A simple mathematical model of the bovine estrous cycle: follicle development and endocrine interactions.

Bovine fertility is the subject of extensive research in animal sciences, especially because fertility of dairy cows has declined during the last decades. The regulation of estrus is controlled by the complex interplay of various organs and hormones. Mathematical modeling of the bovine estrous cycle could help in understanding the dynamics of this complex biological system. In this paper we present a mechanistic mathematical model of the bovine estrous cycle that includes the processes of follicle and corpus luteum development and the key hormones that interact to control these processes. The model generates successive estrous cycles of 21 days, with three waves of follicle growth per cycle. The model contains 12 differential equations and 54 parameters. Focus in this paper is on development of the model, but also some simulation results are presented, showing that a set of equations and parameters is obtained that describes the system consistent with empirical knowledge. Even though the majority of the mechanisms that are included in the model are based on relations that in the literature have only been described qualitatively (i.e. stimulation and inhibition), the output of the model is surprisingly well in line with empirical data. This model of the bovine estrous cycle could be used as a basis for more elaborate models with the ability to study effects of external manipulations and genetic differences.

Anatomy- and physics-based facial animation for craniofacial surgery simulations.

A modelling approach for the realistic simulation of facial expressions of emotion in craniofacial surgery planning is presented. The method is different from conventional, non-physical techniques for character animation in computer graphics. A consistent physiological mechanism for facial expressions was assumed, which was the effect of contracting muscles on soft tissues. For the numerical solution of the linear elastic boundary values, the finite element method on tetrahedral grids was used. The approach was validated on a geometrical model of a human head derived from tomographic data. Using this model, individual facial expressions of emotion were estimated by the superpositioning of precomputed single muscle actions.

Antenna arrays in the SIGMA-eye applicator: interactions and transforming networks.

OBJECTIVES: In multiantenna applicators such as the SIGMA-60 or SIGMA-Eye, which consist of 4 or 12 pairs of antennas shunt to 4 or 12 amplifiers ("antenna couplets"), phases and amplitudes in the feed points of these antennas under certain conditions can significantly differ from the values selected at the multichannel amplifier (forward parameters), mainly due to coupling. In the SIGMA-Eye, this interaction is particularly affected by the transforming networks between the generators and the feed points, thus hampering the control of the feed point parameters. In this work, we perform measurements at existing applicators, present a formalism to describe the facts numerically, and investigate modifications of the transforming networks to improve the performance. METHODS AND MATERIALS: We prepared an experimental setup for the SIGMA-Eye applicator that is fed by forward waves of a 12-channel amplifier system. In this setup, we made the water bolus, the interior of the tissue-equivalent phantom, and the entire transforming network accessible for measuring probes. Then, we constructed various alternative transforming networks such as Pawsey loops, LC matching networks, and power dividers and compared them with the original matching network of the SIGMA-Eye applicator. In particular, we utilized a high-resistive probe to determine the disturbances and influences caused by some channels with respect to some selected feed points of the SIGMA-Eye dipoles. RESULTS: In the original SIGMA-Eye applicator, the influences of coupling channels on the phases and voltages in the feed point of a particular antenna are largest for adjacent longitudinal channels. Here, the +/- 10 degrees phase shift and +/- 30% voltage change were observed if the reference channel (i.e., the disturbed channel) and disturbing channel are equally powered. The changes eminently increased to -30 degrees to + 100 degrees phase shift and -80% to +50% voltage change if the reference channel is fed with much lower power (four to eight-fold) than the disturbing channel. The disturbance from distant channels is less but still significant, reaching shifts of -10 degrees to +50 degrees and -50% to +20%, respectively. Using Pawsey loops instead of the original ferrite rings in the SIGMA-Eye network, the efficacy of the baluns was improved by a more than a factor of 4. Using an LC matching network, dependencies on frequency and external arrangements can be reduced significantly. Applying a power divider circuit, the coupling between antennas combined to one channel is considerably diminished (down to <-25 dB). CONCLUSION: Coupling between resonators (pairs of antennas including the matching network) reduces the control of the SIGMA-Eye applicator, i.e., it causes deviations between the selection of forward parameters at the amplifier and the total actual parameters in the feed points of the antennas. Modified transformation networks can improve the control, in particular by reducing sheath currents and asymmetries. There is a linear but variable relationship between selected (amplifiers) and actually given (feed points) parameters. This linear mapping (described by a matrix) and its characteristics need further investigation.

