MATHEMATICAL MODELING OF MOLECULAR TRANSMEMBRANE TRANSPORT AND CHANGES OF TISSUES´ DIELECTRIC PROPERTIES DUE TO ELECTROPORATION

Visokonapetostni električni pulzi povečajo prepustnost celične membrane (Tsong 1991Weaver 1993Kotnik et al. 2012) skozi pore (Abidor et al. 1979), ki nastanejo na tistih njenih delih, kjer vsiljena transmembranska napetost preseže kritično vrednost (Towhidi et al. 2008Kotnik et al. 2010). Elektroporacija je reverzibilna, če si celica po pulzih opomore, in ireverzibilna, če je škoda preobsežna in celica odmre (Pakhomova et al. 2013bJiang et al. 2015a). Trenutne optične metode por ne morejo zaznati, zato njihov nastanek zaznavamo posredno, bodisi z meritvami vnosa različnih molekul v celice ali z meritvami električnih lastnosti celic (Napotnik in Miklavčič 2017). Uporaba elektroporacije V živilski industriji (Toepfl 2012Toepfl et al. 2014) uporabljamo elektroporacijo oziroma pulzirajoča električna polja (angl. pulsed electric fields), kar je uveljavljen izraz v tej industriji, za uničevanje patogenih organizmov in njihovih produktov (encimov in toksinov). V nasprotju s termično obdelavo hrane električni pulzi ne vplivajo na okus, barvo ali hranilno vrednost. V biotehnologiji uporabljamo elektroporacijo za ekstrakcijo molekul iz mikroorganizmov in rastlin, s čimer se izognemo uporabi kemičnih sredstev in ne uničimo celičnih organelov, torej se izognemo tudi dodatnemu čiščenju končnega produkta (Sack et al. 2010Haberl et al. 2013aMahnič-Kalamiza et al. 2014bKotnik et al. 2015). Primeri: ekstrakcija DNK iz bakterijsladkorja iz sladkorne pese (Haberl et al. 2013b), sokov iz sadjapolifenolov iz grozdja za izboljšanje kvalitete vina (Puértolas et al. 2010)vode pri sušenju zelene biomase, ki služi kot vir za biogorivo (Golberg et al. 2016). Elektroporacija je tudi nova metoda pri zamrzovanju celic in tkiv, angl. cryopreservation (Galindo in Dymek 2016Dovgan et al. 2017). Elektroporacijo uporabljamo tudi v medicini (Miklavčič et al. 2010Yarmush et al. 2014), in sicer pri elektrokemoterapiji (Miklavčič et al. 2012Mali et al. 2013Cadossi et al. 2014Miklavčič et al. 2014Campana et al. 2014Serša et al. 2015), netermičnem odstranjevanju tkiva z ireverzibilno elektroporacijo (Davalos et al. 2005Garcia et al. 2010José et al. 2012Cannon et al. 2013Scheffer et al. 2014bJiang et al. 2015aRossmeisl et al. 2015), genski terapiji (Golzio et al. 2002Vasan et al. 2011Gothelf in Gehl 2012Calvet et al. 2014Heller in Heller 2015Trimble et al. 2015) in vnosu učinkovin v kožo in skoznjo (Denet et al. 2004Zorec et al. 2013b). Pri genski terapiji vnesemo v celice plazmide, v katerih je zapisana sinteza določenega proteina, ki lahko spremeni biološko funkcijo celice (Aihara in Miyazaki 1998Heller in Heller 2015). Z elektroporacijo povišamo varnost genske terapije, saj se izognemo uporabi virusov in kemikalij. Mehanizmi genske terapije z elektroporacijo še niso popolnoma pojasnjeni, osnovni koraki so opisani v literaturi (Rosazza et al. 2016). Z elektroporacijo lahko zlivamo različne celice, s čimer pridobivamo celice, ki proizvajajo monoklonska protitelesa ali inzulin (Ramos in Teissié 2000Trontelj et al. 2008Rems et al. 2013). V doktorski disertaciji sem se osredotočila na uporabo elektroporacije v medicini, predvsem pri elektrokemoterapiji, netermičnem odstranjevanju tkiva z ireverzibilno elektroporacijo in pri vnosu učinkovin v kožo in skoznjo je, zato so ti trije posegi podrobneje opisani v naslednjem poglavju. Medicinski posegi z elektroporacijo – elektrokemoterapija, netermično odstranjevanje tkiva z ireverzibilno elektroporacijo in vnos učinkovin v kožo in skoznjo Elektrokemoterapija je kombinacija kemoterapije in električnih pulzov, dovedenih neposredno na tarčno tkivo. Električni pulzi povečajo prepustnost celične membrane za kemoterapevtike, zato povečamo učinkovitost zdravljenja, obenem pa zmanjšamo dovedeno dozo kemoterapevtika in omilimo stranske učinke. Celoten tumor mora biti pokrit z dovolj visokim električnim poljem, da povečamo prepustnost vseh tumorskih celic (Miklavčič et al. 2006a), zagotoviti pa moramo tudi dovolj visoko koncentracijo kemoterapevtika znotraj tumorja (Miklavčič et al. 2014). Okoliško tkivo ne sme biti uničeno, torej mora biti električno polje