Patent Publication Number: US-2023157759-A1

Title: System and method for estimating a treatment volume for administering electrical-energy based therapies

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application is a Continuation of U.S. patent application Ser. No. 15/011,752 filed Feb. 1, 2016 (which published as U.S. Patent Application Publication No. 2016/0143698 on May 26, 2016). The &#39;752 application is a Continuation of U.S. patent application Ser. No. 14/012,832 filed Aug. 28, 2013 (which published as U.S. Patent Application Publication No. 2013/0345697 on Dec. 26, 2013 and issued as U.S. Pat. No. 9,283,051 on Mar. 15, 2016 and which relies on and claims the benefit of the filing date of U.S. Provisional Application No. 61/694,144, filed on Aug. 28, 2012), which &#39;832 application is a Continuation-in-Part (CIP) application of U.S. application Ser. No. 12/491,151, filed Jun. 24, 2009 (which issued as U.S. Pat. No. 8,992,517 on Mar. 31, 2015 and which claims priority to and the benefit of the filing dates of U.S. Provisional Patent Application No. 61/171,564, filed Apr. 22, 2009, 61/167,997, filed Apr. 9, 2009, and 61/075,216, filed Jun. 24, 2008), and which &#39;151 application is a CIP application of U.S. patent application Ser. No. 12/432,295, filed Apr. 29, 2009 (which published as U.S. Patent Application Publication No. 2009/0269317 on Oct. 29, 2009 and issued as U.S. Pat. No. 9,598,691 on Mar. 21, 2017 and which claims priority to and the benefit of the filing date of U.S. Provisional Patent Application No. 61/125,840, filed Apr. 29, 2008). The disclosures of these patent applications are hereby incorporated by reference herein in their entireties. 
    
    
     FIELD OF THE INVENTION 
     The present invention is related to medical therapies involving the administering of electrical treatment energy. More particularly, embodiments of the present invention provide systems and methods for estimating a target ablation zone for a medical treatment device that applies electrical treatment energy through a plurality of electrodes defining a target treatment area. 
     DESCRIPTION OF RELATED ART 
     Electroporation-based therapies (EBTs) are clinical procedures that utilize pulsed electric fields to induce nanoscale defects in cell membranes. Typically, pulses are applied through minimally invasive needle electrodes inserted directly into the target tissue, and the pulse parameters are tuned to create either reversible or irreversible defects. Reversible electroporation facilitates the transport of molecules into cells without directly compromising cell viability. This has shown great promise for treating cancer when used in combination with chemotherapeutic agents or plasmid DNA (M. Marty et al., “Electrochemotherapy—An easy, highly effective and safe treatment of cutaneous and subcutaneous metastases: Results of ESOPE (European Standard Operating Procedures of Electrochemotherapy) study,” European Journal of Cancer Supplements, 4, 3-13, 2006; A. I. Daud et al., “Phase I Trial of Interleukin-12 Plasmid Electroporation in Patients With Metastatic Melanoma,” Journal of Clinical Oncology, 26, 5896-5903, Dec. 20, 2008). Alternatively, irreversible electroporation (IRE) has been recognized as non-thermal tissue ablation modality that produces a tissue lesion, which is visible in real-time on multiple imaging platforms (R. V. Davalos, L. M. Mir, and B. Rubinsky, “Tissue ablation with irreversible electroporation,” Ann Biomed Eng, 33, 223-31, February 2005; R. V. Davalos, D. M. Otten, L. M. Mir, and B. Rubinsky, “Electrical impedance tomography for imaging tissue electroporation,” IEEE Transactions on Biomedical Engineering, 51, 761-767, 2004; L. Appelbaum, E. Ben-David, J. Sosna, Y. Nissenbaum, and S. N. Goldberg, “US Findings after Irreversible Electroporation Ablation: Radiologic-Pathologic Correlation,” Radiology, 262, 117-125, Jan. 1, 2012). Because the mechanism of cell death does not rely on thermal processes, IRE spares major nerve and blood vessel architecture and is not subject to local heat sink effects (B. Al-Sakere, F. Andre, C. Bernat, E. Connault, P. Opolon, R. V. Davalos, B. Rubinsky, and L. M. Mir, “Tumor ablation with irreversible electroporation,” PLoS ONE, 2, e1135, 2007). These unique benefits have translated to the successful treatment of several surgically “inoperable” tumors (K. R. Thomson et al., “Investigation of the safety of irreversible electroporation in humans,” J Vasc Interv Radiol, 22, 611-21, May 2011; R. E. Neal II et al., “A Case Report on the Successful Treatment of a Large Soft-Tissue Sarcoma with Irreversible Electroporation,” Journal of Clinical Oncology, 29, 1-6, 2011; P. A. Garcia et al., “Non-thermal irreversible electroporation (N-TIRE) and adjuvant fractionated radiotherapeutic multimodal therapy for intracranial malignant glioma in a canine patient,” Technol Cancer Res Treat, 10, 73-83, 2011). 
     In EBTs, the electric field distribution is the primary factor for dictating defect formation and the resulting volume of treated tissue (J. F. Edd and R. V. Davalos, “Mathematical modeling of irreversible electroporation for treatment planning,” Technology in Cancer Research and Treatment, 6, 275-286, 2007 (“Edd and Davalos, 2007”); D. Miklavcic, D. Semrov, H. Mekid, and L. M. Mir, “A validated model of in vivo electric field distribution in tissues for electrochemotherapy and for DNA electrotransfer for gene therapy,” Biochimica et Biophysica Acta, 1523, 73-83, 2000). The electric field is influenced by both the geometry and positioning of the electrodes as well as the dielectric tissue properties. Because the pulse duration (˜100 μs) is much longer than the pulse rise/fall time (˜100 ns), static solutions of the Laplace&#39;s equation incorporating only electric conductivity are sufficient for predicting the electric field distribution. In tissues with uniform conductivity, solutions can be obtained analytically for various needle electrode configurations if the exposure length is much larger than the separation distance (S. Corovic, M. Pavlin, and D. Miklavcic, “Analytical and numerical quantification and comparison of the local electric field in the tissue for different electrode configurations,” Biomed Eng Online, 6, 2007; R. Neal II et al., “Experimental Characterization and Numerical Modeling of Tissue Electrical Conductivity during Pulsed Electric Fields for Irreversible Electroporation Treatment Planning,” Biomedical Engineering, IEEE Transactions on, PP, 1-1, 2012 (“Neal et al., 2012”)). This is not often the case in clinical applications where aberrant masses with a diameter on the order of 1 cm are treated with an electrode exposure length of similar dimensions. Additionally, altered membrane permeability due to electroporation influences the tissue conductivity in a non-linear manner. Therefore numerical techniques may be used to account for any electrode configuration and incorporate a tissue-specific function relating the electrical conductivity to the electric field distribution (i.e. extent of electroporation). 
     Conventional devices for delivering therapeutic energy such as electrical pulses to tissue include a handle and one or more electrodes coupled to the handle. Each electrode is connected to an electrical power source. The power source allows the electrodes to deliver the therapeutic energy to a targeted tissue, thereby causing ablation of the tissue. 
     Once a target treatment area is located within a patient, the electrodes of the device are placed in such a way as to create a treatment zone that surrounds the treatment target area. In some cases, each electrode is placed by hand into a patient to create a treatment zone that surrounds a lesion. The medical professional who is placing the electrodes typically watches an imaging monitor while placing the electrodes to approximate the most efficient and accurate placement. 
     However, if the electrodes are placed by hand in this fashion, it is very difficult to predict whether the locations selected will ablate the entire treatment target area because the treatment region defined by the electrodes vary greatly depending on such parameters as the electric field density, the voltage level of the pulses being applied, size of the electrode and the type of tissue being treated. Further, it is often difficult or sometimes not possible to place the electrodes in the correct location of the tissue to be ablated because the placement involves human error and avoidance of obstructions such as nerves, blood vessels and the like. 
     Conventionally, to assist the medical professional in visualizing a treatment region defined by the electrodes, an estimated treatment region is generated using a numerical model analysis such as complex finite element analysis. One problem with such a method is that even a modest two dimensional treatment region may take at least 30 minutes to several hours to complete even in a relatively fast personal computer. This means that it would be virtually impossible to try to obtain on a real time basis different treatment regions based on different electrode positions. 
     Therefore, it would be desirable to provide an improved system and method to predict a treatment region in order to determine safe and effective pulse protocols for administering electrical energy based therapies, such IRE. 
     SUMMARY OF THE INVENTION 
     The inventors of the present invention have made the surprising discovery that by monitoring current delivery through the electrodes placed for treatment, it is possible to determine the extent of electroporation in the tissue and accurately predict the treatment volume. In addition to current measurements, the prediction can also rely on prior knowledge of the tissue-specific conductivity function and electric field threshold for either reversible electroporation or cell death in the case of IRE. 
     The inventors have characterized this non-linear conductivity behavior in ex vivo porcine kidney tissue. (“Neal et al., 2012”). Using this information, the inventors performed a comprehensive parametric study on electrode exposure length, electrode spacing, voltage-to-distance ratio, and ratio between the baseline conductivity pre-IRE and maximum conductivity post-IRE. Current measurements from all 1440 possible parameter combinations were fitted to a statistical (numerical) model accounting for interaction between the pulse parameters and electrode configuration combinations. The resulting equation is capable of relating pre- and post-treatment current measurements to changes in the electric field distribution for any desired treatment protocol, such as for IRE. 
     The present invention provides a system for estimating a 3-dimensional treatment volume based on the numerical models developed. Various embodiments of this system are summarized below to provide an illustration of the invention. 
     In one embodiment, the invention provides a system for estimating a 3-dimensional treatment volume for a medical treatment device that applies treatment energy through a plurality of electrodes defining a treatment area. The system comprises a memory, a display device, a processor coupled to the memory and the display device, and a treatment planning module stored in the memory and executable by the processor. In embodiments, the treatment planning module is adapted to generate an estimated first 3-dimensional treatment volume for display in the display device based on the ratio of a maximum conductivity of the treatment area to a baseline conductivity of the treatment area. 
     In embodiments of the invention, the treatment planning module can be configured such that it is capable of deriving the baseline conductivity and maximum conductivity of the treatment area based on a relationship of current as a function of W, X, Y and Z, in which: 
     W=voltage to distance ratio; 
     X=edge to edge distance between electrodes; 
     Y=exposure length of electrode. 
     Z=maximum conductivity/baseline conductivity. 
     In embodiments, the treatment planning module is operably configured such that it is capable of deriving baseline conductivity using a pre-treatment pulse. 
     In embodiments of the invention, the treatment planning module generates the estimated first 3-dimensional treatment volume using a numerical model analysis. 
     In embodiments of the invention, the numerical model analysis includes one or more of finite element analysis (FEA), Modified Analytical Solutions to the Laplace Equation, and other Analytical Equations (e.g., Ellipsoid, Cassini curve) that fit the shape of a specific Electric Field isocontour from the FEA models either by a look-up table or interpolating analytical approximations. 
     In embodiments of the invention, the treatment planning module generates the estimated first 3-dimensional treatment volume using a set of second 3-dimensional ablation volumes according to different W, X, Y and Z values, which have been predetermined by a numerical model analysis. 
     The treatment planning module can generate the estimated first 3-dimensional treatment volume using a set of second 3-dimensional ablation volumes according to different W, X, Y and Z values, which have been pre-determined by a numerical model analysis; and generating a set of interpolated third 3-dimensional volumes based on the predetermined set of second 3-dimensional ablation volumes. 
     In embodiments of the invention, the treatment planning module derives by curve fitting of one or more of: a mathematical function of x values of the ablation volume as a function of any one or more of W, X, Y and/or Z; a mathematical function of y values of the ablation volume as a function of any one or more of W, X, Y and/or Z; and a mathematical function of z values of the ablation volume as a function of any one or more of W, X, Y and/or Z. 
     In embodiments, the treatment planning module generates the estimated first 3-dimensional treatment volume using the three mathematical functions. 
     In embodiments of the invention, the treatment planning module is capable of measuring a baseline and maximum conductivity of the treatment area pre-treatment; capable of generating a fourth 3-dimensional treatment volume based on the measured baseline and maximum conductivity; and capable of displaying both the first and fourth 3-dimensional treatment volumes in the display device. 
     In embodiments, the treatment planning module superimposes one of the first and fourth 3-dimensional treatment volumes over the other to enable a physician to compare an estimated result to an actual estimated result. 
     Embodiments of the invention include a method for estimating a 3-dimensional treatment volume for a medical treatment device that applies treatment energy through a plurality of electrodes define a treatment area, wherein the steps of the method are executable through a processor, the method comprising: a) Determining the baseline electric conductivity; b) Determining the maximum electric conductivity; and c) Generating an estimated first 3-dimensional treatment volume based on the ratio of the maximum conductivity of the treatment area to the baseline conductivity of the treatment area. 
     Included within the scope of the invention is a method for estimating a target ablation zone for a medical treatment device that applies electrical treatment energy through a plurality of electrodes defining a target treatment area, the method comprising: determining a baseline electrical flow characteristic (EFC) in response to delivery of a test signal to tissue of a subject to be treated; determining, based on the baseline EFS, a second EFC representing an expected EFC during delivery of the electrical treatment energy to the target treatment area; and estimating the target ablation zone for display in the display device based on the second EFC. Such methods can include where the step of determining a baseline EFC includes determining an electrical conductivity. 
     In embodiments of the invention, the baseline electric conductivity can be determined by: i) Initiating a low voltage pre-IRE pulse through a probe; ii) Measuring the current of the low voltage pre-IRE pulse through the probe; iii) Optionally, scaling the current measured in step ii. to match a voltage-to-distance ratio value; and iv) Solving for factor in the following equation provided in Example 1 to determine the baseline electric conductivity 
         I =factor·[ aW+bX+cY+dZ+e ( W− W   )( X− X   )+ f ( W− W   )( Y− Y   )+ g ( W− W   )( Z− Z   )+ h ( X− X   )( Y− Y   )+ i ( X− X   )( Z− Z   )+ j ( Y− Y   )( Z− Z   )+ k ( W− W   )( X− X   )( Y− Y   )+ l ( X− X   )( Y− Y   )( Z− Z   )+ m ( W− W   )( Y− Y   )( Z− Z   )+ n ( W− W   )( X− X   )( Z− Z   )+ o ( W− W   )( X− X   )( Y− Y   )( Z− Z   )+ p ]
 
     In embodiments of the invention, at least one high voltage IRE pulse is initiated through the probe, such as after step b, to provide an IRE treatment. 
     In embodiments of the invention, the maximum electric conductivity is determined after the IRE treatment by: i) Providing a low voltage post-IRE pulse through the probe; ii) Measuring the current of the low voltage post-IRE pulse through the probe; iii) Optionally, scaling the current measured in step ii. to match a voltage-to-distance value; and iv) Solving for factor in the following equation provided in Example 1 to determine the maximum electric conductivity post-IRE 
         I =factor·[ aW+bX+cY+dZ+e ( W− W   )( X− X   )+ f ( W− W   )( Y− Y   )+ g ( W− W   )( Z− Z   )+ h ( X− X   )( Y− Y   )+ i ( X− X   )( Z− Z   )+ j ( Y− Y   )( Z− Z   )+ k ( W− W   )( X− X   )( Y− Y   )+ l ( X− X   )( Y− Y   )( Z− Z   )+ m ( W− W   )( Y− Y   )( Z− Z   )+ n ( W− W   )( X− X   )( Z− Z   )+ o ( W− W   )( X− X   )( Y− Y   )( Z− Z   )+ p ]
 
