HASapl07.pdf

On noise processes and limits of performance in biosensors

Arjang Hassibi

Electrical and Computer Engineering Department, University of Texas at Austin, Austin, Texas 78712

Haris Vikalo and Ali Hajimiri

Electrical Engineering Department, California Institute of Technology, Pasadena, California 91125

Received 11 January 2007; accepted 4 May 2007; published online 11 July 2007

In this paper, we present a comprehensive stochastic model describing the measurement uncertainty,

output signal, and limits of detection of affinity-based biosensors. The biochemical events within the

biosensor platform are modeled by a Markov stochastic process, describing both the probabilistic

mass transfer and the interactions of analytes with the capturing probes. To generalize this model

and incorporate the detection process, we add noisy signal transduction and amplification stages to

the Markov model. Using this approach, we are able to evaluate not only the output signal and the

statistics of its fluctuation but also the noise contributions of each stage within the biosensor

platform. Furthermore, we apply our formulations to define the signal-to-noise ratio, noise figure,

and detection dynamic range of affinity-based biosensors. Motivated by the platforms encountered

in practice, we construct the noise model of a number of widely used systems. The results of this

study show that our formulations predict the behavioral characteristics of affinity-based biosensors

which indicate the validity of the model. ©

2007 American Institute of Physics

DOI:

10.1063/1.2748624

I. INTRODUCTION

Affinity-based biosensors use selective binding and in-

teraction between certain biomolecules

recognition probes

and specific target analytes to determine the presence of the

latter in the biological samples. The essential role of the

biosensor platform

e.g., microarrays,

and immunoassays

is to facilitate the binding of the probe-target complexes to

produce a

detectable

signal, which correlates with the pres-

ence of the target and conceivably its abundance. The mini-

mal components required for affinity-based detection include

a molecular recognition layer

capturing probe

integrated

within or intimately associated with a signal-generating

physiochemical transducer

and a readout device.

To generate target-specific signal, the target analytes in

the sample volume first need to collide with the recognition

layer, interact with the probes, and ultimately take part in a

transduction process. The analyte motion in typical biosensor

settings

e.g., aqueous biological mediums

is dominated by

diffusion spreading, which from a microscopic point of view

is a probabilistic mass-transfer process

i.e., random-walk

events for a single analyte molecule

. Accordingly, the ana-

lyte collisions with the probes become stochastic processes.

Moreover, because of the quantum-mechanical nature of

chemical bond forming,

–

interactions between the probes

and the analyte molecules are also probabilistic, adding more

uncertainty

to the biosensing procedure. On top of these two

processes, we also have the transducer and readout circuitry,

which likely adds more noise to the already noisy process.

Beside the inevitable uncertainty associated with the

binding, transduction, and readout, in all practical biosen-

sors, binding of other species to the probes

nonspecific

binding

is also possible. Nonspecific binding

e.g., cross-

hybridization in DNA microarrays

is generally less prob-

able than the specific binding when target analyte and the

interfering species have the same abundance. Nonetheless, if

the concentration of the nonspecific analyte becomes much

higher than the target analyte, nonspecific bindings may

dominate the measured signal and hence limit the minimum-

detectable level

MDL

of the biosensor platform.

In this

paper, we examine all of the aforementioned processes and

assess their contributions to the biosensor output signal. Our

goal is to introduce a general methodology to model the

noise in biosensors, and based on that, evaluate the MDL,

highest detection level

HDL

, and detection dynamic range

Initially, in Sec. II, we review the existing stochastic

model of the biosensor platforms,

which is essentially a

Markov model for the analyte capturing. We proceed by add-

ing to it a probabilistic model for the transduction, probe

saturation, and the readout process. In Sec. III, we implement

this model to evaluate the uncertainty of the detected signal

and the relative noise contributions of all underlying pro-

cesses as a function of the analyte concentrations, capturing

probe density and other characteristics of the biosensor sys-

tem. Finally, in Sec. IV, demonstrating the practical use of

our model, we implement the model to describe the stochas-

tic behavior of impedimetric

and fluorescent-based

bio-

sensors.

