Mycoplasma contamination represents a significant problem to the culture of mammalian cells used for research as it can cause disastrous effects on eukaryotic cells by altering cellular parameters leading to unreliable experimental results. Mycoplasma cells are very small bacteria therefore they cannot be detected by visual inspection using a visible light microscope and, thus, can remain unnoticed in the cell cultures for long periods. The detection techniques used nowadays to reveal mycoplasma contamination are time consuming and expensive with each having significant drawbacks. The ideal detection should be simple to perform with minimal preparation time, rapid, inexpensive, and sensitive. To our knowledge, for the first time, we employed Fourier transform infrared (FTIR) microspectroscopy to investigate whether we can differentiate between control cells and the same cells which have been infected with mycoplasmas during the culturing process. Chemometric methods such as HCA and PCA were used for the data analysis in order to detect spectral differences between control and intentionally infected cells, and spectral markers were revealed even at low contamination level. The preliminary results showed that FTIR has the potential to be used in the future as a reliable complementary detection technique for mycoplasma-infected cells. Mollicutes that lack a cell wall which makes them unaffected by common antibiotics. Mycoplasma contamination is a significant problem to the culture of mammalian cells used for research and the rate could be as high as 70% [Mycoplasmas are types of bacteria of class h as 70% . The conh as 70% , 3. In ch as 70% , 5. The Mycoplasma cells are very small (less than 1 μm); therefore, they cannot be detected by visual inspection using a visible light microscope and, thus, can remain unnoticed in the cell cultures for long periods. Most if not all of the detection techniques used as DNA probe, PCR, fluorescent DNA staining (with DAPI or Hoechst), microbiological culture or enzyme-linked immunosorbent assays (ELISA), and immunoblotting are time consuming and expensive with each having significant drawbacks. The ideal detection should be simple to perform with minimal preparation time, rapid, inexpensive, and sensitive and can be used to test different cell cultures on a regular basis. Unfortunately, mycoplasma assays, used nowadays, have some of these characteristics but not all. Even the polymerase chain reaction (PCR) , which is known to be the most reliable assay, has drawbacks: it is complex, time consuming, and suffers from false positive and negative results if performed inadequately .Compared to other molecular and imaging techniques used nowadays, FTIR microspectroscopy has many advantages . The use of FTIR microspectroscopy for studying biological samples is a wide and active area of research and became very common especially in the last two decades thanks to conventional instruments for imaging or synchrotron radiation IR for single cell analysis. IR spectral differences have been reported on many biological samples from cancerous and normal cells, cells in different growth stages or different environments, studying the effect of drugs on cells, and plant cells –20 to baTo our knowledge, there is no previous study reported in the literature that tried to compare/detect mycoplasma contamination in cultured mammalian cells. The only study found to analyze and characterize Mollicutes by IR spectroscopy done by Melin et al.  comparedFTIR imaging via a focal plane array detector (FPA) using a global source was employed in this study as it is advantageous for fast acquisition and good spectral quality necessary for cell analysis, hence considered as an ideal tool for quickly investigating the biochemical difference between infected and non-infected cells with mycoplasma for a first screening step. The results obtained by FTIR are compared with PCR technique and a biochemical test for correlation. The purpose of the present preliminary study was to introduce FTIR spectroscopy method for mycoplasma detection in cell culture as a first go. Further research and samples will have to be studied to test the specificity in different mycoplasma species.A supernatant from a hybridoma culture ) contaminated by mycoplasma was collected and confirmed with the MycoAlert™ PLUS Mycoplasma Detection Kit  to be positive. A test with a serial dilution of this medium was performed and verified that there was a detectable contamination even to a 1:128 dilution. The medium was sterilized with a 0.22-μM filter to make sure to use a bacteria-free medium culture . Ten percent of FBS and 2 mM GlutaMax (Gibco by Life Technologies) were added to this filtered medium and a second MycoAlert™ test was performed to check again the positivity . Cells were maintained in a humidified atmosphere in a 37 °C incubator supplied with 5% CO2 to reach 70–80% confluence. They were then infected with the mycoplasma suspension  during the culturing process. Three cell categories were prepared in different concentrations to have non-infected cells acting as control, low contaminated (Myc+) and high contaminated cells (Myc+++). Cells were kept to grow with or without mycoplasma for 24 h in two separate incubators to avoid contaminating the non-infected cells  were seeded in T25 culture flasks at a density of 1 × 10The MycoAlert™ assay was used to confirm the cells were successfully infected. This is a selective biochemical test that exploits the activity of certain mycoplasmal enzymes that react with the MycoAlert™ substrate catalyzing the conversion of ADP to ATP. By measuring the level of ATP in a sample both before and after the addition of the specific and luminescent substrate, a ratio can be obtained which is indicative of the presence or absence of mycoplasma.One hundred microliters of cleared supernatant were transferred in a single well of 96-wells plate adding 100 μl of MycoAlert™ reagent to each sample. After 5 min, the plate was placed in a luminometer  to measure luminescence (reading A). Then 100 μl of MycoAlert™ substrate were added