 10 μm 2 [8]. Fig 1 shows an example of a sensor array system depicted as
sensors randomly distributed on a small area of neighborhood. With this array system, the
objective is to estimate a time profile of the local concentration of target molecules with high
accuracy.

Stochastic adsorption model. The number of adsorbed molecules is assumed to be read at
every sampling time from the sensors, which are distributed in a sufficiently small area of a
same concentration level. In developing a sensor model, free target molecules A in its surrounding liquid phase are assumed to adsorb onto unoccupied sites of the nanotube segment θ
to form bound molecules Aθ through reversible adsorption:

A þ y

k0A

Ay ð1Þ

where k

0A [s−1] and kD [s−1] are adsorption and desorption rate constants, respectively. The

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

Fig 1. An example of a SWNT-based sensor array system.

doi:10.1371/journal.pone.0141930.g001

corresponding rates are expressed as

0A N y ð2Þ

rA ¼ k0

rD ¼ kDNAy ð3Þ

where Nθ is the number of empty sites and NAθ is the number of occupied sites. The adsorption
rate can be considered to be a first-order function of the local concentration of the surrounding
target molecules C(t) [24],

kA ¼ k A CðtÞ ð4Þ

where kA is a constant factor in the adsorption coefficient.

These equations connect the sensor information (i.e., the number of absorbed molecules) to
the concentration in the surrounding media. If the adsorption/desorption events could be
deterministic, a continuum (or average) model for the sensor can be formed by one differential
equation for the number of adsorbed molecules NAθ 2 [0, NT] as a continuous variable with an
initial value of NAθ,0,

dNAyðtÞ

dt ¼ kACðtÞðNT                              - NAyðtÞÞ � kDNAyðtÞ; NAyð0Þ ¼ NAy;0 ð5Þ

A recursive form of the solution obtained by considering the previous measurement NAθ(tk

−1 ) as an initial condition and integrating the equation for one sample interval assuming C(t k )

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

remains constant over the interval is

NAyðtkÞ ¼

- - - NT þ NAyðtk�1Þ 1 þ kAkCDðtkÞ - NT e �ðkACðtkÞþkDÞDt

1 þ kAkCDðtkÞ

ð6Þ

where k is the index for the time step and Δt is the size of the sample time step size, which is set
sufficiently small for the approximation to be accurate.

In actuality, the adsorption reaction on the sensor surface is highly stochastic because only a
very small number of molecules (~10) are involved. Hence, significant fluctuations occur from
the average behavior described in (6). In this case, use of the chemical master equation (CME)
composed of differential equations describing the evolution of the probabilities for all possible
discrete states of the system is more appropriate [27]. Then, the state of the system is defined as
the discrete number of adsorbed molecules N ~ Ay 2 ½0; NT�, resulting in NT +1 total possible
states. The probability of being in each state is denoted by Pi ¼ PrðN ~ Ay ¼ iÞ 2 ½0; 1�, where i is
the number of adsorbed molecules. The CME, along with the appropriate boundary equation,
can be expressed by NT +1 ordinary differential equations (ODEs):

dP0ðtÞ 0

¼ �P0ðtÞ½kA
dt

0A N T �þ P 1 ðtÞ½k D � ð7Þ

dPiðtÞ 0

¼ Pi�1ðtÞ½k
dt

dPiðtÞ

0A ðN T �ði � 1ÞÞ�� P i ðtÞ½k D i þ k 0

0A ðN T � iÞ�þ P iþ1 ðtÞ½k D ði þ 1Þ�; i

¼ 1; 2; . . . ; NT - 1ð8Þ

dPNT ðtÞ

k0A �� P NT ðtÞ½k D N T � ð9Þ

dt ¼ PNT �1ðtÞ½k0

The monomolecular reaction systems, which were studied by [28], provide a path to an analytical solution of the CME. The adsorption/desorption process can be considered as a monomolecular reaction system with only two species (e.g. adsorbed molecules on sensor surface
and desorbed molecules in bulk). For such a system, the probability distribution of the CME is
described by a binomial distribution with time-varying parameters. More specifically, the number of adsorbed molecules NAθ at a time tk is a random variable distributed as a binomial with
the number of trials equal to NT and probability parameter equal to λ(tk), which is related to
NAθ(tk) calculated from the continuum Eq (6) divided by NT as

N~ AyðtkÞ � BðNT; lðtkÞÞ ð10Þ

ið1 � lðtkÞÞ

ðlðtkÞÞ

PrðN ~ AyðtkÞ ¼ iÞ ¼

NT �i ð11Þ

ð12Þ

lðtkÞ ¼ NAyðtkÞ

The local concentration of target molecule C(tk) enters the probability distribution of Eq
(10) through NAθ(tk) of Eq (6) appearing in Eq (12) for λ(tk).