II. BIOSENSOR STATISTICAL MODEL

A. Analyte motion

Molecules, cells, and many other analytes immersed in

the aqueous mediums of biosensor platforms are subject to

thermal fluctuations and, in particular, sensory platforms,

Previously at Electrical Engineering Department, California Institute of

Technology, Pasadena, CA 91125; electronic mail: arjang@mail.utexas.edu

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subject to electromagnetic or mechanical forces. While me-

chanical movement

e.g., convection

and electromagnetic

forces

e.g., electrophoresis

are in some sense deterministic,

thermal fluctuations are random in nature. Thermal fluctua-

tions of a particle from a microscopic point of view follow

the characteristics of a typical random-walk process

i.e.,

Brownian motion

, which results in a diffusive spreading

phenomenon in macroscopic systems.

It is challenging to statistically describe the motion of

individual analyte molecules in the general case using the

continuity equation formulation. However, we can utilize a

discrete-time Markov process

to model the particle sto-

chastic behavior within the reaction chamber.

In this

model, we discretize the coordinates of the reaction chamber,

where the states of the Markov model correspond to a fixed

set of coordinates within the chamber

Fig.

. The transition

probabilities of the Markov process are defined as the prob-

abilities of an analyte moving from one coordinate

state

another in the given time epoch

. Such a simple model

captures the stochastic nature of the process yet is still trac-

table enough to allow for computation of the probability den-

sity function

pdf

of the analyte particles. Additionally, we

can easily derive the expected behavior

ensemble average

of the system, a quantity that is generally assessed using a

continuity-based equation formulation.

For two arbitrary coordinates

v

and

v

in the system,

associated with states

and

, the state transition probability

from state

, for an analyte particle in the time interval

, is denoted by

=Pr

v

v

v

v

where

v

and

v

specify the coordinate of an analyte

particle at times 0 and

, respectively. The

defined in

represents the

element of the analyte transition matrix

and

is the total number of states

.If

an analyte particle has the probability distribution

time zero across the state space, where for all

then at

Typical biosensor structures have a large number of analyte

particles within the reaction chamber. Provided that their mo-

tion is statistically independent,

and

become in-

dependent, resulting in a homogeneous Markov process.

Note that this is, in fact, a realistic assumption for most bio-

sensors, because statistical motion in such systems is gov-

erned by analyte interaction with the medium molecules and

temperature, and not by analyte-analyte interactions. Hence,

analytes seldom affect the probabilistic motion of one an-

other. Because of the homogeneity of the Markov process,

we are able to calculate the spatial probability distribution of

the analyte particles at all time instants that are multiples of

, given the initial distribution

and

,by

where

is an integer. If we have

analyte particles in the

system we can also define a spatial concentration distribution

of the analyte particles

, where

and hence

where

is the initial concentration of the analyte par-

ticles. It is important to realize that the matrix

can

theoretically be estimated for any small time increment

given the exact statistics of the mass transport process de-

scribed by the continuity equation.

B. Analyte capture

Now that we have a statistical model of analyte motion

within the solution, we need to incorporate the boundary

conditions within the biosensor chamber. The boundary con-

ditions for conventional biosensors are either purely reflec-

tive

e.g., inert chamber walls and solution surfaces

or se-

lectively absorbing

e.g., surfaces where the capturing probes

exist

. Incorporation of reflective boundaries into the Markov

model is carried out by assigning zero probability for par-

ticles to move beyond the boundaries. However, accommo-

dation of selective absorption requires a probabilistic model

for the collision and interaction processes.

Analyte collision with the binding sites may be reactive,

elastic, or inelastic.

We have previously introduced a

probabilistic model to describe the specific binding of an

analyte particle

to a single probe

in an affinity-based

biosensor.

To begin with, we assume that the probe

confined

immobilized

in a certain coordinate. Now we can

argue that any meaningful interaction between

and

time

only occurs if

v

v

i.e., molecule

is in intimate

proximity of

, as shown in Fig.

. If the bulk-phase reac-

tion between the analytes

, and the capturing probes

has

an association rate

, and disassociation rate

−1

, then we

can write

↔

−1

−

−1

where the symbol

indicates concentration of the species.

Now based on

we can apply the following approximation

to find the transition probabilities between the captured state

of the analyte and the collided state 1:

FIG. 1. Markov model for probabilistic motion in the reaction chamber.

