Pharmacokinetic Modeling and Biodistribution Estimation Through the Molecular Communication Paradigm Youssef Chahibi

Pharmacokinetic Modeling and Biodistribution Estimation Through the Molecular Communication Paradigm Youssef Chahibi
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 62, NO. 10, OCTOBER 2015
Pharmacokinetic Modeling and Biodistribution
Estimation Through the Molecular
Communication Paradigm
Youssef Chahibi∗ , Student Member, IEEE, Massimiliano Pierobon, Member, IEEE, and Ian F. Akyildiz, Fellow, IEEE
Abstract—Targeted drug delivery systems (TDDSs) are
therapeutic methods based on the injection and delivery of
drug-loaded particles. The engineering of TDDSs must take into
account both the therapeutic effects of the drug at the target
delivery location and the toxicity of the drug while it accumulates
in other regions of the body. These characteristics are directly
related to how the drug-loaded particles distribute within the body,
i.e., biodistribution, as a consequence of the processes involved
in the particle propagation, i.e., pharmacokinetics. In this paper,
the pharmacokinetics of TDDSs is analytically modeled through
the abstraction of molecular communication, a novel paradigm
in communication theory. Not only is the particle advection and
diffusion, considered in our previous study, included in this model,
but also are other physicochemical processes in the particle propagation, such as absorption, reaction, and adhesion. In addition, the
proposed model includes the impact of cardiovascular diseases,
such as arteriosclerosis and tumor-induced blood vessel leakage.
Based on this model, the biodistribution at the delivery location
is estimated through communication engineering metrics, such as
channel delay and path loss, together with the drug accumulation
in the rest of the body. The proposed pharmacokinetic model
is validated against multiphysics finite-element simulations, and
numerical results are provided for the biodistribution estimation
in different scenarios. Finally, based on the proposed model,
a procedure to optimize the drug injection rate is proposed to
achieve a desired drug delivery rate. The outcome of this study is
a multiscale physics-based analytical pharmacokinetic model.
Index Terms—Biodistribution, inverse problem, molecular
communication, nanonetworks, pharmacokinetics, targeted drug
delivery systems.
I. INTRODUCTION
ARGETED drug delivery systems (TDDSs) [1] are
cutting-edge therapeutic methods, which aim at delivering
the drug exactly where it is needed while minimizing the adverse effects of the drug on the other healthy parts of the body, by
using micro or nanosized drug-loaded particles. The estimation
T
Manuscript received January 17, 2014; revised April 23, 2015; accepted April
25, 2015. Date of publication May 6, 2015; date of current version September
16, 2015. This work was supported in part by the Samsung Advanced Institute of
Technology, Global Research Outreach Program, under project title: Molecular
Communication Fundamentals in Action Potential-Triggered Targeted Drug
Delivery Systems, and the US National Science Foundation under Grant MCB1449014. Asterisk indicates corresponding author
∗ Y. Chahibi is with the Broadband Wireless Networking Laboratory, School
of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: youssef@ece.gatech.edu).
M. Pierobon is with the University of Nebraska-Lincoln.
I. F. Akyildiz is with the Georgia Institute of Technology.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TBME.2015.2430011
of how the drug-loaded particles distribute within the body,
named biodistribution, is essential for TDDS engineering, and it
is directly related to the processes involved in the particle propagation, such as their advection and diffusion in the blood stream,
their absorption from surrounding tissues, and their chemical
and physical interactions with other biomolecules present in the
body. Although drug biodistribution can be estimated empirically through clinical experiments, these are rarely performed
because of the ethical and financial constraints they pose [2] and
their specificity to each individual subject.
Recent advances in biomaterials allow the engineering of drug
particles with very specific chemical and geometric properties in
order to provide a targeted drug delivery. To benefit from these
technological advances and study the properties of drug particles to guarantee an optimal biodistribution, the aforementioned
particle propagation processes have to be modeled through the
study of the so-called drug pharmacokinetics. The most successful existing TDDS pharmacokinetic models are based on
the multicompartmental approach [3], where large portions of
the human body are considered as single compartments, with
homogeneous chemical and physical properties. The pharmacokinetics in one compartment is commonly described through
first-order differential equations, and the evolution of the pharmacokinetics is obtained for a time scale in the order of hours.
These models include: 1) Target-mediated drug disposition [4],
where the equations are based on a very limited number of
parameters that are empirically derived; 2) PK/PD (pharmacokinetics and pharmacodynamics) [5], where the equation parameters are statistically derived from experimental work, and the
pharmacokinetics is modeled only locally within a spatial scale
of a cell; 3) PBPK (physiologically-based pharmacokinetics)
[6], where pharmacokinetics is modeled globally for the whole
body but by considering each organ as a single compartment
where the drug is homogeneously distributed.
Especially, nanomedicine-enabled TDDSs require new pharmacokinetic models where the particle propagation processes
within the body are described in greater precision at a much
smaller time and space resolution, and in a tractable manner,
whereas the aforementioned models account for particle propagation only at the spatial resolution of organs and the time scale
of days. Moreover, the existing models are not sufficiently scalable and are not customizable to the patients and their specific
diseases [3].
To tackle the aforementioned problems, we propose a TDDS
pharmacokinetic model based on the abstraction of molecular communication (MC), a recently developed paradigm in
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CHAHIBI et al.: PHARMACOKINETIC MODELING AND BIODISTRIBUTION ESTIMATION THROUGH THE MOLECULAR COMMUNICATION
communication theory that defines information exchange
through the emission, propagation, and reception of molecules.
In [7], we developed an MC model to calculate the time-varying
blood velocity in any location of the cardiovascular system, and
to predict the propagation of the drug-loaded particles due to
advection and diffusion in the blood flow. In this paper, by stemming from our previous work, we develop a TDDS pharmacokinetic model able to predict the propagation of the particles by
taking into account other specific physicochemical processes, as
well as abnormal health conditions. Through the MC paradigm,
we consider the following physicochemical processes:
1) The advection process, which represents the transport of
particles due to the blood velocity.