     In another embodiment, the invention provides a system for estimating a target ablation zone for a medical treatment device that applies electrical treatment energy through a plurality of electrodes defining a target treatment area, the system comprising a memory, a display device, a processor coupled to the memory and the display device; and a treatment planning module stored in the memory and executable by the processor, the treatment planning module adapted to receive a baseline electrical flow characteristic (EFC) in response to delivery of a test signal to tissue of a subject to be treated, determine a second EFC representing an expected EFC during delivery of the electrical treatment energy to the target treatment area, and estimating the target ablation zone for display in the display device based on the second EFC. 
     In embodiments of the invention, the EFC includes an electrical conductivity. 
     In embodiments of the invention, the second EFC includes a maximum conductivity expected during the delivery of the electrical treatment energy to the target treatment area; and the treatment planning module estimates the target ablation zone based on the ratio of the second EFC to the baseline EFC. 
     In embodiments, the treatment planning module estimates the second EFC to be a multiple of the baseline EFC which is greater than 1 and less than 6. 
     The treatment planning module is capable of estimating the second EFC to be a multiple of the baseline EFC which is greater than 3 and less than 4. 
     Additionally or alternatively, the treatment planning module delivers the test signal that includes a high frequency AC signal in the range of 500 kHz and 10 MHz. 
     In embodiments of the invention, the treatment planning module delivers the test signal that includes an excitation AC voltage signal of 1 mV to 10 mV. The test voltage that can be applied can be in the range of from about 1 mV to about 125 V, such as from 1 V to 5 V, or from 3 V to 50 V, such as from 10 V to 100 V, or 50-125 V. 
     The treatment planning module in embodiments delivers the test signal that includes a low voltage non-electroporating pre-IRE pulse of 10 V/cm to 100 V/cm. 
     The treatment planning module in embodiments delivers the test signal that includes a low voltage non-electroporating pre-IRE pulse of 25 V/cm to 75 V/cm. 
     In embodiments of the invention, the treatment planning module delivers the test signal that includes the high frequency AC signal and a DC pulse. 
     In embodiments of the invention, the treatment planning module determines a third EFC that represents an actual EFC during delivery of the electrical treatment energy for confirmation of the estimated target ablation zone. 
     In embodiments of the invention, the treatment planning module determines the second EFC based on W, X and Y, in which: 
     W=voltage to distance ratio; 
     X=edge to edge distance between electrodes; 
     Y=exposure length of electrode. 
     In embodiments of the invention the treatment planning module estimates the target ablation zone based on a set of predetermined ablation zones according to different W, X, Y and Z values. 
     The treatment planning module can estimate the target ablation zone by curve fitting: a mathematical function of x values of the ablation volume as a function of any one or more of W, X, Y and/or Z; a mathematical function of y values of the ablation volume as a function of any one or more of W, X, Y and/or Z; and/or a mathematical function of z values of the ablation volume as a function of any one or more of W, X, Y and/or Z. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings illustrate certain aspects of embodiments of the present invention, and should not be used to limit or define the invention. Together with the written description the drawings serve to explain certain principles of the invention. 
       Further, the application file contains at least one drawing executed in color. Copies of the patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee. 
         FIG.  1    is a schematic diagram of a representative system of the invention. 
         FIG.  2    is a schematic diagram of a representative treatment control computer of the invention. 
         FIG.  3    is schematic diagram illustrating details of the generator shown in the system of  FIG.  1   , including elements for detecting an over-current condition. 
         FIG.  4    is a flow chart illustrating a method, e.g., algorithm, of the invention. 
         FIG.  5    is a graph of the asymmetrical Gompertz function showing tissue electric conductivity as a function of electric field. 
         FIG.  6    is a graph showing a representative 3D plot of current [A] as a function of Z (σ max /σ o ) and voltage-to-distance ratio (W) for a separation distance of 1.5 cm and an electrode exposure length of 2.0 cm as used by Ben-David et al. 
         FIGS.  7 A and  7 B  are graphs showing representative contour plots of current [A] as a function of electrode exposure and separation distance using 1500 V/cm for Z=1 ( FIG.  7 A ) and Z=4 ( FIG.  7 B ). 
         FIGS.  8 A and  8 B  are tables showing Whole Model Parameter Estimates and Effect Tests, respectively. 
         FIG.  8 C  is a graph showing a plot of Actual Current vs. Predicted Current. 
         FIGS.  9 A- 9 E  are graphs showing the representative (15 mm gap) correlation between current vs. exposure length and electrode radius for maximum conductivities (1×-6×, respectively). 
         FIG.  10 A  is a table showing experimental validation of the code for determining the tissue/potato dynamic from in vitro measurements, referred to as potato experiment #1. 
         FIG.  10 B  is a table showing experimental validation of the code for determining the tissue/potato dynamic from in vitro measurements, referred to as potato experiment #2. 
         FIGS.  11 A and  11 B  are graphs plotting residual current versus data point for analytical shape factor ( FIG.  11 A ) and statistical (numerical) non-linear conductivity ( FIG.  11 B ). 
         FIGS.  12 A- 12 C  are graphs showing representative contour plots of the electric field strength at 1.0 cm from the origin using an edge-to-edge voltage-to-distance ratio of 1500 V/cm assuming z=1, wherein  FIG.  12 A  is a plot of the x-direction,  FIG.  12 B  is a plot of the y-direction, and  FIG.  12 C  is a plot of the z-direction. 
         FIGS.  13 A- 13 C  are 3D plots representing zones of ablation for a 1500 V/cm ratio, electrode exposure of 2 cm, and electrode separation of 1.5 cm, at respectively a 1000 V/cm IRE threshold ( FIG.  13 A ), 750 V/cm IRE threshold ( FIG.  13 B ), and 500 V/cm IRE threshold ( FIG.  13 C ) using the equation for an ellipsoid. 
         FIG.  14 A  is a schematic diagram showing an experimental setup of an embodiment of the invention. 
         FIG.  14 B  is a schematic diagram showing dimension labeling conventions. 
         FIG.  14 C  is a waveform showing 50 V pre-pulse electrical current at 1 cm separation, grid=0.25 A, where the lack of rise in intrapulse conductivity suggests no significant membrane electroporation during pre-pulse delivery. 
         FIG.  14 D  is a waveform showing electrical current for pulses 40-50 of 1750 V at 1 cm separation, grid=5 A, where progressive intrapulse current rise suggests continued conductivity increase and electroporation. 
         FIGS.  15 A and  15 B  are electric field [V/cm] isocontours for non-electroporated tissue ( FIG.  15 A ) and electroporated tissue ( FIG.  15 B ) maps assuming a maximum conductivity to baseline conductivity ratio of 7.0×. 
         FIGS.  16 A and  16 B  are representative Cassini Oval shapes when varying the ‘a=0.5 (red), 0.6 (orange), 0.7 (green), 0.8 (blue), 0.9 (purple), 1.0 (black)’ or ‘b=1.0 (red), 1.05 (orange), 1.1 (green), 1.15 (blue), 1.2 (purple), 1.25 (black)’ parameters individually. Note: If a&gt;1.0 or b&lt;1.0 the lemniscate of Bernoulli (the point where the two ellipses first connect (a=b=1) forming “∞”) disconnects forming non-contiguous shapes. 
         FIG.  17    is a graph showing NonlinearModelFit results for the ‘a’ and ‘b’ parameters used to generate the Cassini curves that represent the experimental IRE zones of ablation in porcine liver. 
         FIG.  18    shows Cassini curves from a ninety 100-μs pulse IRE treatment that represent the average zone of ablation (blue dashed), +SD (red solid), and −SD (black solid) according to a=0.821±0.062 and b=1.256±0.079. 
         FIG.  19    is a representation of the Finite Element Analysis (FEA) model for a 3D Electric Field [V/cm] Distribution in Non-Electroporated (Baseline) Tissue with 1.5-cm Electrodes at a Separation of 2.0 cm and with 3000 V applied. 
         FIGS.  20 A-D  are representations of the Electric Field [V/cm] Distributions from the 3D Non-Electroporated (Baseline) Models of  FIG.  19   , wherein  FIG.  20 A  represents the x-y plane mid-electrode length,  FIG.  20 B  represents the x-z plane mid-electrode diameter,  FIG.  20 C  represents the y-z plane mid-electrode diameter, and  FIG.  20 D  represents the y-z plane between electrodes. 
         FIG.  21    is a representation of the Finite Element Analysis (FEA) model for a 3D Electric Field [V/cm] Distribution in Electroporated Tissue with 1.5-cm Electrodes at a Separation of 2.0 cm and 3000 V applied assuming σ max /σ 0 =3.6. 
         FIGS.  22 A- 22 D  are representations of the Electric Field [V/cm] Distributions from the 3D Electroporated Models with 1.5-cm Electrodes at a Separation of 2.0 cm and 3000 V (cross-sections) assuming σ max /σ 0 =3.6, wherein  FIG.  22 A  represents the x-y plane mid-electrode length,  FIG.  22 B  represents the x-z plane mid-electrode diameter,  FIG.  22 C  represents the y-z plane mid-electrode diameter, and  FIG.  22 D  represents the y-z plane between electrodes. 
         FIG.  23    is a representative Cassini curve representing zones of ablation derived using the pre-pulse procedure to determine the ratio of maximum conductivity to baseline conductivity. For comparison purposes the baseline electric field isocontour is also presented in which no electroporation is taken into account. 
     
    
    