For monomolecular adsorption, the overall population can be divided into two subsets, representing occupied sites and unoccupied sites on the sensor. With some previously measured
value N ^ Ayðtk�1Þ, the distribution at the next time step can be derived as the convolution of two

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

binomial distributions applicable to the “fully occupied” and “empty” subsets, which are of size

N^ Ayðtk�1Þ and 1 � N^ Ayðtk�1Þ respectively:

N~ AyðtkÞ � Bð ^N Ayðtk�1Þ; l F ðtkÞÞ�BðNT - N^ Ayðtk�1Þ; l E ðtkÞÞ ð13Þ

ð14Þ

ð15Þ

l F ðtkÞ ¼

1 þ kAkCDðtkÞ e�ðkACðtkÞþkDÞDt

1 þ kAkCDðtkÞ

l E ðtkÞ ¼ 1 � 1 e þ �ðkA k C A ð kC t D k ðt Þþ kÞ kDÞDt

The first binomial distribution can be derived from (10)–(12) by assuming the sites are fully
occupied initially, and the second binomial distribution can be derived from (10)–(12) by considering the initial state as being empty [28]. If the expression for NAθ(tk) obtained by setting
NAθ(tk−1) = NT (“fully occupied”) in (6) is further substituted into (12), the probability parameter λ(tk) becomes λ F (tk) of (14) and NT cancels out. If the same substitution is carried out by
setting NAθ(tk−1) = 0 (“fully empty”) in (6), λ(tk) becomes λ E (tk) of (15).

Recursive State Estimation Design

Based on the observation model proposed in Section 2, the overall system for state estimation
can be generally described by the discrete-time state space model,

xk ¼ f ðxk�1Þ þ wk ð16Þ

yk;j                                                    - pðyk;jjxk; yk�1;jÞ; 8j ¼ 1; . . . ; Ns ð17Þ

where xk is a single state indicating the local concentration C(tk) in the neighborhood; wk is
zero-mean white noise; yk,j is the measurement of the number of adsorbed molecules N ~ Ay;j ðtkÞ
onto the surface of the jth sensor; Ns is the number of sensors in the neighbourhood, f(�) represents the state transition function which can describe production, degradation, mass transport,
biological reactions, etc. of the signal molecules; and p(�) denotes the probability distribution
represented by the convolution of the two binomial distributions, as in (13), which describes
the stochastic adsorption reaction model. The expression involves both xk and yk−1,j (corresponding to C(tk) and N ^ Ay;j ðtk�1Þ, respectively), which explains the use of the notation p(yk,j|xk,
yk−1,j). The available information at time step k is the set of measurements
Yk ¼ fyi 2 R Ns : i ¼ 1; . . . ; kg. Note that other biological effects on the concentration are not
considered in the model (16) and (17). The “cell state” as an example of such effects will be
included as a hidden Markov state in the later part of this article.

The above model can be extended to a multiple-state (vector x) system where concentrations at different spatial locations are measured by separate sets of CNT sensors, which can be
useful in cases where one deals with a spatially distributed concentration profile and/or multiple signal molecules over a large sensing area. In this case, the concentrations and therefore the
measured data at different locations can be correlated through the mass transfer phenomena,
which can be represented by mass transport models such as diffusion equation [29]. To communicate the essence of the problem in a simple and transparent manner, this article focuses
on estimation of concentration at a single location, using single or multiple sensors.

Kalman filter. The Bayesian approach offers a systematic way to combine prior knowledge, state and observation models, and measurement information into an informative

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

estimate of the state (i.e., a posteriori probability density function (PDF) of the state p(xk|Yk)).
For linear Gaussian systems, the Kalman filter (KF) enables a recursive construction of the
exact PDF of the state estimate, which is parameterized by the mean and covariance. Kalman
filtering can be applied to the exact probability distribution model (13) by approximation of
the exact PDF by a Gaussian distribution function.