2) The diffusion process, which corresponds to the Brownian
motion of particles.
3) The absorption process, which quantifies the particles absorption through tissues surrounding the blood vessels
[8].
4) The reaction process, which is a consequence of the degradation of particles in the blood [9].
5) The adhesion process, which accounts for other
biomolecules binding to the drug-loaded particles. The
adhesion process is one of the main adverse effects to the
performance of the TDDSs [10].
In the proposed pharmacokinetic model, we also account for
the effects on the drug pharmacokinetics of cardiovascular diseases, which include blood vessel leakage, e.g., due to tumors,
and rigidity, e.g., due to arteriosclerosis. These effects are analytically considered in the proposed pharmacokinetic model,
and are shown to greatly affect the drug particle distribution
through numerical evaluations of the pharmacokinetic model
and the biodistribution estimation.
Compared to the aforementioned existing models, the proposed pharmacokinetic model can be used at different spatial
scales (organs, small tissues, and cells) and at a more precise
time scale, where the drug pharmacokinetics is predicted within
fractions of seconds. Moreover, our model does not require
empirically-obtained parameters and it has a lower computational complexity.
By stemming from the proposed MC-based pharmacokinetic
model, we propose a method to estimate the drug biodistribution. We propose to characterize the presence of the drug at
the delivery location through communication engineering metrics, namely, channel delay and path loss, analytically derived
from the proposed pharmacokinetic model. The channel delay
corresponds to the time needed by the drug particles to reach
their peak concentration at the delivery location after they are
injected, while the channel path loss is the ratio of the drug
particles that effectively reach the delivery location over the
drug particles that were initially injected. In addition, we also
demonstrate that the proposed pharmacokinetic model allows
to analytically estimate the drug accumulation in the rest of the
body.
The proposed MC-based pharmacokinetic model is validated through finite-element simulations on COMSOL, which
consider 3-D Navier–Stokes and advection-diffusion-reaction
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equations to simulate the drug propagation in a time-varying
blood flow through a 3-D model of a blood arterial network.
The proposed MC-based pharmacokinetic model proves to be in
good agreement with the results of the simulation, therefore reproducing similar results with analytical expressions, which do
not require the computational complexity of the finite-element
simulations. Additionally, numerical results are provided for the
biodistribution estimation in different health scenarios, namely,
in the presence of arteriosclerosis and tumor-induced blood
blood vessel leakage. Through these results, we show that the
transport and kinetic properties are important factors influencing
the pharmacokinetics of the drug-loaded particles.
Finally, by stemming from the proposed model, we detail a
procedure to analytically express the optimal drug injection rate
given a target drug delivery rate. For this, we suppose that the
healing of the disease requires an objective drug delivery rate,
and that the drug injection and delivery locations are known. The
proposed pharmacokinetic model is then applied to analytically
obtain the optimal drug injection rate.
The rest of the paper is organized as follows. In Section II,
we mathematically describe the pharmacokinetic model based
on the MC abstraction of the physicochemical processes in the
drug-loaded particle propagation, namely, advection, diffusion,
reaction, absorption, and adhesion. Moreover, we incorporate
in the pharmacokinetic model possible cardiovascular diseases
affecting the blood flow. In Section III, we obtain the biodistribution estimation of the particles through the communication
engineering metrics of channel path loss and delay, and the expressions to compute the drug accumulation in the rest of the
body. Numerical results are provided for the biodistribution in
different scenarios. In Section IV, the validation of the MCbased pharmacokinetic model with multiphysics finite element
simulation is presented. In Section V, we apply the MC-based
pharmacokinetic model to find the optimal drug injection rate
that would achieve an objective drug delivery rate at the delivery
location. Finally, Section VI concludes the paper with comments
about the validity of the model and the various factors influencing the performance of TDDSs.
II. MC-BASED PHARMACOKINETIC MODEL
In this section, we mathematically describe the pharmacokinetic model of a TDDS based on the analytical MC channel
abstraction, which considers additional physicochemical processes in the particle propagation from the injection location to
the delivery location, in addition to the advection and diffusion
processes already considered in [7].
The network of blood vessels is abstracted here as an MC
network. Fig. 1 illustrates the physicochemical processes in a
blood network consisting of several blood vessels. un (t) denotes the blood velocity in a blood vessel n, and t is the time
variable. The drug propagates in this blood network subject
to an absorption with rate ρn , reaction with rate μn , adhesion
with an adsorption rate k + and a desorption rate k − , diffusion with a diffusion coefficient D, and advection driven by the
blood velocity. The drug propagation is abstracted as an MC
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reasons, the expression in (1) is different from a convolution.
We derive the analytical expression of the time-varying impulse
response of the MC link n, as follows:
2
n (t,τ ))
exp − (l n −m
−
μ
(t
−
τ
)
2
n
2σ n (t,τ )
(ρ ,μ )
h(nn) n (t, τ ) =
2πσn2 (t, τ )
(2)
where:
1) mn (t, τ ) is a function of apparent velocity vn (t) as
follows:
t
mn (t, τ ) =
vn (t )dt
(3)
τ
where t is the time integration variable.
2) σn2 (t, τ ) is a function of the effective diffusivity Dn (t) as
follows:
Fig. 1.
Scheme of the MC modeling of TDDSs pharmacokinetics.
channel, and completely characterizes the relationship between
the drug injection rate and the drug delivery rate. The drug injection rate is the MC signal transmitted at the inlet of the blood
vessel and the drug delivery rate is the MC signal received at the
outlet of the blood vessel. This is achieved by a time-varying
(ρ ,μ )
impulse response h(nn) n (t, τ ), where τ is a time variable, for
every blood vessel n (n = 1, 2, . . . , 7). The MC link channels
are cascaded to obtain an MC path, which provides the relationship between the drug injection rate x(t) and the drug delivery
rate y(t), through the time-varying impulse response for the
(ρ 1 ,μ 1 ,ρ 2 ,μ 2 ,ρ 4 ,μ 4 )
(t, τ ) for the
path channel, denoted, e.g., by h(1,2,4)
cascade of MC links 1, 2, and 4 as shown in Fig. 1.