     DETAILED DESCRIPTION OF VARIOUS EMBODIMENTS OF THE INVENTION 
     Reference will now be made in detail to various exemplary embodiments of the invention. Embodiments described in the description and shown in the figures are illustrative only and are not intended to limit the scope of the invention. Changes may be made in the specific embodiments described in this specification and accompanying drawings that a person of ordinary skill in the art will recognize are within the scope and spirit of the invention. 
     Throughout the present teachings, any and all of the features and/or components disclosed or suggested herein, explicitly or implicitly, may be practiced and/or implemented in any combination, whenever and wherever appropriate as understood by one of ordinary skill in the art. The various features and/or components disclosed herein are all illustrative for the underlying concepts, and thus are non-limiting to their actual descriptions. Any means for achieving substantially the same functions are considered as foreseeable alternatives and equivalents, and are thus fully described in writing and fully enabled. The various examples, illustrations, and embodiments described herein are by no means, in any degree or extent, limiting the broadest scopes of the claimed inventions presented herein or in any future applications claiming priority to the instant application. 
     One embodiment of the present invention is illustrated in  FIGS.  1  and  2   . Representative components that can be used with the present invention can include one or more of those that are illustrated in  FIG.  1   . For example, in embodiments, one or more probes  22  can be used to deliver therapeutic energy and are powered by a voltage pulse generator  10  that generates high voltage pulses as therapeutic energy such as pulses capable of irreversibly electroporating the tissue cells. In the embodiment shown, the voltage pulse generator  10  includes six separate receptacles for receiving up to six individual probes  22  which are adapted to be plugged into the respective receptacle. The receptacles are each labeled with a number in consecutive order. In other embodiments, the voltage pulse generator can have any number of receptacles for receiving more or less than six probes. 
     For example, a treatment protocol according to the invention could include a plurality of electrodes. According to the desired treatment pattern, the plurality of electrodes can be disposed in various positions relative to one another. In a particular example, a plurality of electrodes can be disposed in a relatively circular pattern with a single electrode disposed in the interior of the circle, such as at approximately the center. Any configuration of electrodes is possible and the arrangement need not be circular but any shape periphery can be used depending on the area to be treated, including any regular or irregular polygon shape, including convex or concave polygon shapes. The single centrally located electrode can be a ground electrode while the other electrodes in the plurality can be energized. Any number of electrodes can be in the plurality such as from about 1 to 20. Indeed, even 3 electrodes can form a plurality of electrodes where one ground electrode is disposed between two electrodes capable of being energized, or 4 electrodes can be disposed in a manner to provide two electrode pairs (each pair comprising one ground and one electrode capable of being energized). During treatment, methods of treating can involve energizing the electrodes in any sequence, such as energizing one or more electrode simultaneously, and/or energizing one or more electrode in a particular sequence, such as sequentially, in an alternating pattern, in a skipping pattern, and/or energizing multiple electrodes but less than all electrodes simultaneously, for example. 
     In the embodiment shown, each probe  22  includes either a monopolar electrode or bipolar electrodes having two electrodes separated by an insulating sleeve. In one embodiment, if the probe includes a monopolar electrode, the amount of exposure of the active portion of the electrode can be adjusted by retracting or advancing an insulating sleeve relative to the electrode. See, for example, U.S. Pat. No. 7,344,533, which is incorporated by reference herein in its entirety. The pulse generator  10  is connected to a treatment control computer  40  having input devices such as keyboard  12  and a pointing device  14 , and an output device such as a display device  11  for viewing an image of a target treatment area such as a lesion  300  surrounded by a safety margin  301 . The therapeutic energy delivery device  22  is used to treat a lesion  300  inside a patient  15 . An imaging device  30  includes a monitor  31  for viewing the lesion  300  inside the patient  15  in real time. Examples of imaging devices  30  include ultrasonic, CT, MRI and fluoroscopic devices as are known in the art. 
     The present invention includes computer software (treatment planning module  54 ) which assists a user to plan for, execute, and review the results of a medical treatment procedure, as will be discussed in more detail below. For example, the treatment planning module  54  assists a user to plan for a medical treatment procedure by enabling a user to more accurately position each of the probes  22  of the therapeutic energy delivery device  20  in relation to the lesion  300  in a way that will generate the most effective treatment zone. The treatment planning module  54  can display the anticipated treatment zone based on the position of the probes and the treatment parameters. The treatment planning module  54  can display the progress of the treatment in real time and can display the results of the treatment procedure after it is completed. This information can be displayed in a manner such that it can be used for example by a treating physician to determine whether the treatment was successful and/or whether it is necessary or desirable to re-treat the patient. 
     For purposes of this application, the terms “code”, “software”, “program”, “application”, “software code”, “computer readable code”, “software module”, “module” and “software program” are used interchangeably to mean software instructions that are executable by a processor. The “user” can be a physician or other medical professional. The treatment planning module  54  executed by a processor outputs various data including text and graphical data to the monitor  11  associated with the generator  10 . 
     Referring now to  FIG.  2   , the treatment control computer  40  of the present invention manages planning of treatment for a patient. The computer  40  is connected to the communication link  52  through an I/O interface  42  such as a USB (universal serial bus) interface, which receives information from and sends information over the communication link  52  to the voltage generator  10 . The computer  40  includes memory storage  44  such as RAM, processor (CPU)  46 , program storage  48  such as ROM or EEPROM, and data storage  50  such as a hard disk, all commonly connected to each other through a bus  53 . The program storage  48  stores, among others, a treatment planning module  54  which includes a user interface module that interacts with the user in planning for, executing and reviewing the result of a treatment. Any of the software program modules in the program storage  48  and data from the data storage  50  can be transferred to the memory  44  as needed and is executed by the CPU  46 . 
     In one embodiment, the computer  40  is built into the voltage generator  10 . In another embodiment, the computer  40  is a separate unit which is connected to the voltage generator through the communications link  52 . In a preferred embodiment, the communication link  52  is a USB link. In one embodiment, the imaging device  30  is a standalone device which is not connected to the computer  40 . In the embodiment as shown in  FIG.  1   , the computer  40  is connected to the imaging device  30  through a communications link  53 . As shown, the communication link  53  is a USB link. In this embodiment, the computer can determine the size and orientation of the lesion  300  by analyzing the data such as the image data received from the imaging device  30 , and the computer  40  can display this information on the monitor  11 . In this embodiment, the lesion image generated by the imaging device  30  can be directly displayed on the grid (not shown) of the display device (monitor)  11  of the computer running the treatment planning module  54 . This embodiment would provide an accurate representation of the lesion image on the grid, and may eliminate the step of manually inputting the dimensions of the lesion in order to create the lesion image on the grid. This embodiment would also be useful to provide an accurate representation of the lesion image if the lesion has an irregular shape. 
     It should be noted that the software can be used independently of the pulse generator  10 . For example, the user can plan the treatment in a different computer as will be explained below and then save the treatment parameters to an external memory device, such as a USB flash drive (not shown). The data from the memory device relating to the treatment parameters can then be downloaded into the computer  40  to be used with the generator  10  for treatment. Additionally, the software can be used for hypothetical illustration of zones of ablation for training purposes to the user on therapies that deliver electrical energy. For example, the data can be evaluated by a human to determine or estimate favorable treatment protocols for a particular patient rather than programmed into a device for implementing the particular protocol. 
       FIG.  3    illustrates one embodiment of a circuitry to detect an abnormality in the applied pulses such as a high current, low current, high voltage or low voltage condition. This circuitry is located within the generator  10  (see  FIG.  1   ). A USB connection  52  carries instructions from the user computer  40  to a controller  71 . The controller can be a computer similar to the computer  40  as shown in  FIG.  2   . The controller  71  can include a processor, ASIC (application-specific integrated circuit), microcontroller or wired logic. The controller  71  then sends the instructions to a pulse generation circuit  72 . The pulse generation circuit  72  generates the pulses and sends electrical energy to the probes. For clarity, only one pair of probes/electrodes are shown. However, the generator  10  can accommodate any number of probes/electrodes (e.g., from 1-10, such as 6 probes) and energizing multiple electrodes simultaneously for customizing the shape of the ablation zone. In the embodiment shown, the pulses are applied one pair of electrodes at a time, and then switched to another pair. The pulse generation circuit  72  includes a switch, preferably an electronic switch, that switches the probe pairs based on the instructions received from the computer  40 . A sensor  73  such as a sensor can sense the current or voltage between each pair of the probes in real time and communicate such information to the controller  71 , which in turn, communicates the information to the computer  40 . If the sensor  73  detects an abnormal condition during treatment such as a high current or low current condition, then it will communicate with the controller  71  and the computer  40  which may cause the controller to send a signal to the pulse generation circuit  72  to discontinue the pulses for that particular pair of probes. The treatment planning module  54  can further include a feature that tracks the treatment progress and provides the user with an option to automatically retreat for low or missing pulses, or over-current pulses (see discussion below). Also, if the generator stops prematurely for any reason, the treatment planning module  54  can restart at the same point where it terminated, and administer the missing treatment pulses as part of the same treatment. In other embodiments, the treatment planning module  54  is able to detect certain errors during treatment, which include, but are not limited to, “charge failure”, “hardware failure”, “high current failure”, and “low current failure”. 
     General treatment protocols for the destruction (ablation) of undesirable tissue through electroporation are known. They involve the insertion (bringing) electroporation electrodes to the vicinity of the undesirable tissue and in good electrical contact with the tissue and the application of electrical pulses that cause irreversible electroporation of the cells throughout the entire area of the undesirable tissue. The cells whose membrane was irreversible permeabilized may be removed or left in situ (not removed) and as such may be gradually removed by the body&#39;s immune system. Cell death is produced by inducing the electrical parameters of irreversible electroporation in the undesirable area. 
     Electroporation protocols involve the generation of electrical fields in tissue and are affected by the Joule heating of the electrical pulses. When designing tissue electroporation protocols it is important to determine the appropriate electrical parameters that will maximize tissue permeabilization without inducing deleterious thermal effects. It has been shown that substantial volumes of tissue can be electroporated with reversible electroporation without inducing damaging thermal effects to cells and has quantified these volumes (Davalos, R. V., B. Rubinsky, and L. M. Mir, Theoretical analysis of the thermal effects during in vivo tissue electroporation. Bioelectrochemistry, 2003. Vol. 61(1-2): p. 99-107). 
     The electrical pulses used to induce irreversible electroporation in tissue are typically larger in magnitude and duration from the electrical pulses required for reversible electroporation. Further, the duration and strength of the pulses for irreversible electroporation are different from other methodologies using electrical pulses such as for intracellular electro-manipulation or thermal ablation. The methods are very different even when the intracellular (nano-seconds) electro-manipulation is used to cause cell death, e.g. ablate the tissue of a tumor or when the thermal effects produce damage to cells causing cell death. 
     Typical values for pulse length for irreversible electroporation are in a range of from about 5 microseconds to about 62,000 milliseconds or about 75 microseconds to about 20,000 milliseconds or about 100 microseconds±10 microseconds. This is significantly longer than the pulse length generally used in intracellular (nano-seconds) electro-manipulation which is 1 microsecond or less—see published U.S. application 2002/0010491 published Jan. 24, 2002. 
     The pulse is typically administered at voltage of about 100 V/cm to 7,000 V/cm or 200 V/cm to 2000 V/cm or 300 V/cm to 1000 V/cm about 600 V/cm for irreversible electroporation. This is substantially lower than that used for intracellular electro-manipulation which is about 10,000 V/cm, see U.S. application 2002/0010491 published Jan. 24, 2002. 
     The voltage expressed above is the voltage gradient (voltage per centimeter). The electrodes may be different shapes and sizes and be positioned at different distances from each other. The shape may be circular, oval, square, rectangular or irregular etc. The distance of one electrode to another may be 0.5 to 10 cm, 1 to 5 cm, or 2-3 cm. The electrode may have a surface area of 0.1-5 sq. cm or 1-2 sq. cm. 
     The size, shape and distances of the electrodes can vary and such can change the voltage and pulse duration used. Those skilled in the art will adjust the parameters in accordance with this disclosure to obtain the desired degree of electroporation and avoid thermal damage to surrounding cells. 
     Additional features of protocols for electroporation therapy are provided in U.S. Patent Application Publication No. US 2007/0043345 A1, the disclosure of which is hereby incorporated by reference in its entirety. 
     The present invention provides systems and methods for estimating a 3-dimensional treatment volume for a medical treatment device that applies treatment energy through a plurality of electrodes defining a treatment area. The systems and methods are based in part on calculation of the ratio of a maximum conductivity of the treatment area to a baseline conductivity of the treatment area, and may be used to determine effective treatment parameters for electroporation-based therapies. The present inventors have recognized that the baseline and maximum conductivities of the tissue should be determined before the therapy in order to determine safe and effective pulse protocols. 
     The numerical models and algorithms of the invention, as provided in the Examples, such as Equation 3 of Example 1 can be implemented in a system for estimating a 3-dimensional treatment volume for a medical treatment device that applies treatment energy through a plurality of electrodes defining a treatment area. In one embodiment, the numerical models and algorithms are implemented in an appropriate computer readable code as part of the treatment planning module  54  of the system of the invention. Computing languages available to the skilled artisan for programming the treatment planning module  54  include general purpose computing languages such as the C and related languages, and statistical programming languages such as the “S” family of languages, including R and S-Plus. The computer readable code may be stored in a memory  44  of the system of the invention. A processor  46  is coupled to the memory  44  and a display device  11  and the treatment planning module  54  stored in the memory  44  is executable by the processor  46 . The treatment planning module  54 , through the implemented numerical models, is adapted to generate an estimated first 3-dimensional treatment volume for display in the display device  11  based on the ratio of a maximum conductivity of the treatment area to a baseline conductivity of the treatment area (Z). 
     In one embodiment, the invention provides for a system for estimating a 3-dimensional treatment volume for a medical treatment device that applies treatment energy through a plurality of electrodes  22  defining a treatment area, the system comprising a memory  44 , a display device  11 , a processor  46  coupled to the memory  44  and the display device  11 , and a treatment planning module  54  stored in the memory  44  and executable by the processor  46 , the treatment planning module  54  adapted to generate an estimated first 3-dimensional treatment volume for display in the display device  11  based on the ratio of a maximum conductivity of the treatment area to a baseline conductivity of the treatment area. 
     The foregoing description provides additional instructions and algorithms for a computer programmer to implement in computer readable code a treatment planning module  54  that may be executable through a processor  46  to generate an estimated 3-dimensional treatment volume for display in the display device  11  based on the ratio of a maximum conductivity of the treatment area to a baseline conductivity of the treatment area. 
     The treatment planning module  54  may derive the baseline conductivity and maximum conductivity of the treatment area based on a relationship of current as a function of W, X, and Y, in which: 
     W=voltage to distance ratio; 
     X=edge to edge distance between electrodes; and 
     Y=exposure length of electrode. 
     The treatment planning module  54  may derive the baseline conductivity using a pre-treatment pulse. 
     The treatment planning module  54  may generate the estimated first 3-dimensional treatment volume using a numerical model analysis such as described in the Examples. The numerical model analysis may include finite element analysis (FEA). 
     The treatment planning module  54  may generate the estimated first 3-dimensional treatment volume using a set of second 3-dimensional ablation volumes according to different W, X, Y and Z values, which have been predetermined by the numerical model analysis. 
     The treatment planning module  54  may generate the estimated first 3-dimensional treatment volume using a set of second 3-dimensional ablation volumes according to different W, X, Y and Z values, which have been pre-determined by a numerical model analysis; and generating a set of interpolated third 3-dimensional volumes based on the predetermined set of second 3-dimensional ablation volumes. 
     The treatment planning module  54  may derive by curve fitting: a mathematical function of x values of the ablation volume as a function of any one or more of W, X, Y and/or Z; a mathematical function of y values of the ablation volume as a function of any one or more of W, X, Y and/or Z; and/or a mathematical function of z values of the ablation volume as a function of any one or more of W, X, Y and/or Z. 
     The treatment planning module  54  may generate the estimated first 3-dimensional treatment volume using the three mathematical functions. 
     The treatment planning module  54  may: measure a baseline and maximum conductivity of the treatment area; generate a fourth 3-dimensional treatment volume based on the measured baseline and maximum conductivity; and optionally display one or both the first and fourth 3-dimensional treatment volumes in the display device  11 . 
     The treatment planning module  54  may superimpose one of the first and fourth 3-dimensional treatment volumes over the other so as to enable a physician to compare an estimated result to an actual estimated result. 
     In another embodiment, the invention provides a system for estimating a target ablation zone for a medical treatment device that applies electrical treatment energy through a plurality of electrodes  22  defining a target treatment area, the system comprising a memory  44 , a display device  11 , a processor  46  coupled to the memory  44  and the display device  11 ; and a treatment planning module  54  stored in the memory  44  and executable by the processor  46 , the treatment planning module  54  adapted to receive a baseline electrical flow characteristic (EFC) in response to delivery of a test signal to tissue of a subject  15  to be treated, determine a second EFC representing an expected EFC during delivery of the electrical treatment energy to the target treatment area, and estimating the target ablation zone for display in the display device  11  based on the second EFC. The EFC may include an electrical conductivity. 
     The second EFC may include a maximum conductivity expected during the delivery of the electrical treatment energy to the target treatment area; and the treatment planning module estimates the target ablation zone based on the ratio of the second EFC to the baseline EFC. 
     The treatment planning module  54  may estimate the second EFC to be a multiple of the baseline EFC which is greater than 1 and less than 6. 
     The treatment planning module  54  may estimate the second EFC to be a multiple of the baseline EFC which is greater than 3 and less than 4. 
     The treatment planning module  54  may deliver the test signal that includes a high frequency AC signal in the range of 500 kHz and 10 MHz. 
     The treatment planning module  54  may deliver the test signal that includes an excitation AC voltage signal of 1 to 10 mV, such as from 1 mV to 125 V, including for example from about 1 to 5 V, or from about 10-50 V, or from about 100-125 V. 
     The treatment planning module may deliver the test signal that includes a low voltage non-electroporating pre-IRE pulse of 10 V/cm to 100 V/cm. 
     The treatment planning module may deliver the test signal that includes a low voltage non-electroporating pre-IRE pulse of 25 V/cm to 75 V/cm. 
     The treatment planning module  54  may deliver the test signal that includes the high frequency AC signal and a DC pulse. 
     The treatment planning module  54  may determines a third EFC that represents an actual EFC during delivery of the electrical treatment energy for confirmation of the estimated target ablation zone. 
     The treatment planning module  54  may determine the second EFC based on W, X and Y, in which: W=voltage to distance ratio; X=edge to edge distance between electrodes; Y=exposure length of electrode. 
     The treatment planning module  54  may be operably configured to estimate the target ablation zone based on a set of predetermined ablation zones according to different W, X, Y and Z values. 
     The treatment planning module may estimate the target ablation zone by curve fitting: a mathematical function of x values of the ablation volume as a function of any one or more of W, X, Y and/or Z; a mathematical function of y values of the ablation volume as a function of any one or more of W, X, Y and/or Z; and/or a mathematical function of z values of the ablation volume as a function of any one or more of W, X, Y and/or Z. 
     The treatment planning module  54  is programmed to execute the algorithms disclosed herein through the processor  46 . In one embodiment, the treatment planning module  54  is programmed to execute the following algorithm  1000 , as shown in  FIG.  4   , in computer readable code through the processor  46 : 
     Initiating a low voltage pre-IRE pulse through a probe  1100 ; 
     Measuring the current of the low voltage pre-IRE pulse through probe  1200 ; 
     Optionally, scaling the current measured in step  1200  to match a voltage-to-distance ratio value  1300 ; 
     Solving  1400  for factor in the following equation provided in Example 1 to determine the baseline electric conductivity 
         I =factor·[ aW+bX+cY+dZ+e ( W− W   )( X− X   )+ f ( W− W   )( Y− Y   )+ g ( W− W   )( Z− Z   )+ h ( X− X   )( Y− Y   )+ i ( X− X   )( Z− Z   )+ j ( Y− Y   )( Z− Z   )+ k ( W− W   )( X− X   )( Y− Y   )+ l ( X− X   )( Y− Y   )( Z− Z   )+ m ( W− W   )( Y− Y   )( Z− Z   )+ n ( W− W   )( X− X   )( Z− Z   )+ o ( W− W   )( X− X   )( Y− Y   )( Z− Z   )+ p ]
 
     Providing at least one high voltage IRE pulse through the probe to provide an IRE treatment  1500 ; 
     Providing a low voltage post-IRE pulse through the probe  1600 ; 
     Measuring the current of the low voltage post-IRE pulse through probe  1700 ; 
     Optionally, scaling the current measured in step  1700  to match a voltage-to-distance value  1800 ; 
     Solving  1900  for factor in the following equation provided in Example 1 to determine the maximum electric conductivity post-IRE 
         I =factor·[ aW+bX+cY+dZ+e ( W− W   )( X− X   )+ f ( W− W   )( Y− Y   )+ g ( W− W   )( Z− Z   )+ h ( X− X   )( Y− Y   )+ i ( X− X   )( Z− Z   )+ j ( Y− Y   )( Z− Z   )+ k ( W− W   )( X− X   )( Y− Y   )+ l ( X− X   )( Y− Y   )( Z− Z   )+ m ( W− W   )( Y− Y   )( Z− Z   )+ n ( W− W   )( X− X   )( Z− Z   )+ o ( W− W   )( X− X   )( Y− Y   )( Z− Z   )+ p ]
 
