Skip to main content

Advertisement

SpringerLink
Log in

In silico study about the influence of electroporation parameters on the cellular internalization, spatial uniformity, and cytotoxic effects of chemotherapeutic drugs using the Method of Fundamental Solutions

  • Original Article
  • Published:
Medical & Biological Engineering & Computing Aims and scope Submit manuscript

Abstract

Reversible electroporation is a suitable technique to aid the internalization of medicaments in cancer tissues without inducing permanent cellular damage, allowing the enhancement of cytotoxic effects without incurring in electric-driven necrotic or apoptotic processes by the presence of non-reversible aqueous pores. An adequate selection of electroporation parameters acquires relevance to reach these goals and avoid opposite effects. This work applies the Method of Fundamental Solutions (MFS) for drug transport simulations in electroporated cancer tissues, using a continuum tumor cord approach and considering both electro-permeabilization and vasoconstriction effects. The MFS algorithm is validated with published results, obtaining satisfactory accuracy and convergence. Then, MFS simulations are executed to study the influence of electric field magnitude \((E)\), number of electroporation treatments \(({N}_{ep})\), and electroporation time \(({T}_{ep})\) on three assessment parameters of electrochemotherapy: the internationalization efficacy accounting for the ability of the therapy to introduce moles into viable cells, cell-kill capacity indicating the faculty to reduce the survival fraction of cancer cells, and distribution uniformity specifying the competence to supply drug homogeneously through the whole tissue domain. According to numerical results, when \(E\) is the reversibility threshold, a positive influence on the first two parameters is only possible once specific values of \({T}_{ep}\) and \({N}_{ep}\) have been exceeded; when \(E\) is just the irreversibility threshold, any combination of \({T}_{ep}\) and \({N}_{ep}\) is beneficial. On the other hand, the drug distribution uniformity is always adversely affected by the application of electric pulses, being this more noticeable as \(E\), \({T}_{ep}\), and \({N}_{ep}\) increases.

Graphical abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig.7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24

Similar content being viewed by others

Data availability

The data supporting the findings of this study are available within the article. For availability of the whole code developed for this work, please contact the corresponding author.

References

  1. Batista Napotnik T, Miklavčič D (2018) In vitro electroporation detection methods – an overview. Bioelectrochemistry 120:166–182. https://doi.org/10.1016/j.bioelechem.2017.12.005

    Article  CAS  PubMed  Google Scholar 

  2. Kotnik T, Rems L, Tarek M, Miklavčič D (2019) Membrane electroporation and electropermeabilization: mechanisms and models. Annu Rev Biophys 48(1):63–91. https://doi.org/10.1146/annurev-biophys-052118-115451

    Article  CAS  PubMed  Google Scholar 

  3. Rems L, Miklavčič D (2016) Tutorial: Electroporation of cells in complex materials and tissue. J Appl Phys 119(20):201101. https://doi.org/10.1063/1.4949264

    Article  ADS  Google Scholar 

  4. Sweeney DC, Douglas TA, Davalos RV (2018) Characterization of cell membrane permeability in vitro part II: computational model of electroporation-mediated membrane transport*. Technol Cancer Res Treat 17:153303381879249. https://doi.org/10.1177/1533033818792490

    Article  CAS  Google Scholar 

  5. Sweeney DC, Weaver JC, Davalos RV (2018) Characterization of cell membrane permeability in vitro part I: transport behavior induced by single-pulse electric fields*. Technol Cancer Res Treat 17:153303381879249. https://doi.org/10.1177/1533033818792491

    Article  CAS  Google Scholar 

  6. Bellard E et al (2012) Intravital microscopy at the single vessel level brings new insights of vascular modification mechanisms induced by electropermeabilization. J Control Release 163(3):396–403. https://doi.org/10.1016/j.jconrel.2012.09.010

    Article  CAS  PubMed  Google Scholar 

  7. Brinton M, Mandel Y, Schachar I, Palanker D (2018) Mechanisms of electrical vasoconstriction. J Neuroeng Rehabil 15(1):1–10. https://doi.org/10.1186/s12984-018-0390-y