to each sample and after 10 min luminescence was measured again (reading B). The ratio of reading B to reading A is used to determine whether a cell culture is contaminated by mycoplasma. If the value is <1, the sample is negative for mycoplasma, if it is between 1 and 1.2, the sample is borderline and the cells must be kept in quarantine and re-test in 24 h. If the ratio is more than 1.2, the sample is positive for mycoplasma contamination.As seen in Table Mycoplasma sp. , Mycoplasma faucium, (both from human), and many strains of Mycoplasma bovis. Total extract from 0.1 × 106 cells/point was prepared lysing in 100 μl of 9% Nonidet-P40, 60 μg/ml proteinase K and treating at 60 °C for 30 min and 95 °C for 10 min for enzyme inactivation. PCR was performed in a SimpliAmp™ Thermal Cycler PCR System (Applied Biosystems by Thermo Fisher Scientific) for 35 cycles  in a volume of 25 μl reaction buffer of KAPA Taq DNA Polymerase (5 U/μl) (Kapa Biosystems by Sigma-Aldrich), 0.2 μM each primer , and 1 μl of total extract.Polymerase chain reaction (PCR) on DBTRG non-infected and infected cell lysates was performed using primers which can detect at least 49 different mycoplasma strains that include uncultured Mycoplasma hominis and Mycoplasma genitalium, in a single real-time PCR reaction.The possibility of contamination of infected medium by human mycoplasma strains was excluded by the negative result of Anyplex™ II STI-7 Seegene test  that enables detection and identification of seven sexually transmitted infections pathogens, including 2). Single FPA shots (field of view 128 × 128 μm2) were taken on many cells at a time using the 20× objective with matching condenser in transmission mode at 4 cm−1 spectral resolution and 256 scans accumulation (for sample and background) for a measurement time of circa 6 min. A large area of the sample was mapped with the FPA to cover many cells in order to get more chance having cells infected with mycoplasma in the contaminated series. All data acquisition was performed using the OPUS 7.2 software (Bruker). Data analysis was performed in the Unscrambler X10.3 software. Statistical analysis was performed on the PCA results using ANOVA test.DBTRG cell spots of each category deposited on ZnS optical substrates were analyzed in the mid-IR range on the equipment available at the MIRIAM Beamline B22 at Diamond Light Source, UK) . This coDue to their small size, contamination of cell culture by mycoplasmas cannot be visualized and also it does not induce medium turbidity or color change usually generated by bacterial or fungal contamination. Moreover, morphological changes and growth rate effect induced by mycoplasmas can be minimal or even absent. In this study, the morphological changes were absent but the growth rate was slightly altered that the cells grew at a slower rate especially within the high contaminated category. Using visible microscopy the contamination of mycoplasma in the infected cells cannot be detected. As seen in Fig. We demonstrated that we have successfully infected DBTRG cell line culturing them in a medium derived from a mycoplasma positive supernatant for 24 h. Both conditions, high contamination (Myc+++) and low contamination , lead to an amplification of a 717-bp amplicon from 16S ribosomal RNA sequence in common with 49 different mycoplasma strains. The contamination in the lower condition approximately quantifies three times less than DBTRG cells growth in the presence of undiluted medium . Data analysis was performed in the Unscrambler X10.3 software, using the second derivative spectra (Savitzky-Golay second order and applying 9 points smoothing) to remove slowly varying baseline effects and applying unit vector normalization. Principal component analysis (PCA) was performed on all categories grouped together, using the non-linear iterative partial least squares (NIPALS) algorithm and leverage correction validation method. Statistical analysis was performed on the PCA results using one-way ANOVA test. Average spectra were taken for each category to have an overview of the major differences in the spectra that allows the separation of the cells. Spectra were divided in separate regions to better illustrate the results and understand which are the most important bands allowing for separation among the categories. The two major regions  were taken into consideration, where the fingerprint region (1800–950 cm−1) was divided into three subregions  to cover all of the membrane phospholipids, the proteins (amides I and II), the nucleic acid, and the carbohydrate absorptions.Spectra of the cells were retrieved from the FPA images (between 2 to 4 images per sample) and one average spectrum per cell was taken, averaging 36 to 64 pixels depending on cell size for all the cells in each category. Cells as shown in Fig. The second derivative of the average spectra of each group are shown in Fig. —HCA was used to classify HC and CN groups, tried on the four spectral regions used for PCA, and the region of 1500–1300 cm−1 gave the best classification separating 100% the spectra.Fig. −1 gave better separation than the other regions. There was some overlapping which is certainly due to the low contamination in the LC group so some cells may have not been infected with the mycoplasma or affected as other cells.Fig. −1 spectral region.Fig. −1 gave the best separation in HCA between HC vs CN and HC vs LC, a test set of 33 spectra in total  were chosen randomly to validate if this region is able to classify blindly the test set in their correct cluster within the training set (201 spectra). The results , an unsupervised classification method, was performed on the second derivative and vector normalized spectra of all the groups using the Ward’s algorithm to classify them into clusters. Results are shown in Fig. Principal component analysis (PCA), a multivariate data analysis method used to reduce dimensionality of the spectral data, was applied on the second derivative and unit vector normalized cell spectra using the non-linear iterative partial least squares (NIPALS) algorithm and leverage