Antibody-Based Molecular Communicationfor Targeted Drug Delivery Systems*

Youssef Chahibi1, Ian F. Akyildiz2, and Sang Ok Song3

Abstract— Antibody-based drug delivery systems (ADDS) areestablished as the most promising therapeutic methods forthe treatment of human cancers and other diseases. ADDSare composed of small molecules (antibodies) that selectivelybind to receptors (antigens) expressed by the diseased cells.In this paper, the Molecular Communication (MC) paradigm,where the delivery of molecules is abstracted as the deliveryof information, is extended to be applied to the design andengineering of ADDS. The authors have previously developeda straightforward framework for the modeling of ParticulateDrug Delivery Systems (PDDS) using nano-sized molecules.Here, the specificities of antibody molecules are taken intoaccount to provide an analytical model of ADDS transport. Theinputs of the MC model of PDDS are the geometric properties ofthe antibodies and the topology of the blood vessels where theyare propagated. Numerical results show that the analytical MCmodel is in good agreement with finite-element simulations, andthat the anisotropy is an important factor influencing ADDS.

I. INTRODUCTION

Antibody-based Drug Delivery Systems (ADDS) are at theforefront of targeted drug delivery research [1]. ADDS aretherapeutic methods that use artificial molecules to mimicthe functioning of naturally produced antibody molecules.ADDS constitute biologically available materials to buildand engineer therapeutic methods. They are inspired by thenaturally occurring autoimmune mechanisms that enable thehuman body to diagnose itself and destroy the exact source ofthe disease, in an adaptive and constructive fashion. ADDSare engineered and inspired by this same advanced immunesystem mechanism. The versatility in engineering ADDSand their attested clinical success open up the possibilityto develop sophisticated therapeutic strategies to effectivelytarget diseases [2].

We propose to use the MC paradigm to model ADDSwhile taking into account the unique features of antibodiesand the new possibilities that are offered through them. TheMolecular Communication (MC) paradigm [3], where theinformation is conveyed through molecules, has been previ-ously used to model and optimize Particulate Drug DeliverySystems (PDDS) [4]. ADDS propagate in the body by advec-tion and diffusion in the network of blood vessels and tissues.

*This work was supported by the Samsung Advanced Institute ofTechnology (SAIT), Global Research Outreach (GRO) Program, projecttitle: “Molecular Communication Fundamentals for Antibody-based DrugDelivery Systems”.

1, 2 Youssef Chahibi and Ian F. Akyildiz are with the BroadbandWireless Networking Laboratory, School of Electrical and ComputerEngineering, Georgia Institute of Technology, Atlanta, Georgia 30332youssef@ece.gatech.edu, ian@ece.gatech.edu.

3Sang Ok Song is with the Well-Aging Research Center, Sam-sung Advanced Institute of Technology, Gyeonggi-Do 449-712, Koreaso4057.song@samsung.com.

Blood transport Blood velocity field

Antibodies Antigens

Tissue transport

Drug injection

Drug delivery Blood vessels

Healthy cells

Diseased cell

v3

v4

v5

Blood flow

v1

v2

Fig. 1. Elements of an ADDS.

As shown in Fig. 1, the drug injection of antibody moleculespropagates in the network of blood vessels. The antibodiesare transported to the drug delivery site by blood transportand tissue transport processes. The antibodies specificallybind to the diseased cells because they express antigens thatmatch to the antibody and do not appear in the healthycells. They trigger their therapeutic effect to cells througha special case of the ligand-binding mechanism, called theantibody-antigen mechanism. Therefore, some aspects of theMC modeling previously developed for PDDS are readilyapplicable to ADDS. However, the transport and mechanismof action of ADDS is more complex and advanced thanPDDS. The scope of our MC modeling of ADDS in thispaper is on the extracellular-cellular transport of ADDS, bymodifying the MC PDDS framework to take into accountthe arbitrary shape of antibody molecules. The MC ADDSmodeling will provide a clearer understanding of the modeof operation of antibodies, and enable the development ofinnovative methods to guide the engineering of verifiable andsafe antibody-based therapies.