Each coordinate corresponds to a state of the process into which target

analytes can move or from which they can leave.

v

represents the location

of the analyte particle at time

, and

v

defines the coordinates of the state

in the system.

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−1

The parameters

and

are also defined as the associa-

tion and disassociation transition probabilities, respectively.

The quantity

basically describes the effects of immobiliza-

tion of probes and the finite reactive distance between the

probe and target. Using

for absorbing boundaries and

for the mass transfer and reflection cases, we can analyze the

binding and motion of the analytes in the biosensor systems

in its entirety.

The formulation in

assumes infinite capturing capac-

ity for probes, which is not always realistic. If there are

immobilized capturing sites in state 1, then the association

and disassociation probabilities for the immobilized

spe-

cies become functions of

, the number of captured ana-

lytes at state

, such that

−

−1

where the term

−

is basically the probability

of finding a free probe at the capturing site

see Fig.

. The

incorporation of

into

clearly results in a nonhomoge-

neous Markov process, since the transition probabilities have

become a function of target analyte distribution, i.e.,

becomes a function of

. Finding a closed form for

the dynamics of such system is challenging; however, we

will demonstrate in Sec. III how in chemical equilibrium the

fluctuations of the system can be quantified.

C. Transduction

We define the transduction process and the output signal

of the biosensor as a function of

, the target concentra-

tion distribution within the system. We can represent trans-

duction process by matrix

, where

and

being

the number of detectable capturing states. Accordingly, the

output signal

is defined by

where

is a random process describing the additive noise

of the transducer. The matrix

signifies the process

wherein binding results in a detectable signal or parameter.

For example, the unit of

in a chemiluminescence-based

biosensor

is photons/s target, while in impedimetric biosen-

sors are

−1

/target

see Sec. IV

. As implied by

, the

transduction process itself may introduce some uncertainty in

the form of an additive noise component which is repre-

sented by

. We know from

that

is a random

process, but if we assume that its expected behavior

en-

semble average

is denoted by

, we can rewrite

, the random process

essentially describes the

deviation of the actual analyte concentration

perturbation

from the expected distribution

, where

=0 for all

The function

therefore describes the observed

bio-

chemical noise

of the system.

Since the transduction mechanism varies tremendously

among biosensor platforms, we need to reexamine

and

derive a formulation which makes their performances com-

parable. One approach that we propose in this paper is to

refer the biosensor output signal to the input of the biosensor

system which is essentially the number of analytes in the

sample. In this approach the biosensor equivalent signal

represents the number of

perceived

target analytes in the

sample plus its uncertainty. To carry out the procedure, we

divide

by the effective gain of the biosensor system

which is

. The parameter

is basi-

cally the expected output signal change per analyte concen-

tration change within the system. Now the biosensor master

equation expressed in terms of the number of analytes is

given by

−1

If there are

stages after transducer

see Fig.

, each with

gains

and additive noise

=1,...,

, we can incor-

porate them in

such that

−1

Equations

and

essentially provide the number of

observed

or existing

target analytes in the biosensor sys-

tem. In the next section we apply these stochastic formula-

FIG. 2. Markov model for probabilistic capture in the reaction chamber.

Only when the target analytes get into intimate proximity of the capturing

probes

state 1

there is a probability of capturing denote by

FIG. 3. Modification of association transition probability

, due to probe

saturation.

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tions to calculate the noise characteristics of biosensor plat-

forms.

III. NOISE AND SIGNAL FLUCTUATION

At this point we can utilize the transition matrix derived

along with the correct boundary conditions of

simulate various characteristics of the biosensor, and use

–

to assess the output signal of the system. The simu-

lation of such a system although possible is generally cum-

bersome, since the number of states is extremely large in

practical biosensors. Hence, in the following subsections we

will derive the closed form expressions describing the fluc-

tuation of biosensor output. In particular, we will focus on

the noise at biochemical equilibrium which, for many prac-

tical reasons, is the state where the measurements are pref-

erably carried out.