    Article  Google Scholar 

  8. Markelc B, Čemažar M, Serša G (2017) Effects of reversible and irreversible electroporation on endothelial cells and tissue blood flow. In: Handbook of electroporation. Springer International Publishing, Cham, pp 607–620. https://doi.org/10.1007/978-3-319-32886-7_70

  9. Markelc B et al (2018) Increased permeability of blood vessels after reversible electroporation is facilitated by alterations in endothelial cell-to-cell junctions. J Control Release 276(9):30–41. https://doi.org/10.1016/j.jconrel.2018.02.032

    Article  CAS  PubMed  Google Scholar 

  10. Markelc B et al (2012) In vivo molecular imaging and histological analysis of changes induced by electric pulses used for plasmid DNA electrotransfer to the skin: a study in a dorsal window chamber in mice. J Membr Biol 245(9):545–554. https://doi.org/10.1007/s00232-012-9435-5

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  11. Boyd B, Becker S (2016) Macroscopic modeling of in vivo drug transport in electroporated tissue. J Biomech Eng 138(3):1–12. https://doi.org/10.1115/1.4032380

    Article  Google Scholar 

  12. Cîndea N, Fabrèges B, de Gournay F, Poignard C (2010) Optimal placement of electrodes in an electroporation process. ESAIM: Proceedings 30. https://doi.org/10.1051/proc/2010004

  13. Adeyanju OO, Al-Angari HM, Sahakian AV (2012) The optimization of needle electrode number and placement for irreversible electroporation of hepatocellular carcinoma. Radiol Oncol 46(2). https://doi.org/10.2478/v10019-012-0026-y

  14. Kramar P, Miklavcic D, Lebar AM (2009) A system for the determination of planar lipid bilayer breakdown voltage and its applications. IEEE Trans Nanobioscience 8(2):132–138. https://doi.org/10.1109/TNB.2009.2022834

    Article  PubMed  Google Scholar 

  15. Shirakashi R, Sukhorukov VL, Tanasawa I, Zimmermann U (2004) Measurement of the permeability and resealing time constant of the electroporated mammalian cell membranes. Int J Heat Mass Transf 47(21):4517–4524. https://doi.org/10.1016/j.ijheatmasstransfer.2004.04.007

    Article  CAS  Google Scholar 

  16. Kingham TP et al (2012) Ablation of perivascular hepatic malignant tumors with irreversible electroporation. J Am Coll Surg 215(3). https://doi.org/10.1016/j.jamcollsurg.2012.04.029

  17. Philips P, Hays D, Martin RCG (2013) Irreversible electroporation ablation (IRE) of unresectable soft tissue tumors: learning curve evaluation in the first 150 patients treated. PLoS One 8(11). https://doi.org/10.1371/journal.pone.0076260

  18. Šel D, Cukjati D, Batiuskaite D, Slivnik T, Mir LM, Miklavčič D (2005) Sequential finite element model of tissue electropermeabilization. IEEE Trans Biomed Eng 52(5):816–827. https://doi.org/10.1109/TBME.2005.845212

    Article  PubMed  Google Scholar 

  19. Argus F, Boyd B, Becker SM (2017) Electroporation of tissue and cells: a three-equation model of drug delivery. Comput Biol Med 84(1):226–234. https://doi.org/10.1016/j.compbiomed.2017.04.001

    Article  CAS  PubMed  Google Scholar 

  20. Boyd B, Becker S (2015) Modeling of in vivo tissue electroporation and cellular uptake enhancement. IFAC-PapersOnLine 48(20):255–260. https://doi.org/10.1016/j.ifacol.2015.10.148

    Article  Google Scholar 

  21. Vélez Salazar FM, Patiño Arcila ID, Ruiz Villa CA (2020) 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. Int J Numer Method Biomed Eng 36(3):e3304. https://doi.org/10.1002/cnm.3304

    Article  MathSciNet  PubMed  Google Scholar 

  22. Vélez Salazar FM, Patiño Arcila ID, Ruiz Villa CA, Hernández-Blanquisett A (2022) In-silico study about the influence of electroporation parameters on the chemotherapeutic drug transport in cancer tissues using the meshless method of approximate particular solutions. Comput Math Appl 125:116–135. https://doi.org/10.1016/j.camwa.2022.08.034