The binomial distribution B(n, p) has the mean of np and the variance of np(1 − p) and can
be approximated by a normal distribution with the same mean and variance,
N ðnp; npð1 � pÞÞ[30]. In this work, the two binomial distributions in the exact observation
model can be approximated by

N~ Ay;jðtkÞ � N ðm F j

F j Þ

Þ 2Þ�N ðmEj ; ðsE j Þ

Þ 2Þ�N ðmEj ; ðsE j

2Þ ð18Þ

F j ; ðsF j

m F j ¼ N^ Ay;j ðt k�1 ÞlFðt k Þ ð19Þ

s F j ¼ N^ Ay;j ðt k�1 ÞlFðt k Þð1 � lFðt k ÞÞ ð20Þ

m E j ¼ ðN T � N^ Ayj ðt k�1 ÞÞlEðt k Þ ð21Þ

s E j ¼ ðN T � N^ Ay;j ðt k�1 ÞÞlEðt k Þð1 � lEðt k ÞÞ ð22Þ

Convolution of the two Gaussian distributions N ðm1; s 2 1 Þ and N ðm 2 ; s2 2 Þ is a Gaussian dis-

tribution with N ðm1 þ m2; s 2 1 þ s 2 2Þ [31], so the observation model can be approximated by

Convolution of the two Gaussian distributions N ðm1; s 2 1

2 1 þ s 2 2Þ [31], so the observation model can be approximated by

N~ Ay;jðtkÞ � N ðm F j

E j ; ðsF j

F j þ mE j

F j Þ

2 þ ðsEj Þ 2Þ ð23Þ

Hence, the Gaussian-approximated observation model for the jth sensor is defined by

yk;j ¼ hðxk;j; yk�1;jÞ þ vk;j; 8j ¼ 1; . . . ; Ns ð24Þ

where h(xk,j, yk−1,j) is same as the mean in (23) and vk,j is zero-mean Gaussian noise with the
variance of (23).

The KF method can be summarized in a recursion of prediction and correction steps, starting from an initial guess defined by the mean ^x 1j1 and the covariance P1|1. Given the posterior
mean and covariance of xk, the mean and covariance of the prior PDF of next state xk+1 is

x^kjk�1 ¼ A � k�1x^k�1jk�1 ð25Þ

Pkjk�1 ¼ A � k�1Pk�1jk�1A � T k�1 þ Q ð26Þ

where P is the covariance of the state and Q is the covariance of the process noise wk, and A � k�1
follows from the linearization

-
@f
A� k�1 ¼
@x

x¼x^k�1jk�1

ð27Þ

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

The mean and covariance of the posterior PDF is

hðx^kjk�1;1; yk�1;1Þ

.. .

hðx^kjk�1;Ns ; yk�1;NsÞ

C
C
C
C
A

B
B

x^kjk ¼ ^x kjk�1 þ LkBByk �

66664

77775

ð28Þ

T k ðR k þ C� k P kjk�1 C� T k

Lk ¼ Pkjk�1C � T k

T k Þ

�1 ð29Þ

Pkjk ¼ Pkjk�1 � LkC � kPkjk�1 ð30Þ

where Lk 2 R Ns is the Kalman gain matrix and C � k 2 R Ns follows from the linearization

" #

@hðxk;j; yk�1;jÞ
½C � k�j ¼

@xk;j

x¼x^kjk�1

ð31Þ

The covariance matrix Rk 2 R Ns�Ns for the measurement noise can be defined by a diagonal
matrix

½Rk�jj ¼ rk;j ð32Þ

where rk,j is same with the variance in (23) for the jth sensor.

Particle filter. To use the non-Gaussian observation model (17) directly, a samplingbased approach known as particle filtering (PF) can be used. PF is based on a discrete weighted
approximation of the true posterior PDF with a set of random samples (particles). If the number of samples becomes extremely large, the approximation converges to the true posterior
PDF.

The sequential importance sampling (SIS) algorithm is considered as the current standard
of PF [26]. The first step of the algorithm is an initialization of N particles and their weights,
denoted by fxk i ; wi k ; i ¼ 1; . . . ; Mg. In this step, each particle is sampled from the initial PDF

denoted by fxk i ; wi k ; i ¼ 1; . . . ; Mg. In this step, each particle is sampled from the initial PDF

pðx1Þ � N ðx�1; Q1Þ and the associated weight is initialized to 1/ M. After the initialization, the
importance sampling step and the weight update step are repeated. In the importance sampling
step, xk i , i = 1,. . ., M are sampled from an importance density q(xk|Yk) which is a user-defined

k i ; wi k

step, xk i , i = 1,. . ., M are sampled from an importance density q(xk|Yk) which is a user-defined