Clinical evaluation and verification of the hyperthermia treatment planning system hyperplan.

PURPOSE: A prototype of the hyperthermia treatment planning system (HTPS) HyperPlan for the SIGMA-60 applicator (BSD Medical Corp., Salt Lake City, Utah, USA) has been evaluated with respect to clinical practicability and correctness. MATERIALS AND METHODS: HyperPlan modules extract tissue boundaries from computed tomography (CT) images to generate regular and tetrahedral grids as patient models, to calculate electric field (E-field) distributions, and to visualize three-dimensional data sets. The finite difference time-domain (FDTD) method is applied to calculate the specific absorption rate (SAR) inside the patient. Temperature distributions are calculated by a finite-element code and can be optimized. HyperPlan was tested on 6 patients with pelvic tumors. For verification, measured SAR values were compared with calculated SAR values. Furthermore, intracorporeal E-field scans were performed and compared with calculated profiles. RESULTS: The HTPS can be applied under clinical conditions. Measured absolute SAR (in W/kg), as well as relative E-field scans, correlated well with calculated values (+/-20%) using the contour-based FDTD method. Values calculated by applying the FDTD method directly on the voxel (CT) grid, were less well correlated with measured data. CONCLUSION: The HyperPlan system proved to be clinically feasible, and the results were quantitatively and qualitatively verified for the contour-based FDTD method.

Influence of patient models and numerical methods on predicted power deposition patterns.

BACKGROUND: A hyperthermia planning system has been developed for generating patient and applicator models as well as calculating and visualizing E-field and temperature distributions. Significant dependencies on models and algorithms have been found. METHODS: Computerized tomography (CT) data sets are first transformed into so called 'labelled CT-volume'-data sets of equal resolution, which are used for segmentation. The first type of patient model obtained subsequently is based on regions with specified electrical properties representing tissues or organs (so called 'region-based model'). The second patient model renders a direct transformation of Hounsfield Units (HU) to electrical constants (so called 'HU-based model'). The FDTD-method (finite difference time domain) is then applied on a cubic lattice employing either an auxiliary 'sub-cubic lattice' (for HU-based segmentation) or a tetrahedron grid (for region-based segmentation) to assign the electrical properties, both representing the anatomy of the patient. E-field distributions are corrected by a post-processing procedure with respect to the geometry of interfaces defined by the tetrahedron grid. For comparison, the VSIE method (volume surface integral equation) is performed on the same tetrahedron grid. The applicator model assumes eight half-wavelength dipole antennas fed with constant voltages with water as background medium. RESULTS: For both numerical methods (FDTD, VSIE) the resulting antenna input impedances as well as the current distributions along the antennas were quite similar and almost insensitive to the particular geometry model (region-based, HU-based). In contrast to that, the power deposition patterns in the interior of the patient depended strongly on those models. Major differences can be related to different labels of the tissue type bone in the HU-based model in comparison to the definition via regions. Conversely, comparable results were obtained using the VSIE method and the FDTD method on the region-based patient model with a posteriori correction at the tetrahedron grid points. SAR (specific absorption rate) elevations up to a factor of 10 were predicted when employing region-based models. Those peaks might correspond to specific toxicity of electromagnetic radiation clinically known as hot spot phenomena or musculo-skeletal syndromes. Conversely, HU-based models generated quite homogeneous power deposition patterns with fluctuations of at most factor 2. CONCLUSION: The methods employing region-based geometry models such as the VSIE method and FDTD method in conjunction with a posteriori correction at tissue interfaces result in comparable E-field distributions for regional hyperthermia. Due to its shorter calculation time, the FDTD method is currently used in the clinic. Predictions derived from HU-based models without prior corrections of tissue specifications are not always supported by clinical experience.

Evaluation of segmentation algorithms for generation of patient models in radiofrequency hyperthermia.