The methods proposed in related works [5] are unable todifferentiate drugs reaching the target site from the otherdrugs. The path takes by drugs remains unknown in existingmodels. The system parameters are statistical estimated,and, generally, the most complex compartmental modelsconsist of two of three compartments assuming constantblood flow perfusion. The MC-ADDS model solves thisproblem by providing mechanistic models, based on lawof biophysics instead of empirical observations, without theneed of parameters estimation, and by giving higher spatial

978-1-4244-7929-0/14/$26.00 ©2014 IEEE 5707

and temporal resolution tracking of the drug propagation inthe micro-scale and millisecond scale, and being scalableto lower and higher resolutions with small changes to thesystem model.

The remainder of the paper is organized as follows: inSec. II, the analytical MC model of ADDS and the methodto obtain the diffusion parameters of antibody moleculesfrom their 3D geometry are presented. Second, in Sec. III,finite-element simulations results and numerical evaluationsof the analytical model are shown to validate the MC modelof ADDS. Finally, Sec. IV concludes the paper with theoutcomes of this work.

II. MC ADDS MODEL

In this section, we derive an analytical model of ADDSextracellular transport using the MC paradigm. As illustratedin Fig. 2, the drug injection is regarded as an MC Transmitter,the body is regarded as an MC Channel, and the drug deliveryis regarded as an MC Receiver. The signals of the MCsystem are the concentrations of the antibody molecules.The antibody molecule is characterized by two diffusionparameters, namely: the translational diffusion coefficientDt, and the radial diffusion coefficient Dr. The MC modelprovides a time-varying impulse response h(t, τ) whichrelates the transmitted signal x(t), which is the drug injectionconcentration, to the received signal y(t), which is the drugdelivery concentration at the diseased cell. The MC modelconsists of two parts. First, the transport model is presentedin Sec. II-A. Second, the calculation of the diffusion param-eters on which the MC ADDS transport model depends inpresented in Sec. II-B.

A. MC ADDS Transport Model

We develop here the MC ADDS transport model thatenables the prediction of the propagation of antibodies inthe blood vessels and tissue segments. The impulse responseh(t, τ) is obtained by cascading the impulse responses ofeach segment between the drug injection site and the drugdelivery site, which can be expressed as follows:

h(t, τ) = hi1(t, τ) ⊗ . . . hin(t, τ) · · · ⊗ hiL(t, τ) , (1)

where ⊗ denotes the cascading operation of two linear peri-odically time-varying systems as presented in [4], hin(t, τ)is the impulse response of the k-th blood vessel or segment., k is the index of the blood vessel or tissue segment, and Lis the number of blood vessels and/or tissue segments fromthe drug injection to the drug delivery.

The transport process in the tissue compartment isdiffusion-dominated since the hydrodynamic dispersion isso small that it can be neglected. Based on that, we de-rive the impulse response hin(t, τ) for each segment fromthe generalized anisotropic Taylor dispersion equation withabsorption [6] as follows:

hin(t, τ) =1√

2πσ2in

(t, τ)exp

(− (l −min(t, τ) )

2

2σ2in

(t, τ)

),

(2)

where:• The mean antibody velocity is time-varying and ex-

pressed as:

min(t, τ) =

∫ t

τ

ueffin(r, t) dt′ , (3)

• The variance of the antibody is time-varying and isequal to:

σ2in(t, τ) = 2

∫ t

τ

Deffin

(t′) dt′ , (4)

where t and t′ are time parameters, The effective diffusioncoefficient of antibodies Deff

in(t) is expressed as follows [6]

:

Deffin

(t) = DtPfin +DwinPwin

+ ∆Deffin

(t) , (5)

where Dt is the translational diffusion coefficient of theADDS in the blood expressed in (8), Dwin

is the diffusioncoefficient in the vessel or cell walls in i, and ∆Deff

inis the diffusion coefficient increment of antibodies due toanisotropy, the effective blood velocity ueffin

