A statistical model for multidimensional irreversible electroporation cell death in tissue

Abstract

Background: Irreversible electroporation (IRE) is a minimally invasive tissue ablation technique which utilizes electric pulses delivered by electrodes to a targeted area of tissue to produce high amplitude electric fields, thus inducing irreversible damage to the cell membrane lipid bilayer. An important application of this technique is for cancer tissue ablation. Mathematical modelling is considered important in IRE treatment planning. In the past, IRE mathematical modelling used a deterministic single value for the amplitude of the electric field required for causing cell death. However, tissue, particularly cancerous tissue, is comprised of a population of different cells of different sizes and orientations, which in conventional IRE are exposed to complex electric fields; therefore, using a deterministic single value is overly simplistic.

Methods: We introduce and describe a new methodology for evaluating IRE induced cell death in tissue. Our approach employs a statistical Peleg-Fermi model to correlate probability of cell death in heterogeneous tissue to the parameters of electroporation pulses such as the number of pulses, electric field amplitude and pulse length. For treatment planning, the Peleg-Fermi model is combined with a numerical solution of the multidimensional electric field equation cast in a dimensionless form. This is the first time in which this concept is used for evaluating IRE cell death in multidimensional situations.

Results: We illustrate the methodology using data reported in literature for prostate cancer cell death by IRE. We show how to fit this data to a Fermi function in order to calculate the critical statistic parameters. To illustrate the use of the methodology, we simulated 2-D irreversible electroporation protocols and produced 2-D maps of the statistical distribution of cell death in the treated region. These plots were compared to plots produced using a deterministic model of cell death by IRE and the differences were noted.

Conclusions: In this work we introduce a new methodology for evaluation of tissue ablation by IRE using statistical models of cell death. We believe that the use of a statistical model rather than a deterministic model for IRE cell death will improve the accuracy of treatment planning for cancer treatment with IRE.

Figure 1

Dependence of Ec and A on the number of pulses as developed from the work of Canatella et al [54]. A. 50 μsec pulse lenth. B.100 μsec pulse length, C. 1 msec pulse length and D. 10 msec pulse length.

Figure 2

Dependance of Ec and A on the number of applied pulses, normalized to Eco. A. 50 μsec pulse lenth. B.100 μsec pulse length, C. 1 msec pulse length and D. 10 msec pulse length.

Figure 3

Dimensionless electric field distribution solution in the treated tissue for A. C = 0.5, B. C = 1.5 C. C = 2.5.

Figure 4

Viability plots for IRE in prostate tissue in 2D for different electroporation protocols that have various number of pulses (n), voltages on the electrodes C, and pulse length, (t). A. n = 10 C = 1.5 t = 100 μsec B. n = 50 C = 1.5 t = 100 μsec C. n = 100 C = 1.5 t = 100 μsec. D. n = 50 pulses, C = 0.5, t = 100 μsec E. n = 50 pulses c = 1.5 t = 100 μsec F. n = 50 pulses c = 2.5 t = 100 μsec. G. n = 50 C = 1.5 t = 100 μsec H. n = 50 C = 1.5 t = 1 msec.