correction validation method. The PC loading plots are useful to understand the correlations between the variables and interpret the samples grouping as they contain the spectral features of the constituents. Peaks in the loading plots indicate important spectral regions for the particular component which are responsible for the classification. PCA shows different efficiency of separation based on the spectral region used among the four spectral regions.In Fig. PC1 separating between CN and HC mostly reveals the following bands: 3012, 2951, 2918, and 2848 on the negative side of PC1 so these intensities are lower in the CN group compared to HC (CN < LC < HC). PC2 separating between HC and LC mostly reveals bands around 3010, 2952, 2925, 2910, and 2842 on the negative side of PC2 scores which implies less intensities in the negative scores group (LC < HC).This sub-region of the fingerprint showed also a good separation Fig. c especiaPC1 separating between CN/LC vs HC mostly reveals the following bands: 1743, 1631, 1533, and 1512 on the negative side so these intensities are lower in the CN/LC group compared to HC. PC1 loadings on the positive side reveal bands at 1658 and 1558 which are lower in HC than in CN group. PC2 mostly separating CN vs HC reveals the bands at 1745, 1700, and 1635 on the negative side so these intensities are lower in the CN group compared to HC. PC2 loadings on the positive side reveal the bands around 1647, 1625, and 1571 and imply that the intensities of these bands are lower in LC/HC groups compared to CN.PCA scores based on this region gave the best separation among the three groups Fig. e. The PCPC1 separating between HC and CN/LC mostly reveals the following bands: 1471, 1444, 1403, 1348, and 1328 on the negative side so these intensities are lower in the HC group compared to CN/LC. The positive side of PC1 loadings reveals the bands at 1434 and 1338 to be lower in CN/LC groups compared to HC. PC2 separating between LC and CN groups mostly reveals bands: 1473, 1456, 1419, and 1338 on the negative side, so the CN group which is on the negative side of PC2 scores should have less intensities of these bands than in the LC group (CN < LC). The positive side of PC2 loadings reveals the band at 1446 to be lower in LC group compared to CN group.This sub-region of the fingerprint corresponding to carbohydrates and phosphate absorption showed a good separation Fig. g betweenPC1 and PC2 mostly reveal the following bands: 1218, 1043, 987, 981, and 964 on the negative sides of both PC loadings so these intensities are lower in the CN/LC group compared to HC. The positive side of PC1 loading reveals the bands at 1255 and 1091 to be lower in HC group compared to CN/LC group.P < 0.05). The t test performed on PC1 and PC2 scores for every two groups gave values < 0.01.Since PCA results of the region 1500–1300 gave the best results in separating the three groups, more data analysis was performed on this region to evaluate the significance level statistically and understand better what is causing the differences. PCA score results in this spectral region were proven significantly different among all groups by statistical tests one way  vs. actual classes was obtained with 87.06% accuracy; it shows the percentage of the spectra correctly classified  and misclassified (the other numbers) as illustrated in Table Linear discriminant analysis (LDA), a supervised multivariate analysis method, is used to classify samples into groups based on features that can be used to describe the objects. Three different methods  for the LDA can be used and the method chosen will depend on the similarity of the different classes to be discriminated. Here PCA-LDA using the Mahalanobis method was used as it may model the classes better. Since the data set contains more variables than samples , PCA-LDA was chosen  to overcome the dimensionality problems. The model was performed on a training set (201 spectra) using the PC scores of PC1 and PC2 of the PCA based on the spectral region 1500–1300 cmOnce the LDA model is built, it can be used to classify unknown samples or predict samples (by reflecting probability of belonging to a certain class). For this, the test samples  was used and the results obtained are the following: 11 out of 11 spectra of the CN group were classified with the CN class, 11 out of 12 spectra of the HC group were classified with HC class and 1 classified with LC class, and 8 out of 10 spectra of LC group were classified with LC class and 2 with CN class.It is noteworthy to say that the data in Table Spectral data collected from control cells and cells which have been infected with mycoplasma led to some identification of spectral markers that could indicate the presence of mycoplasma and the effects on the host cells induced by the bacterial infection. The bands affected along with their assignments are summarized in Table To be able to identify if the spectral differences in the three groups are due to mycoplasma presence or to its effects on the host cells, an average spectrum of a positive control (dried spot taken from the mycoplasma suspension which served to infect the cells) was used to compare with the differential spectrum i.e. the subtracted spectrum of CN average from HC average . In principle, this spectrum subtraction can highlight the bands which belong to mycoplasma presence in the highly contaminated cell samples as well as the bands in the cells which were affected by induced cellular changes due to the infection.As seen in Fig. In Fig. −1 could be due to the change of the fatty acid components of membranes or their structural deformation. It is known that cholesterol is abundant in mycoplasmas [−1 are especially correlated with the composition of the cellular membrane being altered by the bacterial infection. The changes in the protein bands mean that the protein secondary structure is altered and deformed following the infection. Lipid and proteins content changes in the region of 1500–1300 cm−1 could be due to the alteration of the lipid and protein level/synthesis individually as well as the lipoprotein