choice. The importance density is commonly chosen as the prior PDF,

qðxkjYkÞ ¼ pðxkjxk�1Þ ð33Þ

In the weight update step, the weight w i

YNs

i k for each i = 1,. . ., M is updated with

k i ;j ; y k�1;j Þpðx k jx k�1 Þ

w i

i

k / w k i

k�1

j¼1 pðyk;jjx k i

ð34Þ
qðxkjYkÞ

If (33) is substituted into (34), the weight update equation is described as

k i ;j ; y k�1;j Þ ð35Þ

w i k

i

k / w k i

k�1

YNs

j¼1 pðyk;jjx k i

Based on the samples and normalized weights, the posterior PDF can be approximated as

k i dðx k � x k i

k i Þ ð36Þ

pðxkjYk�1Þ ¼

i¼1 w k i

where δ(�) is the Dirac delta function. The estimated value is commonly calculated as a

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

weighted mean,

k x k i

i

k i ð37Þ

x^kjk ¼

i¼1 w k i

In addition, a resampling step can be added to mitigate the degeneracy problem [26]. The
degeneracy problem refers to the growing number of samples having negligible weights with
iterations. The resampling step eliminates the samples with small weights and concentrates the
calculation on those samples with large weights whenever a significant degeneracy problem is
detected. After generating a new set of xk i for i = 1,. . ., M by resampling, the weights are reset to

detected. After generating a new set of xk i for i = 1,. . ., M by resampling, the weights are reset to

1/ M as in the initialization step. After resampling, the estimated value ^x kjk is calculated as a
mean of xk i for i = 1,. . ., M.

k i for i = 1,. . ., M.

Results and Discussion

This case study for testing the two approaches is motivated by the problem of detecting tumor
cells through NO and H2O2 signal molecules. NO generated from vascular endothelial NO
synthase (eNOS) correlates with stimulation of angiogenesis. This activity is intimately linked
with metastasis of tumor cells since their survival and proliferation are highly dependent on
adequate supply of O2 and nutrients from blood vessels by diffusion [32–34]. Membrane-associated NADPH oxidases are also found in vascular endothelial as well as smooth muscle cells,
and generate H2O2 as an important signal molecule in angiogenesis. Produced H2O2 can activate signalling pathways to stimulate tumor cell proliferation and migration [35–37]. Knowledge of how concentrations of these signal molecules change as a cell changes its state can help
understand their biological roles in tumor cell growth, which in turn can lead to advances in
medical treatments.

The estimation of the concentration of signal molecules from a normal cell is examined
first, and then the more complex case of a cell transitioning from a normal state to a tumor
state is considered.

Estimating the concentration of signal molecules from a normal vascular
endothelial cell

This section develops a state estimation problem for the signal molecules (NO or H2O2) from
vascular endothelial cells. The width and length of the endothelial cells is more than 10 μm

[38], which indicates that dozens of SWNT-based sensors can be placed on a single vascular
endothelial cell and send multiple stochastic monomolecular adsorption data [7] (Fig 2).
Among them, sensors near the enzymes generating signal molecules, where frequent adsorption/desorption events are detected, can be selected and used in the estimation of the local concentration of the signal molecules. The small area proximate to the generator of the signal
molecules can be considered as a neighborhood sharing same local concentration that represents the cell state as a whole.

It is difficult to obtain from an experimental setup a large dataset that includes sufficient,
representative stochastic variations to render a fair and thorough evaluation of estimation performance. Alternatively, representative stochastic adsorption datasets can be generated from
kinetic Monte Carlo (KMC) simulations. Each KMC simulation run can be viewed as a realization of the stochastic system that is described by the CME [39]. The adsorption/desorption
process involves fairly simple molecular level events and Zhang et al. 2010 [8] showed that
experimental data for this system was well described by the KMC simulation.

In this particular simulation study, the number of adsorbed molecules on the sensor is
allowed to range from 0 to 10, so the number of possible discrete states is 11. The length of

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

Fig 2. Generation of signal molecules from a membrane-associated enzyme and their detection from a sensor array system on a single cell.

doi:10.1371/journal.pone.0141930.g002

each run is 2000 s and the sampling time interval is 1 s. The starting state is assumed to be 0
(empty of molecules). Adsorption/desorption parameters, kA and kD, are chosen as 100 M −1 s −1

and 0.001 s −1 respectively, which are taken from [6]. In the normal vascular endothelial cell,
the signal molecules are released consistently from the enzyme at a low concentration level
(~10 μM) [40]. These dynamics can be simply described as an integrated white noise process,

xk ¼ xk�1 þ wk ð38Þ

Fig 3A shows an example time-varying concentration with wk - N ð0; 10 �16 Þ in (38) and
Fig 3B shows associated five representative realizations of the time profile of the number of
adsorbed molecules.