In Section II-A, we present how a blood vessel is abstracted
as an MC link. In Section II-B, we describe how the physicochemical processes between the drug particles and the body can
be modeled by combining MC links. Finally, in Section II-C, the
modeling of cardiovascular diseases using equivalent circuits is
proposed.
A. MC Link Model
We found in [7] that the drug injection rate x(t) and the drug
delivery rate y(t) in the blood vessel n are related mathematically by the following expression:
+∞
(ρ ,μ )
x(τ )h(nn) n (t, τ )dτ .
(1)
y(t) =
−∞
Due to the fluctuations in the blood flow, the impulse response
of the system depends on the state of the blood flow at the time
of the injection, therefore the system is not linear time-invariant
(LTI). The response of non-LTI systems cannot be expressed
in the form of a convolution operation. For the aforementioned
t
σn2 (t, τ ) = 2
Dn (t )dt .
(4)
τ
3) μn is a characteristic of the reaction process, and represents the rate of reaction between the particles and the
blood.
In the following, we provide the expression of the apparent
velocity vn (t) and the effective diffusivity Dn (t) for advectiondiffusion (see Section II-A1), absorption (see Section II-A2),
and adhesion (see Section II-A3).
1) Advection-Diffusion Case (No Reaction): When the reaction process is absent, and only the advection-diffusion is occurring, the apparent velocity in the case of no reaction vnnone (t)
and the effective diffusivity in the case of no reaction Dnnone (t)
are
⎧ none
⎪
⎨ vn (t) = un (t)
2
2
⎪
⎩ Dnone (t) = D + un (t)rn
n
192D
(5)
which is a result we derived in [7].
2) Absorption Case: When there is absorption due to the
tissues that surround the blood network, the apparent velocity in
the case of absorption vnabsorption (t) and the effective diffusivity
in the case of absorption Dnabsorption (t) are [11]
⎧
2
absorption
⎪
⎪
v
ρ
(t)
=
1
+
n un (t)
⎪
⎨ n
15
⎪
⎪
u 2 (t)rn2
4
⎪
⎩ Dnabsorption (t) = D + n
1 − ρn .
192D
15
(6)
3) Adhesion Case: When adhesion to the proteins in the
blood plasma or to the blood vessel walls is occurring, the
apparent velocity in the case of adhesion vnadhesion (t) and the
CHAHIBI et al.: PHARMACOKINETIC MODELING AND BIODISTRIBUTION ESTIMATION THROUGH THE MOLECULAR COMMUNICATION
effective diffusivity in the case of adhesion Dnadhesion (t) are [12]
⎧
1
⎪
vnadhesion (t) =
⎪
+ un (t)
⎪
⎪
1 + kk −
⎪
⎪
⎪
⎪
⎪
2
⎪
+
⎪
2 k+
⎪
+ 12rn2 kk − + rn3
44r
2 2
⎨
n k−
rn un (t)
adhesion
(t) =
Dn
3
+
48D
⎪
⎪
rn + 2 kk −
⎪
⎪
⎪
+
⎪
⎪
2u2n (t)rn2 kk −
⎪
⎪
⎪
+ 3 .
⎪
⎪
+
⎩
k − rn + 2 kk −
(7)
Section IV-B provides numerical values for the crosssectional average blood velocities of three blood vessels, obtained using the transmission line method described in [7].
B. MC Path Model
The MC channel model of a path (n; n = 1 . . . N ) where n is
the index of a link n, is obtained by using the harmonic transfer
matrix function HTM{·} and its inverse HTM−1 {·} [7]
+∞
(ρ ,μ n ;n =1...N )
y(t) =
x(τ )h(nn;n =1...N
(t, τ )dτ
(8)
)
−∞
where the time-varying impulse response of the path
(ρ ,μ n ;n =1...N )
is expressed as follows:
h(nn;n =1...N
)
(ρ ,μ ;n =1...N )
n
h(nn;n =1...N
=
)
n =1
(ρ n ,μ n )
−1
HTM h(n )
(t, τ )
HTM
.
(9)
n =N
Through the HTM method [13], we can find analytical solutions
of the end-to-end impulse response of TDDSs, as opposed to
numerical solutions by finite-element models.
C. Disease Models With Equivalent Circuits
In this section, we present an equivalent circuit modeling of
cardiovascular diseases, including arteriosclerosis (rigid blood
vessel model), and blood vessel leakage (leaky blood vessel
model).
A blood vessel is considered as a cylindrical elastic tube with
radius rn and length ln , and modeled as an electrical circuit,
whose electrical components are related to the geometry of
the blood vessels. A healthy blood vessel n possesses three
electrical components. First, a resistance Rn , which is related
to the blood viscosity and the diameter of the blood vessel.
Second, an inductance Ln , which is related to the blood inertia,
that is how a difference in blood pressure causes a difference in
blood flow. Third, a capacitance Cn , which measures the blood
vessel elasticity. We give below the expression of the electrical
components based on their physiology [7].
The resistance of the blood vessel n is expressed as follows:
Rn =
where ν is the blood viscosity.
8ν
πln rn4
(10)
Fig. 2.
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Equivalent electrical circuits for a blood vessel in different conditions.
The inductance of the blood vessel n is expressed as follows:
η
(11)
Ln =
πln rn2
where η is the blood density.
1) Rigid Vessel Model: The elasticity of a blood vessel is
an important parameter in the success of drug delivery. There
have been studies to show how abnormal elasticity affects drug
propagation [14]. Blood blood vessels can become rigid because
of aging and diseases such as arteriosclerosis [15].
