Cecilia Suárez

When the Optimal Is Not the Best: Parameter Estimation in Complex Biological Models

Background The vast computational resources that became available during the past decade enabled the development and simulation of increasingly complex mathematical models of cancer growth. These models typically involve many free parameters whose determination is a substantial obstacle to model development. Direct measurement of biochemical parameters in vivo is often difficult and sometimes impracticable, while fitting them under data-poor conditions may result in biologically implausible values. Results We discuss different methodological approaches to estimate parameters in complex biological models. We make use of the high computational power of the Blue Gene technology to perform an extensive study of the parameter space in a model of avascular tumor growth. We explicitly show that the landscape of the cost function used to optimize the model to the data has a very rugged surface in parameter space. This cost function has many local minima with unrealistic solutions, including the global minimum corresponding to the best fit. Conclusions The case studied in this paper shows one example in which model parameters that optimally fit the data are not necessarily the best ones from a biological point of view. To avoid force-fitting a model to a dataset, we propose that the best model parameters should be found by choosing, among suboptimal parameters, those that match criteria other than the ones used to fit the model. We also conclude that the model, data and optimization approach form a new complex system and point to the need of a theory that addresses this problem more generally.

The Role of pH Fronts in Reversible Electroporation

We present experimental measurements and theoretical predictions of ion transport in agar gels during reversible electroporation (ECT) for conditions typical to many clinical studies found in the literature, revealing the presence of pH fronts emerging from both electrodes. These results suggest that pH fronts are immediate and substantial. Since they might give rise to tissue necrosis, an unwanted condition in clinical applications of ECT as well as in irreversible electroporation (IRE) and in electrogenetherapy (EGT), it is important to quantify their extent and evolution. Here, a tracking technique is used to follow the space-time evolution of these pH fronts. It is found that they scale in time as , characteristic of a predominantly diffusive process. Comparing ECT pH fronts with those arising in electrotherapy (EChT), another treatment applying constant electric fields whose main goal is tissue necrosis, a striking result is observed: anodic acidification is larger in ECT than in EChT, suggesting that tissue necrosis could also be greater. Ways to minimize these adverse effects in ECT are suggested.

The Role of Ph Fronts in Tissue Electroporation Based Treatments

Treatments based on electroporation (EP) induce the formation of pores in cell membranes due to the application of pulsed electric fields. We present experimental evidence of the existence of pH fronts emerging from both electrodes during treatments based on tissue EP, for conditions found in many studies, and that these fronts are immediate and substantial. pH fronts are indirectly measured through the evanescence time (ET), defined as the time required for the tissue buffer to neutralize them. The ET was measured through a pH indicator imaged at a series of time intervals using a four-cluster hard fuzzy-c-means algorithm to segment pixels corresponding to the pH indicator at every frame. The ET was calculated as the time during which the number of pixels was 10% of those in the initial frame. While in EP-based treatments such as reversible (ECT) and irreversible electroporation (IRE) the ET is very short (though enough to cause minor injuries) due to electric pulse characteristics and biological buffers present in the tissue, in gene electrotransfer (GET), ET is much longer, enough to denaturate plasmids and produce cell damage. When any of the electric pulse parameters is doubled or tripled the ET grows and, remarkably, when any of the pulse parameters in GET is halved, the ET drops significantly. Reducing pH fronts has relevant implications for GET treatment efficiency, due to a substantial reduction of plasmid damage and cell loss.

The role of additional pulses in electropermeabilization protocols.

Electropermeabilization (EP) based protocols such as those applied in medicine, food processing or environmental management, are well established and widely used. The applied voltage, as well as tissue electric conductivity, are of utmost importance for assessing final electropermeabilized area and thus EP effectiveness. Experimental results from literature report that, under certain EP protocols, consecutive pulses increase tissue electric conductivity and even the permeabilization amount. Here we introduce a theoretical model that takes into account this effect in the application of an EP-based protocol, and its validation with experimental measurements. The theoretical model describes the electric field distribution by a nonlinear Laplace equation with a variable conductivity coefficient depending on the electric field, the temperature and the quantity of pulses, and the Pennes Bioheat equation for temperature variations. In the experiments, a vegetable tissue model (potato slice) is used for measuring electric currents and tissue electropermeabilized area in different EP protocols. Experimental measurements show that, during sequential pulses and keeping constant the applied voltage, the electric current density and the blackened (electropermeabilized) area increase. This behavior can only be attributed to a rise in the electric conductivity due to a higher number of pulses. Accordingly, we present a theoretical modeling of an EP protocol that predicts correctly the increment in the electric current density observed experimentally during the addition of pulses. The model also demonstrates that the electric current increase is due to a rise in the electric conductivity, in turn induced by temperature and pulse number, with no significant changes in the electric field distribution. The EP model introduced, based on a novel formulation of the electric conductivity, leads to a more realistic description of the EP phenomenon, hopefully providing more accurate predictions of treatment outcomes.

Tissue damage modeling in gene electrotransfer: the role of pH.