okoli tumorja pod mejo za ireverzibilno elektroporacijo. Pri elektrokemoterapiji običajno dovajamo osem pulzov dolžine 100 μs s ponavljalno frekvenco 1 Hz. S poskusi določena meja za povišanje prepustnosti tumorskega tkiva je 0,4 kV/cm (Miklavčič et al. 2010). Osem pulzov je bilo določenih kot optimalno število pulzov (Marty et al. 2006Mir et al. 2006), večje število dovedenih pulzov namreč že zmanjšuje preživetje (Dermol in Miklavčič 2015). Za zdravljenje tumorjev z elektrokemoterapijo so bili definirani standardni postopki (angl. standard operating procedures) (Marty et al. 2006Mir et al. 2006), kjer so glede na število tumorjev, njihovo velikost in lokacijo (na koži ali pod kožo) določeni tip elektrod, kemoterapevtik, anestezija in način dovajanja kemoterapevtika. Kemoterapevtik lahko dovedemo lokalno ali sistemsko. V elektrokemoterapiji oz. terapiji z električnimi pulzi sta najbolj razširjena kemoterapevtika cisplatin in bleomicin. Z elektrokemoterapijo je možno zdraviti tudi globlje ležeče tumorje (Miklavčič et al. 2010Pavliha et al. 2013Edhemović et al. 2014Miklavčič in Davalos 2015). V zadnjem času se uveljavlja tudi uničevanje tumorskih celic z visokimi koncentracijami kalcija in električnimi pulzi (Frandsen et al. 2015Frandsen et al. 2016Frandsen et al. 2017). Pri elektrokemoterapiji se pojavijo še dodatni učinki, ki povišajo učinkovitost elektroporacije. Vazokonstrikcija zmanjša spiranje kemoterapevtika iz tumorja in s tem ohranja visoko koncentracijo kemoterapevtika v tumorju, obenem se zmanjša pretok krvi skozi tumor, kar povzroči hipoksijo in pomanjkanje hranilnih snovi (Mir 2006Serša et al. 2008). Elektrokemoterapija sproži tudi odziv imunskega sistema, ki nato odstrani preostale tumorske celice (Serša et al. 2015). Z ireverzibilno elektroporacijo netermično odstranjujemo tumorje brez uporabe kemoterapevtika (Jiang et al. 2015a). Tako se popolnoma izognemo stranskim učinkom kemoterapevtikov, vendar na račun več dovedene energije in posledično Joulovega gretja. Pri ireverzibilni elektroporaciji dovajamo več (okoli 90) električnih pulzov, dolgih od 50 μs do 100 μs, s ponavljalno frekvenco 1 Hz. Dovedeno električno polje je v rangu nekaj kV/cm, kar je dosti več kot pri elektrokemoterapiji. Pri ireverzibilni elektroporaciji lahko z visoko natančnostjo odstranimo želeno tkivo – območje med uničenim in nepoškodovanim tkivom je široko le nekaj premerov celic (Rubinsky et al. 2007). Za odstranjevanje tumorjev tradicionalno uporabljamo termične metode (Hall et al. 2014) – radiofrekvenčno odstranjevanje in odstranjevanje s tekočim dušikom, kjer tkivo uničujemo z visoko oz. z nizko temperaturo. Prednost ireverzibilne elektroporacije pred uveljavljenimi termičnimi metodami je krajši čas zdravljenja, izognemo se učinkom hlajenja oz. gretja tkiva zaradi bližine žil (Golberg et al. 2015), pri čemer ostanejo okoliške pomembne strukture (žile, živci) nedotaknjene (Jiang et al. 2015a). Tudi pri ireverzibilni elektroporaciji je v dokončno odstranitev tumorskih celic vpleten imunski sistem (Neal et al. 2013). Pri elektrokemoterapiji in ireverzibilni elektroporaciji se zaradi daljših pulzov in ponavljalne frekvence 1 Hz pojavljajo težave zaradi krčenja mišic (Miklavčič et al. 2005), bolečine med dovajanjem pulzov, heterogenosti električnih lastnosti tkiv v tem frekvenčnem področju ter zaradi možnosti srčnih aritmij (Ball et al. 2010). Bolečini in krčenju mišic se lahko izognemo, če pulze dovajamo z višjo frekvenco, npr. 5 kHz (Županič et al. 2007Serša et al. 2010). Srčnim aritmijam se izognemo tako, da s sinhroniziramo dovedene električne pulze z električno aktivnostjo srčne mišice (Mali et al. 2008Deodhar et al. 2011aMali et al. 2015). Bolečini, krčenju mišic in heterogenosti električnih lastnosti tkiv se lahko izognemo z dovajanjem 1 μs bipolarnih pulzov (Arena et al. 2011Arena in Davalos 2012Sano et al. 2015). V zadnjem času so se pojavile tudi metode, s katerimi so vnos barvil v celico dosegli brezkontaktno s t. i. magnetoporacijo (Chen et al. 2010Towhidi et al. 2012Kardos in Rabussay 2012Novickij et al. 2015Kranjc et al. 2016Novickij et al. 2017bNovickij et al. 2017a). Elektroporacijo lahko uporabljamo ne le za zdravljenje tumorjev, temveč tudi za vnos učinkovin v kožo in skoznjo. Vnos