The adsorption data show five distinct time profiles for the same local concentration profile,
indicating significant stochastic characteristics of the sensor system at the nano-scale. Other
features of the stochastic data shown in Fig 3B are the stepwise (discretized) and reversible variations. In addition, each time profile shows several stationary regions after unpredicted transitions even as the local concentration changes continuously at the bulk phase, which indicates
the information-poor characteristic of the dataset, for which accurate modelling is essential for
accurate state estimation.

We consider the state estimation problem with increasing number of sensors in the neighborhood measuring the same local concentration. KF and PF (with 200 particles) are compared
with increasing number of sensors. KF is based on the Gaussian-approximated observation
model, while PF considers the full non-Gaussian stochastic model resulting in the non-Gaussian posterior PDFs as shown in Fig 4. The posterior PDFs are kernel densities reconstructed
from the particles of PF for a specific dataset at t = 100 s, 500 s, 1000 s, and 1500 s. The overall
PDFs show non-Gaussian distributions that are mostly positively skewed.

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

Fig 3. Time plots for (a) the local concentration and (b) associated stochastic adsorption data for a single sensor, generated by KMC simulations.

doi:10.1371/journal.pone.0141930.g003

Performance of the two estimation methods can be compared by observing how well the
estimates track the true concentration profile throughout the run time from a wrong initial
guess (x�1 ¼ 2 � 10 �5 ; Q1 ¼ 1 � 10 �5 ). Plots of the estimates for the 1-sensor and 5-sensor
cases are shown in Fig 5. For the 1-sensor case, the PF estimates follow the true concentration
more closely than the KF (Fig 5A). For the 5-sensor case, the gap between the PF and KF estimates is reduced as long time (Fig 5B).

For quantitative comparison, the root-mean-square-errors (RMSEs) of the estimated concentrations per run are averaged over 100 runs that generated different adsorption/desorption
data from different local concentration profiles (2000-sample dataset per one run). The averaged RMSE is defined by

s f f f f f f f f f f f f f f f f f f f f f f

f f i f f i f f i f f i f f i f f i f f i f f i f f i f f i f f i f f i f f i f f i f f i f f i f f i f f i f f i f f i f f i f f i ffi
XNT

k¼1 ðCtrue;i;k�105�C^ i;k�105Þ2

ð39Þ

RMSE ¼

XNR

i¼1

where NS is the number of samples in one run, NR is the number of runs, Ctrue,i,k is the true
local concentration value, and C ^ i;k is the estimate for the kth sample time of the ith run.

In Table 1, for all cases, the RMSEs of the estimates from PF are smaller than from KF. The
difference in the RMSEs of the two methods slowly decreases with increasing number of sensors in the neighborhood, while the computation time of PF is higher and increases more rapidly than that of KF. If the objective is only a nominal state estimate, the benefit of rigorous
stochastic modelling in the state estimation is reduced when more information is contained in
the dataset (through the use of multiple sensors). Of course, a disadvantage of the KF for any
number of sensors is that it is not capable of estimating the non-Gaussian character of the distribution of the state estimates.

Estimating the signal molecules from a cell having two states

In normal vascular endothelial cells, signal molecules generated from the enzymes are at a low
concentration level (~10 μM). In tumor vascular endothelial cells, on the other hand, the
expression levels and activities of eNOS are abnormally increased compared to the normal
endothelial cells (Fig 6), and the elevated level of NO promotes tumor progression and metastasis by inducing angiogenesis as well as tumor cell invasion, proliferation, and migration [41,

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

Fig 4. Kernel density reconstruction of the posterior PDFs from the PF at (a) 100 s, (b) 500 s, (c) 1000 s, and (d) 1500 s.

doi:10.1371/journal.pone.0141930.g004

Fig 5. Time plots of the concentration estimates from the KF (red - �� -) and PF (blue - � -) for (a) 1 sensor and (b) 5 sensors.

doi:10.1371/journal.pone.0141930.g005

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

Table 1. Averaged RMSEs of estimates from the PF and KF and associated computation time (in seconds) with increasing number of sensors.