Optimal gene electrotransfer (GET) requires a compromise between maximum transgene expression and minimal tissue damage. GET in skeletal muscle can be improved by pretreatment with hyaluronidase which contributes to maximize transgene uptake and expression. Nevertheless, tissue damage remains severe close to the electrodes, with a concomitant loss of GET efficiency. Here we analyze the role of pH in tissue damage in GET protocols through in vivo modeling using a transparent chamber implanted into the dorsal skinfold of a mouse (DSC) and intravital microscopy, and in silico modeling using the Poisson-Nernst-Planck equations for ion transport. DSC intravital microscopy reveals the existence of pH fronts emerging from both electrodes and that these fronts are immediate and substantial thus giving rise to tissue necrosis. Theoretical modeling confirms experimental measurements and shows that in GET protocols whether with or without hyaluronidase pretreatment, pH fronts are the principal cause of muscle damage near the electrodes. It also predicts that an optimal efficiency in GET protocols, that is a compromise between obtaining maximum electroporated area and minimal tissue damage, is achieved when the electric field applied is near 183 V/cm in a GET protocol and 158 V/cm in a hyaluronidase+GET protocol.

Mathematical Modeling of Human Glioma Growth Based on Brain Topological Structures: Study of Two Clinical Cases

Gliomas are the most common primary brain tumors and yet almost incurable due mainly to their great invasion capability. This represents a challenge to present clinical oncology. Here, we introduce a mathematical model aiming to improve tumor spreading capability definition. The model consists in a time dependent reaction-diffusion equation in a three-dimensional spatial domain that distinguishes between different brain topological structures. The model uses a series of digitized images from brain slices covering the whole human brain. The Talairach atlas included in the model describes brain structures at different levels. Also, the inclusion of the Brodmann areas allows prediction of the brain functions affected during tumor evolution and the estimation of correlated symptoms. The model is solved numerically using patient-specific parametrization and finite differences. Simulations consider an initial state with cellular proliferation alone (benign tumor), and an advanced state when infiltration starts (malign tumor). Survival time is estimated on the basis of tumor size and location. The model is used to predict tumor evolution in two clinical cases. In the first case, predictions show that real infiltrative areas are underestimated by current diagnostic imaging. In the second case, tumor spreading predictions were shown to be more accurate than those derived from previous models in the literature. Our results suggest that the inclusion of differential migration in glioma growth models constitutes another step towards a better prediction of tumor infiltration at the moment of surgical or radiosurgical target definition. Also, the addition of physiological/psychological considerations to classical anatomical models will provide a better and integral understanding of the patient disease at the moment of deciding therapeutic options, taking into account not only survival but also life quality.

Numerical simulation of avascular tumor growth

A mathematical and numerical model for the description of different aspects of microtumor development is presented. The model is based in the solution of a system of partial differential equations describing an avascular tumor growth. A detailed second-order numeric algorithm for solving this system is described. Parameters are swiped to cover a range of feasible physiological values. While previous published works used a single set of parameters values, here we present a wide range of feasible solutions for tumor growth, covering a more realistic scenario. The model is validated by experimental data obtained with a multicellular spheroid model, a specific type of in vitro biological model which is at present considered to be optimum for the study of complex aspects of avascular microtumor physiology. Moreover, a dynamical analysis and local behaviour of the system is presented, showing chaotic situations for particular sets of parameter values at some fixed points. Further biological experiments related to those specific points may give potentially interesting results.

Integrated analysis of the potential, electric field, temperature, pH and tissue damage generated by different electrode arrays in a tumor under electrochemical treatment

Fil: Soba, Alejandro. Comision Nacional de Energia Atomica; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina

Cell membrane electroporation modeling: A multiphysics approach

Electroporation-based techniques, i.e. techniques based on the perturbation of the cell membrane through the application of electric pulses, are widely used at present in medicine and biotechnology. However, the electric pulse - cell membrane interaction is not yet completely understood neither explicitly formalized. Here we introduce a Multiphysics (MP) model describing electric pulse - cell membrane interaction consisting on the Poisson equation for the electric field, the Nernst-Planck equations for ion transport (protons, hydroxides, sodium or calcium, and chloride), the Maxwell tensor and mechanical equilibrium equation for membrane deformations (with an explicit discretization of the cell membrane), and the Smoluchowski equation for membrane permeabilization. The MP model predicts that during the application of an electric pulse to a spherical cell an elastic deformation of its membrane takes place affecting the induced transmembrane potential, the pore creation dynamics and the ionic transport. Moreover, the coincidence among maximum membrane deformation, maximum pore aperture, and maximum ion uptake is predicted. Such behavior has been corroborated experimentally by previously published results in red blood and CHO cells as well as in supramolecular lipid vesicles.

Effects of Pulse Addition in Electropermeabilization: Theoretical Insights on the Electric Conductivity

The electrochemical treatment (ECT) of solid tumors is an electropermeabilization technique firmly established and widely used. In ECT protocols, pulse intensity as well as tissue electric conductivity are of utmost importance for assessing the final electropermeabilized area. Present ECT mathematical modeling based on the solution of the nonlinear Laplace equation for the electric field with a conductivity coefficient depending on the electric field and the temperature have greatly contributed to ECT protocol optimization. However, experimental results from literature report that a succession of pulses may increase tissue electric conductivity and the extent of tissue permeabilization, a phenomenon that present models fail to describe. Here we present new insights of a recently introduced ECT theoretical model that takes into account the effect of pulse addition on tissue electric conductivity. The model describes the electric field with the nonlinear Laplace equation with a conductivity coefficient depending on the electric field, the temperature and the quantity of pulses applied. ECT theoretical predictions show that the rise in the electric current density during the addition of pulses is due solely to an increment in the tissue electric conductivity with no significant changes in the electric field. A potential consequence of these results is that, under certain conditions, it would be possible to obtain larger electropermeabilized areas with the same pulse amplitude simply by increasing the number of pulses. The theoretical implications of this new model lead to a more realistic description of the EP phenomenon, hopefully providing more accurate predictions of ECT treatment outcomes.