učinkovin skozi kožo je neinvaziven, poleg tega pa se izognemo degradaciji učinkovin pri prehodu skozi prebavni trakt. Skozi kožo lahko preide le malo molekul, zato uporabljamo različne metode za povečanje prehoda učinkovin – iontoforezo, radiofrekvenčno mikroablacijo, laser, mikroigle, ultrazvok in elektroporacijo (Zorec et al. 2013b). Proces elektroporacije kože je slabo razumljen. Predpostavljamo, da pri dovajanju visokonapetostnih električnih pulzov v roženi plasti nastanejo lokalna transportna območja, kjer sta povišani električna prevodnost in prepustnost (Pliquett et al. 1996Pliquett et al. 1998Pliquett et al. 1998Pavšelj in Miklavčič 2008a). Skozi lokalna transportna območja lahko nato učinkovine še nekaj ur po dovedenih pulzih vstopajo skozi kožo v krvni obtok (Zorec et al. 2013a). Gostota teh območij je odvisna od električnega polja v koži – višje električno polje jih povzroči več. Velikost lokalnih transportnih območij je odvisna od trajanja pulza. Med samim pulzom se zaradi Joulovega gretja topijo lipidi v roženi plasti, kar povzroči njihovo širjenje (Pliquett et al. 1996Prausnitz et al. 1996Pliquett et al. 1998Weaver et al. 1999Vanbever et al. 1999Gowrishankar et al. 1999b). Načrtovanje posegov elektrokemoterapije in netermičnega odstranjevanja tkiva z ireverzibilno elektroporacijo Pri zdravljenju tumorjev z elektroporacijo lahko uporabimo standardne oblike in postavitve elektrod z že določenimi parametri električnih pulzov (Marty et al. 2006Mir et al. 2006Campana et al. 2014). Če zdravimo velike tumorje ali tumorje nepravilnih oblik, ki pogosto ležijo globlje, s standardno postavitvijo elektrod ne moremo zagotoviti ustrezne pokritosti tumorja z dovolj visokim električnim poljem. V tem primeru lahko elektrode med samim posegom večkrat premaknemo ali pa prilagodimo njihovo število in postavitev. Pri tem moramo prej pripraviti načrt posega (Kos et al. 2010Miklavčič et al. 2010Pavliha et al. 2012Linnert et al. 2012Edhemović et al. 2014). V njem zagotovimo, da bo cel tumor izpostavljen dovolj visokemu električnemu polju (Miklavčič et al. 2006a), obenem pa škoda na okoliškem tkivu minimalna. Načrtovanje posega poteka v več korakih: 1. zajem medicinskih slik (računalniška tomografija, magnetna resonanca) tumorja in okoliškega tkiva2. obdelava slik3. razgradnja slik in določitev geometrije tkiva4. vzpostavitev tridimenzionalnega modela5. optimizacija postavitve elektrod glede na obliko in velikost tumorja6. izdelava modela elektroporacije (izračun električnega polja in spremembe električne prevodnosti tkiva)7. optimizacija napetosti med elektrodami in položaja elektrod (Pavliha et al. 2012). Na sliki 1 lahko vidimo izračunano električno polje v tumorju in okoliškem tkivu pri eni izmed možnih postavitev elektrod.Electroporation is a phenomenon, which occurs when short high voltage pulses are applied to cells and tissues resulting in a transient increase in membrane permeability or cell death, presumably due to pore formation. If cells recover after pulse application, this is reversible electroporation. If cells die, this is irreversible electroporation. Electroporation is used in biotechnology for biocompound extraction and cryopreservation, in food processing for sterilization and pasteurization of liquid food and in medicine for treating tumors by electrochemotherapy or irreversible electroporation as an ablation technique, for gene electrotransfer, transdermal drug delivery, DNA vaccination, and cell fusion. In electroporation-based medical treatments, we can treat tumors with predefined electrode geometry and parameters of electric pulses. When we treat larger tumors of irregular shape treatment plan of the position of the electrodes and parameters of the electric pulses has to be calculated before each treatment to assure coverage of the tumor with a sufficient electric field. In treatment plans, currently, 1) we assume that above an experimentally determined critical electric field all cells are affected and below not, although, in reality, the transition between non-electroporated and electroporated state is continuous. 2) We do not take into account the excitability of some tissues. 3) The increase in tissues’ conductivity is described phenomenologically and does not include mechanisms of electroporation. 