# of sensors 1 5 10 20

Average RMSE 1 PF 0.5038 0.3129 0.2457 0.2074

KF 0.5356 0.3333 0.2521 0.1972

Average computational time 1,2 PF 0.0035 0.0135 0.0275 0.0519

KF <0.0001 0.0001 0.0001 0.0001

1The values are averaged over 100 runs.
2The computation time was recorded in seconds using a workstation with 3.40 GHz CPU and 8GB RAM.

doi:10.1371/journal.pone.0141930.t001

42]. For H2O2, there is also a considerable variation among cells in the concentration level
required to initiate a particular biological process. Moreover, it has been observed that different
levels of H2O2 can induce distinct responses within a cell. For example, overproduction of
H2O2 results in proliferation and migration of smooth muscle cells, contributing to atherogenesis and restenosis [43].

Hidden Markov model. The concentration of the signal molecules affects and is affected
by the state of the cell. For real-time state estimation, consideration of all the complex biological processes associated with different cell states is very challenging and linking with concentration variations of the signal molecules can easily become intractable. In addition, the
signal molecules are small gaseous molecules showing very fast diffusion (with diffusion coefficients of around 10 −5 [cm 2 /s]) compared to cell activities in tissues [40]. In this context, we
simplify the system to having two states: a normal state and an abnormal state. In the normal
state, the signal molecules are released consistently at a low concentration level. In the abnormal state, the concentration of the signal molecules increases (drifts) rapidly to a new elevated

Fig 6. Production of signal molecules from vascular endothelial cells.

doi:10.1371/journal.pone.0141930.g006

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

Fig 7. Variation of the concentration of signal molecules with different cell states.

doi:10.1371/journal.pone.0141930.g007

level. Fig 7 illustrates a possible concentration variation in the vascular endothelial cell as it
transitions to the abnormal state (based on data generated from the artificial Hidden Markov
model).

Such a pattern in Fig 7 can be characterized as a mixture of quiescent and drifting phases,
which is called “intermittent drift.” A hidden Markov model (HMM) can be used for modelling
such shifts in the disturbance pattern [44, 45]. HMM represents a useful class of statistical
models where a hidden state, H 2 f1; 2; . . . ; Hg transitions probabilistically among possible
states in a Markovian fashion. In this work, each member of the set H represents a particular
system state, for example, “normal cell” or “tumor cell” states. Mathematically, a finite-state
Markov chain is a sequence of random integers, rk, where the transition probability matrix P
has elements defined by

pij ¼ Prðrk ¼ jjrk ¼ iÞ; i; j 2 H ð40Þ

j¼1 pij ¼ 1 ð41Þ

Based on the transition probability matrix, the intermittent drift in the concentration of the
signal molecule, xk can be described by

xk ¼ xk�1 þ wrk ; rk 2 1; 2 ð42Þ

p11 ¼ p22                                          - 1 ð43Þ

p12 ¼ p21 < 1 ð44Þ

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

where “1” indicates the normal cell and “2” indicates the tumor cell. The wrk is a white Gaussian
noise with covariance Qrk defined by

Qrk¼1                     - 0 ð45Þ

Qrk¼2                    - Qrk¼1 ð46Þ

Since there is only a low probability of switching once the system enters a particular regime,
a diagonally dominant P is employed, as reflected in (43) and (44). Note that the actual regime
is usually not known with complete certainty and must be inferred from measurements. Additional behavior could be incorporated into the model by introducing more hidden states (e.g.
other transitional cell states or environmental effects on the local concentration) with appropriate accompanying stochastic models for them.

Second-order generalized pseudo-Bayesian algorithm. For using KF for a Markov jump
system represented by (42), the generalized pseudo-Bayesian estimation algorithm of order 2
(GPB2) has been suggested as an effective sub-optimal filter [46]. Let ^x kjkðrk�1; rkÞ denote the
estimate conditioned on the two most recent hidden state realizations. Similarly, the corresponding estimation error covariance is represented as Pk|k(rk−1, rk). The main idea is to generate multiple Gaussian distributions from KF for all possible trajectories of the last two hidden
states, and combine them into a single Gaussian distribution, parameterized by fx^kjk; Pkjkg. A
recursive scheme is characterized by two steps: “branching” and “merging.”