     Generating an estimated first 3-dimensional treatment volume based on the ratio of the maximum conductivity of the treatment area to the baseline conductivity of the treatment area  2000 . 
     The steps in the algorithm  1000  shown in  FIG.  4    need not be followed exactly as shown. For example, it may be desirable to eliminate one or both scaling steps  1300 ,  1800 . It may also be desirable to introduce or substitute steps, such as, for example, providing a mathematical function of x, y, z values of the 3-dimensional treatment volume as a function of any one or more of W, X, Y, and/or Z by curve fitting, and estimating the 3-dimensional treatment volume using the three mathematical functions, or introducing additional steps disclosed herein. 
     For example, specific method embodiments of the invention include a method for estimating a target ablation zone for a medical treatment device that applies electrical treatment energy through a plurality of electrodes defining a target treatment area, the method comprising: determining a baseline electrical flow characteristic (EFC) in response to delivery of a test signal to tissue of a subject to be treated; determining, based on the baseline EFS, a second EFC representing an expected EFC during delivery of the electrical treatment energy to the target treatment area; and estimating the target ablation zone for display in the display device based on the second EFC. 
     Such methods can include where the step of determining a baseline EFC includes determining an electrical conductivity. 
     Additionally or alternatively, such methods can include where the step of determining a second EFC includes determining an expected maximum electrical conductivity during delivery of the electrical treatment energy to the target treatment area; and the step of estimating includes estimating the target ablation zone based on the ratio of the second EFC to the baseline EFC. 
     Even further, the treatment planning module in method and system embodiments can estimate the second EFC to be a multiple of the baseline EFC which is greater than 1 and less than 6. For example, the treatment planning module can estimate the second EFC to be a multiple of the baseline EFC which is greater than 3 and less than 4. 
     Methods of the invention are provided wherein the treatment planning module delivers the test signal that includes a high frequency AC signal in the range of 500 kHz and 10 MHz. Alternatively or in addition methods can include wherein the treatment planning module delivers the test signal that includes the high frequency AC signal and a DC pulse. 
     In method embodiments of the invention, the treatment planning module determines a third EFC that represents an actual EFC during delivery of the electrical treatment energy for confirmation of the estimated target ablation zone. 
     Methods of the invention can comprise displaying results of treatment in a manner which indicates whether treatment was successful or whether further treatment is needed. Method steps can include further delivering electrical treatment energy to a target tissue or object in response to data obtained in the determining and/or estimating steps of methods of the invention. 
     According to embodiments, a method is provided for estimating a target ablation zone, the method comprising: determining a baseline electrical flow characteristic (EFC) in response to delivery of a test signal to a target area of an object; determining, based on the baseline EFS, a second EFC representing an expected EFC during delivery of the electrical treatment energy to the target area; estimating the target ablation zone for display in the display device based on the second EFC. For example, according to such methods the object can be a biological object, such as tissue, a non-biological object, such as a phantom, or any object or material such as plant material. 
     Systems and methods of the invention can comprise a treatment planning module adapted to estimate the target ablation zone based in part on electrode radius and/or a step of estimating the target ablation zone based in part on electrode radius. 
     In embodiments of the methods, the treatment planning module determines the second EFC based on W, X and Y, in which: W=voltage to distance ratio; X=edge to edge distance between electrodes; and Y=exposure length of electrode. 
     The treatment planning module of method embodiments can estimate the target ablation zone based on a set of predetermined ablation zones according to different W, X and Y values. 
     Method embodiments further include that the treatment planning module estimates the target ablation zone by curve fitting: a mathematical function of x values of the ablation volume as a function of any one or more of W, X and/or Y; a mathematical function of y values of the ablation volume as a function of any one or more of W, X and/or Y; and a mathematical function of z values of the ablation volume as a function of any one or more of W, X and/or Y. 
     The system may be further configured to include software for displaying a Graphical User Interface in the display device with various screens for input and display of information, including those for Information, Probe Selection, Probe Placement Process, and Pulse Generation as described in International Patent Application Publication WO 2010/117806 A1, the disclosure of which is hereby incorporated by reference in its entirety. 
     Additional details of the algorithms and numerical models disclosed herein will be provided in the following Examples, which are intended to further illustrate rather than limit the invention. 
     In Example 1, the present inventors provide a numerical model that uses an asymmetrical Gompertz function to describe the response of porcine renal tissue to electroporation pulses. However, other functions could be used to represent the electrical response of tissue under exposure to pulsed electric fields such as a sigmoid function, ramp, and/or interpolation table. This model can be used to determine baseline conductivity of tissue based on any combination of electrode exposure length, separation distance, and non-electroporating electric pulses. In addition, the model can be scaled to the baseline conductivity and used to determine the maximum electric conductivity after the electroporation-based treatment. By determining the ratio of conductivities pre- and post-treatment, it is possible to predict the shape of the electric field distribution and thus the treatment volume based on electrical measurements. An advantage of this numerical model is that it is easy to implement in computer software code in the system of the invention and no additional electronics or numerical simulations are needed to determine the electric conductivities. The system and method of the invention can also be adapted for other electrode geometries (sharp electrodes, bipolar probes), electrode diameter, and other tissues/tumors once their response to different electric fields has been fully characterized. 
     The present inventors provide further details of this numerical modeling as well as experiments that confirm this numerical modeling in Example 2. In developing this work, the present inventors were motivated to develop an IRE treatment planning method and system that accounts for real-time voltage/current measurements. As a result of this work, the system and method of the invention requires no electronics or electrodes in addition to the NANOKNIFE® System, a commercial embodiment of a system for electroporation-based therapies. The work shown in Example 2 is based on parametric study using blunt tip electrodes, but can be customized to any other geometry (sharp, bipolar). The numerical modeling in Example 2 provides the ability to determine a baseline tissue conductivity based on a low voltage pre-IRE pulse (non-electroporating ˜50 V/cm), as well as the maximum tissue conductivity based on high voltage IRE pulses (during electroporation) and low voltage post-IRE pulse (non-electroporating ˜50 V/cm). Two numerical models were developed that examined 720 or 1440 parameter combinations. Results on IRE lesion were based on in vitro measurements. A major finding of the modeling in Example 2 is that the electric field distribution depends on conductivity ratio pre- and post-IRE. Experimental and clinical IRE studies may be used to determine this ratio. As a result, one can determine e-field thresholds for tissue and tumor based on measurements. The 3-D model of Example 2 captures depth, width, and height e-field distributions. 
     In Example 3, as a further extension of the inventors work, the inventors show prediction of IRE treatment volume based on 1000 V/cm, 750 v/cm, and 500 V/cm IRE thresholds as well as other factors as a representative case of the numerical modeling of the invention. 
     In Example 4, the inventors describe features of the Specific Conductivity and procedures for implementing it in the invention. 
     In Example 5, the inventors describe in vivo experiments as a reduction to practice of the invention. 
     In Example 6, the inventors describe how to use the ratio of maximum conductivity to baseline conductivity in modifying the electric field distribution and thus the Cassini oval equation. 
     In Example 7, the inventors describe the Cassini oval equation and its implementation in the invention. 
     EXAMPLES 
     Example 1 
     Materials And Methods 
     The tissue was modeled as a 10-cm diameter spherical domain using a finite element package (Comsol 4.2a, Stockholm, Sweden). Electrodes were modeled as two 1.0-mm diameter blunt tip needles with exposure lengths (Y) and edge-to-edge separation distances (X) given in Table 1. The electrode domains were subtracted from the tissue domain, effectively modeling the electrodes as boundary conditions. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Electrode configuration and relevant electroporation- 
               
               
                 based treatment values used in study. 
               
            
           
           
               
               
               
            
               
                   
                 PARAMETER VALUES 
                 MEAN 
               
               
                   
                   
               
            
           
           
               
               
               
               
            
               
                   
                 W [V/cm] 
                 500, 1000, 1500, 2000, 
                 1750 
               
               
                   
                   
                 2500, 3000 
               
               
                   
                 X [cm] 
                 0.5, 1.0, 1.5, 2.0, 2.5 
                 1.5 
               
               
                   
                 Y [cm] 
                 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 
                 1.75 
               
               
                   
                 Z [cm] 
                 1.0, 1.25, 1.5, 2.0, 3.0, 4.0, 
                 2.96875 
               
               
                   
                   
                 5.0, 6.0 
               
               
                   
                   
               
            
           
         
       
     
     The electric field distribution associated with the applied pulse is given by solving the Laplace equation: 
       ∇·(σ(| E |)∇φ)=0   (1)
 
     where σ is the electrical conductivity of the tissue, E is the electric field in V/cm, and φ is the electrical potential (Edd and Davalos, 2007). Boundaries along the tissue in contact with the energized electrode were defined as φ=V o , and boundaries at the interface of the other electrode were set to ground. The applied voltages were manipulated to ensure that the voltage-to-distance ratios (W) corresponded to those in Table 1. The remaining boundaries were treated as electrically insulating, ∂φ/∂n=0. 
     The analyzed domain extends far enough from the area of interest (i.e. the area near the electrodes) that the electrically insulating boundaries at the edges of the domain do not significantly influence the results in the treatment zone. The physics-controlled finer mesh with ˜100,000 elements was used. The numerical models have been adapted to account for a dynamic tissue conductivity that occurs as a result of electroporation, which is described by an asymmetrical Gompertz curve for renal porcine tissue (Neal et al., 2012): 
       σ(| E |)=σ o +(σ max −σ o )exp[− A ·exp[− B·E ]  (2)
 
     where σ o  is the non-electroporated tissue conductivity and σ max  is the maximum conductivity for thoroughly permeabilized cells, A and B are coefficients for the displacement and growth rate of the curve, respectively. Here, it is assumed that σ o =0.1 S/m but this value can be scaled by a factor to match any other non-electroporated tissue conductivity or material as determined by a pre-treatment pulse. In this work the effect of the ratio of maximum conductivity to baseline conductivity in the resulting electric current was examined using the 50-μs pulse parameters (A=3.05271; B=0.00233) reported by Neal et al. (Neal et. al., 2012). The asymmetrical Gompertz function showing the tissue electric conductivity as a function of electric field is for example shown in  FIG.  5   . 
     The current density was integrated over the surface of the ground electrode to determine the total current delivered. A regression analysis on the resulting current was performed to determine the effect of the parameters investigated and their interactions using the NonlinearModelFit function in Wolfram Mathematica 8.0. Current data from the numerical simulations were fit to a mathematical expression that accounted for all possible interactions between the parameters: 
         I =factor·[ aW+bX+cY+dZ+e ( W− W   )( X− X   )+ f ( W− W   )( Y− Y   )+ g ( W− W   )( Z− Z   )+ h ( X− X   )( Y− Y   )+ i ( X− X   )( Z− Z   )+ j ( Y− Y   )( Z− Z   )+ k ( W− W   )( X− X   )( Y− Y   )+ l ( X− X   )( Y− Y   )( Z− Z   )+ m ( W− W   )( Y− Y   )( Z− Z   )+ n ( W− W   )( X− X   )( Z− Z   )+ o ( W− W   )( X− X   )( Y− Y   )( Z− Z   )+ p ]  (3)
 
     where I is the current in amps, W is the voltage-to-distance ratio [V/cm], X is the edge-to-edge distance [cm], Y is the exposure length [cm], and Z is the unitless ratio σ max /σ o . The  W ,  X ,  Y , and  Z  are means for each of their corresponding parameters (Table 1) and the coefficients (a, b, c, . . . , n, o, p) were determined from the regression analysis (Table 2). 
     Results. 
     A method to determine electric conductivity change following treatment based on current measurements and electrode configuration is provided. The best-fit statistical (numerical) model between the W, X, Y, and Z parameters resulted in Eqn. 3 with the coefficients in Table 2 (R 2 =0.999646). Every coefficient and their interactions had statistical significant effects on the resulting current (P&lt;0.0001*). With this equation one can predict the current for any combination of the W, Y, X, Z parameters studied within their ranges (500 V/cm≤W≤3000 V/cm, 0.5 cm≤X≤2.5 cm, 0.5 cm≤Y≤3.0 cm, and 1.0≤Z≤6.0). Additionally, by using the linear results (Z=1), the baseline tissue conductivity can be extrapolated for any blunt-tip electrode configuration by delivering and measuring the current of a non-electroporating pre-treatment pulse. The techniques described in this specification could also be used to determine the conductivity of other materials, such as non-biological materials, or phantoms. 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Coefficients (P &lt; 0.0001*) from the Least Square analysis 
               
               
                 using the NonlinearModelFit function in Mathematica. 
               
            
           
           
               
               
            
               
                   
                 ESTIMATE 
               
               
                   
                   
               
            
           
           
               
               
               
            
               
                   
                 a → 
                 0.00820 
               
               
                   
                 b → 
                 7.18533 
               
               
                   
                 c → 
                 5.80997 
               
               
                   
                 d → 
                 3.73939 
               
               
                   
                 e → 
                 0.00459 
               
               
                   
                 f → 
                 0.00390 
               
               
                   
                 g → 
                 0.00271 
               
               
                   
                 h → 
                 3.05537 
               
               
                   
                 i → 
                 2.18763 
               
               
                   
                 j → 
                 1.73269 
               
               
                   
                 k → 
                 0.00201 
               
               
                   
                 l → 
                 0.92272 
               
               
                   
                 m → 
                 0.00129 
               
               
                   
                 n → 
                 0.00152 
               
               
                   
                 o → 
                 0.00067 
               
               
                   
                 p → 
                 −33.92640 
               
               
                   
                   
               
            
           
         
       