A. Biochemical fluctuation in equilibrium

†

„

...‡

We can show that when there is no saturation in the

biosensor, the general solution for

and

which de-

fines the single analyte probability distribution at time

given an initial condition

, is described by the

following:

= exp

−

where

is the identity matrix of the same dimensions as

. Matrix

advances the initial distribution in time

and is calculated using the matrix exponential function. It

can be shown that

has a single eigenvalue equal to 1,

and all other eigenvalues of

have magnitudes less

than 1, ensuring a single equilibrium distribution,

which is

the eigenvector associated with an eigenvalue of 1. Further-

more, it can be shown that

and

, i.e., it

holds that

To quantify the

biochemical

noise of the system

,we

need to characterize the analyte fluctuation in every state,

and particularly in the states for which the transduction oc-

curs

i.e., the states corresponding to nonzero entries of

To perform this calculation for the biochemical equilibrium,

we first need to evaluate the autocorrelation function of the

stationary random processes describing the occupancy of the

states denoted by

, where 1

describes the state

of the system and 1

represents the target analytes. The

occupancy

is essentially an indicator function of the

form

v

v

otherwise,

which means that

=1 if the

th particle occupies state

at time

, and zero otherwise. It can be shown that

, the

autocorrelation of

, is identical for all

analytes and

can be found using the definition of

to be

where

is the

th component of

and

is the

diagonal entry of

Since we have

independent par-

ticles in the system, we define the stochastic process

counting the total number of particles occupying state

and hence

. Now by implement-

ing

, we can derive

, the autocorrelation function of

,as

−

Accordingly, the unilateral

single-sided

power spectral den-

sity

PSD

defined by

which describes the fluc-

tuations as a function of frequency becomes

−

where

−

To find

, the variance of

, we have

−

1−

It is evident from

and

that the expected value of

independent particles in state

becomes

. Thus,

the

expected concentration at equilibrium becomes

FIG. 4. Block diagram of different stages of a general biosensor platform.

Various noise processes originating from different sources may corrupt the

signal; however, all of these noise sources can be referred to as the number

of analytes in the sample.

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= lim

→

= lim

→

Based on the definition of

, now for the equilibrium

fluctuation

=lim

→

we can derive the expected

value

, and the variance

, where

=0,

1−

Now by using

, we easily compute for the expected value

of the output noise fluctuation

and its variance

equilibrium

=0,

where

is the element-by-element

square of the transduction matrix

.Itis

important to realize that

is the variance of the unavoidable

noise of the system, which is basically the biochemical shot

noise originating from the quantum nature of binding events.

This particular noise is inherent to affinity-based biosensors

and the uncertainty

and certainty

limitation which it estab-

lishes is often referred to as the quantum-limit

detec-

tion level.

Note that while

–

are derived assuming a homo-

geneous process

i.e., no probe saturation and no analyte-

analyte molecular interactions

, the same methodology can

be adopted for nonhomogeneous processes, given that an

equilibrium distribution exists for the system. The only nec-

essary modification is that a transition matrix at the equilib-

rium for nonhomogeneous process must be derived to calcu-

late

. Essentially in this approach we perform

linearization

, and the method is thus valid when the fluctua-

tions of the concentration are small compared to the en-

semble average concentration

i.e., small perturbation

B. Output signal fluctuation at equilibrium

†

„

...

„

...‡

Although the biochemical noise can easily be quantified

by the capturing probability and vector

, presenting a gen-

eral stochastic model for the transducer noise

appears

to be more difficult. The main reason is that the additive

noise of the transduction process is platform dependent and

may originate from fundamentally different physical pro-

cesses for different transducers. Nevertheless, we can still

evaluate the characteristics of the system for a number of

specific cases.

The simplest case is when

is an additive signal

independent of the binding process, with expected value of

zero and variance of

. In this case the observed output

noise of the biosensor becomes

with the fol-

lowing expected value

and variance

=0,

Now that we have the fluctuation characteristics of the bio-

chemical noise at equilibrium, we can revisit the biosensor

master equation. Based on

, we have

Therefore, the master equation at equilibrium, constraining

our considerations to only biochemical noise and additive

transducer noise, becomes

= lim

→

−1

C. Signal-to-noise ratio and noise figure

definitions

We define the signal-to-noise ratio

SNR

of a biosensor

platform as the ratio of the square of the detected target

analyte

i.e., signal power

over its noise variance. Using

, we can write the general expression for the SNR as

SNR =

−

By employing

and

, we can rewrite

SNR =

−2

SNR =

1−

The first term in the denominator of

and

describes

the biochemical shot noise

Poisson noise

of the biosensor,

while the second term is the added noise of the transducer

. It is again important to emphasize that the biochemical shot

noise is unavoidable and essentially inherent to the system.