    Article  MathSciNet  Google Scholar 

  23. Puc M, Kotnik T, Mir LM, Miklavčič D (2003) Quantitative model of small molecules uptake after in vitro cell electropermeabilization. Bioelectrochemistry 60(1–2):1–10. https://doi.org/10.1016/S1567-5394(03)00021-5

    Article  CAS  PubMed  Google Scholar 

  24. Groh CM et al (2014) Mathematical and computational models of drug transport in tumours. J R Soc Interface 11(94):20131173. https://doi.org/10.1098/rsif.2013.1173

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  25. Hubbard ME, Jove M, Loadman PM, Phillips RM, Twelves CJ, Smye SW (2017) Drug delivery in a tumour cord model: a computational simulation. R Soc Open Sci 4(5):170014. https://doi.org/10.1098/rsos.170014

    Article  MathSciNet  CAS  PubMed  PubMed Central  ADS  Google Scholar 

  26. Thurber GM, Weissleder R (2011) A systems approach for tumor pharmacokinetics. PLoS One 6(9):e24696. https://doi.org/10.1371/journal.pone.0024696

    Article  CAS  PubMed  PubMed Central  ADS  Google Scholar 

  27. Zhan W, Alamer M, Xu XY (2018) Computational modelling of drug delivery to solid tumour: understanding the interplay between chemotherapeutics and biological system for optimised delivery systems. Adv Drug Deliv Rev 132:81–103. https://doi.org/10.1016/j.addr.2018.07.013

    Article  CAS  PubMed  Google Scholar 

  28. El-Kareh AW, Secomb TW (2000) A mathematical model for comparison of bolus injection, continuous infusion, and liposomal delivery of doxorubicin to tumor cells. Neoplasia 2(4):325–338. https://doi.org/10.1038/sj.neo.7900096

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  29. Eikenberry S (2009) A tumor cord model for doxorubicin delivery and dose optimization in solid tumors. Theor Biol Med Model 6(1):16. https://doi.org/10.1186/1742-4682-6-16

    Article  MathSciNet  CAS  PubMed  PubMed Central  Google Scholar 

  30. Jackson TL (2003) Intracellular accumulation and mechanism of action of doxorubicin in a spatio-temporal tumor model. J Theor Biol 220(2):201–213. https://doi.org/10.1006/jtbi.2003.3156

    Article  MathSciNet  CAS  PubMed  ADS  Google Scholar 

  31. Palanker D, Vankov A, Freyvert Y, Huie P (2008) Pulsed electrical stimulation for control of vasculature: temporary vasoconstriction and permanent thrombosis. Bioelectromagnetics 29(2):100–107. https://doi.org/10.1002/bem.20368

    Article  PubMed  Google Scholar 

  32. Mandel Y et al (2013) Vasoconstriction by electrical stimulation: new approach to control of non-compressible hemorrhage. Sci Rep 3(1):2111. https://doi.org/10.1038/srep02111

    Article  PubMed  PubMed Central  Google Scholar 

  33. Corovic S, Markelc B, Dolinar M, Cemazar M, Jarm T (2015) Modeling of microvascular permeability changes after electroporation. PLoS One 10(3):e0121370. https://doi.org/10.1371/journal.pone.0121370

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  34. Gehl J, Skovsgaard T, Mir LM (2002) Vascular reactions to in vivo electroporation: characterization and consequences for drug and gene delivery. Biochim Biophys Acta Gen Subj 1569(1–3):51–58. https://doi.org/10.1016/S0304-4165(01)00233-1

    Article  CAS  Google Scholar 

  35. Meulenberg CJW, Todorovic V, Cemazar M (2012) Differential cellular effects of electroporation and electrochemotherapy in monolayers of human microvascular endothelial cells. PLoS ONE 7(12):1–9. https://doi.org/10.1371/journal.pone.0052713

    Article  CAS  Google Scholar 

  36. Golberg A, Rubinsky B (2010) A statistical model for multidimensional irreversible electroporation cell death in tissue. Biomed Eng Online 9(1):13. https://doi.org/10.1186/1475-925X-9-13