4) Transport of chemotherapeutics into the tumor cells in electrochemotherapy treatments is not included in the treatment plan although it is vital for a successful treatment. We focused on the mathematical and numerical models of electroporation with the aim of including them in the treatment planning of electroporation-based medical treatments. We aimed to model processes happening during electroporation of tissues, relevant in the clinical procedures, by taking into account processes happening at the single cell level. First, we used mathematical models of cell membrane permeability and cell death which are phenomenological descriptions of experimental data. The models were chosen on the basis of the best fit with the experimental data. However, they did not include mechanisms of electroporation, and their transferability to tissues was questionable. We modeled time dynamics of dye uptake due to increased cell membrane permeability in several electroporation buffers with regard to the electrosensitization, i.e., delayed hypersensitivity to electric pulses caused by pretreating cells with electric pulses. We also modeled the strength-duration depolarization curve and cell membrane permeability curve of excitable and non-excitable cell lines which could be used to optimize pulse parameters to achieve maximal drug uptake at minimal tissue excitation. Second, we modeled change in dielectric properties of tissues during electroporation. Model of change in dielectric properties of tissues was built for skin and validated with current-voltage measurements. Dielectric properties of separate layers of skin before electroporation were determined by taking into account geometric and dielectric properties of single cells, i.e., keratinocytes, corneocytes. Dielectric properties of separate layers during electroporation were obtained from cell-level models of pore formation on single cells of lower skin layers (keratinocytes in epidermis and lipid spheres in papillary dermis) and local transport region formation in the stratum corneum. Current-voltage measurements of long low-voltage pulses were accurately described taking into account local transport region formation, pore formation in the cells of lower layers and electrode polarization. Voltage measurements of short high-voltage pulses were also accurately described in a similar way as with long low-voltage pulseshowever, the model underestimated the current, probably due to electrochemical reactions taking place at the electrode-electrolyte interface. Third, we modeled the transport of chemotherapeutics during electrochemotherapy in vivo. In electrochemotherapy treatments, transport of chemotherapeutics in sufficient amounts into the cell is vital for a successful treatment. We performed experiments in vitro and measured the intracellular platinum mass as a function of pulse number and electric field by inductively coupled plasma – mass spectrometry. Using the dualporosity model, we calculated the in vitro permeability coefficient as a function of electric field and number of applied pulses. The in vitro determined permeability coefficient was then used in the numerical model of mouse melanoma tumor to describe the transport of cisplatin to the tumor cells. We took into account the differences in the transport of cisplatin in vitro and in vivo caused by the decreased mobility of molecules and decreased membrane area available for the uptake in vivo due to the high volume fraction of cells, the presence of cell matrix and close cell connections. Our model accurately described the experimental results obtained in electrochemotherapy of tumors and could be used to predict the efficiency of electrochemotherapy in vitro thus reducing the number of needed animal experiments. In the thesis, we connected the models at the cell level to the models at the tissue level with respect to cell membrane permeability and depolarization, cell death, change in dielectric properties and transport. Our models offer a step forward in modeling and understanding electroporation at the tissue level. In future, our models could be used to improve treatment planning of electroporation-based medical treatments