(t) is expressedas follows:

ueffin(t) = Pfinuin(t) , (6)

where Pfin = 11+Kin

and Pfin =Kin

1+Kinare kinetic ratios,

Kin = k+

k− is the equilibrium constant, uin(t) is the realblood velocity, k+in and k−in are, respectively, the associationand disassociation kinetic coefficients of antibodies with thewalls of the vessel in. The diffusion coefficient increment ofantibodies due to anisotropy, ∆Deff

in, is expressed as follows:

∆Deffin

(t) = P3fin

((vfin (t) − vwin)Kin

k−in2 (7)

+Kin

k−in

r2in48Dr

(1 + 6Kin + 11K2in)v2fin

− 4Kin(Kin + 1)vfin (t)vwin+ 6K2

in + v2win) ,

where Dr is the radial diffusion coefficient of the ADDS inthe blood expressed in (8) rin is the radius of the vessel in,vwin

is the blood velocity on the wall.Finally, from (1) and (2), we obtain the MC end-to-end

impulse response of the ADDS.

B. MC ADDS Diffusion Parameters

This section presents a method to obtain the ADDSdiffusion parameters that are used by the MC ADDS trans-port model presented in Sec. II-A, namely the translationaldiffusion coefficient Dt and the radial diffusion coefficientDr.

5708

MC Transmitter

MC Receiver

MC Channel

h3(t,τ)

h4(t,τ)

h5(t,τ)

h1(t,τ)

h2(t,τ)

Antibody (Dt, Dr)

h5(t,τ)

t  

MC Channel h(t,τ)

Transmitted signal x(t)

Received signal y(t)

Antigen-binding (k+, k-)

Fig. 2. MC Abstraction of an ADDS

As illustrated in Fig. 3, the antibody-antigen has a uniqueshape. In fact, antibodies are generally Y -shaped moleculesthat consist of different heterogeneous regions (light chainand heavy chain). Antibodies come in different arbitraryshapes and structures as can be seen in X-Ray structureanalysis of this type of molecules [7]. The geometry ofthe antibody necessarily affects its motion. The irregularshape can create arbitrary motions and fluctuations thatare different from the case of spherical nanoparticles thatwere considered in PDDS. In the literature, all MC andpharmacokinetic models have supposed spherical or, at best,ellipsoid particles, for the modeling of antibody propagation.Therefore, there is a need for a model that takes into accountthe antibody shape and structure to predict the diffusionparameters of this small molecule without any empiricalchoices.

According to Brenner’s general theory of diffusion [8],the transport of irregularly shaped molecules leads to theanisotropic coupling of radial and translational diffusionparameters of the molecules [9]. In classical MC, suchmotion is governed through Fick’s law, with one parameterD, called the diffusion coefficient, but in the case of ADDS,we will consider two parameters, namely the translationaldiffusion coefficient Dt, and the radial diffusion coefficientDr.

As illustrated in Fig. 3, we approximate the antibodymolecule as a set of N beads Bn. A bead Bn is characterizedby a radius ρn, and is located at the Cartesian coordinates(xn, yn, zn) from an arbitrary origin O. Two beads Bm andBn are located at the distance Rm,n from each other. Rm,n

denotes the distance vector between the beads Bm and Bn.The translational diffusion coefficient Dt is calculated as

follows [10]:

Dt =kBT

µ

1

3tr(At) , (8)

where kB is Boltzmann coefficient, T is the temperature ofthe blood, supposed constant, and At is the top-left com-ponent of the mobility tensor [9] of the antibody molecule.The radial diffusion coefficient Dr is expressed by a similar

m

Rm,n

ρm

ρn

Rm,n

n

xnynzn

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

xmymzm

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Fig. 3. Bead model of an antibody.

expression as follows [10]:

Dr =kBT

µ

1

3tr(Ar) , (9)

where Ar is the top-left component of the mobility tensor [9]of the antibody molecule.