     
       FIG.  6    shows a representative case in which the effect of the W and Z are studied for electroporation-based therapies with 2.0 cm electrodes separated by 1.5 cm. The 3D plot corroborates the quality of the model which shows every data point from the numerical simulation (green spheres) being intersected by the best-fit statistical (numerical) model. This 3D plot also shows that when Z is kept constant, the current increases linearly with the voltage-to-distance ratio (W). Similarly, the current increases linearly with Z when the voltage-to-distance ratio is constant. However, for all the other scenarios there is a non-linear response in the current that becomes more drastic with simultaneous increases in W and Z. 
     In order to fully understand the predictive capability of the statistical (numerical) model, two cases in which the current is presented as a function of the exposure length and electrode separation are provided.  FIG.  7 A  shows the linear case (Z=1) in which the current can be scaled to predict any other combination of pulse parameters as long as the pulses do not achieve electroporation. For example, one can deliver a non-electroporation pulse (˜50 V/cm) and measure current. The current can then be scaled to match one of the W values investigated in this study. By using Eqn. 3 and solving for the factor, the baseline electric conductivity of the tissue can be determined and used for treatment planning.  FIG.  7 B  is the case in which the maximum electric conductivity was 0.4 S/m (Z=4) after electroporation. The trends are similar to the ones described in  FIG.  5    in that if exposure length is constant, the current increases linearly with increasing electrode separation and vice versa. However, even though the conductivity within the treated region increases by a factor of 4, the current increases non-linearly only by a factor of 3. This can be seen by comparing the contours in  FIG.  7 A  with those in  FIG.  7 B  which consistently show that the curves are increased by a factor of 3. 
     Example 2 
     Determining the Relationship Between Blunt Tip Electrode Configuration and Resulting Current After IRE Treatment 
     Model Assumptions: 
     Gompertz Conductivity: Pulse duration=50 μs, Ex-vivo kidney tissue 
     Baseline Conductivity: σ=0.1 S/m 
     Spherical Domain: diameter=10 cm 
     Applied Voltage: Voltage=1000 V 
     Parametric Study: 
     Total Combinations: 720 models 
     Maximum Conductivity: 1.0×, 1.25×, 1.5×, 2×, 3×, 4×, 5×, 6× the baseline 
     Edge-to-edge Distance: 5, 10, 15, 20, 25 mm 
     Electrode Exposure: 5, 10, 15, 20, 25, 30 mm 
     Electrode Radius: 0.5, 0.75, 1.0 mm 
     The output of statistical analysis software (JMP 9.0) used to fit model and determine the coefficients for all parameter combinations is shown in the tables of  FIGS.  8 A and  8 B  and the plot of  FIG.  8 C . 
     Parameters of Best Fit for Dynamic Conductivity Changes Between 1×-6× the Baseline Conductivity (R 2 =0.96): 
     a=−1.428057; (*Intercept Estimate*) 
     b=−0.168944; (*Gap Estimate*) 
     c=2.1250608; (*Radius Estimate*) 
     d=0.2101464; (*Exposure Estimate*) 
     e=1.1114726; (*Factor Estimate*) 
     f=−0.115352; (*Gap-Radius Estimate*) 
     g=−0.010131; (*Gap-Exposure Estimate*) 
     h=−0.067208; (*Gap-Factor*) 
     i=0.0822932; (*Radius-Exposure Estimate*) 
     j=0.4364513; (*Radius-Factor Estimate*) 
     k=0.0493234; (*Exposure-Factor Estimate*) 
     l=−0.006104; (*Gap-Radius-Exposure Estimate*) 
     m=0.0165237; (*Radius-Exposure-Factor Estimate*)*) 
     n=−0.003861; (*Gap-Exposure-Factor Estimate*) 
     o=−0.041303; (*Gap-Radius-Factor Estimate*) 
     p=−0.002042; (*Gap-Radius-Exposure-Factor Estimate*) 
     Analytical Function for Dynamic Conductivity Changes Between 1×-6× the Baseline Conductivity (R 2 =0.96): 
     5 mm&lt;gap=x&lt;25 mm, 0.5 mm&lt;radius=y&lt;1.0 mm, 
     5 mm&lt;exposure=z&lt;30 mm, 1&lt;factor=w&lt;6 
     Default conductivity of 0.1 S/m and 1000 V which can be scaled for dynamic conductivities. The function is a linear combination of all iterations examined in the parametric study: 
     Current(w,x,y,z)=a+bx+cy+dz+ew+f(x+bb)(y+cc)+g(x+bb)(z+dd)+h(x+bb)(w+ee)+i(y+cc)(z+dd)+j(y+cc)(w+ee)+k(z+dd)(w+ee)+I(x+bb)(y+cc)+m(y+cc)(z+dd)(w+ee)+n(x+bb)(z+dd)(w+ee)+o(x+bb)(y+cc)(w+ee)+p(x+bb)(y+cc)(z+dd)(w+ee) 
       FIGS.  9 A- 9 E  show the representative (15 mm gap) correlation between current vs. exposure length and electrode radius for maximum conductivities (1×-6×, respectively). 
       FIGS.  10 A and  10 B  are tables showing experimental validation of the code for determining the tissue/potato dynamic conductivity from in vitro measurements. 
     Determining the Relationship Between Blunt Tip Electrode Configuration and E-Field Distribution After IRE Treatment 
     Model Assumptions: 
     Gompertz Conductivity: Pulse duration=50 μs, Ex-vivo kidney tissue 
     Baseline Conductivity: σ=0.1 S/m 
     Spherical Domain: diameter=10 cm 
     Electrode Radius: r=0.5 mm 
     Parametric Study: 
     Total Combinations: 1440 models 
     Maximum Conductivity: 1.0×, 1.25×, 1.5×, 2×, 3×, 4×, 5×, 6× the baseline 
     Edge-to-edge Distance: 5, 10, 15, 20, 25 mm 
     Electrode Exposure: 5, 10, 15, 20, 25, 30 mm 
     Voltage-to-distance Ratio: 500, 1000, 1500, 2000, 2500, 3000 V/cm 
     Example 3 
     Comparison of analytical solutions with statistical (numerical) model to calculate current and explanation of procedure that results in 3D IRE volume. 
     The process of backing-out the electrical conductivity using the analytical solutions and the one proposed in the “Towards a Predictive Model of Electroporation-Based Therapies using Pre-Pulse Electrical Measurements” abstract presented in the IEEE Engineering in Medicine and Biology Conference in Aug. 28, 2012 in San Diego, Calif. were compared. A method to determine the predictive power of the equations to calculate current is analyzing the residuals of the 1440 combinations of parameters examined. In the context of this specification, a residual is the difference between the predicted current and the actual current. As can be seen in  FIGS.  11 A and  11 B  with increasing non-linear change in conductivity due to electroporation and increasing applied electric field there is an increase in the residual for both cases. The main message though is that using the shape factor (analytical) method the maximum residual is 11.3502 A and with the statistical (numerical) model the maximum is 1.55583 A. This analysis suggests that the shape factor method may be inadequate to predict the non-linear changes in current that occur during electroporation and for reliable predictions the statistical (numerical) method may be better. 
     In terms of the prediction of the volume treated a representative method is to map out the electric field 5 cm in the directions along the (x,0,0), (0,y,0), and (0,0,z) axes from the origin. In addition, the electric field can be extracted along a line that starts at the origin and ends at 3 cm along each of the axes. These plots contain the information for determining the distances at which a particular IRE threshold occurs. In embodiments, 1440 different parameter combinations were simulated that resulted in data sets of 28,692 (x-direction), 20,538 (y-direction), 27,306 (z-direction), and 25,116 (xyz-direction) for homogeneous conductivity. Even though these simulations only include dynamic conductivity changes due to electroporation, it is believed that an identical analysis for simulations that also include the changes in conductivity due to temperature could also be performed. In this manner, it would be possible to determine irreversible electroporation thresholds as a function of temperature and electroporation. Manipulating these large data sets is challenging but it provides all the necessary information to study the effect of electrode separation, electrode length, dynamic conductivity factor, and voltage-to-distance ratio for any position along the described paths. In order to be able to manipulate the data and extract the distance for different IRE thresholds, the function NonlinearModelFit (Mathematica) was used in order to come up with analytical expressions that would closely match the electric field. A different function was used for each of the directions studied in the positive directions along the Cartesian coordinate system. The Micheilis Menten function was used along the x-direction (R 2 =0.978978), the analytical solution to the Laplace equation along the y-direction (R 2 =0.993262), and the Logistic equation in the z-direction (R 2 =0.983204). Each of those functions was scaled by a 3rd order polynomial function that enabled the fit to incorporate the electrode separation and electrode exposure as well. Even though the described functions were used to fit the data from the numerical data, there might be other functions that are also appropriate and this will be explored further in order to use the most reliable fit. In  FIGS.  12 A- 12 C  provided are representative contour plots of the electric field strength at 1.0 cm from the origin using an edge-to-edge voltage-to-distance ratio of 1500 V/cm assuming a z=1 which is the case for non-electroporated electrical conductivity. It is important to note that in this case the y and z data are starting from (0, 0, 0) and the x-data starts outside the external electrode-tissue boundary. One representative case is presented, but any of the 1440 parameters combinations that were disclosed in the conference proceeding could be plotted as well. 
     The following functions describe the electric field [V/cm] distributions along the x-axis (E x ), y-axis (E y ), and z-axis (E z ) as a function of voltage-to-distance (W), edge-to-edge separation between the electrodes (X), exposure length (Y), maximum conductivity to baseline conductivity (Z), and distance in the x-direction (xx), y-direction (yy), and z-direction (zz). 
     Micheilis Menten Equation (Electric Field in the x-Direction) 
         E   x ( W, X, Y, Z, xx )= W *( a *Exp[− b·xx ]+ c )*( dX   3   +eX   2   +fX+gY   3   +hY   2   +iY+j )+ k  
 
     The coefficients for the NonlinearModelFit are given below: 
     a=−0.447392, b=8.98279, c=−0.0156167, d=−0.0654974, e=0.468234, f=−6.17716, g=0.326307, h=−2.33953, I=5.90586, j=−4.83018, k−9.44083 
     Laplace Equation (Electric Field in the y-Direction) 
     
       
         
           
             
               
                 E 
                 y 
               
               ( 
               
                 W 
                 , 
                 X 
                 , 
                 Y 
                 , 
                 Z 
                 , 
                 yy 
               
               ) 
             
             = 
             
               a 
               + 
               
                 
                   ( 
                   
                     
                       X 
                       3 
                     
                     + 
                     
                       X 
                       2 
                     
                     + 
                     bX 
                     + 
                     
                       cY 
                       3 
                     
                     + 
                     
                       dY 
                       2 
                     
                     + 
                     eY 
                     + 
                     f 
                   
                   ) 
                 
                 * 
                 
                   ( 
                   
                     h 
                     + 
                     
                       
                         
                           ( 
                           gWXZ 
                           ) 
                         
                         2 
                       
                       * 
                       
                         ( 
                         
                           1 
                           
                             
                               
                                 Log 
                                 [ 
                                 
                                   
                                     X 
                                     + 
                                     0.1 
                                   
                                   0.05 
                                 
                                 ] 
                               
                               ) 
                             
                             * 
                           
                         
                         ) 
                       
                       * 
                       
                         Abs 
                         [ 
                         
                           
                             1 
                             
                               
                                 1 
                                 · 
                                 yy 
                               
                               - 
                               
                                 X 
                                 2 
                               
                               - 
                               
                                 0. 
                                 .5 
                               
                             
                           
                           - 
                           
                             1 
                             
                               
                                 1 
                                 · 
                                 yy 
                               
                               + 
                               
                                 X 
                                 2 
                               
                               + 
                               0.05 
                             
                           
                         
                         ] 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     The coefficients for the NonlinearModelFit are given below: 
     a=−56.6597, b=−42.9322, c=6.66389, d=−50.8391, e=141.263, f=138.934, g=0.00417123, h=0.184109 
     Logistic Equation (Electric Field in the z-Direction) 
     
       
         
           
             
               
                 E 
                 z 
               
               ( 
               
                 W 
                 , 
                 X 
                 , 
                 Y 
                 , 
                 Z 
                 , 
                 zz 
               
               ) 
             
             = 
             
               a 
               + 
               
                 
                   bWZ 
                   
                     1 
                     + 
                     
                       c 
                       · 
                       
                         Exp 
                         [ 
                         
                           d 
                           · 
                           
                             ( 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   zz 
                                 
                                 y 
                               
                               - 
                               e 
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
                 · 
                 
                   ( 
                   
                     
                       fX 
                       3 
                     
                     + 
                     
                       gX 
                       2 
                     
                     + 
                     hX 
                     + 
                     i 
                   
                   ) 
                 
                 · 
                 
                   ( 
                   
                     
                       jY 
                       3 
                     
                     + 
                     
                       kY 
                       2 
                     
                     + 
                     lY 
                     + 
                     m 
                   
                   ) 
                 
               
             
           
         
       
     
     The coefficients for the NonlinearModelFit are given below: 
     a=49.0995, b=−0.00309563, c=1.39341, d=4.02546, e=1.24714, f=0.276404, g=−1.84076, h=4.93473, I=−9.13219, j=0.699588, k=−5.0242, l=12.8624, m=19.9113. 
     In order to visualize the predicted IRE shape the equation of an ellipsoid was used and the semi-axes were forced to intersect with the locations at which the IRE threshold wants to be examined. Therefore, the provided functions can be adjusted in real-time to display the IRE volume for any electric field threshold. This is important since different tissues have different IRE thresholds that depend on the temperature, dielectric properties of the tissue, the electrode configuration, and the pulse parameters used. Once again, even though the equation for an ellipsoid is used to represent the IRE volume, other functions may be evaluated that may also be appropriate to replicate the morphology of the zones of ablation being achieved experimentally such as the Cassini curve. A 1500 V/cm was used as the voltage-to-distance ratio, electrode exposure 2 cm, and electrode separation 1.5 cm to generate 3 different IRE zones using 1000 V/cm, 750 V/cm, and 500 V/cm as the IRE thresholds with z=1. 
     From the 3D plots representing the zones of ablation shown in  FIGS.  13 A- 13 C  it can be seen that if the IRE threshold is reduced from 1000 V/cm to either 750 V/cm or 500 V/cm, the volume becomes larger. This is representative of how different tissues may have different thresholds and this code may provide the ability to simulate the fields in a broad/generic manner that can then be applied to any tissue. Incorporating the xyz-data that was extracted from the parametric study will help modify the “roundness” of the current depictions of the zone of IRE ablation in order to more realistically replicate the experimental results. However, to the best of the inventors&#39; knowledge there is no such adaptable code currently available to provide a 3D IRE volume as a function of measured current, electrode length, electrode exposure, applied voltage-to-distance ratio, and customizable electric field threshold so it is believed that this will greatly help the medical community in planning and verifying the clinical treatments of patients being treated with the IRE technology. 
     Example 4 
     Specific Conductivity 
     Specific conductivity can be important in embodiments for treatment planning of irreversible electroporation (IRE). For many applications, especially when treating tumors in the brain, the volume (area) of IRE should be predicted to maximize the ablation of the tumorous tissue while minimizing the damage to surrounding healthy tissue. The specific electrical conductivity of tissue during an irreversible electroporation (IRE) procedure allows the physicians to: determine the current threshold; minimize the electric current dose; decrease the Joule heating; and reduce damage to surrounding healthy tissue. To measure the specific conductivity of tissue prior to an IRE procedure the physician typically performs one or more of the following: establishes the electrode geometry (shape factor); determines the physical dimensions of the tissue; applies a small excitation AC voltage signal (1 to 10 mV); measures the AC current response; calculates the specific conductivity (σ) using results from the prior steps. This procedure tends to not generate tissue damage (low amplitude AC signals) and will supply the physician (software) with the required information to optimize IRE treatment planning, especially in sensitive organs like the brain which is susceptible to high electrical currents and temperatures. Thus, the IRE procedure is well monitored and can also serve as a feedback system in between series of pulses and even after the treatment to evaluate the area of ablation. 
     Special Cases for Electrode Geometry 
     Nomenclature (units in brackets): 
     V e =voltage on the hot electrode (the highest voltage), [V] 
     G=electroporation voltage gradient (required for electroporation), [V/m] 
     R 1 =radius of electrode with highest voltage (inner radius), [m] 
     R 2 =radius at which the outer electrodes are arranged (outer radius), [m] 
     i=total current, [A] 
     L=length of cylindrical electrode, [m] 
     A=area of plate electrode, [m 2 ] 
     σ=electrical conductivity of tissue, [S/m] 
     ρ=density 
     c=heat capacity 
     Case 1 
     Electrical conduction between a two-cylinder (needle) arrangement of length L in an infinite medium (tissue). It is important to note that this formulation is most accurate when L»R 1 , R 2  and L»w. The electrical conductivity can be calculated from, 
     
       
         
           
             σ 
             = 
             
               
                 i 
                 · 
                 S 
               
               
                 V 
                 e 
               
             
           
         
       
     
     where the shape factor (S) corresponding to the electrode dimensions and configuration is given by, 
     
       
         
           
             
               2 
               · 
               π 
               · 
               L 
             
             
               
                 cosh 
                 
                   - 
                   1 
                 
               
               ( 
               
                 
                   
                     
                       4 
                       ] 
                     
                     ⁢ 
                     
                       w 
                       2 
                     
                   
                   - 
                   
                     
                       ( 
                       
                         2 
                         · 
                         
                           R 
                           1 
                         
                       
                       ) 
                     