Hence, the SNR of the biosensor excluding any transducer

noise

i.e., ideal detector

has an upper bound defined by

this shot noise which is generally referred to as the QL SNR.

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We can further generalize

for the case where the

biosensor comprises of other amplification stages as in

In this case, the SNR becomes

SNR =

−1

where

represents the variance of the noise added by the

th amplification stage.

In addition to SNR, we can define the noise figure

of the biosensor as another figure of merit for the sensing

system. This figure of merit essentially describes the ratio of

the added noise to the input noise which is the unavoidable

shot noise. We define NF as

total output noise power

output noise power due to input source

and by applying

–

and for simplicity referring every-

thing to the transducer output we have

NF=1+

−1

=1+

−1

1−

Equations

and

both indicate that by increasing

the concentration

or quantity

of the analytes, the system

generates more noise although SNR increases and NF de-

creases. This phenomenon, which one expects from the in-

herent shot noise of binding events, sets no upper bound for

the concentration levels that the biosensor is able to detect.

However, from

we know that this is not practically jus-

tifiable since the probe saturation will eventually occur as we

keep increasing the amount of analytes. In the next subsec-

tion, we will examine saturation to assess the upper bound of

the detection and evaluate the dynamic range of the biosen-

sor.

D. Saturation and dynamic range

To find the distribution of analyte particles in the case

where the number of capturing probes is finite, we need to

incorporate

into

and compute the equilibrium con-

centration from

. This basically results in finding the

equilibrium state of a nonhomogeneous Markov process,

which is numerically possible but difficult. One alternative

approach is to use the simplifying assumption that the ana-

lyte molecules in the solution are not depleted by the binding

events

i.e., only a very small fraction of analytes is captured

by the probes

. Although this limits the generality of the

solution, it is still applicable to most biosensor platforms. In

this case, for each capturing site in state 1 in our model

described in Sec. II A, we write the following balance equa-

tion at the equilibrium:

−

where

are the transition probabilities from

and

is the number of the captured particles when the system

is in saturation. Solving

to find

, we have

12,

Note that the parameter

is basically equivalent to

, which is the analyte capturing probability with-

out saturation. Hence, we use

to find

, the capturing

probability with saturation, as a function of

and

The value of

is basically an entry of

, the

equilibrium distribution vector of the system in the presence

of saturation. The limits of

and

when

→

0 are

and

, respectively, which are the nonsaturated equilibrium dis-

tribution and capturing probability. The other limit of

zero, obtained by letting

→

. It is important to realize that

the value of

in the general case is different for different

states of the system and can be represented by vector

which has finite entries corresponding to the states

with capturing probes, and infinite otherwise.

The most important impact of saturation on our formu-

lations is the breakdown of the linearity assumption between

the input

analyte concentration

and the biosensor output

signal which enabled the calculation of the SNR and NF in

the previous section. Equation

basically describes a non-

linear and monotonic relationship between the biosensor in-

put and output. Because of the monotonicity, we can argue

that the input can always be evaluated by observing the out-

put of the system, but the concern for the validity of SNR

expression is caused by observing the effect of saturation

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nonlinearity

on the detection uncertainty. In order to evalu-

ate this effect, we use the same approach that we had in

evaluating SNR of a linear system, which is referring the

output signal plus its noise to the input. The difference here

is their nonlinear relationship, which alters the function

First we derive the output fluctuation of the biosensor

system in the presence of saturation. Equation

indicates

that the fluctuation of the number of captured analytes is a

function of the capturing probability and

. This is indepen-

dent of the relationship between

and

. Thus, we can

derive the expected value and variance of the fluctuation of

the captured analytes in saturation as

=0,

1−

and we can use

for the biosensor output fluctuation in

In saturation, the parameter

which we defined as

is derived by calculating the derivative of

with

respect to

for all states where the transduction occurs. Us-

ing the derivative of

we have

and therefore, by considering the transduction process,

which essentially denotes the value of

in saturation.

