THEORETICAL GAS CONCENTRATIONS ACHIEVING 100% FILL OF THE VITREOUS CAVITY IN THE POSTOPERATIVE PERIOD: A Gas Eye Model Study.

PURPOSE: To determine the concentrations of different gas tamponades in air to achieve 100% fill of the vitreous cavity postoperatively and to examine the influence of eye volume on these concentrations. METHODS: A mathematical model of the mass transfer dynamics of tamponade and blood gases (O2, N2, and CO2) when injected into the eye was used. Mass transfer surface areas were calculated from published anatomical data. The model has been calibrated from published volumetric decay and composition results for three gases sulphahexafluoride (SF6), hexafluoroethane (C2F6), or perfluoropropane (C3F8). The concentrations of these gases (in air) required to achieve 100% fill of the vitreous cavity postoperatively without an intraocular pressure rise were determined. The concentrations were calculated for three volumes of the vitreous cavity to test whether ocular size influenced the results. RESULTS: A table of gas concentrations was produced. In a simulation of pars plana vitrectomy operations in which an 80% to 85% fill of the vitreous cavity with gas was achieved at surgery, the concentrations of the 3 gases in air to achieve 100% fill postoperatively were 10% to 13% for C3F8, 12% to 15% for C2F6, and 19% to 25% for SF6. These were similar to the so-called "nonexpansive" concentrations used in the clinical setting. The calculations were repeated for three different sizes of eye. Aiming for an 80% fill at surgery and 100% postoperatively, an eye with a 4-mL vitreous cavity required 24% SF6, 15% C2F6, or 13% C3F8; 7.2 mL required 25% SF6, 15% C2F6, or 13% C3F8; and 10 mL required 25% SF6, 16% C2F6, or 13% C3F8. When using 100% gas (e.g., used in pneumatic retinopexy), to achieve 100% fill postoperatively, the minimum vitreous cavity fill at surgery was 43% for SF6, 29% for C2F6, and 25% for C3F8 and was only minimally changed by variation in the size of the eye. CONCLUSION: A table has been produced, which could be used for surgical innovation in gas usage in the vitreous cavity. It provides concentrations for different percentage fills, which will achieve a moment postoperatively with a full fill of the cavity without a pressure rise. Variation in axial length and size of the eye does not seem to alter the values in the table significantly. Those using pneumatic retinopexy need to increase the volume of gas injected with increased size of the eye to match the percentage fill of the vitreous cavity recommended for a given tamponade agent.

A thermoelastic deformation model of tissue contraction during thermal ablation.

PURPOSE: Thermal ablation is an energy-based ablation technique widely used during minimally invasive cancer treatment. Simulations are used to predict the dead tissue post therapy. However, one difficulty with the simulations is accurately predicting the ablation zone in post-procedural images due to the contraction of tissue as a result of exposure to elevated temperatures. MATERIALS AND METHODS: A mathematical model of the thermoelastic deformation for an elastic isotropic material was coupled with a three state thermal denaturation model to determine the contraction of tissue during thermal ablation. A finite difference method was considered to quantify the tissue contraction for a typical temperature distribution during thermal ablation. RESULTS: The simulations show that tissue displacement during thermal ablation was not bound to the tissue heated regions only. Both tissue expansion and contraction were observed at the different stages of the heating process. Tissue contraction of up to 42% was obtained with an applicator temperature of 90 °C. A recovery of around 2% was observed with heating removed as a result of unfolded state proteins returning back to its native state. Poisson's ratio and the applicator temperature have both been shown to affect the tissue displacement significantly. The maximum tissue contraction was found to increase with both increasing Poisson's ratio and temperature. CONCLUSIONS: The model presented here will allow predictions of thermal ablation to be corrected for tissue contraction, which is an important effect, during comparison with post-procedural images, thus improving the accuracy of mathematical simulations for treatment planning.

Mathematical model of the post-ablation enhancement zone as a tissue-level oedematic response.