For a rigid blood vessel, we model the change in elasticity
using an arterial elasticity factor, which measures the ratio
between normal elasticity and rigid elasticity. We retain the
same electrical components as in the healthy blood vessel model,
except for the capacitance, which is now equal to
Cn =
πrn 2
FC (a1 exp(−a2 rn ) + a3 )
(12)
where FC is the arterial elasticity factor (FC = 1 for a
healthy blood vessel, FC = 0 for a completely rigid blood vessel). a1 = 1.34 × 107 g/(s2 · cm), a2 = 22.53 cm−1 , and a3 =
5.77 × 105 g/(s2 · cm) are statistical parameters obtained from
physiological measurements[16].
2) Leaky Vessel Model: The leakage of a blood vessel is
modeled by an equivalent conductance, which is related to how
easy it is for a fluid to leak from the blood vessel. We retain the
same electrical components as for the healthy blood vessel case,
but we add an additional conductance Gn to model the blood
vessel leakiness
FL
(13)
Gn =
Rn
where FL is leakiness factor, which compares the leakage to the
conductance of the healthy blood vessel (inverse of the resistance), and Rn is the resistance of the blood vessel.
Fig. 2 shows the equivalent electrical circuit components for
a blood vessel in different conditions such as a healthy condition, arteriosclerosis, and blood vessel leakage. By defining
electrical equivalents of diseased blood vessels, the blood velocities are calculated by using the transmission line theory method
presented in [7], after substituting the expressions of the conductances and the capacitances for healthy blood vessels with
the expressions in (12) and (13), respectively. For the numerical results, the inner iliac blood vessel [17] was chosen, and
the properties of three of its children blood vessels, denoted as
(3, 6, 7) in Figs. 3–5, respectively, have been modified according
to the considered disease condition.
In Fig. 3, we observe that in a healthy arterial tree, the blood
velocity tends to dampen slowly as we go farther from the root
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 62, NO. 10, OCTOBER 2015
Blood velocities at a tree of small blood vessels in a healthy condition.
Fig. 5. Blood velocities at a tree of small blood vessels with a branch suffering
from arteriosclerosis.
tissues, or reacting with elements of the blood plasma. In this
section, we estimate the biodistribution of TDDSs using the MC
paradigm through the definition of two MC metrics, namely, the
channel delay and the channel path loss. In Section III-A, the
channel delay is the time needed by the drug particles to reach
their peak concentration at the delivery location after they are
injected in the body. In Section III-B, the channel path loss is
the proportion of the injected particles that reach the delivery
location despite the blood vessels branching, reaction, adhesion,
and absorption. Finally, in Section III-C, the drug accumulation
in the rest of the body is expressed analytically using the MC
model.
A. Channel Delay to the Delivery Location
Fig. 4. Blood velocities at a tree of small blood vessels with a branch suffering
from blood vessel leakage.
of the blood vessel. In the case of a blood vessel leakage, as
illustrated in Fig. 4, this trend is not observed, where we can
see that the blood velocity may increase in some daughter blood
vessels, since the resistance is reduced. Fig. 5 shows the extreme
case where a portion of the arterial tree is affected by a severe
arteriosclerosis. In that case, the diseased blood vessels exhibit
a highly oscillatory blood flow.
The method introduced in this section can be applied to model
the drug propagation in any location of the arterial network.
We define the delay for a TDDS as the time required by injected molecules to reach their peak concentration at the delivery
location, which is a definition typically used in biodistribution.
Another definition of delay used in biodistribution studies is
the half-life of a drug [18], which is only meaningful for drugs
undergoing an exponential decay. The definition we choose is
more general than half-life, and can provide more information
about the toxicity, potency, and elimination rate of the drug,
since these properties depend on the overall time spent by the
majority of the molecules between the injection location and the
delivery location.
We express the channel delay tdelay for the path (n; n =
1 . . . N ) as
tdelay =
III. BIODISTRIBUTION ESTIMATION
The biodistribution is the study of the location and the quantity of the drug that is accumulated in the delivery location and
the rest of the body, whether in the blood vessels, their surround
1
T
(ρ ,μ )
0
T
(ρ ,μ ;n =1...N )
n
arg max h(nn;n =1...N
)
t> τ
(t + τ, τ )dτ
(14)
n
where h(nn;n =1...N
) (t, τ ) is the time-varying impulse response
with injection starting at the time τ , and T is the heartbeat
period.
CHAHIBI et al.: PHARMACOKINETIC MODELING AND BIODISTRIBUTION ESTIMATION THROUGH THE MOLECULAR COMMUNICATION
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Fig. 6. Effect of the diffusion coefficient D and the absorption rate ρn on the
channel delay.
Fig. 7. Effect of the absorption rate (ρn ) and the reaction rate (μ n ) on the
path loss.
Since the channel is time-varying and the blood flow changes
periodically, the injected drug particles will be delivered with
a different channel delay at the delivery location depending on
the blood velocity that was experienced by the body when they
were injected. We consider the ambiguity in knowing the blood
velocity at the time of injection by averaging over the channel
delays for all possible blood velocity values that the body may
experience.
The definition of the delay as the average is only acceptable for long propagation times. However, it is acceptable to
use the delay as the average value to compare several drug delivery systems that are within the same flow, and propagation
length conditions. The standard deviation (or error) in the delay
calculation can be highly variable for the scenario where the
propagation time is low. If the blood velocity period is higher
than the time it takes for the molecules to reach the delivery
location, then the error can be as much as in the order of 100%.
However, if the blood velocity period is small compared with
the delay, then the error is negligible, which means that the
injection time is not critical.
duction in delay for increased absorption is that the absorption
reduces the number of particles in the blood that are in proximity of the walls, which are the slowest moving particles, thus
increasing the average velocity of all the particles.
B. Channel Path Loss at the Delivery Location
We define the channel path loss for the path (n; n = 1 . . . N )
as
L = 10 log10
1−
0
(ρ ,μ )
+∞
(ρ ,μ n ;n =1...N )
h(nn;n =1...N
(t, 0)dt
)
C. Drug Accumulation in the Rest of the Body
Using the time-varying impulse response, we can calculate
the proportions of the drug particles that are either still in the
blood, have been absorbed by the surrounding tissues, or have
reacted with the blood plasma.