                     2 
                   
                   - 
                   
                     
                       ( 
                       
                         2 
                         · 
                         
                           R 
                           2 
                         
                       
                       ) 
                     
                     2 
                   
                 
                 
                   8 
                   · 
                   
                     R 
                     1 
                   
                   · 
                   
                     R 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
     Case 2 
     Cylindrical arrangement in which the central electrode is a cylinder (needle) with radius R 1  and the outer electrodes are arranged in a cylindrical shell with a shell radius of R 2  (not the radius of the electrodes). The voltage on the central electrode is V e . The voltage distribution in the tissue may be determined as a function of radius, r: 
     
       
         
           
             V 
             = 
             
               
                 V 
                 e 
               
               ⁢ 
               
                 
                   ln 
                   ⁢ 
                   
                     r 
                     
                       R 
                       2 
                     
                   
                 
                 
                   ln 
                   ⁢ 
                   
                     
                       R 
                       1 
                     
                     
                       R 
                       2 
                     
                   
                 
               
             
           
         
       
     
     The required voltage on the central electrode to achieve IRE: 
     
       
         
           
             
               V 
               e 
             
             = 
             
               
                 GR 
                 2 
               
               ⁢ 
                   
               ln 
               ⁢ 
               
                 
                   R 
                   2 
                 
                 
                   R 
                   1 
                 
               
             
           
         
       
     
     The required current on the central electrode: 
     
       
         
           
             i 
             = 
             
               
                 2 
                 ⁢ 
                 π 
                 ⁢ 
                 L 
                 ⁢ 
                 σ 
                 ⁢ 
                 
                   V 
                   e 
                 
               
               
                 ln 
                 ⁢ 
                 
                   
                     R 
                     2 
                   
                   
                     R 
                     1 
                   
                 
               
             
           
         
       
     
     The specific conductivity (σ) of the tissue can be calculated since the voltage signal (V e ) and the current responses (i) are known. 
     Explanation of Electrical Concepts. 
     By using the bipolar electrode described previously in US Patent Application Publication No. 2010/0030211 A1, one can apply a small excitation AC voltage signal (for example from about 1 to 10 mV), 
         V ( t )= V   0  Sin(ω t )
 
     where V(t) is the potential at time t, V 0  is the amplitude of the excitation signal and ω is the frequency in radians/s. The reason for using a small excitation signal is to get a response that is pseudo-linear since in this manner the value for the impedance can be determined indicating the ability of a system (tissue) to resist the flow of electrical current. The measured AC current (response) that is generated by the excitation signal is described by 
         I ( t )= I   0  Sin(ω t +θ)
 
     where I(t) is the response signal, I 0  is the amplitude of the response (I 0 ≠V 0 ) and θ is the phase shift of the signal. The impedance (Z) of the system (tissue) is described by, 
         Z ( V ( t ))/( I ( t ))=( V   0  Sin(ω t ))/( I   0  Sin(ω t +θ))= Z   0 (Sin(ω t ))/(Sin(ω t +θ))
 
     It is important to note that the measurement of the response is at the same excitation frequency as the AC voltage signal to prevent interfering signals that could compromise the results. The magnitude of the impedance |Z 0 | is the electrical resistance of the tissue. The electrical resistivity (Ωm) can be determined from the resistance and the physical dimensions of the tissue in addition to the electrode geometry (shape factor). The reciprocal of the electrical resistivity is the electrical conductivity (S/m). Therefore, after deriving the electrical resistivity from the methods described above, the conductivity may be determined. 
     As described in U.S. Patent Application No. 61/694,144 the analytical solution (Table 4) assumes that the length of the electrodes is much larger than the electrode radius or separation distance between the electrodes. Additionally, the analytical solution is not capable of capturing the non-linear electrical response of the tissue during electroporation procedures. The proposed statistical algorithm (Table 3) is preferably used in order to capture the response in treatments that are being conducted clinically and show how the analytical overestimates the baseline and maximum current that uses the experimental data. 
     
       
         
           
               
             
               
                 TABLE 3 
               
             
            
               
                   
               
               
                 Determination of conductivity using the statistical model 
               
               
                 and in vivo data from pre-pulse and IRE pulses in canine 
               
               
                 kidney tissue using identical electrode configuration 
               
               
                 that the experimental one described below. 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 Current 
                 Voltage 
                 Volt-2-Dist 
                 Conductivity 
                 Z = 
               
               
                   
                 [A] 
                 [V] 
                 [V/cm] 
                 [S/m] 
                 σ max /σ min   
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 Pre-Pulse 
                 0.258 
                 48 
                 53 
                 0.365 
                 — 
               
               
                 IRE-Pulse 
                 20.6 
                 1758 
                 1953 
                 1.037 
                 2.841 
               
               
                 IRE-Pulse 
                 23.7 
                 1758 
                 1953 
                 1.212 
                 3.320 
               
               
                 IRE-Pulse 
                 23.6 
                 1758 
                 1953 
                 1.207 
                 3.305 
               
               
                 Avg. IRE 
                 22.6 
                 1758 
                 1953 
                 1.150 
                 3.150 
               
               
                 IRE-Pulse 
                 10.4 
                 1259 
                 1399 
                 0.727 
                 1.990 
               
               
                 IRE-Pulse 
                 11.1 
                 1257 
                 1397 
                 0.789 
                 2.162 
               
               
                 IRE-Pulse 
                 11 
                 1257 
                 1397 
                 0.781 
                 2.138 
               
               
                 Avg. IRE 
                 10.8 
                 1257 
                 1397 
                 0.763 
                 2.090 
               
               
                 Pre-Pulse 
                 0.343 
                 73.3 
                 52 
                 0.341 
                 — 
               
               
                 IRE-Pulse 
                 23.6 
                 2262 
                 1616 
                 1.007 
                 2.952 
               
               
                 IRE-Pulse 
                 24.3 
                 2262 
                 1616 
                 1.041 
                 3.051 
               
               
                 IRE-Pulse 
                 25.4 
                 2262 
                 1616 
                 1.094 
                 3.207 
               
               
                 Avg. IRE 
                 24.5 
                 2262 
                 1616 
                 1.050 
                 3.080 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 4 
               
             
            
               
                   
               
               
                 Determination of conductivity using the analytical model and in vivo 
               
               
                 data from pre-pulse and IRE pulses in canine kidney tissue using identical 
               
               
                 electrode configuration than the experimental one described below. 
               
               
                 Assumption: Length &gt;&gt; radius, Length &gt;&gt; width, 2 
               
               
                 cylindrical electrodes in an infinite medium. 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 Current 
                 Voltage 
                 Volt-2-Dist 
                 Shape Factor 
                 Conductivity 
               
               
                   
                 [A] 
                 [V] 
                 [V/cm] 
                 [m] 
                 [S/m] 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 Pre-Pulse 
                 0.258 
                 48 
                 53 
                 0.01050 
                 0.512 
               
               
                 IRE-Pulse 
                 20.6 
                 1758 
                 1953 
                 0.01050 
                 1.116 
               
               
                 IRE-Pulse 
                 23.7 
                 1758 
                 1953 
                 0.01050 
                 1.284 
               
               
                 IRE-Pulse 
                 23.6 
                 1758 
                 1953 
                 0.01050 
                 1.279 
               
               
                 Avg. IRE 
                 22.6 
                 1758 
                 1953 
                 0.01050 
                 1.225 
               
               
                 IRE-Pulse 
                 10.4 
                 1259 
                 1399 
                 0.01050 
                 0.787 
               
               
                 IRE-Pulse 
                 11.1 
                 1257 
                 1397 
                 0.01050 
                 0.841 
               
               
                 IRE-Pulse 
                 11 
                 1257 
                 1397 
                 0.01050 
                 0.834 
               
               
                 Avg. IRE 
                 10.8 
                 1257 
                 1397 
                 0.01050 
                 0.819 
               
               
                 Pre-Pulse 
                 0.343 
                 73.3 
                 52 
                 0.00924 
                 0.506 
               
               
                 IRE-Pulse 
                 23.6 
                 2262 
                 1616 
                 0.00924 
                 1.129 
               
               
                 IRE-Pulse 
                 24.3 
                 2262 
                 1616 
                 0.00924 
                 1.163 
               
               
                 IRE-Pulse 
                 25.4 
                 2262 
                 1616 
                 0.00924 
                 1.215 
               
               
                 Avg. IRE 
                 24.5 
                 2262 
                 1616 
                 0.00924 
                 1.172 
               
               
                   
               
            
           
         
       
     
     Example 5 
     In Vivo Experiments 
     1) Animals. 
     IRE ablations were performed in canine kidneys in a procedure approved by the local animal ethics committee. Male canines weighing approximately 30 kg were premedicated with acetylpromazine (0.1 mg/kg), atropine (0.05 mg/kg), and morphine (0.2 mg/kg) prior to general anesthesia induced with propofol (6 mg/kg, then 0.5 mg/kg/min) and maintained with inhaled isofluorane (1-2%). Anesthetic depth was monitored by bispectral index monitoring (Covidien, Dublin, Ireland) of EEG brain activity. After ensuring adequate anesthesia, a midline incision was made and mesenchymal tissue was maneuvered to access the kidney. Pancuronium was delivered intravenously to mitigate electrically mediated muscle contraction, with an initial dose of 0.2 mg/kg, and adjusted if contractions increased. 
     2) Experimental Procedure. 
     Two modified 18 gauge needle electrodes (1.0 mm diameter and 1.0 cm in exposure) were inserted as pairs into the superior, middle, or inferior lobe of the kidney, with lobes being randomly selected. A BTX ECM830 pulse generator (Harvard Apparatus, Cambridge, Mass.) was used to deliver an initial 100 μs pre-pulse of 50 V/cm voltage-to-distance ratio (center-to-center) between the electrodes to get an initial current able to be used to determine baseline conductivity. Electrical current was measured with a Tektronix TCP305 electromagnetic induction current probe connected to a TCPA300 amplifier (both Tektronix, Beaverton, Oreg.). A Protek DS0-2090 USB computer-interface oscilloscope provided current measurements on a laptop using the included DSO-2090 software (both GS Instruments, Incheon, Korea). A schematic of the experimental setup can be found in  FIG.  14 A . Following the pre-pulse, a series of 100 pulses, each 100 μs long, at a rate of 1 pulse per second was delivered, reversing polarity after 50 pulses. A five second pause was encountered after pulses 10 and 50 to save data. A schematic diagram showing dimension labeling conventions is shown in  FIG.  14 B . Representative current waveforms from a pre-pulse and experimental pulse can be found in  FIGS.  14 C and  14 D , respectively. Electrode exposure lengths were set to 1 cm for all trials. The separation distance between electrodes and applied voltage may be found in Table 5. After completing pulse delivery, the electrodes were removed. Two additional ablations were performed in the remaining lobes before repeating the procedure on the contralateral kidney, resulting in a total of three ablations per kidney and six per canine. 
     
       
         
           
               
             
               
                 TABLE 5 
               
             
            
               
                   
               
               
                 KIDNEY EXPERIMENT PROTOCOLS IN CANINE SUBJECTS 
               
            
           
           
               
               
               
               
               
            
               
                   
                   
                   
                 Voltage- 
                   
               
               
                   
                 Separation, 
                 Voltage, 
                 Distance 
               
               
                 Setup 
                 cm 
                 V 
                 Ratio, V/cm 
                 n 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 1 
                 1 
                 1250 
                 1250 
                 4 
               
               
                 2 
                 1 
                 1750 
                 1750 
                 4 
               
               
                 3 
                 1.5 
                 2250 
                 1500 
                 6 
               
               
                   
               
            
           
         
       
     
     3) Kidney Segmentation and 3D Reconstruction. 
     Numerical models provide an advantageous platform for predicting electroporation treatment effects by simulating electric field, electrical conductivity, and temperature distributions. By understanding the electric field distribution, one can apply an effective lethal electric field threshold for IRE, E IRE , to predict ablation lesion dimensions under varying pulse protocols (electrode arrangements and applied voltages). However, in order to do so, these models should first be calibrated with experimental data. Here, the numerical simulation algorithm developed from porcine kidneys was expanded that accounts for conductivity changes using an asymmetrical sigmoid function (R. E. Neal, 2nd, et al., “Experimental characterization and numerical modeling of tissue electrical conductivity during pulsed electric fields for irreversible electroporation treatment planning,” IEEE Trans Biomed Eng., vol. 59, pp. 1076-85. Epub 2012 Jan. 6, 2012 (“R. E. Neal, 2 nd , et al., 2012”)). The model is calibrated to the experimental lesions to determine an effective electric field threshold under the three experimental setups used. In addition, static and linear conductivity functions are also correlated to the lesion dimensions. The three functions are used to evaluate which numerical technique will result in better accuracy in matching lesion shapes and resulting current from actual IRE ablations in mammalian tissue, particularly for kidney. 
     The imaging-based computational model domains were constructed from a magnetic resonance imaging (MRI) scan of a kidney from a canine subject of similar size to those in the study. The scans were scaled by 1.21 times in all directions to better match the experimental kidney dimensions while maintaining the anatomical characteristics. Mimics 14.1 image analysis software (Materialise, Leuven, BG) was used to segment the kidney geometry from the surrounding tissues. The kidney was traced in each of the two-dimensional (2D) MRI axial slices, which were then integrated into a three-dimensional (3D) solid representation of the kidney volume which was refined and exported to 3-matic version 6.1 (Materialise, Leuven, BG) to generate a volumetric mesh compatible with Comsol Multiphysics finite element modeling software (Comsol Multiphysics, v.4.2a, Stockholm, Sweden). 
     Electrodes were simulated as paired cylinders, each 1 cm long and 1 mm in diameter, and separated by 1 or 1.5 cm to represent the two experimental conditions. The pairs were inserted into the 3D kidney mesh in two configurations, representing both experimental approaches that used either the superior/inferior (vertical) or middle (horizontal) lobe of the kidney, both with tips 1.5 cm deep. The finite element model simulated the electric field distribution in the kidney, which was used to determine cell death EIRE by correlating the electric field values with the average in vivo lesion height and width dimensions. 
     4) Electric Field Distribution and Lethal EIRE Determination. 
     The electric field distribution is determined according to 
       −∇·(σ(| E |)∇ϕ)=0   (1)
 
     where σ is the electrical conductivity of the tissue, E is the electric field in V/cm, and ϕ is the electrical potential. Tissue-electrode boundaries for the cathode and anode were defined as ϕ=V o  and ground, respectively. The remaining boundaries were treated as electrically insulating, dϕ/dn=0, since the kidneys were isolated from the surrounding mesenchymal tissue during the experimental procedures. The current density was integrated over a mid-plane parallel to both electrodes to determine simulated electric current. 
     The model was solved for the vertical and horizontal electrode configurations, each considering three electrical conductivity tissue responses. These responses included a homogeneous static conductivity (σ 0 ) as well as two that accounted for electroporation based conductivity changes in tissue that result from cell membrane permeabilization. The dynamic models are based on a relationship between a minimum baseline and a maximum conductivity. The static conductivity model was used to determine the baseline conductivity, σ 0 , by matching simulated electrical current with the pre-pulse experimental data, where the field strength should be below that able to permeabilize any cells in the tissue. The maximum conductivity, σ max , occurs when the number of cells electroporated in the tissue has saturated, and the cellular membranes no longer restrict the extent of interstitial electrolyte mobility. The statistical model discussed in (P. A. Garcia, et al., “Towards a predictive model of electroporation-based therapies using pre-pulse electrical measurements,” Conf Proc IEEE Eng Med Biol Soc, vol. 2012, pp. 2575-8, 2012 (“P. A. Garcia, et al., 2012”)) was used to predict σ max  from previously characterized tissue response to pre-pulse σ 0  and electrical data. 
     The σ 0  and σ max  values provide the required parameters to define the electric field-dependent conductivity, σ(|E|), of renal tissue in vivo. One model assumed a linear relationship that grew between the minimum and maximum conductivities over a range from 200 to 2000 V/cm, σ L (|E|), and the second used an asymmetrical sigmoid Gompertz curve, σ S (|E|), derived from the work described in (R. E. Neal, 2nd, et al., 2012) using the equation: 
       σ S (| E |)=σ 0 +(σ max −σ 0 )·exp[− A ·exp(− B·E )]  (2)
 
     where A and B are unitless coefficients that vary with pulse length, t(s). This function was fit using curve parameters for a 100 μs long pulse, where A=3.053 and B=0.00233 (R. E. Neal, 2 nd , et al., 2012) 
     The electric field distribution along a width and height projection based at the midpoint length of the electrodes was used to determine the electric field magnitude that matched experimental lesion dimensions. This was performed for all three conductivity scenarios in all three experimental protocol setups in order to determine which model best matched the IRE ablations, providing the optimum conductivity modeling technique for mammalian tissue. 
     5) Results: In Vivo Experiments. 
     Electrical Currents. 
     All animals survived the procedures without adverse event until euthanasia. Electrical pre-pulse currents were 0.258±0.036 A (mean±SD) for the 1 cm electrode separation trials and 0.343±0.050 A for the 1.5 cm separation trials. Electrical currents from the trials for pulses 1-10, 40-50, and 90-100 are reported in Table 6. Although currents are typically reported to increase with consecutive pulses, there is no statistically significant correlation between pulse number and measured current. Therefore, all numerical calibrations to match electrical current and determine σ max  used the average current from all captured pulses for each experimental setup. 
     