Starting with fx^k�1jk�1ðrk�1Þ; Pk�1jk�1ðrk�1Þg, the branching step is to obtain the set
fx^kjkðrk�1; rkÞ; Pkjkðrk�1; rkÞg through the prediction and correction steps of KF. The merging
step involves the law of total probability and Bayes’ rule to collapse the products from the
branching step as

x^kjkðrkÞ ¼

rk�1¼1 x^kjkðrk�1; rkÞ pðrk�1jrk; YkÞ ð47Þ

PkjkðrkÞ ¼

rk�1¼1 ½fx^kjkðrk�1; rkÞ � x^kjkðrkÞg

2 þ Pkjkðrk�1; rkÞ�pðrk�1jrk; YkÞ ð48Þ

pðrk�1jrk; YkÞ ¼ 1

pðykjrk�1; rk; Yk�1Þ pðrkjrk�1Þ pðrk�1jYk�1Þ ð49Þ

where c1 is a constant ensuring that p(rk−1|rk, Yk) sums to unity, and p(yk|rk−1, rk, Yk−1) is
related to the correction step of KF in the branching step. A point estimate is obtained from

x^kjk ¼

rk�1¼1 x^kjkðrkÞ pðrkjYkÞ ð50Þ

2 þ PkjkðrkÞ�pðrkjYkÞ ð51Þ

Pkjk ¼

rk�1¼1 ½fx^kjkðrkÞ � x^kjkg

pðrkjYkÞ ¼ 1

rk�1¼1 pðykjrk�1; rk; Yk�1Þ pðrkjrk�1Þ pðrk�1jYk�1Þ ð52Þ

where c2 is a constant ensuring that p(rk|Yk) sums to unity.

Markov chain Monte Carlo algorithm. Adapting PF to the Markov jump system is relatively simpler than KF. Starting with fxk i �1 ðr k�2 ; r k�1 Þ; wi k�1 ðr k�2 ; r k�1 Þg, samples x k i ðr k�1 ; r k Þ for

tively simpler than KF. Starting with fxk i �1 ðr k�2 ; r k�1 Þ; wi k�1 ðr k�2 ; r k�1 Þg, samples x k i ðr k�1 ; r k Þ for

i = 1,. . ., M are generated from the same importance density of (33) for all possible trajectory
for the recent hidden Markov states, rk−1 = 1, 2 and rk = 1, 2. This approach is called the

k i �1 ðr k�2 ; r k�1 Þ; wi k

i k�1 ðr k�2 ; r k�1 Þg, samples x k i

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

Markov chain Monte Carlo (MCMC) algorithm [47]. The weight update Eq (35) is modified by
including, p(rk, rk−1|Yk),

k i ;j ðr k�1 ; r k Þ; y k�1;j Þ ð53Þ

w i

i k ðr k�1 ; r k Þ / wi k

i k�1 ðr k�1 ; r k Þ pðr k ; r k�1 jY k Þ

YNs

j¼1 pðyk;jjx k i

Finally, the point estimate can be obtained by

k i ðr k�1 ; r k Þx k i

k i ðr k�1 ; r k Þ ð54Þ

x^kjk ¼

rk�1¼1

rk¼1

i¼1 w k i

Detection of tumor cell activity. As stated before, two regimes are considered in the system: the normal cell state and the tumor cell state. The objective is to detect a regime change
through the local concentration variations of the signal molecules seen from the nano sensors.
The reference work [7] investigates the effect of a growth factor, which stimulates cell growth,
proliferation, and differentiation on the H2O2 generation in living cells. From the 3000 s observation after the stimulation with the growth factor at t = 0, it was observed that the H2O2 concentration level increased immediately and reached a maximum in the time range between 600
s and 1800 s. This observation indicates that the tumor cell activity and its effect on the local
concentration of signal molecules can be prolonged for a long time (~ 30 min).

Based on this data, stochastic adsorption/desorption profiles were generated from KMC
simulation. The number of adsorption sites on the sensor is 10 and the length of each run is
4000 s with the sampling time interval of 1 s. The starting state is assumed to be a random integer less than 10 (partly occupied) and the kA and kD are assumed to be 100 M −1 s −1 and 0.001
s −1 . Eq (42) is used in the state estimation as the state model. At the ‘normal cell’ state, the local
concentration is stable and affected only by low-level noise (wk - N ð0; 10 �16 Þ). When the cell
becomes a tumor cell, the local concentration becomes elevated by high-level noise
(wk - N ð0; 5 � 10 �14 Þ). The plots in Fig 8 show a representative concentration variation with
the tumor cell activity for the time period from 2000 s to 4000 s and five different realizations
of the associated time profile of the number of adsorbed molecules.