    Article  PubMed  PubMed Central  Google Scholar 

  37. Ozawa S, Sugiyama Y, Mitsuhashi Y, Kobayashi T, Inaba M (1988) Cell killing action of cell cycle phase-non-specific antitumor agents is dependent on concentration-time product. Cancer Chemother Pharmacol 21(3):185–190. https://doi.org/10.1007/BF00262767

    Article  CAS  PubMed  Google Scholar 

  38. Lankelma J, Fernández Luque R, Dekker H, Pinedo HM (2003) Simulation model of doxorubicin activity in islets of human breast cancer cells. Biochim Biophys Acta Gen Subj 1622(3):169–178. https://doi.org/10.1016/S0304-4165(03)00139-9

    Article  CAS  Google Scholar 

  39. El-Kareh AW, Secomb TW (2005) Two-mechanism peak concentration model for cellular pharmacodynamics of doxorubicin. Neoplasia 7(7):705–713. https://doi.org/10.1593/neo.05118

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  40. Millenbaugh NJ, Wientjes MG, Au JLS (2000) A pharmacodynamic analysis method to determine the relative importance of drug concentration and treatment time on effect. Cancer Chemother Pharmacol 45(4):265–272. https://doi.org/10.1007/s002800050039

    Article  CAS  PubMed  Google Scholar 

  41. Karageorghis A, Fairweather G (1999) The method of fundamental solutions for axisymmetric potential problems. Int J Numer Methods Eng 44(11). https://doi.org/10.1002/(SICI)1097-0207(19990420)44:11<1653::AID-NME558>3.0.CO;2-1

  42. Chen CS, Muleshkov AS, Golberg MA, Mattheij RMM (2005) A mesh-free approach to solving the axisymmetric Poisson’s equation. Numer Methods Partial Differ Equ 21(2):349–367. https://doi.org/10.1002/num.20040

    Article  MathSciNet  Google Scholar 

  43. Cheng AHD, Hong Y (2020) An overview of the method of fundamental solutions—solvability, uniqueness, convergence, and stability. Eng Anal Bound Elem 120:118–152. https://doi.org/10.1016/j.enganabound.2020.08.013

    Article  MathSciNet  Google Scholar 

  44. Shigeta T, Young DL, Liu CS (2012) Adaptive multilayer method of fundamental solutions using a weighted greedy QR decomposition for the Laplace equation. J Comput Phys 231(21):7118–7132. https://doi.org/10.1016/j.jcp.2012.05.036

    Article  MathSciNet  ADS  Google Scholar 

  45. Ramachandran PA (2002) Method of fundamental solutions: singular value decomposition analysis. Commun Numer Methods Eng 18(11):789–801. https://doi.org/10.1002/cnm.537

    Article  Google Scholar 

  46. Shigeta T, Young DL (2009) Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points. J Comput Phys 228(6):1903–1915. https://doi.org/10.1016/j.jcp.2008.11.018

    Article  MathSciNet  ADS  Google Scholar 

  47. Chen JT, Han H, Kuo SR, Kao SK (2014) Regularization methods for ill-conditioned system of the integral equation of the first kind with the logarithmic kernel. Inverse Probl Sci Eng 22(7):1176–1195. https://doi.org/10.1080/17415977.2013.856900

    Article  MathSciNet  CAS  Google Scholar 

  48. Fairweather G, Karageorghis A (1998) The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 9(1–2):69–95. https://doi.org/10.1023/a:1018981221740

    Article  MathSciNet  Google Scholar 

  49. Golberg MA (1995) The method of fundamental solutions for Poisson’s equation. Eng Anal Bound Elem 16(3):205–213. https://doi.org/10.1016/0955-7997(95)00062-3

    Article  Google Scholar 

  50. Onishi K (1996) Boundary inverse problems in seepage and viscous fluid flows. In: Proceedings of the International Conference on Computer Methods in Water Resources, CMWR

  51. Kobayashi K, Onishi K, Ohura Y (1996) On identifying Dirichlet condition for 2D Laplace equation by BEM. Eng Anal Bound Elem 17(3):223–230. https://doi.org/10.1016/S0955-7997(96)00016-1

