Sequential Finite Element Model of Tissue Electropermeabilisation

Miklavčič, Damijan; Šel, Davorka; Cukjati, D.; Batiuškaitė, Danutė; Slivnik, T.; Mir, Lluis M.

doi:10.1109/iembs.2004.1403998

Cited by 21 publications

(27 citation statements)

References 44 publications

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“…The extent of electroporation of cells accounts for the increase of the permeability in the cellular membrane and is directly related to the magnitude of the electric field, E , which in turn, can be computed as follows:

E = (||, \nabla \emptyset),

where ∅ is the electric potential that can be calculated solving a Laplace‐type equation, given by

\nabla ∙ ((), σ ∙ \nabla \emptyset) = 0 .

As the electroporation process leads to the formation of pores in the cellular membrane, the electric conductivity of that membrane increases as well. The effective electric conductivity, σ , that appears in (3) is estimated in terms of the magnitude of the electric field, E , using the relationship established in Šel et al:

σ ((), E) = \frac{σ_{\max} - σ_{\min}}{1 + α . e^{(- \frac{E - a}{b})}} + σ_{\min},

a = \frac{E_{rev} + E_{irrev}}{2},

b = \frac{E_{irrev} - E_{rev}}{β},

where α and β are sigmoidal function parameters determined in a combined experimental and computational study by Šel et al and E rev and E irrev represent the thresholds of E corresponding to reversible and irreversible electroporation, respectively. On the other hand, σ max and σ min are the maximum and minimum electric conductivities of the tissue, respectively, with σ max corresponding to the maximum threshold of σ due to the electroporation and σ min to electric conductivity without electroporation.…”

Section: Mathematical Modeling and Numerical Methodsmentioning

confidence: 99%

“…On the other hand, D tissue , ε and μ R represent the effective diffusivity of the drug in the tissue, the porosity of the tissue and the mass transport coefficient, respectively. The calculation of μ R is carried out following the model proposed by Boyd and Becker, in which μ R is posed in terms of the magnitude of the electric field, E , and the time elapsed from the previous electroporation pulse ( t p ), as given by

μ_{R} = {DOE}_{R} ((), E, t) . ((), μ_{\max} - μ_{\min}) + μ_{\min},

where μ max and μ min are the maximum and minimum membrane mass transfer coefficient, respectively while DOE R ( E , t ) is the transient degree of electroporation (DOE), which accounts for the transient resealing of reversibly electroporated cells and is computed as follows:

{DOE}_{R} ((), E, t) = italicDOE ((), E) . ([], ((), 1 - italicDIE ((), E)) . e^{(- \frac{t_{p}}{τ})} + italicDIE ((), E)),

with τ as the permeability decay time coefficient and DOE is the degree of electroporation that takes into account the increase in the electric conductivity of the tissue as a result of electroporation, as given by

italicDOE ((), E) = \frac{σ ((), E) - σ_{\min}}{σ_{\max} - σ_{\min}} .

The time constant for cell permeability, τ , is dependent on the pulse characteristics. The constant used in this work was experimentally determined in, which used a pulse duration of 100 μs and is therefore the value considered here.…”

Section: Mathematical Modeling and Numerical Methodsmentioning

confidence: 99%

“…In Sel et al, reversible and irreversible limits were obtained as fitting parameters of a sigmoid function that relates the local electric conductivity of the tissue to the electric field intensity; these parameters were estimated by optimizing the current calculated by the model to be close to that measured experimentally, which in turn, is a reliable parameter to quantify the extent of electroporation in the cell membrane. The values of E rev and E irrev obtained in that work were used in and in the present research.…”

Section: Introductionmentioning

confidence: 99%

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Simulation of the influence of voltage level and pulse spacing on the efficiency, aggressiveness and uniformity of the electroporation process in tissues using meshless techniques