level as a hole. This could be easily interpreted since mycoplasmas are known to contain highly immunogenic lipoproteins anchored on the outer face of the plasma membrane [−1 for instance is lower in the control group. The changes in the phosphate bands of the nucleic acids reveal the alteration of the nucleic acid synthesis and may be chromosomal aberrations as well as structural alterations. Finally, the changes in the carbohydrates reveal mostly the alteration of the cellular metabolism.It was stated by Degeling et al.  that theoplasmas , and thimembrane , 32; henIt is noteworthy to mention that mostly the cellular membrane changes  and the protein level alteration are seen before the possible damage of nucleic acids and the alteration of the cellular metabolism. This could be concluded from the different spectral regions separating or not the LC group (not only the HC group) from the CN group. The nucleic acid region separated the HC from CN but the LC was not very well separated which also proves that the DNA region alone is not capable to identify the mycoplasma contamination. This could be one of the reasons why PCR has false negative and positive results sometimes, or why the staining with DAPI is not 100% reliable as also based on the DNA staining.The results obtained by FTIR shows the sensitivity of the technique to detect the presence of the mycoplasma and/or the induced cellular changes by being able to differentiate the LC group from the CN group, whereas the MycoAlert™ test shows negative results for the LC group . Further investigations are in progress to better interpret these spectroscopic markers. The preliminary results showed that micro-FTIR has the potential to be used in the future as a complementary detection technique for mycoplasma-infected cells and could be very useful in detecting if cells are cleared from mycoplasmas following a treatment to eliminate the bacterial contamination.


One recent area of cancer research is irreversible electroporation (IRE). Irreversible electroporation is a minimally invasive procedure where needle electrodes are inserted into the body to ablate tumor cells with electricity. The aim of this paper is to propose a mathematical model that incorporates a tissue’s conductivity increasing more in the direction of the electrical field as this has been shown to occur in experiments.It was necessary to mathematically derive a valid form of the conductivity tensor such that it is dependent on the electrical field direction and can be easily implemented into numerical software. The derivation of a conductivity tensor that can take arbitrary functions for the conductivity in the directions tangent and normal to the electrical field is the main contribution of this paper. Numerical simulations were performed for isotropic-varying and anisotropic-varying conductivities to evaluate the importance of including the electrical field’s direction in the formulation for conductivity.By starting from previously published experimental results, this paper derived a general formulation for an anistropic-varying tensor for implementation into irreversible electroporation modeling software. The anistropic-varying tensor formulation allows the conductivity to take into consideration both electrical field direction and magnitude, as opposed to previous published works that only took into account electrical field magnitude.The anisotropic formulation predicts roughly a five percent decrease in ablation size for the monopolar simulation and approximately a ten percent decrease in ablation size for the bipolar simulations. This is a positive result as previously reported results found the isotropic formulation to overpredict ablation size for both monopolar and bipolar simulations. Furthermore, it was also reported that the isotropic formulation overpredicts the ablation size more for the bipolar case than the monopolar case. Thus, our results are following the experimental trend by having a larger percentage change in volume for the bipolar case than the monopolar case.The predicted volume of ablated cells decreased, and could be a possible explanation for the slight over-prediction seen by isotropic-varying formulations. Electroporation is an electrically-driven biological process that uses externally applied electric fields to briefly open the nano-sized pores in a cell’s membrane. On one hand, electroporation can be used to transport drugs and genetic materials into cells . On the A cell’s viability is directly related to the electrical field it is exposed to. Each cell is characterized by a critical value so that when the cell is exposed to an electrical field above this critical value it loses its ability to reseal the pores in its membrane and eventually dies. Any electrical field below this critical value, either does not form pores or the pores close when the electrical field is removed. The critical value depends on various factors such as tissue type, tumor stage, electrode configuration and size, and the direction of the applied electric field. The success of an IRE-based therapy for tumors relies on this critical value and thus knowledge about the optimal configuration and size of the electrodes as well as the applied voltage needed to completely ablate the tumor while minimizing the damage to healthy tissue is relevant to clinicians.Mathematical models of electroporation and corresponding numerical simulations able to predict the electric field distribution in a tissue of interest for various geometries and positions of electrodes could play an important, complementary role in the design of individual treatment protocols , 4. In pTherefore, in this paper we propose a mathematical model that incorporates a tissue’s conductivity increasing more in the direction of the electrical field. To do so, it was required for us to derive a formulation for the conductivity tensor such that conductivity increases by different amounts in the direction tangent and normal to the electrical field and can be easily implemented in numerical methods such as finite elements. Numerical simulations were