Impedance Analysis of Tissues in nsPEF Treatment for Cancer Therapy

Nanosecond pulsed electric field (nsPEF) for cancer therapy is characterized by applications of high voltage pulses with low pulsed energy to induce non-thermal effects on tissues such as tumor ablation. It nonthermally treats tissues via electroporation. Electroporation is the increase in permeabilization of a cell membrane due to the application of high pulsed electric field. The objective of this study was to investigate the effect of nsPEF on tissue by monitoring the tissue’s impedance in real-time. Potato slices (both untreated and electroporated), and tumors extracted from female BALBc mice were studied. 100ns, 1-10kV pulses were applied to the tissues using a four-pin electrode at a pulse repetition frequency of 3Hz. The impedance change during the treatment was recorded by a custom made V-I monitor, and a network analyzer measured the impedance before and after treatment over a frequency range of 100kHz to 30MHz. In addition, system calibration was conducted to ensure the accuracy of the measurements. This includes determination of the attenuation offered by the V-I monitor measured to be 60dB and the cell constant K which represents the geometry of the four-pin electrode measured to be 0.8453����−1 (±0.02cm). Results show that the impedance of tissue reduced with increasing number of pulses and voltage applied, up to 44.4% and 22.3% decrease in the impedance of potato and tumor tissues were respectively observed. Also, the impedance values were higher at lower frequencies compared to those at higher frequencies. This is due to the high resistance of the membrane at low frequencies

The influence of skeletal muscle anisotropy on electroporation: in vivo study and numerical modeling

The aim of this study was to theoretically and experimentally investigate electroporation of mouse tibialis cranialis and to determine the reversible electroporation threshold values needed for parallel and perpendicular orientation of the applied electric field with respect to the muscle fibers. Our study was based on local electric field calculated with three-dimensional realistic numerical models, that we built, and in vivo visualization of electroporated muscle tissue. We established that electroporation of muscle cells in tissue depends on the orientation of the applied electric field; the local electric field threshold values were determined (pulse parameters: 8 × 100 μs, 1 Hz) to be 80 V/cm and 200 V/cm for parallel and perpendicular orientation, respectively. Our results could be useful electric field parameters in the control of skeletal muscle electroporation, which can be used in treatment planning of electroporation based therapies such as gene therapy, genetic vaccination, and electrochemotherapy

Numerical optimization of gene electrotransfer into muscle tissue

<p>Abstract</p> <p>Background</p> <p>Electroporation-based gene therapy and DNA vaccination are promising medical applications that depend on transfer of pDNA into target tissues with use of electric pulses. Gene electrotransfer efficiency depends on electrode configuration and electric pulse parameters, which determine the electric field distribution. Numerical modeling represents a fast and convenient method for optimization of gene electrotransfer parameters. We used numerical modeling, parameterization and numerical optimization to determine the optimum parameters for gene electrotransfer in muscle tissue.</p> <p>Methods</p> <p>We built a 3D geometry of muscle tissue with two or six needle electrodes (two rows of three needle electrodes) inserted. We performed a parametric study and optimization based on a genetic algorithm to analyze the effects of distances between the electrodes, depth of insertion, orientation of electrodes with respect to muscle fibers and applied voltage on the electric field distribution. The quality of solutions were evaluated in terms of volumes of reversibly (desired) and irreversibly (undesired) electroporated muscle tissue and total electric current through the tissue.</p> <p>Results</p> <p>Large volumes of reversibly electroporated muscle with relatively little damage can be achieved by using large distances between electrodes and large electrode insertion depths. Orienting the electrodes perpendicular to muscle fibers is significantly better than the parallel orientation for six needle electrodes, while for two electrodes the effect of orientation is not so pronounced. For each set of geometrical parameters, the window of optimal voltages is quite narrow, with lower voltages resulting in low volumes of reversibly electroporated tissue and higher voltages in high volumes of irreversibly electroporated tissue. Furthermore, we determined which applied voltages are needed to achieve the optimal field distribution for different distances between electrodes.</p> <p>Conclusion</p> <p>The presented numerical study of gene electrotransfer is the first that demonstrates optimization of parameters for gene electrotransfer on tissue level. Our method of modeling and optimization is generic and can be applied to different electrode configurations, pulsing protocols and different tissues. Such numerical models, together with knowledge of tissue properties can provide useful guidelines for researchers and physicians in selecting optimal parameters for <it>in vivo </it>gene electrotransfer, thus reducing the number of animals used in studies of gene therapy and DNA vaccination.</p