Salazar

Arcila

Villa

2020

Numer Methods Biomed Eng

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Electroporation is a widely used method consisting of application of high‐voltage, short‐duration electric pulses to increase cell membrane permeability, allowing cellular internalization of medications. In this work, the influence of two primary parameters, voltage level (V) and pulse spacing (N), on electroporation efficiency, uniformity and aggressiveness, as quantified by the total mass transport to viable cells, intracellular concentration gradients and an aggressiveness factor introduced here, is studied by means of numerical simulations of drug transport in electroporated tissues. The global method of approximate particular solutions (Global MAPS) is used to solve the governing equations, together with domain scaling, singular value decomposition and smoothing algorithms, to address the ill‐conditioning of the final system and suppress small scale oscillations. The accuracy of Global MAPS is evaluated by comparing the initial extracellular concentration, Ce, and final intracellular concentration, Ci, with previous finite volume method results, obtaining similar behavior of Ce and Ci along the tissue domain, with some differences for Ci in high‐gradient zones. According to the Global MAPS results, the influence of V and N on Ci is only significant over a certain range, within which the largest drug transport to viable cells occurs. In general, both electroporation efficiency and aggressiveness change in nonuniform manner with V and decrease with N, whereas the electroporation uniformity decreases as V increases and N decreases. The contour plots obtained here can be considered useful tools to compare electroporation‐based treatments in terms of their efficiency, aggressiveness and uniformity, assisting in the selection of a suitable treatment plan for cancer.

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E = (||, \nabla \emptyset),

where ∅ is the electric potential that can be calculated solving a Laplace‐type equation, given by

\nabla ∙ ((), σ ∙ \nabla \emptyset) = 0 .

σ ((), E) = \frac{σ_{\max} - σ_{\min}}{1 + α . e^{(- \frac{E - a}{b})}} + σ_{\min},

a = \frac{E_{rev} + E_{irrev}}{2},

b = \frac{E_{irrev} - E_{rev}}{β},

Section: Mathematical Modeling and Numerical Methodsmentioning

confidence: 99%

μ_{R} = {DOE}_{R} ((), E, t) . ((), μ_{\max} - μ_{\min}) + μ_{\min},

{DOE}_{R} ((), E, t) = italicDOE ((), E) . ([], ((), 1 - italicDIE ((), E)) . e^{(- \frac{t_{p}}{τ})} + italicDIE ((), E)),

italicDOE ((), E) = \frac{σ ((), E) - σ_{\min}}{σ_{\max} - σ_{\min}} .

Section: Mathematical Modeling and Numerical Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Simulation of the influence of voltage level and pulse spacing on the efficiency, aggressiveness and uniformity of the electroporation process in tissues using meshless techniques

Salazar

Arcila

Villa

2020

Numer Methods Biomed Eng

View full text Add to dashboard Cite

show abstract

“…We were unable to assume constant conductivity since it has been shown that conductivity of tissue increases more dramatically after electroporation than cells in suspension due to the large ratio of cell volume to extracellular fluid volume 25 . Ions leak out of the cells when they are electroporated 38 therefore increasing conductivity. Joule heating effects also increase conductivity and act as a volumetric heat source.…”

Section: Methodsmentioning

confidence: 99%

“…In using this function, we assumed that conductivity increases by a factor of three since this was reported for other organs during electroporation 38 . We must also account for metabolic heat generation, conduction and convection due to blood perfusion.…”

Section: Methodsmentioning

confidence: 99%

The Feasibility of Enhancing Susceptibility of Glioblastoma Cells to IRE Using a Calcium Adjuvant

et al. 2017

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Irreversible electroporation (IRE) is a cellular ablation method used to treat a variety of cancers. IRE works by exposing tissues to pulsed electric fields which cause cell membrane disruption. Cells exposed to lower energies become temporarily permeable while greater energy exposure results in cell death. For IRE to be used safely in the brain, methods are needed to extend the area of ablation without increasing applied voltage, and thus, thermal damage. We present evidence that IRE used with adjuvant calcium (5mM CaCl2) results in a nearly two-fold increase in ablation area in vitro compared to IRE alone. Adjuvant 5mM CaCl2 induces death in cells reversibly electroporated by IRE, thereby lowering the electric field thresholds required for cell death to nearly half that of IRE alone. The calcium-induced death response of reversibly electroporated cells is confirmed by electrochemotherapy pulses which also induced cell death with calcium but not without. These findings, combined with our numerical modeling, suggest the ability to ablate up to 3.2× larger volumes of tissue in vivo when combining IRE and calcium. The ability to ablate a larger volume with lowered energies would improve the efficacy and safety of IRE therapy.

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