performed for isotropic and anisotropic varying conductivities to evaluate the importance of including the electrical field’s direction in the formulation for conductivity.This section will discuss the details of the mathematical models used for comparing the well established isotropically increasing-conductivity formulation and this paper’s anistropically increasing-conductivity formulation. The simulations used for comparison will be based off the irreversible electroporation experiments and simulations performed by Castellvi et al. . This woU, satisfies the differential form of Gauss’s law at steady state: In numerical simulations for treatment planning, it is assumed that the applied direct current electric field is at steady state . The eleσ is the tissue’s conductivity. Then the electric field E is given by: E=−∇U. If σ is a constant, then Eq. (σ=σ(E).where then Eq.  becomes We model the isotropic varying conductivity as a sigmoid Gompertz curve , 13, 14:A and B are unitless coefficients, σ0 is the the base conductivity before electroporation, σmax is the maximum conductivity a tissue can achieve after electroporation, and ∥E∥ is the l2− norm of the electrical field. By definition, the l2− norm of the electrical field is Ex, Ey, Ez is the x, y, z component of the electrical field respectively. The values of A and B are found by fitting the curve for conductivity to experimental data.where σ is a scalar. However, the conductivity is anisotropic when the conductivity increases more in the direction of the electrical field. This results in the conductivity, σ, being represented by a rank-2 tensor.If the conductivity is isotropic, then σ does not take into account the direction of the electrical field (isotropic-varying) and when σ does take into account the direction of the electrical field (anisotropic-varying). Both cases will solve Eq. σ is defined.We will compare two different cases. The first being when This is the well established model that is often used for IRE simulation predictions. It will be used for comparison with our anisotropic-varying formulation. The isotropic-varying formulation has the conductivity increase equally in all directions. Hence, it being refereed to as the isotropic formulation in this paper. The advantage of using an isotropic formulation is that the conductivity is represented by a single scalar, and the direction of the electrical field does not need to be considered. It is also less computationally intensive.In the isotropic formulation, the changes in conductivity from electroporation can be taken into account with a sigmoid Gompertz curve for the conductivity , 14, 15.will be used for the isotropic formulation.In this paper, we propose a conductivity that takes into account the electrical field’s direction and magnitude. We wish to formulate the conductivity tensor such that the conductivity increases more in the direction of the electrical field. Since the electrical field does not increase equally in all directions, we will refer to this formulation as the anisotropic formulation in this paper.We now wish to derive the matrix representation for the conductivity tensor for the anisotropic-varying case such that it can be implemented into a numerical scheme. This is the main contribution of the paper.We begin by assuming the conductivity tensor is a real valued symmetric matrix. This is common assumption and is justified by assuming conductivity is independent of current flow being positive or negative. All symmetric matrices are diagnolizable. Therefore, the conductivity tensor is diagonalizable.The anisotropic case can have a different conductivity in the direction of the electrical field than perpendicular to it. If we make the assumption that the conductivity in the direction of the electrical field is always equal or greater than any other direction, and that the conductivity in a direction perpendicular to the electrical field is always equal or less than any non-normal direction then the directions tangent and normal to the electrical field are principal directions. These assumptions are justified according to the experimental work found in .σt be an arbitrary function representing the conductivity in the direction tangent to the electrical field. Also let σn and σb be arbitrary functions representing conductivity in the direction normal and bi-normal to the electrical field respectively. By treating σt, σn, and σb as arbitrary functions, we are providing a derivation that is valid for any expression for σt, σn, and σb which may someday be ascertained through experiments.Let A tensor is diagonal when the principal directions are chosen as the basis representation. We will therefore consider a coordinate system comprised of a unit vector tangent to the electrical field and two unit vectors perpendicular to the electrical field. A coordinate system consisting of a tangent vector, normal vector, and bi-normal vector, is sometimes referred to as a Frenet-Serret frame. These three vectors form an orthonormal basis in which the basis vectors point in the principal directions of the conductivity. Since the Frenet-Serret basis vectors are aligned with the principal directions then the conductivity tensor matrix representation is diagonal in the Frenet-Serret coordinate system and can be represented as σf is the matrix representation of the conductivity tensor in the Frenet-Serret coordinate system. It will be assumed that the conductivity is the same for any direction normal to the electrical field. This implies that both the normal and bi-normal direction conductivities are equal, where Equation σf represents a linear transformation that takes a vector from the Frenet-Serret frame and outputs a vector in the Frenet-Serret frame. This can be restated as where where σc is the matrix representation of the conductivity tensor in the Cartesian coordinate system, and where θ about the z-axis and a rotation ϕ about the y-axis. Note only two rotations instead of three are necessary because the conductivity has been assumed to be equal in all normal directions.Since the Frenet-Serret frame and