PURPOSE: A hyperdense rim is commonly observed at the periphery of ablation zones during post-ablation imaging (e.g. ultrasound) in tumours. A mathematical model has been developed here to investigate the occurrence of this enhanced rim, caused by the ablated cells, giving an indication of the location of the final ablation region. MATERIALS AND METHODS: The enhanced rim has been assumed here to be due to a tissue-level oedematic response of viable cells, which necessitated coupling multiple modelling elements in a spatially distributed system: thermal cell death, tissue-state dependent ion concentration dynamics, ion transport in the extracellular space, and osmotic cell volume regulation. RESULTS: In response to the imposed temperature function, an ablation zone was predicted, distinguishing the tissue state between 'dead' and 'alive'. A disturbance in intracellular/extracellular ion concentrations was induced due to ion redistribution, which acted as an osmotic stress and contributed to significant cell swelling in a thin rim at the periphery of the ablation zone. It was also found that the rim size only changed slightly with varying lesion size, in response to different temperature profiles. CONCLUSIONS: The study presents a novel mathematical model to understand the enhanced rim surrounding the ablation zone by assuming tissue-level cell oedema as the primary potential cause. The model links the direct response to thermal injury to an observable secondary response, which could be of clinical value in that the location of this bright ring could potentially be used for more accurate determination of the extent of the ablation zone.

A model of tissue contraction during thermal ablation.

A model of a globular protein is used to describe the contraction of tissue exposed to elevated temperatures. This will be useful in predicting the contraction of tissue that is observed during thermal ablation of tumours, which is a problem when trying to determine the ablation zone in post-operative images. The transitions between the states of the protein can be related to a change in the length of the molecule, which can be directly observed as a change in the length of the tissue. A three state model of a globular protein is used to describe the contraction of tissue exposed to elevated temperatures. A nonlinear fitting algorithm is considered here to fit available experimental data and thus to obtain the values of the model parameters. A sensitivity analysis of the proposed mathematical model is performed to determine the most important parameters in the model. The model parameters were obtained from experimental data of isothermal free shrinkage experiments. The predictions of the complete model show similar agreement with the data, well within the experimental error of 10%. The overall activation energy and frequency factor were found to be 201 kJ mol(-1) and [Formula: see text] s(-1) respectively. The results show that the experimental data were well described by the three state model considered here. Furthermore, it was possible to determine the most sensitive parameters in the model. The model presented here will allow predictions of thermal ablation to be corrected for tissue shrinkage, thus improving mathematical simulations for treatment planning, although clinical translation will require adapting the model from experimentally obtained tendon data to soft tissue data.

Cell death, perfusion and electrical parameters are critical in models of hepatic radiofrequency ablation.

PURPOSE: A sensitivity analysis has been performed on a mathematical model of radiofrequency ablation (RFA) in the liver. The purpose of this is to identify the most important parameters in the model, defined as those that produce the largest changes in the prediction. This is important in understanding the role of uncertainty and when comparing the model predictions to experimental data. MATERIALS AND METHODS: The Morris method was chosen to perform the sensitivity analysis because it is ideal for models with many parameters or that take a significant length of time to obtain solutions. A comprehensive literature review was performed to obtain ranges over which the model parameters are expected to vary, crucial input information. RESULTS: The most important parameters in predicting the ablation zone size in our model of RFA are those representing the blood perfusion, electrical conductivity and the cell death model. The size of the 50 °C isotherm is sensitive to the electrical properties of tissue while the heat source is active, and to the thermal parameters during cooling. CONCLUSIONS: The parameter ranges chosen for the sensitivity analysis are believed to represent all that is currently known about their values in combination. The Morris method is able to compute global parameter sensitivities taking into account the interaction of all parameters, something that has not been done before. Research is needed to better understand the uncertainties in the cell death, electrical conductivity and perfusion models, but the other parameters are only of second order, providing a significant simplification.

A mathematical framework for minimally invasive tumor ablation therapies.

Minimally invasive tumor ablations (MITAs) are an increasingly important tool in the treatment of solid tumors across multiple organs. The problems experienced in modeling different types of MITAs are very similar, but the development of mathematical models is mostly performed in isolation according to modality. Fundamental research into the modeling of specific types of MITAs is indeed required, but to choose the optimal treatment for an individual the primary clinical requirement is to have reliable predictions for a range of MITAs. In this review of the mathematical modeling of MITAs 4 modalities are considered: radiofrequency ablation, microwave ablation, cryoablation, and irreversible electroporation. The similarities in the mathematical modeling of these treatments are highlighted, and the analysis of the models within a general framework is discussed. This will aid in developing a deeper understanding of the sensitivity of MITA models to physiological parameters and the impact of uncertainty on predictions of the ablation zone. Through robust validation and analysis of the models it will be possible to choose the best model for a given application. This is important because many different models exist with no objective comparison of their performance. The collection of relevant in vivo experimental data is also critical to parameterize such models accurately. This approach will be necessary to translate the field into clinical practice.