We can express the proportion of drug particles that have been
absorbed as follows:
+∞
r2
(0,0;n =1...N )
h(n ;n =1...N ) (t, 0)dt
dabsorb ed = N2
r1
0
+∞
(0,μ ;n =1...N )
h(n ;nn=1...N ) (t, 0)dt .
(16)
−
0
Similarly, the proportion of drug particles that have reacted
can be expressed as follows:
+∞
r2
(0,0;n =1...N )
dreacted = N2
h(n ;n =1...N ) (t, 0)dt
r1
0
+∞
(0,μ ;n =1...N )
h(n ;nn=1...N ) (t, 0)dt .
(17)
−
0
(15)
n
where h(nn;n =1...N
) (t, 0) is the time-varying impulse response,
which we defined in Section II-B with injection starting at the
time τ = 0. This relationship comes from the fact that the impulse response is the probability density of a single particle
arriving at a specific location and time. The log-scale is used
because about half of the particles are lost at every blood vessel
bifurcation, which makes the particle loss follow an exponential
trend. In Fig. 6, we see the effect of the blood velocity, the drug
diffusion coefficient and the reaction rate on the channel delay. In the numerically evaluated scenario in Fig. 7, we observe
that the increase in the drug diffusion coefficient contributes
in increasing the delay of the channel, while the effect of the
absorption rate contributes in decreasing the delay.
In Fig. 7, we observe that reaction and absorption have similar consequences on the channel path loss. For the absorption,
we see that the higher the absorption rate the smaller the delay,
which may seem counterintuitive. The reason behind the re-
Finally, the proportion of drug particles that remain in the
blood is equal to the following:
+∞
r2
(0,0;n =1...N )
dblo o d = N2
h(n ;n =1...N ) (t, 0)dt
(18)
r1
0
+∞
(ρ ,μ n ;n =1...N )
h(nn;n =1...N
(t,
0)dt
.
(19)
−
)
0
Therefore, we can use the MC paradigm to predict where the
drug is going to accumulate based on the physiological parameters of the drug delivery system and the body. As presented in
Fig. 8, the blood vessel conditions cause some variance in the
biodistribution. This is moderately important in leaky blood vessels, but is very important in the case of blood vessels affected
by arteriosclerosis.
IV. MULTIPHYSICS FINITE-ELEMENT VALIDATION
In order to obtain a pharmacokinetic model of TDDSs, we
made the following assumptions: continuous concentration at
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TABLE I
BLOOD NETWORK BOUNDARY CONDITIONS NUMERICAL VALUES
k
Fig. 8.
Effect of cardiovascular diseases on drug distribution.
the bifurcation, Poiseuille flow, Taylor dispersion approximation, perfectly cylindrical geometry, and infinite-length blood
vessels. Using finite-element analysis, the developed model is
validated realistically in a 3-D geometry and assuming physical equations in their full forms. In this section, we present
the validation of the MC model of TDDSs by simulation using finite-element analysis. We describe the geometry of the
simulated system, its governing physical equations, and how
the parameters of the analytical model have been mapped to
parameters of the finite-element analysis.
Finite element analysis is a numerical method used to solve
partial differential equations [9] that underlie the behavior of
complex physical systems, including mechanical and chemical
transport systems. Finite element analysis has several advantages compared with analytical models. First, finite element
analysis allows to simulate objects of arbitrarily complex 3-D
geometry. This is especially required for biological objects such
as blood vessels which have an imperfectly cylindrical shape and
bifurcation shapes. Second, finite element analysis makes it possible to simulate the interaction of different physical phenomena,
such as the interaction of the blood vessel walls, the blood flow,
which is governed by fluid mechanics, and the chemical transport of drugs. The validation is carried out using COMSOL,1 a
finite element simulation software package.
The following aspects of a drug delivery systems are considered in the simulation as follows:
1) Blood Flow: The validation is performed using a 3-D
model of a blood arterial network under realistic conditions. The blood flow, which is the main driving force of
the drug propagation, is simulated using the 3-D Navier–
Stokes equations in the stationary domain. In contrast
with existing pharmacokinetic models which are based
on unrealistic assumption of having a constant blood flow
[19], the drug is propagated through a time-varying blood
flow. The blood flow boundary conditions in the arterial
networks are estimated based on the realistic transmission
line theory which provides results in very good agreement
with MRI measurements of blood flow in a human [7].
1 COMSOL
is a registered trademarks of COMSOL AB.
qk , 1
pk , 1
qk , 1
pk , 2
qk , 2
pk , 3
qk , 3
pk , 4
qk , 4
pk , 5
qk , 5
pk , 6
qk , 6
pk , 7
qk , 7
0
1
2
3
1.3 × 10 −4
1.3 × 10 −4
1.3 × 10 −4
1.3 × 10 −4
1.3 × 10 −4
7.3 × 10 −5
7.3 × 10 −5
1.3 × 10 −4
1.3 × 10 −4
7.2 × 10 −5
7.2 × 10 −5
7.2 × 10 −5
7.2 × 10 −5
4.0 × 10 −5
4.0 × 10 −5
2.9 × 10 −3
1.7 × 10 −4
2.9 × 10 −3
1.7 × 10 −4
2.8 × 10 −3
9.6 × 10 −5
1.6 × 10 −3
1.7 × 10 −4
2.8 × 10 −3
9.5 × 10 −5
1.6 × 10 −3
9.5 × 10 −5
1.6 × 10 −3
5.3 × 10 −5
8.9 × 10 −4
−1.8 × 10 −4
5.0 × 10 −5
−1.8 × 10 −4
4.9 × 10 −5
−1.7 × 10 −4
2.8 × 10 −5
−9.8 × 10 −5
4.9 × 10 −5
−1.7 × 10 −4
2.7 × 10 −5
−9.6 × 10 −5
2.7 × 10 −5
−9.6 × 10 −5
1.5 × 10 −5
−5.4 × 10 −5
1.6 × 10 −5
−6.3 × 10 −5
1.6 × 10 −5
−6.2 × 10 −5
1.6 × 10 −5
−3.5 × 10 −5
8.8 × 10 −6
−6.1 × 10 −5
1.5 × 10 −5
−3.5 × 10 −5
8.6 × 10 −6
−3.5 × 10 −5
8.6 × 10 −6
−1.9 × 10 −5
4.9 × 10 −6
2) Geometry: In the simulation, we assume cylindricallyshaped small blood vessels, which is in agreement with
the physiological observations [20]. Large blood vessels
and anomalously shaped blood vessels can be considered
with little modifications.