the Cartesian coordinate system are both orthonormal, a change of basis between the two can be represented by a rotation through an angle R, is a linear transformation such that The rotation tensor, σc, σf, and R can be found in Fig. A function diagram outlining the relationship between R into two subsequent rotations. The first rotation will be about the z-axis, Rz(θ), and can be expressed by the matrix, We can decompose θ is obtained through the relationship where the angle Ex and Ey represent the x and y components of the electrical field respectively. Similarly, the second rotation will be about the y-axis, Ry(ϕ), and is expressed by the matrix, The quantities ϕ is the angle described by the function where Ez represents the z component of the electrical field. The rotation tensor can be then be expressed as The quantity R−1 is Using properties of rotation matrices, We are now able to express the conductivity tensor in matrix form with the domain and codomain being the Cartesian frame. This is accomplished by applying the rotation matrices in the following order: where σt and σn, including those dependent on electrical field strength.The form of an anisotropic varying conductivity tensor has been derived with arbitrary functions for the conductivity in the tangent and normal directions. Therefore, this formulation is valid for any functions of conductivity for σt, will have the same form as the conductivity for the isotropic case: For the anisotropic case simulations, the conductivity used for the tangent direction, This was chosen because the measurements used to experimentally determine the form of the conductivity is often done using electrodes that are aligned with the electrical field.σΔ, where As shown in , the conσΔ. But since the derivation was done in a general setting, it will be possible to plug in different formulations at a later time as they become available from experimental data. The approximation can be combined with the previous expression for σt to obtain the following expression for the conductivity in the perpendicular direction: is an adaptation of the data obtained in the experiments reported in . Due to σΔ results in σt and σn being equal until the onset of electroporation. In other words, the conductivity is isotropic until the onset of electroporation. During electroproation, the conductivity tensor becomes more anisotropic with increasing electrical fields up until a value of σt being 30% larger than σn.It is worth noting, that our form for Boundary conditions need to be specified before Eq. V0 is the applied voltage of the electrode. For a grounded electrode, V0 would be zero and the boundary condition would read where The boundaries not in contact with an electrode, are often modeled as electrically insulating, n is the outward pointing normal [where g normal .The geometry used for the monopolar simulations was the same as used by Castellvi et al. It consisted of two monopolar electrodes spaced 15 mm apart embedded in a 60 mm sphere. A schematic of the simulation geometry is shown in Fig. n is the outward pointing normal is enforced. The remaining boundary conditions is to set the active area of one electrode to ground by enforcing the condition where and to enforce the condition that the active area of the other electrode is held at a constant voltage by enforcing the equation The domain used for the bipolar simulations was a 60 mm sphere. The electrode has a diameter of 1.5 mm and a length of 40.5 mm. The last 7 mm of the electrode are set to the positive voltage. The next 7 mm of the electrode is insulated, and then followed by 7 mm of the electrode set to ground. The remaining portion of the electrode is set to an insulating boundary condition as is the outer sphere. A schematic of the domain is shown in Fig. All simulations were performed using the commercial finite element software COMSOL. The meshes used for the simulations consisted of approximately 500,000 cells, and were run using 2.2 GHz Intel Xeon processors.Both formulations were run with various voltages. A comparison of the the ablation volume can be found in Fig. The shape between the anisotropic and the isotropic formulation are similar except for the anisotropic formulation resulting in a slightly less spherical ablation zone than the isotropic formulation. Similar results were seen for other voltages.For the bipolar geometry, the volume predicted to be ablated is much lower for the anisotropic simulations than for the isotropic simulations. A summary of the results are displayed in Fig. σt and σn, and is therefore applicable irregardless of the form chosen for σt and σn. The formulation was compared to more commonly used isotropically varying formulations, and was found to decrease the predicted ablation zone for both monopolar and bipolar electrode setups.This paper derived a general formulation for an anistropic-varying tensor for implementation into irreversible electroporation modeling software. The anistropic-varying tensor formulation allows the conductivity to take into consideration both electrical field direction and magnitude, as opposed to previous published works that only took into account electrical field magnitude. It was derived for arbitrary functions for Further experimental and numerical work is necessary before any definitive conclusions can be drawn on the importance of including an anisotropic-varying formulation in ablation zone predictions. The hope of this work is to encourage further research on how electrical fields can affect the conductivity tensor in different directions because more experimental data is necessary before any definitive conclusions can be drawn.


