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Research Papers

Macroscopic Modeling of In Vivo Drug Transport in Electroporated Tissue

[+] Author and Article Information
Bradley Boyd

Department of Mechanical Engineering,
University of Canterbury,
Private Bag 4800,
Christchurch 8014, New Zealand
e-mail: bradley.boyd@pg.canterbury.ac.nz

Sid Becker

Department of Mechanical Engineering,
University of Canterbury,
Private Bag 4800,
Christchurch 8014, New Zealand
e-mail: sid.becker@canterbury.ac.nz

1Corresponding author.

Manuscript received July 19, 2015; final manuscript received December 15, 2015; published online February 5, 2016. Assoc. Editor: Ram Devireddy.

J Biomech Eng 138(3), 031008 (Feb 05, 2016) (11 pages) Paper No: BIO-15-1359; doi: 10.1115/1.4032380 History: Received July 19, 2015; Revised December 15, 2015

This study develops a macroscopic model of mass transport in electroporated biological tissue in order to predict the cellular drug uptake. The change in the macroscopic mass transport coefficient is related to the increase in electrical conductivity resulting from the applied electric field. Additionally, the model considers the influences of both irreversible electroporation (IRE) and the transient resealing of the cell membrane associated with reversible electroporation. Two case studies are conducted to illustrate the applicability of this model by comparing transport associated with two electrode arrangements: side-by-side arrangement and the clamp arrangement. The results show increased drug transmission to viable cells is possible using the clamp arrangement due to the more uniform electric field.

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Figures

Grahic Jump Location
Fig. 1

(a) The dependence of the electrical conductivity (σ) on the magnitude of the electric field (E) represented in Eq. (3) and simultaneously the DOE represented in Eq. (4). (b) The dependence of the DIE (thick line) on the magnitude of electric field (E). The DIE is approximated in Eq. (5) using the electrical conductivity (σ) at values above the electrical conductivity associated with the onset of IE (σirrev).

Grahic Jump Location
Fig. 2

A representation of the extracellular and intracellular space, the cell membrane, and the membrane pores

Grahic Jump Location
Fig. 3

Geometry of the two cases being modeled: Case (1) with both electrodes positioned on the surface and Case (2) with a clamped electrode configuration. (a) and (c) The electric pulse setup and (b) and (d) show the drug gel applied to the top surface for Cases (1) and (2), respectively.

Grahic Jump Location
Fig. 5

Prepulse extracellular drug concentration (Ce∗) after leaving the drug to diffuse into the tissue for a duration of an hour for both Cases (1) and (2)

Grahic Jump Location
Fig. 4

The transient behavior of the resealing component of the transient DOE (Eq. (13)) for a single application of electroporation and for a pulse train where the tissue is electroporated periodically. The pulse train considered is applied for the entire 10 min with a pulse spacing of 10 s.

Grahic Jump Location
Fig. 7

Single pulse: after the single electroporation pulse the drug transport is considered for another 10 min with the applied drug gel removed. Pulse train: after the first electroporation pulse of the pulse train the drug transport is considered for another 10 min with the applied drug gel removed. A pulse train with a pulse spacing of 10 s was applied for the entire 10 min in which drug transport was considered (Fig. 4). The increased rate of cellular uptake due to electroporation results in an increased intracellular drug concentration (Ci∗). The sections depicted correspond to the dashed boxes presented in Fig.6.

Grahic Jump Location
Fig. 6

An electric potential is applied between the electrode (V0) and the ground for a duration of 100 μs resulting in electroporation of the liver tissue. (a) and (c) The magnitude of electric field (E) (kV/m) and (b) and (d) show the subsequent DOE. The dashed boxes correspond to the sections depicted in (b) and (d). Symmetry occurs about the z axis at x=0 for both Cases (1) and (2) and about the x axis at z=2.5 mm for Case (2).

Grahic Jump Location
Fig. 8

The computed total mass in living cellular space (mi∗) at different applied voltages using periodically applied electroporation every 10 s for a duration of 10 min (pulse train). The voltages resulting in the maximum cellular uptake to living cells for both cases (Case (1) ∼ 480 V and Case (2) ∼ 350 V) are used for the previously modeled results (Table 1).

Grahic Jump Location
Fig. 9

The simplified case for verification of the numerical solver using the analytical solution. Where (a) is for the steady state electric potential (ϕ) distribution and (b) is for the transient concentration (C) distribution (where μR=0). The analytical solutions to these problems are provided in Ref. [58].

Grahic Jump Location
Fig. 10

The transient behavior of the relative normalized error (L2) of Eq. (8) (where μR=0) compared to the corresponding exact solution presented in Table 3.26 of Ref. [58]

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