Now that we calculated both

and

we can utilize them in

and

to find the SNR and NF

of the biosensor in saturation. Based on

, the SNR of a

biosensor system without saturation should increase when

increases. However, in saturation, according to the results of

and

, we observe that

becomes proportional to

when

is large, while

becomes a constant. Hence,

the SNR of the system becomes proportional to 1/

becomes large, which suggest an increase in the uncertainty

noise

of the input referred signal when the biosensor satu-

rates. This is an extremely important outcome of our formu-

lations, since it establishes the fact that the biosensor SNR is

not a monotonic function but fundamentally a concave func-

tion where SNR

→

0 in both limits of the input

i.e.,

→

and

→

To evaluate the range of detection of biosensors, similar

to all sensing systems, we are required to find the MDL as

well as the HDL. On the lower side of sensing, we define the

MDL or

MDL

as the analyte level where the biosensor SNR

is equal to some minimum acceptable value SNR

min

. How-

ever, based on the results of

–

, for the same SNR

min

we will have another analyte level which is larger than

MDL

This value is basically HDL and the analyte concentration

level associated with it is

HDL

. Consequently, the dynamic

range of the biosensor is defined as

DR =

HDL

MDL

The closed form for DR is generally complicated and is de-

rived by solving

for SNR

min

. As we will show in Sec.

IV, the value of DR for biosensors can easily be evaluated

numerically for any given value of SNR

min

E. Effects of interference, non-specific binding, and

background

The expressions for noise, SNR, and DR in the previous

subsections all assumed that there is only one analyte species

the target

that can bind to the capturing probes. However,

in all practical biosensors, other species coexist with the tar-

get in the sample, and they may or may not bind to the

probes. We call such species interferers, and their capturing

incidents nonspecific bindings. Accordingly, the signal gen-

erated by the transducer which originates from nonspecific

bindings is called nonspecific signal, or in certain biological

detection platforms it is referred to as the background signal.

A nonspecific binding event

e.g., cross-hybridization in

DNA microarrays

is generally less probable than the spe-

cific binding when target analyte and the interfering species

have the same abundance in the sample. Nevertheless, when

the ratio of the target analyte to the nonspecific analyte is

small, contributions of nonspecific binding may dominate the

measured signal. To incorporate the effects of nonspecific

binding in the previously discussed formulations, we first

need to assess the uncertainty associated with the concentra-

tion of interferers. This basically defines whether we should

consider nonspecific binding incidents coming from a deter-

ministic or a random concentration.

To examine the general case of detection in the presence

of interference, we assume that

interferers coexist with the

target in the sample, and denote their amounts by

,..., and

, where for the

th specie the expected

number of analytes is

, with variance of

. Obviously,

when the interferer concentration is known

deterministic in-

terference

, we will have

and

=0. Now in the

general case we can rewrite

, the equilibrium master

equation with interference, as

−1

where

and

are the equilibrium distribution and

biochemical noise of the

th interferer specie, respectively. It

014909-7 Hassibi, Vikalo, and Hajimiri

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2007

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is straightforward to show that

, the variance of

can be calculated as

where

and

1−

By employing

, we can repeat the same procedure as

described in Sec. III C to find the SNR and NF of the system,

given that the values of all

and

are known. In Table

we have listed the formulation for SNR and NF with inter-

ference described by

To find the DR of the system with interference, we need

to find

min

and

max

in the presence of interference. Finding

min

requires solving the modified version of

, as shown

in Table

. To derive

max

, we should do the same thing but

solve the balance equation

in the presence of interferers,

which can be very complicated. It is important to realize that

in most biosensors the concentration levels of analytes ex-

posed to capturing probes are kept low enough to ensure that

the background and nonspecific binding do not drive the sys-

tem into saturation. As a result, the upper limit of detection

max

can still be dominated by the captured target analyte

rather than the interferers. Thus, to find

max

in biosensors we

generally do not need to take into account the interferer ana-

lytes, which makes the calculation less complicated.