    Article  Google Scholar 

  52. Chen CS, Fan CM, Wen PH (2012) The method of approximate particular solutions for solving certain partial differential equations. Numer Methods Partial Differ Equ 28(2):506–522. https://doi.org/10.1002/num.20631

    Article  MathSciNet  Google Scholar 

  53. Lee CK, Liu X, Fan SC (2003) Local multiquadric approximation for solving boundary value problems. Comput Mech 30(5–6):396–409. https://doi.org/10.1007/s00466-003-0416-5

    Article  MathSciNet  Google Scholar 

  54. Yao G, Kolibal J, Chen CS (2011) A localized approach for the method of approximate particular solutions. Comput Math Appl 61(9):2376–2387. https://doi.org/10.1016/j.camwa.2011.02.007

    Article  MathSciNet  Google Scholar 

  55. Yao G, Šarler B, Chen CS (2011) A comparison of three explicit local meshless methods using radial basis functions. Eng Anal Bound Elem 35(3):600–609. https://doi.org/10.1016/j.enganabound.2010.06.022

    Article  MathSciNet  Google Scholar 

  56. Harrouni KEL, Ouazar D, Wrobel LC, Cheng AHD (1995) Global interpolation function based DRBEM applied to Darcy’s flow in heterogeneous media. Eng Anal Bound Elem 16(3):281–285. https://doi.org/10.1016/0955-7997(95)00072-0

    Article  Google Scholar 

  57. Harrouni KEL, Ouazar D, Wrobel LC, Brebbia CA, UMV, Mohammadia E (1992) Method for heterogeneous porous media. Computational Mechanics Publications, no. C, pp. 5–6

  58. Partridge PW, Brebbia CA (1990) Computer implementation of the BEM dual reciprocity method for the solution of general field equations. Commun Appl Numer Methods 6(2):83–92. https://doi.org/10.1002/cnm.1630060204

    Article  MathSciNet  Google Scholar 

  59. Šarler B (1998) Axisymmetric augmented thin plate splines. Eng Anal Bound Elem 21(1):81–85. https://doi.org/10.1016/s0955-7997(98)00004-6

    Article  Google Scholar 

  60. Wang K (2002) BEM simulation for glass parisons door. https://doi.org/10.6100/IR554076

  61. Golberg MA, Muleshkov AS, Chen CS, Cheng AHD (2003) Polynomial particular solutions for certain partial differential operators. Numer Methods Partial Differ Equ 19(1):112–133. https://doi.org/10.1002/num.10033

    Article  MathSciNet  Google Scholar 

  62. Nader E et al (2019) Blood rheology: key parameters, impact on blood flow, role in sickle cell disease and effects of exercise. Front Physiol 10(OCT):1–14. https://doi.org/10.3389/fphys.2019.01329

    Article  Google Scholar 

  63. Karode SK (2001) Laminar flow in channels with porous walls, revisited. J Memb Sci 191(1–2):237–241. https://doi.org/10.1016/S0376-7388(01)00546-4

    Article  CAS  Google Scholar 

  64. Mohammed T, Singh M, Tiu JG, Kim AS (2021) Etiology and management of hypertension in patients with cancer. Cardio-Oncology 7(1):1–13. https://doi.org/10.1186/s40959-021-00101-2

    Article  CAS  Google Scholar 

  65. Ramirez FD, Reddy VY, Viswanathan R, Hocini M, Jaïs P (2020) Emerging technologies for pulmonary vein isolation. Circ Res 127(1):170–183. https://doi.org/10.1161/CIRCRESAHA.120.316402

    Article  CAS  PubMed  Google Scholar 

  66. Fiederer LDJ et al (2016) The role of blood vessels in high-resolution volume conductor head modeling of EEG. Neuroimage 128:193–208. https://doi.org/10.1016/j.neuroimage.2015.12.041

    Article  CAS  PubMed  Google Scholar 

  67. Evans CJ et al (2009) A mathematical model of doxorubicin penetration through multicellular layers. J Theor Biol 257(4):598–608. https://doi.org/10.1016/j.jtbi.2008.11.031