The top-left component of the mobility tensor At of theantibody molecule is expressed as follows:

At =

N∑m=1

N∑n=1

[δm,nI

6πηRm,n+ (1 − δm,n)Tm,n

]−1, (10)

Similarly, the bottom-right component of the mobility tensorAr of the antibody molecule is expressed as:

Ar = −N∑m=1

N∑n=1

Um

[δm,nI

6πηRm,n+ (1 − δm,n)Tm,n

]−1Un

+ 6η

(4π

3

N∑n=1

ρ3n

)I , (11)

where Tm,n is the hydrodynamic tensor of the antibodycalculated as follows from the geometric parameters of theantibody molecule:

Tm,n =1

8πηRm,n

[(I +

Rm,nRTm,n

R2m,n

)

+ρ2m + ρ2nR2m,n

(I

3−

Rm,nRTm,n

R2m,n

)], (12)

and Un is the skew matrix of the bead n, expressed asfollows:

Un =

0 −zn ynzn 0 −xn−yn xn 0

. (13)

Finally, from (8) and (9), we have the analytical expressionof the diffusion parameters of an antibody molecule basedsolely on its geometric arrangement and diameters.

III. NUMERICAL RESULTS

In this section, we present numerical results to validatethe anisotropic transport with a realistic 3D model and toshow the significance of anisotropy. We used COMSOL R©to

5709

0 1 2 3 4 5 6 7 8−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time [s]

Impulsere

sponse

COMSOL Anisotropic

MC Model

Fig. 4. Validation of the analytical impulse response with COMSOL sim-ulation results.

0 2 4 6 8 10 12 14 16 180

20

40

60

80

100

120

140

Time [s]

Impulsere

sponse

D r = 10−7 [m 2/s ]

D r = 10−6 [m 2/s ]

D r = 10−5 [m 2/s ]

D r = 10−4 [m 2/s ]

Fig. 5. Impulse responses for different values of the radial diffusioncoefficients.

simulate antibody propagation based on the full advection-diffusion equation in a 3D geometry, and we evaluatedthe effect on anisotropy on the impulse response of thesystem. Fig. 4 shows the comparison between the math-ematical model stemming from the MC paradigm incor-porating the anisotropy effect and the full 3D simulationwith COMSOL R©. We have used the translational and radialdiffusion coefficients calculated from the bead model inboth COMSOL R©and the MC models. The figure shows andexcellent agreement between the two results. Fig. 5 illustrateshow the impulse response varies highly depending on theradial diffusion coefficient. Fig. 6 shows the dependence ofthe anisotropic diffusion parameters on the angle betweenthe hands of the antibody.

IV. CONCLUSION

The Molecular Communication (MC) framework was usedas an abstraction of Antibody-based Drug Delivery Systems(ADDS) which is at the forefront of drug delivery systemresearch. The proposed MC model is based on the bio-physical laws of antibody transport in the human body. Ananalytical expression of the impulse response characterizingADDS propagation was given, and the distribution of ADDS

−30 −20 −10 0 10 20 306

7

8

9

10

11x 10

−8

Angle [deg]

Diff

usioncoefficients

−30 −20 −10 0 10 20 303

4

5

6

7

8x 10

10

Translat ionalRotat ional

Fig. 6. Dependence of the translational and rotational diffusion coefficientson the antibody angle.

molecules is predicted through the generalized Fick’s lawwhere the diffusion parameters of the ADDS are anisotropicand expressed as a function the geometric structure of themolecule. The results show good agreement with finite-element simulations, and numerical evaluations show thesignificant of anisotropy in ADDS systemic distribution.The finite-element simulations requires hours to completewhile the MC model requires a few seconds to providethe impulse response of the system. We propose to studyas future work non-specific binding of antibodies in theblood serum and extra-cellular matrix, and the variabilityof tortuosity of the paths separating cells. This work canbe used to create versatile, scalable, low-computational cost,and precise pharmacokinetic and pharmacodynamic modelsof antibody-based therapies.

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