This paper contains information relating to the setting up of standard blackbodies for use through the temperature range of about 1,400° to 2,400° K and their use in the calibration of tungsten strip lamps as laboratory standards of spectral radiance for the wavelength region of 0.25 to 2.6 microns. A graphite blackbody is described and representative data are given on the spectral characteristics of the new lamp standard as compared to blackbodies at several selected temperatures. Fifty years ago the principal need in the evaluation of radiant energy was a convenient standard against which a radiometer might be calibrated. Although several crude standards of total radiance in the form of oil lamps or candles existed about 1900, the establishment of the carbon filament lamp , 31 provIt is to fulfill this latter requirement that a new secondary standard of spectral radiance in the form of a tungsten strip lamp has been set up. Again the blackbody has been employed as the reference source in the development of the new standard.Nλ at a particular wavelength to the absolute temperature T by the relationshipC1 and C2 are the first and second radiation constants having the dimensions watts/cm2 and cm/degree, respectively. The exact values attached to these depend not only on the units employed in expressing Nλ, but also upon the most probable values of the various fundamental atomic constants.It has been established that the total and spectral radiation characteristics of a blackbody may be defined in terms of certain equations or laws. Planck’s radiation law relates the radiance In the present case the following values have been employed for the various parameters in the Planck equation:Much research has been carried out on tungsten lamps, in particular regarding the spectral emissivity of tungsten , 17, 18.Before discussing the apparatus and method employed in comparing a group of tungsten ribbon strip lamps with blackbodies at various temperatures a discussion is in order relative to the type of lamp chosen.Investigations by Worthing  many yeaμ. The lamp and the blackbody were mounted side by side on an optical bench (constructed from a lathe bed and table) so they could alternately be placed at the focal point of the auxiliary optical system. In μ wherein a high-temperature graphite blackbody was used. It was possible to move the monochromator so that the lamp and the blackbody would be alternately at the correct object distance of the auxiliary optics.The principal apparatus employed in comparing the radiant energy from the lamp with that of the blackbody was set up as shown in The auxiliary optics consisted of a plane mirror and a spherical mirror having a 71-cm radius of curvature. The placing of the lamp and the blackbody alternately at the same position insured equal light paths and the use of identical optics.The blackbody indicated in lsewhere .The blackbody employed for the shorter wavelengths and shown in long see . The verThis graphite blackbody is heated by induction inside a water-cooled coil by a radiofrequency generator operating at 450 kc. The blackbody tube is insulated by firmly packed boron nitride powder inside a high temperature porcelain tube (closed at one end) 4 in. in outside diameter and 6 in. in length. An alundum ceramic tube placed midway between the graphite core and the high temperature porcelain tube increased the mechanical stability of the unit.Depreciation of the graphite at high temperature was reduced by enclosing the blackbody unit in an airtight chamber see  through The spectral transmittance of the fused silica window of the enclosure was measured and the necessary corrections were made for temperature measurements and spectral radiance.This blackbody is many times larger than the small tungsten enclosures often used in high temperature work. A much larger opening can be employed thereby making possible the use of the entire slit of the usual spectroradiometer.F/6.4 to F/7.2 were employed as shown in μ was an uncooled lead sulfide cell, while for the wavelength range of 0.25 to 0.75 μ a photomultiplier, type 1P28 was employed [Leiss double quartz prism monochromators with relative aperture ranges of employed . In all The radiant energy beam was mechanically chopped at 510 cps and the detector signal amplified by a special tuned amplifier . UsuallyThe temperature of the blackbody used at 1,400° K (and below) was determined by a platinum versus platinum 10 percent rhodium thermocouple calibrated by the heat division of the National Bureau of Standards. The same laboratory calibrated the optical pyrometer employed to determine the temperatures from 1,400° to 2,400° K in the graphite blackbody. Check measurements between the optical pyrometer and the thermocouple usually were in agreement within 2° to 3° K. Calibrations of the optical pyrometer by the heat division before and following the measurements reported in this paper were in agreement to about 1° K.Tests using thermocouples placed at various positions and by observations with the optical pyrometer indicated closely uniform temperatures within the low temperature blackbody. Similar tests with the optical pyrometer indicated a similar uniform condition within the high-temperature enclosure.Each lamp was seasoned by operation on alternating current at 35 amp  for 2 hr. During calibration the lamp current was controlled manually through the use of variable and stepdown transformers as described in a later paragraph see .Each lamp was set up with the envelope vertical and with the section of the filament viewed horizontally through the center of the lamp window being employed in the measurements (no note was made of a notch placed in one edge of the lamp filament for another use). Through the use of the external optics  a full size image of this section of the lamp filament was focused upon the entrance slit of the spectroradiometer. Hence, the physical dimensions of the spectrometer slit (set at about 0.5 mm by 5 mm) determined the source area employed in the measurements. Since the optical arrangement was identical for the lamp and the blackdody equal areas of the two sources were always being compared.Readings were alternately taken on the blackbody and on the lamp for each wavelength setting throughout the spectral range. The blackbodies were operated at several selected temperatures so that the radiance from the lamp and the blackbody were approximately equal in a given region of the spectrum. At each wavelength a number of readings were taken (with changes in the temperature of the blackbody) such that in some cases the radiance of the lamp was greater than that of the blackbody while in others it was less than that of the blackbody. This procedure enabled us to keep the effect of any departure from linearity within the electronics in the detector circuitry as small as practicable.