IV. MODEL IMPLEMENTATION

To implement our model, in this section, we look at two

different biosensor platforms. The systems under analysis

have different transducers

impedimetric and fluorescence

based

; however, we will demonstrate that for both we can

use the described modeling method to evaluate and compare

their limits of detection.

A. Impedimetric biosensor

In impedimetric biosensors the detection relies on the

impedance of an electrode-electrolyte interface which can

monotonically increase

or decrease

by analyte bindings. In

some arrangement, the intrinsic characteristics of the ana-

lytes

e.g., charge

are used to the capacitance

and accord-

ingly the impedance

of the electrode-electrolyte

interface,

–

while in other arrangements, electroactive

labels are used

–

to enhance the impedance changes.

In Fig.

, we have illustrated an example of a microfab-

ricated impedimetric biosensor where an interdigitated

electrode-electrolyte structure is used as the transducer. The

capturing probes are immobilized between the fingers of the

interdigitated structure and binding of analytes changes the

conductivity of the solution near the surface by changing the

mobility of the free ions. This subsequently modifies the im-

pedance of the electrode-electrolyte transducer system

i.e.,

ports 1 and 2 in Fig.

, making the impedance an indication

of the target analyte concentration.

In Fig.

, we have shown the detection circuitry of this

impedimetric biosensor which consist of an excitation volt-

FIG. 5. Impedimetric biosensor where binding of analytes alters the imped-

ance

or admittance

between the fingers of an interdigitated electrode

structure. The change in overall impedance corresponds to the amount of

captured analytes.

TABLE I. Signal-to-noise ratio

SNR

and noise figure

of biosensor systems when

only the target

analyte is in the system,

the target analyte and the

interfering species coexist in the system, and

when

the system in addition has

noisy amplification stages.

Signal-to-noise ratio

SNR

Noise figure

−1

FIG. 6. Detection circuitry for the impedimetric biosensor.

014909-8 Hassibi, Vikalo, and Hajimiri

J. Appl. Phys.

102

, 014909

2007

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age source at 1 MHz, transimpedance amplifier

TIA

, band-

pass filter, and an envelope detector. Any change in imped-

ance of the transducer changes the current going through

structure and subsequently the output of the amplifier which

is finally detected by the ideal envelope detector. In Table

we list the characteristics of this biosensor which is required

to calculate the limits of performance of this system. The

capturing probabilities are a function of target-probe interac-

tion and also the capturing area. These values might seem

quite small; however, they are a good approximate for typi-

cal biosensor systems where the surface to volume ratio is

small. In Table

, we have also listed the transduction ma-

trix, probe density, and the transducer

TIA

additive noise. It

is important to realize that the entries of the transduction

matrix basically correspond to the different capturing areas

of the transducer structure which have different numbers of

capturing probes and capturing probability.

In Fig.

, based on our modeling method we have illus-

trated the performance of this impedimetric biosensor. In

Fig.

, we show the output of the biosensor as a function

target analyte concentration. As expected the output signal of

this system initially increases linearly with the analyte con-

centration level but it plateaus eventually due to the satura-

tion. The probabilistic model not only predicts this behavior

but also quantifies the background signal of the system

which originates from the nonspecific bindings. In Fig.

we show the SNR of this biosensor based on

–

Without saturation we expect the SNR to increase monotoni-

cally, our formulations quantify how SNR decreases as the

analyte concentration level increases. One important obser-

vation here is the fact that SNR is more sensitive to interfer-

ence at low concentration levels, an important empirical ob-

servation which has been previously reported in different

biosensor systems.

The HDL of this system is limited by

saturation and finite number of capturing probe density.

However, our results show that the MDL is interference de-

pendent. The overall DR of this system for SNR

min

=30 dB

varies between 84 and 104 dB depending on the interference

concentration. The noise of the transducer and the detection

circuitry has little effect on the overall performance of the

system which indicates that the biochemical noise and probe

saturation are performance-limiting factors in this impedi-

metric biosensor.

B. Fluorescent-based biosensor

In this example we implement our model for a

fluorescent-based biosensor system. We have chosen this par-

ticular biosensor system for two main reasons. One is the

generality of this system which is currently a widely used

detection method in different assays. The second reason is to

illustrate the versatility of our modeling method which can

unify the performance metrics of different biosensors

The example biosensor platform is illustrated in Fig.