    Article  CAS  PubMed  ADS  Google Scholar 

  68. Goh YMF, Kong HL, Wang CH (2001) Simulation of the delivery of doxorubicin to hepatoma. Pharm Res 18(6):761–770. https://doi.org/10.1023/A:1011076110317

    Article  CAS  PubMed  Google Scholar 

  69. Zhan W, Gedroyc W, Xu X (2014) Mathematical modelling of drug transport and uptake in a realistic model of solid tumour. Protein Pept Lett 21(11):1146–1156. https://doi.org/10.2174/0929866521666140807115629

    Article  CAS  PubMed  Google Scholar 

  70. Lee C, Kim JS, Waldman T (2004) PTEN gene targeting reveals a radiation-induced size checkpoint in human cancer cells. Cancer Res 64(19):6906–6914. https://doi.org/10.1158/0008-5472.CAN-04-1767

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  71. Harada H et al (2008) Diameter of tumor blood vessels is a good parameter to estimate HIF-1-active regions in solid tumors. Biochem Biophys Res Commun 373(4):533–538. https://doi.org/10.1016/j.bbrc.2008.06.062

    Article  CAS  PubMed  Google Scholar 

  72. Vestvik IK, Egeland TAM, Gaustad JV, Mathiesen B, Rofstad EK (2007) Assessment of microvascular density, extracellular volume fraction, and radiobiological hypoxia in human melanoma xenografts by dynamic contrast-enhanced MRI. J Magn Reson Imaging 26(4):1033–1042. https://doi.org/10.1002/jmri.21110

    Article  PubMed  Google Scholar 

  73. National Library of Medicine (2022) Compound summary doxorubicin. https://pubchem.ncbi.nlm.nih.gov/compound/Doxorubicin. Accessed 13 May 2022

  74. Khorasani A (2020) A numerical study on the effect of conductivity change in cell kill distribution in irreversible electroporation. Polish J Med Phys Eng 26(2):69–76. https://doi.org/10.2478/pjmpe-2020-0008

    Article  Google Scholar 

  75. Robert J, Illiadis A, Hoerni B, Cano JP, Durand M, Lagarde C (1982) Pharmacokinetics of adriamycin in patients with breast cancer: correlation between pharmacokinetic parameters and clinical short-term response. Eur J Cancer Clin Oncol 18(8):739–745. https://doi.org/10.1016/0277-5379(82)90072-4

    Article  CAS  PubMed  Google Scholar 

Download references

Acknowledgements

The technical support of Institución Universitaria Pascual Bravo for the development of this work is acknowledged.

Funding

This work was supported by Institución Universitaria Pascual Bravo.

Author information

Author notes

  1. Fabián Mauricio Vélez Salazar and Iván David Patiño Arcila contributed equally to this work.

Authors and Affiliations

  1. Grupo de Investigación E Innovación Ambiental (GIIAM), Institución Universitaria Pascual Bravo, Cl. 73 No 73A-226 (Bloque 7), Medellín, Colombia

    Fabián Mauricio Vélez Salazar & Iván David Patiño Arcila

  2. Grupo de Investigación de Ciencias Administrativas, Instituto Tecnológico Metropolitano - ITM, Medellín, Colombia

    Fabián Mauricio Vélez Salazar

Authors
  1. Fabián Mauricio Vélez Salazar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Iván David Patiño Arcila
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

I. D. P. A. developed the computational model and contributed to results analysis. F. M. V. S. prepared the whole manuscript and contributed to results analysis. The manuscript was written through the contribution of all authors. All authors discussed the results, reviewed, and approved the final version of the manuscript. All authors contributed equally to this work.

Corresponding author

Correspondence to Iván David Patiño Arcila.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix. Code for calculation of interpolation coefficients \({\alpha }_{jk}\)

Appendix. Code for calculation of interpolation coefficients \({\alpha }_{jk}\)

figure dfigure dfigure d

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Salazar, F.M.V., Arcila, I.D.P. In silico study about the influence of electroporation parameters on the cellular internalization, spatial uniformity, and cytotoxic effects of chemotherapeutic drugs using the Method of Fundamental Solutions. Med Biol Eng Comput 62, 713–749 (2024). https://doi.org/10.1007/s11517-023-02964-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11517-023-02964-2

Keywords

Search

Navigation