3) Drug Transport: Through the COMSOL simulation, we
observe that the MC model based on Taylor dispersion is
a good approximation of particle transport in blood and
that, therefore, higher-order approximations [21] which
will make the expression of the analytical solution more
complex are not needed.
4) Drug Kinetic Interactions: The binding is considered
by adding a linear reaction term to the 3-D advectiondiffusion equation. The absorption is simulated as a
boundary condition on the blood vessel walls where the
particles are not perfectly bouncing but proportionally lost
at the surface. The linear first order kinetics for binding
and absorption are common for particles [22]. We assume
that no other kind of binding occurs and that particles
are at a sufficiently low concentration to avoid nonlinear
binding kinetics.
A. Topology
For the numerical evaluation of the model, the topology information was derived from the MRI scan of a young male
individual, which is available from [17]. However, the available
MRI scan anatomical information only covers the large blood
vessels. An algorithm that represents the small blood vessels as
a fractal tree rooted in the extremity of the large blood vessels
was used to obtain the topology of the studied area, in a similar
way as in [7]. The numerical values and structure of the topology are listed in this paper and included in Table I to simplify
the reproduction of the results. In fact, a blood network was considered, consisting of interconnected blood vessels n, where n
is the blood vessel index (n = 1 . . . 7). The parent blood vessel
1 bifurcates into two blood vessels, the daughter blood vessel
2 and the daughter blood vessel 3, and so on. The blood vessel n has a radius rn and a length ln , for n = 1 . . . 7. We have
CHAHIBI et al.: PHARMACOKINETIC MODELING AND BIODISTRIBUTION ESTIMATION THROUGH THE MOLECULAR COMMUNICATION
2417
where ω0 = 2π/T is the radial sampling frequency, K is the
number of samples, and the coefficients {pk ,n ; k = 0 . . . K −
1} and {qk ,n ; k = 0 . . . K − 1} are the even and odd Fourier
coefficients, respectively.
C. Drug Propagation Initial Conditions
The drug propagation initial conditions describe the initial
values of the drug concentration in the blood network at time t.
We express the initial drug concentration c(x1 , y1 , z1 , t) in the
blood vessel n as a function of the Cartesian coordinates, with
the origin at the center of the inlet of the blood vessel 1, and the
x1 axis along the longitude of the blood vessel. We approximate
the drug injection impulse with a Gaussian function with a very
small variance, which we can write as follows:
−
e
c(x1 , y1 , z1 , t) = Fig. 9. Evolution of the drug propagation in a tree of blood vessels showing
the transport of the injected drug particles from the inlet of the tree of blood
vessels to the outlets of the branches, at different times t (a) t = 46 ms (b) t =
92 ms (c) t = 138 ms (d) t = 184 ms.
r1 = 0.5 lmm, r2 = 0.45 mm, r3 = 0.3 mm, r4 = 0.40 mm,
r5 = 0.23 mm, r6 = 0.27 mm, and r7 = 0.18 mm for the radii,
and l1 = 25 mm, l2 = 22.5 mm, l3 = 15 mm, l4 = 20 mm,
l5 = 11.5 mm, l6 = 13.5 mm, l7 = 9 mm. These dimensions
are chosen to be physiologically plausible [17]. According to the
physiological data about the size of blood vessels, all types of
veins and blood vessels have an interior radius of the blood vessels that is very small compared to the length. This is supported
quantitatively in the human and animal physiology literature
such as in [17]. In particular, the work in [17] mentions that the
length of blood vessels is 25 times the size of their diameters
with a standard deviation equal to 5. This study also uses straight
cylinders to model blood propagation in blood vessels, which
occurs at a faster scale than drug diffusion.
B. Blood Velocity Boundary Conditions
The multiphysics finite-element simulation requires the definition of boundary conditions, which are values defined at the
surfaces of the blood network, to find the numerical solutions
that satisfy the physical equations. We use five boundary conditions which are defined at the inlet (n = 1) and the outlets
(n = 4, 5, 6, 7) of the blood network as shown in Fig. 9. Thus,
there are five boundary conditions which are the blood velocity
u1 (t) at the inlet of the network, and the blood velocities un (t)
for the blood vessels n, for n = 4, 5, 6, 7, respectively. The numerical values for the boundary conditions have been obtained
using the transmission line model developed in [7]. Since the
boundary conditions are time-varying and periodic, we express
them in terms of their Fourier series decomposition as follows:
un (t) =
K
−1
k =0
pk ,n sin(kω0 t) + qk ,n cos(kω0 t)
(20)
x2
1
2σ 2
1
2πσ12
c0
(21)
where x1 is the Cartesian coordinate along the longitude of the
blood vessel 1, σ1 is the standard deviation of the impulse, and
c0 is the initial concentration of particles. The justification of a
drug injection as a Gaussian function rather than a Dirac delta
function is essential to obtain the resolution of partial differential
equations using using a finite-element methods solver [23].