Electropermeabilization of inner and outer cell membranes with microsecond pulsed electric field. Quantitative study with calcium ions

Microsecond pulsed electric fields (mu sPEF) permeabilize the plasma membrane (PM) and are widely used in research, medicine and biotechnology. For internal membranes permeabilization, nanosecond pulsed electric fields (nsPEF) are applied but this technology is complex to use. Here we report that the endoplasmic reticulum (ER) membrane can also be electropermeabilized by one 100 mu s pulse without affecting the cell viability. Indeed, using Ca2+ as a permeabilization marker, we observed cytosolic Ca2+ peaks in two different cell types after one 100 mu s pulse in a medium without Ca2+. Thapsigargin abolished these Ca2+ peaks demonstrating that the calcium is released from the ER. Moreover, IP3R and RyR inhibitors did not modify these peaks showing that they are due to the electropermeabilization of the ER membrane and not to ER Ca2+ channels activation. Finally, the comparison of the two cell types suggests that the PM and the ER permeabilization thresholds are affected by the sizes of the cell and the ER. In conclusion, this study demonstrates that mu sPEF, which are easier to control than nsPEF, can permeabilize internal membranes. Besides, mu sPEF interaction with either the PM or ER, can be an efficient tool to modulate the cytosolic calcium concentration and study Ca2+ roles in cell physiology

Induced transmembrane voltage and electropermeabilization of cells in cultures in vitro