For this switching system, KF with the GPB2 algorithm (shortly, KF-GPB2) and PF with the
MCMC algorithm (shortly, PF-MCMC) are designed and compared. The hidden Markov

Fig 8. Time plots of (a) the local concentration affected by the tumor cell activity and (b) associated stochastic adsorption data, generated by
running KMC simulations.

doi:10.1371/journal.pone.0141930.g008

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

Fig 9. Time plots of the concentration estimates from (a) PF (red - � -) and PF-MCMC (blue - �� -) and (b) KF (orange - � -) and KF-GPB2 (green - �� -) for
one sensor.

doi:10.1371/journal.pone.0141930.g009

chain used the state transition probability matrix

P ¼

" #

0:99 0:01

0:01 0:99

ð55Þ

Fig 9 shows the state estimates of PF, PF-MCMC, KF, and KF-GPB2 for one sensor. The
basic PF and KF used in Section 4 cannot effectively follow the concentration drift caused by
the tumor cell activity, while the PF-MCMC and KF-GPB2 estimators follow the drift much
better. Among the methods, the estimates from PF-MCMC are closer to the true concentration
dynamics and show less fluctuations compared to KF-GPB2.

In both PF-MCMC and KF-GPB2, the state estimates are improved further when more stochastic adsorption/desorption data are available, obtained from multiple sensors, as shown in
Fig 10.

Fig 10. Time plots of the concentration estimates from (a) PF-MCMC (blue - �� -) and (b) KF-GPB2 (green - � -) for five sensors.

doi:10.1371/journal.pone.0141930.g010

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

Table 2. Averaged RMSEs and of the estimates from PF-MCMC and KF-GPB2 and associated computation time for increasing number of sensors.

# of sensors 1 5 10 20

Average RMSE 1 PF-MCMC 2.3748 1.9180 1.7544 1.6190

KF-GPB2 2.6095 2.0816 1.8984 1.7542

Average computational time 1,2 PF-MCMC 0.0120 0.0474 0.0839 0.1637

KF-GPB2 0.0016 0.0076 0.0126 0.0250

1The values are averaged over 100 runs.
2The computation time was recorded in seconds using a workstation with 3.40 GHz CPU, 8GB RAM.

doi:10.1371/journal.pone.0141930.t002

Table 2 shows the averaged RMSEs and computational time for PF-MCMC and KF-GPB2
based on 100 runs that generated different adsorption/desorption data from different tumor
cell activities. In both methods, the RMSEs of the estimates decrease with increasing number of
sensors in the neighborhood with PF-MCMC having smaller RMSE values than KF-GPB2.
Though the computation time of PF-MCMC is larger than KF-GPB2, it is far less than the sampling time of 1 s.

In a real application, the neighborhood region proximate to the enzyme should be very
small considering the short-life time and high diffusivity of the signal molecules. Therefore,
less than 5 sensors might be valid in the state estimation for a single cell [7]. In this context,
PF-MCMC can be recommended if accurate estimates of the local concentration of the signal
molecules are needed with such limited information.

Conclusions

Two stochastic state estimation methods–Kalman filtering (KF) and particle filtering (PF)–
were investigated for estimating the time-varying local concentration of signal molecules from
stochastic monomolecular adsorption/desorption data on the surface of the carbon-nanotube
(CNT)-based sensors. In addition, the second-order generalized pseudo Bayesian estimation
(GPB2) algorithm and the Markov chain Monte Carlo (MCMC) algorithm were incorporated
into KF and PF respectively, for detecting latent drift in the concentration affected by different
states of a cell. The stochastic nature of the adsorption data from each CNT-based sensor was
fully modelled by using the chemical master equation (CME). In addition, intermittent concentration variations of the signal molecules were modelled by a hidden Markov model. Performances of the state estimators with the sensor array system were compared through a case
study employing KMC simulation. The PF-MCMC combination showed the highest accuracy
while having reasonable computation time.

Use of drugs affecting the production of signal molecules by inhibiting the associated
enzyme or directly scavenging the signal molecules appears to be a promising strategy to
inhibit angiogenesis and therefore tumor growth [48, 49]. In order to control the modification
of signal molecules in a precise manner, further understanding of various factors involved such
as the timing, concentration, and location is required. The proposed state estimators have
promise in this endeavor.

Supporting Information

[S1 Dataset. Original data including true concentration profiles, stochastic adsorption pro-](http://www.plosone.org/article/fetchSingleRepresentation.action?uri=info:doi/10.1371/journal.pone.0141930.s001)
files and concentration estimates for plotting Figs 3, 5, 8, 9 and 10.
(XLSX)

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State Estimation with CNT-Based Sensors for Tumor Cell Detection

Author Contributions

Analyzed the data: HJ JHL. Wrote the paper: HJ JHL RDB. Formulated the state estimation
methods: HJ. Performed the simulation and estimation: HJ. Performed the numerical analysis:
HJ.

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