       
         
           
               
             
               
                 TABLE 6 
               
             
            
               
                   
               
               
                 EXPERIMENTAL ELECTRIC CURRENTS 
               
               
                 TO CALIBRATE NUMERICAL MODELS 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Separation, 
                 Average Delivered 
                 Pulse 
                 Average Electric 
               
               
                 Setup 
                 cm 
                 Voltage, V 
                 Number 
                 Current, A* 
               
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 Pre 1 
                 1 
                 48 
                 1750 
                 0.258 
                 (0.036) 
               
               
                 Pre 2 
                 1.5 
                 73 
                 1250 
                 0.343 
                 (0.050) 
               
               
                 1 
                 1 
                 1258 
                 1-10 
                 10.4 
                 (1.7) 
               
               
                   
                   
                   
                 40-50  
                 11.1 
                 (1.1) 
               
               
                   
                   
                   
                 90-100 
                 11.0 
                 (1.7) 
               
               
                 2 
                 2 
                 1758 
                 1-10 
                 20.6 
                 (3.2) 
               
               
                   
                   
                   
                 40-50  
                 23.7 
                 (5.1) 
               
               
                   
                   
                   
                 90-100 
                 23.6 
                 (3.8) 
               
               
                 3 
                 1.5 
                 2262 
                 1-10 
                 23.6 
                 (1.47) 
               
               
                   
                   
                   
                 40-50  
                 24.3 
                 (3.25) 
               
               
                   
                   
                   
                 90-100 
                 25.4 
                 (3.27) 
               
               
                   
               
               
                 *Currents given as “average (standard deviation)” 
               
            
           
         
       
     
     6) Determination of Dynamic Conductivity Function. 
     Pre-pulse electrical current was used to calculate the baseline conductivity, σ 0 , used in the static numerical simulation. In addition, the baseline and maximum, σ max , electrical conductivities required for generating the asymmetrical sigmoid and linear dynamic conductivity functions were calculated according to the procedure outlined in (P. A. Garcia, et al., 2012) and are provided in Table 7. The ratio between these conductivities was calculated and demonstrates an increase in conductivity between 2.09 and 3.15 times, consistent with values determined in the literature for other organs (N. Payselj, et al., “The course of tissue permeabilization studied on a mathematical model of a subcutaneous tumor in small animals,” IEEE Trans Biomed Eng, vol. 52, pp. 1373-81, August 2005). 
     
       
         
           
               
             
               
                 TABLE 7 
               
             
            
               
                   
               
               
                 BASELINE AND MAXIMUM ELECTRIC CONDUCTIVITIES 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                   
                 Gap, 
                 V/d Ratio, 
                   
                   
                   
               
               
                   
                 Setup 
                 cm 
                 V/cm 
                 σ 0   
                 σ max   
                 σ max /σ 0   
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 1 
                 1 
                 1250 
                 0.365 
                 0.763 
                 2.09 
               
               
                   
                 2 
                 1 
                 1750 
                 0.365 
                 1.150 
                 3.15 
               
               
                   
                 3 
                 1.5 
                 1500 
                 0.341 
                 1.050 
                 3.08 
               
               
                   
                   
               
            
           
         
       
     
     Example 6 
     How to Use the Ratio of Maximum Conductivity to Baseline Conductivity in Modifying the Electric Field Distribution and thus the Cassini Oval Equation. 
     Irreversible electroporation (IRE) is a promising new method for the focal ablation of undesirable tissue and tumors. The minimally invasive procedure involves placing electrodes into the region of interest and delivering a series of low energy electric pulses to induce irrecoverable structural changes in cell membranes, thus achieving tissue death. To achieve IRE, the electric field in the region of interest needs to be above a critical threshold, which is dependent on a variety of conditions such as the physical properties of the tissue, electrode geometry and pulse parameters. Additionally, the electric conductivity of the tissue changes as a result of the pulses, redistributing the electric field and thus the treatment area. The effect of a dynamic conductivity around the electrodes where the highest electric fields are generated was investigated in order to better predict the IRE treatment for clinical use. 
     The electric field distribution associated with the electric pulse is given by solving the governing Laplace equation, ∇·(σ∇φ)=0, where σ is the tissue electrical conductivity (baseline 0.2 S/m) and φ the electrical potential (3000 V). The dynamic changes in electrical conductivity due to electroporation were modeled with the flc2hs Heaviside function within the finite element modeling software used in the study (Comsol Multiphysics 3.5a, Stockholm, Sweden). The dynamic conductivity factor ranged between 2.0-7.0 times the baseline value in the regions exceeding 3000 V/cm. The total electrical current, volumes, and lesion shapes from the IRE treatment were evaluated. 
       FIGS.  15 A and  15 B  display the electric field distributions for the non-electroporated (baseline conductivity) and electroporated (maximum/baseline conductivity) maps, respectively. The electric field from using the baseline conductivity resulted in a “peanut” shape distribution ( FIG.  15 A ). By incorporating the conductivity ratio between σ max /σ 0 , there is a redistribution of the electric field and thus the volumes, currents and lesion shapes are modified as well. The electric field distribution for a 7.0× factor ( FIG.  15 B ), shows a more gradual dissipation of the electric field and a rounder predicted IRE lesion. 
     A method to predict IRE lesions and incorporate the dynamic changes in conductivity due to electroporation around the electrodes is presented in this example. This procedure provides additional tools to better approximate the electric field distributions in tissue and thus help to generate more reliable IRE treatment planning for clinical use using Finite Element Analysis (FEA) models. 
     Specifically in order to adapt the Cassini Oval to match experimental lesions or electric field distributions the following procedure should be used: 
     In IRE treatments, the electric field distribution is the primary factor for dictating defect formation and the resulting volume of treated tissue (J. F. Edd and R. V. Davalos, “Mathematical modeling of irreversible electroporation for treatment planning,” Technol Cancer Res Treat, vol. 6, pp. 275-286, 2007; D. Sel, et al., “Sequential finite element model of tissue electropermeabilization,” IEEE Trans Biomed Eng, vol. 52, pp. 816-27, May 2005; S. Mahnic-Kalamiza, et al., “Educational application for visualization and analysis of electric field strength in multiple electrode electroporation,” BMC Med Educ, vol. 12, p. 102, 2012 (“S. Mahnic-Kalamiza, et al., 2012”)). The electric field is influenced by both the geometry and positioning of the electrodes as well as the dielectric tissue properties. Additionally, altered membrane permeability due to electroporation influences the tissue conductivity in a non-linear manner. Therefore numerical techniques are preferably used to account for different electrode configurations and incorporate tissue-specific functions relating the electrical conductivity to the electric field distribution (i.e. extent of electroporation). The inventors are currently using imaging-based computational models for IRE treatment planning that use the physical properties of the tissue and patient-specific 3D anatomical reconstructions to generate electric field distributions (P. A. Garcia, et al., “Non-thermal irreversible electroporation (N-TIRE) and adjuvant fractionated radiotherapeutic multimodal therapy for intracranial malignant glioma in a canine patient,” Technol Cancer Res Treat, vol. 10, pp. 73-83, 2011 (“P. A. Garcia, et al, 2011”)). 
     Oftentimes in clinical practice, there is need to rapidly visualize the estimated zone of ablation without relying in complex and time consuming numerical simulations. As an alternative, analytical solutions are powerful techniques that provide valuable insight and offer the ability to rapidly visualize electric field distributions (S. Mahnic-Kalamiza, et al., 2012). However, these analytical solutions assume infinitely long electrodes which are not the case in clinical practice and do not incorporate the non-linear changes in tissue conductivity due to electroporation. Therefore, there is a need for simple, quick, and accurate methods to provide physicians with predicted IRE zones of ablation during surgery when one of the pulse parameters needs to be adjusted. To this end, the inventors have adapted the Cassini curve in an effort to provide researchers and physicians with a graphical representation of IRE zones of ablation, for example, in in vivo porcine liver. The goal of this work is to provide a correlation between experimentally produced zones of ablations in in vivo porcine liver tissue with the corresponding IRE pulse parameters and electrode configuration. These Cassini curves are calibrated to experimental IRE ablations, and incorporate the dynamic changes in tissue conductivity, a limitation of the analytical approach. 
     The Cassini oval is a plane curve that derives its set of values based on the distance of any given point, a, from the fixed location of two foci, q 1  and q 2 , located at (x 1 , y 1 ) and (x 2 , y 2 ). The equation is similar to that of an ellipse, except that it is based on the product of distances from the foci, rather than the sum. This makes the equation for such an oval 
       └( x   1   −a ) 2 +( y   1   −a ) 2 ┘·└( x   2   −a ) 2 +( y   2   −a ) 2   ┘=b   4    (3)
 
     where b 4  is a scaling factor to determine the value at any given point. For incorporation of this equation into shapes that mimic the electric field distribution, it is assumed that the two foci were equidistantly located on the x-axis at (±x,0). The flexibility of the Cassini curve is crucial since it allows for fitting a wide range of shapes by adjusting the ‘a’ and/or ‘b’ parameters from Equation 3 simultaneously and fitting them to the experimental lesion dimensions or the locations at which a particular electric field value results from the computational simulations. The new approach in this analysis is that it is not assumed that the parameter ‘a’ is related to the separation distance between the electrodes used in IRE treatments for example but will be a second parameter to match the width/depth of any distribution thus allowing for more flexibility between the shapes achieved with the Cassini Oval as can be seen in  FIGS.  16 A and  16 B . 
     The in vivo experimental data in porcine liver was provided from published studies performed at the Applied Radiology Laboratory of Hadassah Hebrew University Medical Center (P. A. Garcia, et al., 2011). All experiments were performed with Institutional Animal Care and Use Committee approval from the Hebrew University Medical Center. The treatments were performed with a two-needle electrode configuration, 1.5 cm center-to-center separation, 2.0 cm electrode exposure, and an applied voltage of 2250 V. In this paper we only evaluate the effect of pulse number and pulse duration on the resulting ‘a’ and ‘b’ parameters required to fit the IRE zones of ablation with the Cassini curve. The NonlinearModelFit function in Wolfram Mathematica 9 was used to determine the ‘a’ and ‘b’ parameters (average±standard deviation) for each pulse parameter resulting in three curves for each condition. This same technique can be used to fit the ‘a’ and ‘b’ parameters to match the electric field shape at any particular electric field value as well thus providing an avenue to capture the shape for any IRE lesion independent of the tissue or patient. 
     The NonlinearModelFit results for the ‘a’ and ‘b’ parameters to generate the Cassini curves are provided in  FIG.  17   . The ‘a’ parameter ranged from 0.75-1.04 and the ‘b’ from 1.06-1.35 for the average IRE zones of ablation in the in vivo porcine liver. From these data it can be seen that each pulse parameter used results in a unique ‘a’ and ‘b’ combination except for the twenty 100-μs pulses and ninety 20-μs pulses which overlap since they had identical IRE ablations. Therefore, consideration should be given to pulse length and total number of pulses when planning treatments to ensure maximum accuracy when using Cassini curves to rapidly predict treatment zones. 
       FIG.  18    provides a representation of the average IRE zone of ablation and also includes the experimentally achieved standard deviations. This Cassini curve is the most clinically relevant as ninety 100-μs pulses is the recommended setting by the manufacturer that is currently being used by physicians to treat several types of cancer. The Cassini curves in  FIG.  18    were generated with a=0.821±0.062 and b=1.256±0.079 that corresponded to IRE ablations that were 3.0±0.2 cm in width and 1.9±0.1 cm in depth (P. A. Garcia, et al., 2011). The results suggest that the Cassini curve is a viable method to represent experimentally achieved IRE zones of ablation. These curves can be used to provide physicians with simple, quick, and accurate prediction of IRE treatments. The parameters generated in this study were achieved from porcine liver ablations data. Therefore, future work needs to determine the parameters for other tissues and/or tumors. Cassini curve parameters should be re-calibrated if the pulse parameters or electrode configuration (i.e. separation or exposure) deviate from the typical protocols in Ben-David et al. Additionally, there is a need to calibrate these Cassini curves to electric and temperature distributions in order to take advantage of the relatively simple curves in representing simulated solutions that account for other pulse parameters and electrode configuration including different electrode separations, diameter, exposure, and voltages. A method to represent IRE zones of ablation in a computationally efficient manner and based on experimental data is thus presented. Such methods can be used to predict IRE ablation in liver in order to provide physicians with an immediate tool for treatment planning. 
       FIG.  19    is a representation of the 3D Electric Field [V/cm] Distribution in Non-Electroporated (Baseline) Tissue with 1.5-cm Electrodes at a Separation of 2.0 cm and 3000 V applied. 
       FIGS.  20 A-D  are representations of the Electric Field [V/cm] Distributions from the 3D Non-Electroporated (Baseline) Models with 1.5-cm Electrodes at a Separation of 2.0 cm and 3000 V (cross-sections), wherein  FIG.  20 A  is a representation of the x-y plane mid-electrode length,  FIG.  20 B  is a representation of the x-z plane mid-electrode diameter,  FIG.  20 C  is a representation of the y-z plane mid electrode diameter, and  FIG.  20 D  is a representation of the y-z plane between electrodes. 
       FIG.  21    is a representation of the 3D Electric Field [V/cm] Distribution in Electroporated Tissue with 1.5-cm Electrodes at a Separation of 2.0 cm and 3000 V applied assuming σ max /σ 0 =3.6. 
       FIGS.  22 A- 22 D  are representations of the Electric Field [V/cm] Distributions from the 3D Electroporated Models with 1.5-cm Electrodes at a Separation of 2.0 cm and 3000 V (cross-sections) assuming σ max /σ 0 =3.6, wherein  FIG.  22 A  is a representation of the x-y plane mid-electrode length,  FIG.  22 B  is a representation of the x-z plane mid-electrode diameter,  FIG.  22 C  is a representation of the y-z plane mid electrode diameter, and  FIG.  22 D  is a representation of the y-z plane between electrodes. 
     Example 7 
     The Cassini Oval Equation 
     In mathematics, a Cassini oval is a set (or locus) of points in the plane such that each point p on the oval bears a special relation to two other, fixed points q 1  and q 2 : the product of the distance from p to q 1  and the distance from p to q 2  is constant. That is, if the function dist(x,y) is defined to be the distance from a point x to a point y, then all points p on a Cassini oval satisfy the equation: 
       dist( q   1   , p )×dist( q   2   , p )= b   2    (2)
 
     where b is a constant. 
     Nevertheless, in embodiments the ‘b’ parameter can be modified to manipulate the shape of the Cassini curve and illustrate the desired electric field distribution. Therefore, the ‘b’ is a variable parameter that is determined based on the specific location (distance) of a particular electric field threshold to be displayed. 
     The points q 1  and q 2  are called the foci of the oval. 
     Suppose q 1  is the point (a,0), and q 2  is the point (−a,0). Then the points on the curve satisfy the equation: 
     The equivalent polar equation is: 
         r   4 −2 a   2   r   2  cos 2θ= b   4   −a   4    (4)
 