D. Validation Results
(ρ ,μ ;n =1...N )
n
(t, τ ) are evaluated at
The impulse responses h(nn;n =1...N
)
the outlets of the blood vessels n where n = 1 . . . 3. We evaluate
(ρ ,μ )
the impulse response h(nn) n (t, τ ) as
(ρ ,μ ;n =1...N )
n
h(nn;n =1...N
)
(t, τ )
=
1
SO n
c(x, y, zn , t)dxdydz
M (x,y ,z n )∈O n
(22)
where On denotes the outlet of the blood vessel n, SO n is the surface area of On , M (x, y, zn ) is a point in On , and c(x, y, zn , t)
is the concentration at the time instant t and the point with the
coordinates (x, y, zn ).
The simulations were performed using COMSOL on a desktop machine with a total computation time of 2 h 57 min to
build the map of blood velocity and for the propagation of drug
particles, for a simulation duration Tsim = 0.25 s. Table I lists
the Fourier coefficients that have been used in the multiphysics
finite-element calculations.
In Fig. 10, we compare the impulse responses obtained by
multiphysics finite-element simulation with the analytical results obtained using the MC model described in Section II, where
we use the following values for the diffusion coefficient D =
10−8 m2 /s and the absorption rates (ρn = 1e − 5; n = 1 . . . 7).
We compare the results for all three blood vessels 1, 4, and 5,
and we notice in the three cases that there is good agreement
between the values generated through the simulation and the
model.
2418
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 62, NO. 10, OCTOBER 2015
(ρ ,μ ;n =1...N )
n
(t, τ ) which relates the drug injecresponse h(nn;n =1...N
)
tion rate x(t) to the drug delivery rate y(t), by the following
relationship:
+∞
(ρ ,μ n ;n =1...N )
y(t) =
h(nn;n =1...N
(t, τ )x(τ )dτ .
(23)
)
−∞
Here, our objective is to find the optimal drug injection rate
x∗ (t), such that the obtained drug delivery rate y ∗ (t) is as close
as possible to the drug delivery rate y(t). This is expressed by
x∗ (t) = arg min y(t) − y ∗ (t) .
(24)
x(t)
Fig. 10. Comparison between the impulse responses obtained by the MC
model and the impulse responses obtained by the multiphysics finite-element
simulation technique for different delivery locations at the outlet of the blood
vessels 1, 4, and 5, respectively.
Using the time-varying impulse response, the previous expression becomes
x∗ (t) =
arg min x(t) +∞
−∞
(ρ ,μ n ;n =1...N )
h(nn;n =1...N
(t, τ )x(τ )dτ
)
− y (t)
(25)
∗
(ρ ,μ n ;n =1...N )
h(nn;n =1...N
(t, τ )
)
where
is the time-varying impulse response that characterizes the drug propagation from the injection
location to the delivery location.
yj is defined as follows:
j Ts
(ρ ,μ n ;n =1...N )
yj =
h(nn;n =1...N
(tj , τ )x(τ )dτ .
(26)
)
0
xi is defined as
Fig. 11. Scheme of the injection rate rate optimization for a desired drug
delivery rate.
V. DRUG INJECTION OPTIMIZATION
In this section, we aim to propose a solution to the optimization of the drug injection in order to achieve a desired drug
delivery rate, based on the MC-based pharmacokinetic model
presented in Section II. In order to obtain efficient drug delivery
systems, the timing and location of the drug particles are crucial.
The diseased region needs to receive the particles at the right
time and in the right quantity. When the particles are injected by
systemic administration, the drug particles can be lost in blood
vessel bifurcations, absorbed by blood vessels, and mixed with
the blood due to diffusion. Fig. 11 shows a scheme of the injection rate optimization, where, starting from the desired delivery
rate, an optimal injection rate is found giving exactly the desired
delivery rate with minimal error.
In the following, we present a method to find the optimal
inject rate based on the desired drug delivery rate, the physiological parameters of the body, the drug properties, the injection
location and the delivery location.
We consider a disease that requires a specific drug delivery
rate that will make the healing effective, with just a minimal
number of drug particles, and below the level that causes toxicity.
We suppose that a desired drug injection rate is given by a timevarying function x(t), which describes the drug concentration
rate at every time t in the injection location.
Using the pharmacokinetic model in Section II, we obtain a channel model characterized by a time-varying impulse
xi = x(iTs )
(27)
where i, j = 1 . . . K, K is the number of samples, and Ts is the
sampling period. With this notation, we can write
iT s
j
(ρ ,μ n ;n =1...N )
xi
h(nn;n =1...N
(jTs , τ )dτ .
(28)
yj =
)
i=1
(i−1)T s
(ρ ,μ ;n =1...N )
We define the channel coefficients hi,jn n
as follows:
iT s
(ρ ,μ ;n =1...N )
(ρ ,μ n ;n =1...N )
hi,jn n
=
h(nn;n =1...N
(ti , τ )dτ . (29)
)
(i−1)T s
Therefore, we get the following expression:
yj =
j
(ρ ,μ n ;n =1...N )
xi hi,jn
.
(30)
i=1
Thus, the problem can be written in matrix notation as
(ρ ,μ ;n =1...N )
n
y = H(nn;n =1...N
)
x
(31)
where y = [yj ; j = 1 . . . K] is a K-dimensional vector whose
elements are samples of the desired delivery rate, x =
[xi ; i = 1 . . . K] is a K-dimensional vector whose elements
(ρ ,μ n ;n =1...N )
are samples of the optimal injection rate, and H(nn;n =1...N
)
is the square matrix of size K-by-K, whose components are
defined in (29), and [.] is the vector transpose operator.
(ρ ,μ n ;n =1...N )
The matrix H(nn;n =1...N
is supposed to be invertible. In
)
case the matrix is not invertible, the linear matrix inequality
approach as proposed in [24] can be directly adapted to the MC
model to find the optimal injection rate.
CHAHIBI et al.: PHARMACOKINETIC MODELING AND BIODISTRIBUTION ESTIMATION THROUGH THE MOLECULAR COMMUNICATION
We define the vector x∗ = [xi ; i = 1 . . . K] as follows:
−1
(ρ ,μ n ;n =1...N )
x∗ = H(nn;n =1...N
y
(32)
)
where
−1
(ρ ,μ n ;n =1...N )
H(nn;n =1...N
is the inverse of the matrix
)
(ρ ,μ ;n =1...N )
n
H(nn;n =1...N
.