Quand une cellule est exposée à un champ électrique externe, la tension électrique transmembranaire (ITV) est modifiée. Pendant l'exposition, l' ITV se superimpose au potentiel de repos (RTV) et quand la somme des deux tensions excède une valeur critique, la perméabilité de la membrane cellulaire augmente transitoirement localement . Ce phénomène est désigné comme electropermeabilisation. Dans beaucoup d'applications de l'electropermeabilisation une permeabilisation efficace et en même temps réversible est essentielle. Ainsi, une prédiction de l'expérience, qui implique l'évaluation de l'amplitude de l'ITV pour déclencher la permeabilisation, est exigée. Le problème est critique dans des tissus, où la géométrie cellulaire est plus compliquée, les cellules sont assez proches pour affecter le champ électrique autour d'elles et elles sont souvent connectées entre elles. Dans tous ces cas, une description analytique de l'ITV n'est en général pas accessible et des méthodes numériques sont ainsi souvent la seule approche envisageable. En raison de la complexité de la structure tissulaire, les modèles sont macroscopiques, et on ne considère pas la structure cellulaire détaillée, ou en cas des modèles microscopiques, les modèles sont construits utilisant des formes géométriques simples (des semi-sphères, des cubes). Pour mieux comprendre comment le champ électrique interagit avec des tissus, nous avons construit les modèles microscopiques réalistes de cellules irrégulièrement formées, des groupes de telles cellules et des suspensions denses. Le travail a alors été développé sur le plan expérimental au niveau de la cellule isolée. Les mesures de cinétique de transport de membrane ont montré que l'electropermeabilisation avec des amplitudes d'impulsion ou des durées d'impulsion progressivement croissantes conduit à des transports accrus dans des cellules. Une large augmentation a été observée dans les milisecondes après le début d'une impulsion, suivie par une augmentation de fluorescence progressive. Les résultats mesurés sur un intervalle de temps de 400 µ S ont révélé que le transport à travers la membrane permeabilisée ne peut être détecté que 100 µ S après le début de l'impulsion. En plus, une dynamique différente d'augmentation de fluorescence pendant et après l'impulsion a été observée.When a biological cell is exposed to an external electric field, induced transmembrane voltage (ITV) forms on its membrane. During the exposure, ITV superimposes to the native or resting transmembrane voltage (RTV) and when the sum of both voltages exceeds some threshold value, the permeability of the cell membrane in these regions transiently increases. This phenomenon is termed electropermeabilization. In many applications of electropermeabilization an efficient and at the same time reversible permeabilization is essential (e.g. DNA electrotransfer). Thus, a careful planning of the experiment, which involves the estimation of the amplitude of ITV leading to cell permeabilization, is required. The problem arises in case of tissues, where cell geometry is more complicated, cells are close enough to affect the electric field around each other, and they are often connected with pathways between them. In all these cases, an analytical description of ITV is in general not attainable and numerical methods are often the only feasible approach. Due to the complexity of tissue structure, numerical models are either macroscopic, where detailed cell structure is notconsidered, or in case of microscopic models, the models are constructed using simple geometrical shapes (semi-spheres, cubes). To better understand how the electric field interacts with tissues on a microscopic (single cell) level, which in turn determines the macroscopic behavior of the tissue, we constructed realistic microscopic models of irregularly shaped cells, clusters of such cells, and dense suspensions. Regarding the shape, density and connections between cells, these cell assemblies are in their complexity close to tissues. First, the amplitude of resting transmembrane voltage of cells used in the study was determined. Next, calculations of ITV were performed on models of single spherical, single attached cells, and cell clusters and they were compared to measurements of ITV on the same cells, from which the models were constructed. The course of electropermeabilization of these cells was then monitored and the results were compared with measurements and calculations of ITV. In a separate experiment, a detailed investigation of kinetics of molecular transport into cells after permeabilization was performed. Similarly, for dense cell suspensions, the ITV calculated on a model of suspension was compared with the fraction of permeabilized cells measured in suspensions with increasing cell densities. Measurements of resting transmembrane voltage (RTV) were performed by means of a slow potentiometric fluorescent dye TMRM on different cell lines in culture media and media with progressively decreasing conductivities. ITV was measured on single spherical cells, single irregularly shaped cells, and cell clusters with a fast potentiometric fluorescent dye di-8- ANEPPS. The cross-section fluorescence images of the same cells on which the measurements of ITV were performed, were used to construct realistic numerical models of cells and the ITV on these models was then calculated with finite elements method. Finitethickness, nonzero conductivity cell membrane in the model was replaced by a boundary condition in which a specific surface conductivity was assigned to the interface between the cell interior and the exterior. Electropermeabilization of cells was followed by monitoring thechanges in intracellular fluorescence of membrane-impermeant fluorescent dye Propidium Iodide. Measurements of RTV showed that in physiological conditions (cells in culture medium) and in the presence of pulsing buffer, RTV on investigated cell lines is low (between -4 and -35 mV for suspended cells and between -18 and -27 mV for attached cells). Therefore, in experiments involving electropermeabilization ITV can be used as a rough approximate of the total voltage on the membrane, while RTV can be neglected. RTV in cells in media with decreasing conductivities gradually decreased, but less than expected from theoretical calculations. This was partly attributed to overestimated intracellular concentration of potassium. However, it is also possible that the method for measuring RTV, although reported as efficient, was not suitable for these experiments. Measurements of ITV on single spherical cells, single attached cells, and cell clusters were in qualitative agreement with results of numerical calculations, while in some cases discrepancies in measured and calculated amplitudes could be observed. This was attributed to variations of the slope of calibration curve, the differences between the actual and implemented parameters of the model, physiological state of cells, and experimental setup. In addition, we observed that at pulse parameters used in measurements of ITV, cells in clusters behaved as electrically connected, i.e. a cluster acted as one giant cell. Numerical calculations on models of cells where cell membrane was replaced with a boundary condition resulted in considerably lower number of mesh elements and consequently shorter time needed to solve the problem. We also demonstrated that calculations of ITV on simplified models of irregularly shaped cells can lead to considerable deviations from ITV calculated on a realistic model. Electric field orientation affects the amplitude and distribution of calculated ITV and consequently permeabilization. Namely, cells oriented with their longer axis parallel to the field are more likely to get permeabilized than the same cells oriented perpendicularly to the field. Comparison of measured and calculated ITVs with observations of electropermeabilization on single spherical and single attached cells confirmed that permeabilization occurs in those regions of the membrane, where the absolute value of ITV is the highest (the regions facing the electrodes). Additional experiments performed on single spherical cells showed that during and immediately after the pulse, the fluorescence from cells increases asymmetrically if unipolar pulses were delivered, while symmetrical fluorescence was observed for bipolar pulses. These observations were attributed to electrophoretical effect of the pulse. On a longer time scale, asymmetry in fluorescence was still observed, even for bipolar pulses, and we did not find any reasonable explanation for that. Critical value of ITV, at which permeabilization occurs, was calculated from the polar angle of permeabilization measured immediately after the pulse and was found to be approximately 450 mV, in agreement with reported critical thresholds. Permeabilization results obtained on cell clusters showed that cells in clusters, atpulse parameters used in these experiments, behaved as electrically insulated and were permeabilized individually. This is in contradiction to what we observed during measurements of ITV (i.e. with longer, low voltage pulses), where cells in clusters behaved as electrically connected, and was assumed to be the result of opening and closing of gap junctions at different pulse parameters. Measurements of kinetics of membrane transport showed that electropermeabilization with progressively increasing pulse amplitudes or pulse durations results in increased dye transport into cells. A sharp increase was observed miliseconds after the onset of a pulse, followed by a moderate additional fluorescence increase. Results measured on a time interval of 400 µs revealed that the transport across the permeabilized membrane can be detected within 100 µs after the onset of the pulse. Besides, different dynamics of fluorescence increase was observed during and immediately after the pulse. Experiments carried out on dense cell suspensions showed that with increasing cell density (from 10×106 cells/ml to 400×106 cells/ml) the fraction of permeabilized cells decreased by approximately 50%. We attributed this to the changes in the local electric field, which lead to a decrease in the amplitude of ITV. The uptake of Propidium Iodide also decreased with cell density, but by a larger amount than expected from permeabilization results. We supposed that the additional decrease in fluorescence was mainly due to cell swelling after permeabilization, which reduced extracellular dye availability to the permeabilized membrane and hindered the dye diffusion into the cells. Resealing of cells appeared to be slower in dense suspensions, which can also be attributed to cell swelling resulting from electropermeabilization