     The shape of the oval depends on the ratio b/a. When b/a is greater than 1, the locus is a single, connected loop. When b/a is less than 1, the locus comprises two disconnected loops. When b/a is equal to 1, the locus is a lemniscate of Bernoulli. 
     The Cassini equation provides a very efficient algorithm for plotting the boundary line of the treatment zone that was created between two probes on grid  200 . By taking pairs of probes for each firing sequence, the first probe is set as qi being the point (a,0) and the second probe is set as q 2  being the point (−a,0). This original Cassini oval formulation was revised by modifying the assumption of the ‘a’ parameter being related to the position of the electrodes. In the revised formulation the ‘a’ is a variable parameter that is adjusted depending on the width and length of the Cassini oval in order to intercept the zone of ablation in the x- and y-directions. 
     In summary, the ‘a’ and ‘b’ variable parameters should be determined in order to have the ability to generate a Cassini curve that could fit the shape of any electric field isocontour. Specifically from the electric field simulations or experimental irreversible electroporation zones of ablation the user should determine the distance along the x-axis and y-axis that the Cassini curve should intersect. 
     For example in the case of a Finite Element Analysis (FEA) simulation using two 1-mm in diameter electrodes, separated by a center-to-center distance of 2.0 cm, 1.5 cm in exposure, and an applied voltage of 3000 V to one electrode and ground to the other electrode the distances from the point in between the electrodes to a specific electric field contour is given below (Table 8 for the baseline (non-electroporated) and σ max /σ 0 =3.6 (electroporated) models. 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE 8 
               
               
                   
               
               
                 E-field 
                 Baseline 
                 Baseline 
                 σ max /σ 0  = 3.6 
                 σ max /σ 0  = 3.6 
               
               
                 [V/cm] 
                 (p 1x , 0) [cm] 
                 (0, p 2y ) [cm] 
                 (p 3x , 0) [cm] 
                 (0, p 4y ) [cm] 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 300 
                 1.97 
                 0.92 
                 2.38 
                 1.39 
               
               
                 400 
                 1.81 
                 0.69 
                 2.17 
                 1.18 
               
               
                 500 
                 1.70 
                 0.49 
                 1.99 
                 1.01 
               
               
                   
               
            
           
         
       
     
     Using the 500 V/cm electric field isocontour as an example it can be determined that the Cassini oval using the baseline model will intersect the points (1.70,0) and (0,0.49) and the model using σ max /σ 0 =3.6 will intersect the point (1.99,0) and (0,1.01). Using the two points that will be intersected by the Cassini oval of each specific model type (non-electroporated vs. electroporated) allows for determination of the ‘a’ and ‘b’ variable parameter and still satisfy the mathematical condition outlined above in the first paragraph of this section by way of least square fits such as the NonlinearModelFit function in Mathematica or via interpolation tables as the one presented below. 
     The interpolation method involves assuming values for the ‘a’ parameter from 0.00 cm to 3.00 cm in steps of 0.01 cm and calculating the ‘b’ parameter using the specific points from the previous paragraph. The distance and steps were arbitrarily chosen and can vary depending on the specific Cassini oval that is being developed. In the case of Table 9 the point p1x=(1.70 cm, 0 cm) and the point p2y=(0 cm, 0.49 cm) and the corresponding distances to either q1 (−a,0) or q2 (a,0) are calculated. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                 TABLE 9 
               
               
                   
               
               
                 ‘a’ 
                 d(q1, p1x) = d1 
                 d(q2, p1x) = d2 
                 d1*d2 
                 d(q1, p2y) = d3 
                 d(q2, p2y) = d4 
                 d3*d4 
                 d1*d2/d3*d4 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 1.04 
                 0.66 
                 2.74 
                 1.808 
                 1.150 
                 1.150 
                 1.322 
                 1.37 
               
               
                 1.05 
                 0.65 
                 2.75 
                 1.788 
                 1.159 
                 1.159 
                 1.343 
                 1.33 
               
               
                 1.06 
                 0.64 
                 2.76 
                 1.766 
                 1.168 
                 1.168 
                 1.364 
                 1.30 
               
               
                 1.07 
                 0.63 
                 2.77 
                 1.745 
                 1.177 
                 1.177 
                 1.385 
                 1.26 
               
               
                 1.08 
                 0.62 
                 2.78 
                 1.724 
                 1.186 
                 1.186 
                 1.407 
                 1.23 
               
               
                 1.09 
                 0.61 
                 2.79 
                 1.702 
                 1.195 
                 1.195 
                 1.428 
                 1.19 
               
               
                 1.1 
                 0.60 
                 2.80 
                 1.680 
                 1.204 
                 1.204 
                 1.450 
                 1.16 
               
               
                 1.11 
                 0.59 
                 2.81 
                 1.658 
                 1.213 
                 1.213 
                 1.472 
                 1.13 
               
               
                 1.12 
                 0.58 
                 2.82 
                 1.636 
                 1.222 
                 1.222 
                 1.495 
                 1.09 
               
               
                 1.13 
                 0.57 
                 2.83 
                 1.613 
                 1.232 
                 1.232 
                 1.517 
                 1.06 
               
               
                 1.14 
                 0.56 
                 2.84 
                 1.590 
                 1.241 
                 1.241 
                 1.540 
                 1.03 
               
               
                 1.15 
                 0.55 
                 2.85 
                 1.568 
                 1.250 
                 1.250 
                 1.563 
                 1.00 
               
               
                 1.16 
                 0.54 
                 2.86 
                 1.544 
                 1.259 
                 1.259 
                 1.586 
                 0.97 
               
               
                 1.17 
                 0.53 
                 2.87 
                 1.521 
                 1.268 
                 1.268 
                 1.609 
                 0.95 
               
               
                 1.18 
                 0.52 
                 2.88 
                 1.498 
                 1.278 
                 1.278 
                 1.633 
                 0.92 
               
               
                 1.19 
                 0.51 
                 2.89 
                 1.474 
                 1.287 
                 1.287 
                 1.656 
                 0.89 
               
               
                 1.2 
                 0.50 
                 2.90 
                 1.450 
                 1.296 
                 1.296 
                 1.680 
                 0.86 
               
               
                 1.21 
                 0.49 
                 2.91 
                 1.426 
                 1.305 
                 1.305 
                 1.704 
                 0.84 
               
               
                 1.22 
                 0.48 
                 2.92 
                 1.402 
                 1.315 
                 1.315 
                 1.729 
                 0.81 
               
               
                 1.23 
                 0.47 
                 2.93 
                 1.377 
                 1.324 
                 1.324 
                 1.753 
                 0.79 
               
               
                 1.24 
                 0.46 
                 2.94 
                 1.352 
                 1.333 
                 1.333 
                 1.778 
                 0.76 
               
               
                   
               
            
           
         
       
     
     In the baseline case analyzed above when the variable parameter ‘a’ was 1.15 cm the calculated b 2  were 1.568 and 1.563 for the d1*d2 and d3*d4, respectively. The last column calculates the ratio of both b 2  values in order to determine the location at which they are the same (or closest) which happens when (d1*d2)/(d3*d4)=1.00. 
     Once it is determined that ‘a’=1.15 cm provides the closest ratio to one, the average of the d1*d2 (1.568) and d3*d4 (1.563) quantities is calculated and used to determine the corresponding ‘b’ parameter by taking the square root as shown in the equation below. 
     
       
         
           
             
               
                 
                   b 
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               d 
                               ⁢ 
                               1 
                               * 
                               d 
                               ⁢ 
                               2 
                             
                             + 
                             
                               ( 
                               
                                 d 
                                 ⁢ 
                                 3 
                                 * 
                                 d 
                                 ⁢ 
                                 4 
                               
                               ) 
                             
                           
                         
                         2 
                       
                     
                     = 
                     
                       
                         
                           
                             1.568 
                             + 
                             1.563 
                           
                           2 
                         
                       
                       = 
                       
                         
                           1.5655 
                         
                         = 
                         1.2512 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Once the ‘a’ and ‘b’ parameters are determined then any plotting software can be used to illustrate the Cassini curve in Cartesian coordinates using the modified equation 
     
       
         
           
             
               
                 
                   y 
                   = 
                   
                     ± 
                     
                       
                         
                           - 
                           
                             a 
                             2 
                           
                         
                         - 
                         
                           
                             x 
                             2 
                           
                           ± 
                           
                             
                               
                                 b 
                                 4 
                               
                               + 
                               
                                 4 
                                 ⁢ 
                                 
                                   a 
                                   2 
                                 
                                 ⁢ 
                                 
                                   x 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     The steps outlined in the previous paragraphs just above can also be used to determine the ‘a’ and ‘b’ parameters using the same methodology and with points p3x=(1.99 cm, 0 cm) and p4y=(0 cm, 1.01 cm) and results in ‘a’=1.21 cm and ‘b’=1.578 cm as the Cassini parameters for the electroporated model when σ max /σ 0 =3.6. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                 TABLE 10 
               
               
                   
               
               
                 ‘a’ 
                 d(q1, p3x) = d5 
                 d(q2, p3x) = d6 
                 d5*d6 
                 d(q1, p4y) = d7 
                 d(q2, p4y) = d8 
                 d7*d8 
                 d5*d6/d7*d8 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 1.1 
                 0.89 
                 3.09 
                 2.750 
                 1.493 
                 1.493 
                 2.230 
                 1.23 
               
               
                 1.11 
                 0.88 
                 3.10 
                 2.728 
                 1.501 
                 1.501 
                 2.252 
                 1.21 
               
               
                 1.12 
                 0.87 
                 3.11 
                 2.706 
                 1.508 
                 1.508 
                 2.275 
                 1.19 
               
               
                 1.13 
                 0.86 
                 3.12 
                 2.683 
                 1.516 
                 1.516 
                 2.297 
                 1.17 
               
               
                 1.14 
                 0.85 
                 3.13 
                 2.661 
                 1.523 
                 1.523 
                 2.320 
                 1.15 
               
               
                 1.15 
                 0.84 
                 3.14 
                 2.638 
                 1.531 
                 1.531 
                 2.343 
                 1.13 
               
               
                 1.16 
                 0.83 
                 3.15 
                 2.615 
                 1.538 
                 1.538 
                 2.366 
                 1.11 
               
               
                 1.17 
                 0.82 
                 3.16 
                 2.591 
                 1.546 
                 1.546 
                 2.389 
                 1.08 
               
               
                 1.18 
                 0.81 
                 3.17 
                 2.568 
                 1.553 
                 1.553 
                 2.413 
                 1.06 
               
               
                 1.19 
                 0.80 
                 3.18 
                 2.544 
                 1.561 
                 1.561 
                 2.436 
                 1.04 
               
               
                 1.2 
                 0.79 
                 3.19 
                 2.520 
                 1.568 
                 1.568 
                 2.460 
                 1.02 
               
               
                 1.21 
                 0.78 
                 3.20 
                 2.496 
                 1.576 
                 1.576 
                 2.484 
                 1.00 
               
               
                 1.22 
                 0.77 
                 3.21 
                 2.472 
                 1.584 
                 1.584 
                 2.509 
                 0.99 
               
               
                 1.23 
                 0.76 
                 3.22 
                 2.447 
                 1.592 
                 1.592 
                 2.533 
                 0.97 
               
               
                 1.24 
                 0.75 
                 3.23 
                 2.423 
                 1.599 
                 1.599 
                 2.558 
                 0.95 
               
               
                 1.25 
                 0.74 
                 3.24 
                 2.398 
                 1.607 
                 1.607 
                 2.583 
                 0.93 
               
               
                 1.26 
                 0.73 
                 3.25 
                 2.373 
                 1.615 
                 1.615 
                 2.608 
                 0.91 
               
               
                 1.27 
                 0.72 
                 3.26 
                 2.347 
                 1.623 
                 1.623 
                 2.633 
                 0.89 
               
               
                 1.28 
                 0.71 
                 3.27 
                 2.322 
                 1.630 
                 1.630 
                 2.659 
                 0.87 
               
               
                 1.29 
                 0.70 
                 3.28 
                 2.296 
                 1.638 
                 1.638 
                 2.684 
                 0.86 
               
               
                 1.3 
                 0.69 
                 3.29 
                 2.270 
                 1.646 
                 1.646 
                 2.710 
                 0.84 
               
               
                   
               
            
           
         
       
     
     In  FIG.  23   , it can be seen that with the implementation of the pre-pulse concept to determine the ratio of maximum conductivity to baseline conductivity one can derive a Cassini curve representing zones of ablation. In this case the 500 V/cm isocontour was specified but this technique could be used for any other isocontour that perhaps could represent the lethal IRE threshold for any other tissue/tumor type. 
     The polar equation for the Cassini curve could also be used because since it provides an alternate method for computation. The current Cartesian coordinate algorithm can work equally as well by using the polar equation of the Cassini curve. By solving for r 2  from eq. (4) above, the following polar equation was developed: 
         r   2   =a   2  cos(2*theta)+/−sqrt( b   4   −a   4  sin 2 (2*theta))   (5)
 
     and the ‘a’ and ‘b’ parameters should be determined as previously described in this application. 
     The present invention has been described with reference to particular embodiments having various features. In light of the disclosure provided, it will be apparent to those skilled in the art that various modifications and variations can be made in the practice of the present invention without departing from the scope or spirit of the invention. One skilled in the art will recognize that the disclosed features may be used singularly, in any combination, or omitted based on the requirements and specifications of a given application or design. Other embodiments of the invention will be apparent to those skilled in the art from consideration of the specification and practice of the invention. 
     It is noted in particular that where a range of values is provided in this specification, each value between the upper and lower limits of that range is also specifically disclosed. The upper and lower limits of these smaller ranges may independently be included or excluded in the range as well. The singular forms “a,” “an,” and “the” include plural referents unless the context clearly dictates otherwise. It is intended that the specification and examples be considered as exemplary in nature and that variations that do not depart from the essence of the invention fall within the scope of the invention. Further, all of the references cited in this disclosure are each individually incorporated by reference herein in their entireties and as such are intended to provide an efficient way of supplementing the enabling disclosure of this invention as well as provide background detailing the level of ordinary skill in the art.