In this system, the capturing probes are immobilized at the

bottom of the microtiter plate and the analytes are all fluo-

rescent labeled. After an incubation phase, the solution con-

taining analytes is removed from the well and the captured

analytes

specific and nonspecific

are quantified using a

fluorescence intensity detector. The detector in this example

consists of an excitation light source which uniformly excites

the captured fluorescent labels, an emission filter which

blocks the excitation source while passing the emitted light

of the labels, and a photodetector. The photodetector consists

of a photodiode and a charge-integrating TIA

CTIA

.In

Table

III

we have listed the specification of this system in-

cluding the transduction gain and signal-dependent noise

FIG. 7. Performance result of the impedimetric biosensor. In

the biosen-

sor output signal is plotted vs the target analyte concentration. In

the

SNR of the system is plotted and the DR is calculated for the SNR

min

30 dB.

TABLE II. Characteristics of the impedimetric biosensor.

Capturing probability

0.3 0.45 0.6 0.6 0.45 0.3

−5

Probe density

257752

Reaction chamber volume

=10

Admittance transfer matrix

2 1.5 1.2 1.2 1.5 2

−7

−1

target

Transduction matrix

2 1.5 1.2 1.2 1.5 2

V/target

Transducer noise

180 nV

Interferer capturing probability

0.3 0.45 0.6 0.6 0.45 0.3

−8

Interferer variation

014909-9 Hassibi, Vikalo, and Hajimiri

J. Appl. Phys.

102

, 014909

2007

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variance of the transducer. It is important to realize that in

this system we have intentionally selected the assay specifi-

cations similar to the impedimetric biosensor example, al-

though the transduction mechanism and detector are totally

different.

In Fig.

, we have the performance results of this bio-

sensor. The output of the system as a function of analyte

concentration

see Fig.

is similar to the impedimetric

biosensor; however, the voltage level is lower because of the

lower transduction gain. Nevertheless, the different output

voltage level does not specify the performance of this sys-

tem. In Fig.

, we plot the SNR of this system as a func-

tion of analyte concentration. The graph shows that the HDL

of the system, similar to the previous example, is limited by

the saturation. The MDL, on the other hand, is almost con-

stant and it seems to be independent of interference. By look-

ing carefully at Fig.

, we can see that the SNR at low

analyte concentration deviates very much from the QL-SNR,

which is an indication that the additive noise of the trans-

ducer is dominating the SNR at low concentration levels. To

explore this hypothesis, in Fig.

, we have plotted the

simulation results of the same assay but with a high-

performance photodetector transducer system which has an

integration time of 100 s

ten times more transduction gain

and 100 times less dark current. As illustrated, on the HDL

side of SNR for the modified system is not improved, yet the

MDL is lowered drastically, making the low concentration

SNR interference limited and closer to QL-SNR. The overall

improvement in DR with the modified transducer is as high

as 40 dB.

By examining Figs.

and

we can compare the perfor-

mance of both. The fluorescent-based biosensor with the ini-

FIG. 8. Fluorescence-based biosensor where binding of labeled analytes is

detected after the incubation phase. The emitted light from the fluorescent

labels is detected by the photodiode connected to a charge integrating tran-

simpedance amplifier

CTIA

TABLE III. Characteristics of the fluorescence-based biosensor.

Capturing probability

=2.7

−5

Probe density

=3.2

Reaction chamber volume

=10

Photon flux transfer function

flux

=100 photons/target s

Transduction matrix

=0.02

V/target

Transducer noise

=10

V+2

−8

targets

Interferer capturing probability

=2.7

−8

Interferer variation

FIG. 9. Simulation result of fluorescence-based biosensor. In

the biosen-

sor output signal is plotted vs the target analyte concentration. In

the

SNR of the system is plotted and the DR is shown for the SNR

min

of 30 dB.

the SNR of the system is shown where the integration time

i.e.,

transduction gain

is increased by one order of magnitude while the photo-

detector dark-current noise power is decreased ten times.

014909-10 Hassibi, Vikalo, and Hajimiri

J. Appl. Phys.

102

, 014909

2007

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