Spatio temporal dynamics of direct current in treated anisotropic tumors

The inclusion of a diffusion term in the modified Gompertz equation (Cabrales et al., 2018) allows to describe the spatiotemporal growth of direct current treated tumors. The aim of this study is to extend the previous model to the case of anisotropic tumors, simulating the spatiotemporal behavior of direct current treated anisotropic tumors, also carrying out a theoretical analysis of the proposed model. Growths in the mass, volume and density of the solid tumors are shown for each response type after direct current application (disease progression, partial response, stationary partial response and complete remission). For this purpose, the Method of Lines and different diffusion tensors are used. The results show that the growth of the tumor treated with direct current is faster for the shorter duration of the net antitumor effect and the higher diffusion coefficient and anisotropy degree of the solid tumor. It is concluded that the greatest direct current antitumor effectiveness occurs for the highly heterogeneous, anisotropic, aggressive and hypodense malignant solid tumors

Simulation Studies of Pulsed Voltage Effects on Cells

This dissertation research focuses on the new field of pulsed electric field interactions with biological cells. In particular, Intracellular Electromanipulation which has important biomedical applications, is probed. Among the various aspects studied, nanosecond, high-intensity pulse induced electroporation is one phenomena. It is simulated based on a coupled scheme involving the current continuity and Smoluchowski equations. A dynamic pore model can be achieved by including a dependence on the pore population density and a variable membrane tension. These changes make the pore formation energy E(r) self-adjusting and dynamic in response to pore formation. Additionally, molecular dynamics (MD) simulations are also discussed as a more accurate, though computationally intensive, alternative. Besides inducing pores in cells, external voltages could also be used, in principle, to modulate action potential generation in nerves. The electric-field induced poration could block action potential propagation. This aspect has been studied by modifying the traditional cable model for nerves, by accounting for the increased membrane conductance and the altered membrane capacitance. This conduction block in nerves due to an electroporation related local short-circuit would be similar in concept to stopping the propagation of an air-pressure wave down a leaky pipe. This study also focuses on threshold process in cellular apoptosis induced by nanosecond, high-intensity electric pulses. In particular, the pulse number dependent cell survival trends are quantified based on a biophysical model of the cellular apoptotic processes. Time-dependent evolution of the caspase concentrations and the various molecular species are simulated. The numerical evaluations provide qualitative predictions of pulse number cell survival, the relative assessment of extrinsic and intrinsic pathways, and rough predictions of the time duration over which irreversible activation at the molecular level could be initiated by the electric pulses. Time dependent kinetics of the caspases as well as the various molecular species within the apoptotic pathway, were simulated using the rate equation model originally proposed by Bagci et al. Finally, an asymmetric electroporation model is presented. Electric pulsing pore energy and mechanical pore energy are studied. This has relevance to the flow of ions in and out of cells