)
The desired drug injection rate is found by
i=K
t − iTs
x∗ (t) =
x∗i · sinc
.
Ts
i=1
(33)
According to the Nyquist criterion [25], the sampling period
1
, where B is the bandwidth of the timeshould satisfy Ts < 2B
varying impulse response of the system. The sampling period
depends on the blood velocity and the characteristic time scale
of the advection-diffusion. For the simulations, a value of Ts =
15.645 ms, which is the sampling period of the measured blood
cardiac flow input was chosen. This is much shorter than the
characteristic time scale of the advection-diffusion.
VI. CONCLUSION
In this paper, we propose to apply the abstraction of the MC
paradigm to address important problems in TDDSs, namely,
modeling the drug pharmacokinetics, estimating the biodistribution, and optimizing the drug injection rate. The MC abstraction allowed to obtain an analytical pharmacokinetic model that
accounts for various physicochemical processes in the particle
propagation, and takes into account the impact of cardiovascular diseases. By stemming from the pharmacokinetic model, we
proposed to use communication engineering metrics to estimate
the drug biodistribution at the delivery location, while analytical
expressions are obtained to estimate the drug accumulation in
the rest of the body. We have favorably compared our pharmacokinetic model with multiphysics finite-element simulations of
the drug propagation in the arterial system, and provided numerical results for the drug biodistribution in different scenarios. We
also proposed a procedure to optimize the drug injection rate
according to a desired drug delivery rate through the pharmacokinetic model when the injection location and delivery are
known.
The pharmacokinetic model presented in this paper does not
take into account particles that continue their propagation after having circulated the entire cardiovascular system. This is
justified by the fact that heart and veins tend to significantly disperse the particles, therefore favoring their accumulation over
their recirculation in the cardiovascular system. A possible future extension of this study could also include these effects in
the pharmacokinetics through a stochastic model derived from
an MC noise abstraction, as presented in [26].
The results presented in this paper can support the future
design of intrabody MC networks [27]. In fact, the developed
pharmacokinetic model has the potential to be used to predict
the propagation of MC signals in the human body undergoing several transport and kinetic processes. With regards to the
communication performance of such a system, the theoretical
limits of the amount information that can be reliably transmit-
2419
ted by MC over the blood vessels has been studied in [26].
By defining the encoding and modulation schemes for MC in
the cardiovascular system, the achievable bit error rates can be
evaluated.
In conclusion, the proposed abstraction of a TDDS with the
MC paradigm provides a new way to model the TDDSs and
support their engineering with tractable, yet complete, analytical
models.
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Youssef Chahibi (S’13) received the Diplôme
d’Ingénieur in telecommunications and networks
from Institut National Polytechnique de Toulouse,
Toulouse, France, in 2011, and the M.S. degree from
the Georgia Institute of Technology, Atlanta, GA,
USA, in 2012.
During 2011, he was a Physical-Layer Engineer
at Alcatel-Lucent, Antwerp, Belgium. In the summer of 2014, he was a Guest Research Scholar at the
Nano Communication Center, Tampere University of
Technology, and during 2015, he was a Fellow of the
Research Council of Norway, Norwegian University of Science and Technology, Trondheim, Norway. His research interests are in nanoscale biologicallyinspired communications and drug delivery systems.
Massimiliano Pierobon (M’05) received the M.S.
degree in telecommunication engineering from the
Politecnico di Milano, Milan, Italy, in 2005, and the
Ph.D. degree in electrical and computer engineering
from the Georgia Institute of Technology, Atlanta,
GA, USA, in 2013.
He is currently an Assistant Professor at the Department of Computer Science and Engineering, University of Nebraska-Lincoln, Lincoln, NE, USA. He
received the BWN Lab Researcher of the Year Award
at the Georgia Institute of Technology for his outstanding research achievements in 2011. He was also named IEEE Communications Letters 2013 Exemplary Reviewer in appreciation for his service as
referee. He is an Editor of the IEEE TRANSACTIONS IN COMMUNICATIONS. He
is a Member of ACM and ACS. His current research interests are in molecular
communication theory for nanonetworks, communication engineering applied
to intelligent drug delivery systems, and telecommunication engineering applied
to cell-to-cell communications.
Ian F. Akyildiz (M’86–SM’89–F’96) received the
B.S., M.S., and Ph.D. degrees in computer engineering from the University of Erlangen-Nurnberg, Erlangen, Germany, in 1978, 1981, and 1984, respectively.
He is currently the Ken Byers Chair Professor in
telecommunications at the School of Electrical and
Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USA, the Director of the Broadband Wireless Networking Laboratory and Chair of
the Telecommunication Group, Georgia Tech. He is
an Honorary Professor at the School of Electrical
Engineering, Universitat Politecnica de Catalunya, Barcelona, Spain, and the
Founder of N3Cat (NaNoNetworking Center in Catalunya). He is also an Honorary Professor at the Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria, South Africa, and the Founder of
the Advanced Sensor Networks Lab. Since 2011, he has been a Consulting
Chair Professor at the Department of Information Technology, King Abdulaziz
University, Jeddah, Saudi Arabia. Since January 2013, he is also a FiDiPro Professor (Finland Distinguished Professor Program supported by the Academy of
Finland) at the Department of Communications Engineering, Tampere University of Technology, Tampere, Finland. He is the Editor-in-Chief of Computer
Networks (Elsevier), and the founding Editor-in-Chief of the Ad Hoc Networks
(Elsevier), Physical Communication (Elsevier), and the Nano Communication
Networks (Elsevier). He is an ACM Fellow. He has received numerous awards
from IEEE and ACM. According to Google Scholar, as of May 2014, his h-index
is 90 and the total number of citations he received is 70+K. His current research
interests are in molecular communication, nano-scale machine communication,
5G cellular systems, software-defined networking, and underground wireless
sensor networks.
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