μ). About 10 to 12 independent comparisons were made at the longer wavelengths. Due to the small magnitued of scatter of the results, little is to be gained by additional measurements.Data obtained upon two lamps as compared with the blackbodies are given in μ. Hence the radiant energy values are in a form directly applicable for use with a spectroradiometer of fixed entrance slit. Those who desire to make solid angle measurements may change the tabulated values to erg cm−2 ster−1μ−1sec−1 as reported by DeVos [8. Similarly, the data recorded in the DeVos tables may be reduced to the units employed herein by multiplying by 10−8 .The data of by DeVos  simply bμ, in the experimental setup the actual instrumental spectral slit widths ranged from about 0.001 μ at 0.25 μ to 0.060 μ at 2.6 μ (being about 0.020 μ at 1.0 μ). The use of the 0.1 μ interval affects the absolute values hut little for wavelengths longer than about 1 μ. However, for the shorter wavelengths where the spectral energy curve exhibits a greater degree of curvature, higher accuracy may be obtained if the tabulated values be converted to a shorter wavelength interval corresponding more nearly to that employed in obtaining the data.Although the data for all wavelengths are given on the same basis as regards the wavelength interval of 0.1 In the original comparisons with the blackbody the cone of radiant energy from the lamp was limited to approximately 5° and near normal to the plane of the lamp filament. If, in use, a larger angle is required it should be ascertained that the flux density is uniform and free of polarization throughout the added aperture.2 for the various wavelengths at a distance of 1 m, the lamps may be used at any convenient distance provided proper correction is made for the new distance. However, if there is excessive water vapor in the laboratory atmosphere, errors may result at the wavelengths of water vapor absorption. In the original calibrations any water vapor absorption effects cancelled out for the greater part since the measurements were made using essentially identical air paths.Although the calibrations of the standards are given in terms of radiant power per cmThe quality of the two blackbodies used in this work was determined by the method outlined by André Gouffé  and the The certification of the optical pyrometer was given as ±2° from 1,000° to 1,400° K and ±6° from 1,400° to 2,300° K. This gives rise to an uncertainty of ½ to 2 percent in the longer wavelengths while in the shorter wavelengths the uncertainty increases to 1 to 4 percent. The uncertainty for the thermocouple was less than ±3° which would result in an uncertainty in the radiance not in excess of 2 percent.If one assumes an ability to read the meters employed as two-fifths of the smallest division, this would lead to an uncertainty never more than 1.0 percent. The ability of the electronics to reproduce an output when a given signal is applied to the input is of the order of 1.0 percent.It is estimated that the maximum uncertainty in the results ranges from about 8 percent at the shortest wavelengths to about 3 percent at the longest wavelengths.The auxiliary optics employed with this standard may be composed of two units such as those employed in the original calibrations see , and 6, In general, the radius of curvature of this auxiliary spherical mirror should be greater than the focal length of the spectrometer employed so that no loss of radiant energy will result through overfilling the spectrometer optics. Furthermore, as noted above, the aperture of this mirror should be kept within a total conical angle of about 5°.No diaphragm or other shielding is required in the use of these standards, except for a shield to prevent direct radiation from the lamp, not falling on the concave mirror from entering the spectrometer, since in their use an optical image of the filament is focused upon the spectrometer slit.μ but which increases slightly with wavelength except for a slight dip in the region of 0.8 to 1.0μ. In practice the proper reflectance losses can best be determined through the use of a third mirror (a second plane mirror) which may be temporarily incorporated into the optical setup from time to time.  having a ratio of 10 to 1 or a 50-amp variable transformer may be employed. See . Then toThese lamp standards are expensive laboratory equipment and it is suggested that they be operated at the lower current value in order to prolong their usual life. Only for short intervals should they ever be operated above 30 amp , and then only to calibrate a similar lamp as a working standard. In general even at lower currents a working standard should be prepared and used, except for purposes of checking the operation of such working standard.μ. At wavelengths shorter than 0.25μ the available radiant energy from heated tungsten is too low for practical uses. At longer wavelengths than about 2.6μ the low emissivity of tungsten, together with window absorption and re-radiation effects render the use of this type of lamp impractical. Some other source, in air or employing a special window, will be found more practical. The recent measurements on the spectral emissivity of platinum from 2.0 to 15.0μ provides one possible standard for radiance at long wavelengths [The tungsten strip lamp is a useful working standard for use in spectral radiance measurements within the region of 0.25 to 2.6elengths .The method of calibration against a blackbody is direct and leaves no question relating to filament temperature or tungsten emissivity. The energy of the lamp is equated to that of the blackbody. The principal uncertainty in the results lie within the accurate determination of the blackbody temperatures and the measurement of the current through the lamp filament.