Time-efficient and easy-to-use segmentation algorithms (contour generation) are a precondition for various applications in radiation oncology, especially for planning purposes in hyperthermia. We have developed the three following algorithms for contour generation and implemented them in an editor of the HyperPlan hyperthermia planning system. Firstly, a manual contour input with numerous correction and editing options. Secondly, a volume growing algorithm with adjustable threshold range and minimal region size. Thirdly, a watershed transformation in two and three dimensions. In addition, the region input function of the Helax commercial radiation therapy planning system was available for comparison. All four approaches were applied under routine conditions to two-dimensional computed tomographic slices of the superior thoracic aperture, mid-chest, upper abdomen, mid-abdomen, pelvis and thigh; they were also applied to a 3D CT sequence of 72 slices using the three-dimensional extension of the algorithms. Time to generate the contours and their quality with respect to a reference model were determined. Manual input for a complete patient model required approximately 5 to 6 h for 72 CT slices (4.5 min/slice). If slight irregularities at object boundaries are accepted, this time can be reduced to 3.5 min/slice using the volume growing algorithm. However, generating a tetrahedron mesh from such a contour sequence for hyperthermia planning (the basis for finite-element algorithms) requires a significant amount of postediting. With the watershed algorithm extended to three dimensions, processing time can be further reduced to 3 min/slice while achieving satisfactory contour quality. Therefore, this method is currently regarded as offering some potential for efficient automated model generation in hyperthermia. In summary, the 3D volume growing algorithm and watershed transformation are both suitable for segmentation of even low-contrast objects. However, they are not always superior to user-friendly manual programs for contour generation. When the volume growing algorithm is used, the contours have to be postprocessed with suitable filters. The watershed transformation has a large potential if appropriately developed to 3D sequences and 3D interaction features. After all, the practicality and feasibility of every segmentation method critically depend on various details of the user software as pointed out in this article.

Simulation studies promote technological development of radiofrequency phased array hyperthermia.

A treatment planning program package for radiofrequency hyperthermia has been developed. It consists of software modules for processing three-dimensional computerized tomography (CT) data sets, manual segmentation, generation of tetrahedral grids, numerical calculation and optimisation of three-dimensional E field distributions using a volume surface integral equation algorithm as well as temperature distributions using an adaptive multilevel finite-elements code, and graphical tools for simultaneous representation of CT data and simulation results. Heat treatments are limited by hot spots in healthy tissues caused by E field maxima at electrical interfaces (bone/muscle). In order to reduce or avoid hot spots suitable objective functions are derived from power deposition patterns and temperature distributions, and are utilised to optimise antenna parameters (phases, amplitudes). The simulation and optimisation tools have been applied to estimate the improvements that could be reached by upgrades of the clinically used SIGMA-60 applicator (consisting of a single ring of four antenna pairs). The investigated upgrades are increased number of antennas and channels (triple-ring of 3 x 8 antennas and variation of antenna inclination. Significant improvement of index temperatures (1-2 degrees C) is achieved by upgrading the single ring to a triple ring with free phase selection for every antenna or antenna pair. Antenna amplitudes and inclinations proved as less important parameters.

Strategies for optimized application of annular-phased-array systems in clinical hyperthermia.

A theoretical framework is presented for optimized heating of deep-seated tumours by phase and amplitude steering. The optimization problem for a specific tumour and perfusion case results in a functional dependency between power-level and maximum obtainable therapeutic efficiency. Different optimization criteria and strategies are outlined, which cause an increase of power or thermal dose in the tumour. Three tumour models (central pelvic tumour, eccentric abdominal tumour with or without necrosis) are analysed in detail. The simulation studies predict that appreciable parts of these tumours (50-100%) can be heated efficiently (42.5-43 degrees C) within the range of available and clinically tolerated power levels (1-5 kW/m), if tumour perfusion is less than 20-25 ml/100 g min. Some improvements are obtained by increasing the number of independent channels (from four to eight) and by the application of time-dependent (complementary) power-deposition patterns.

Thermal recovery after passage of the pulmonary circulation assessed by deconvolution.

For indicator-dilution studies, complete thermal recovery after passage of heat through the pulmonary circulation would be desirable. However, the results in the literature obtained by extrapolation techniques are inconsistent. To overcome problems of the extrapolation approach, transport functions of the pulmonary circulation (including the left heart) were computed by deconvolution of pulmonary arterial and aortic pairs of thermodilution curves after central venous indicator injection (10 ml of an ice-cold blood indocyanine green dye mixture). Thermal recovery was determined as the finite integral of the transport function. Thirteen mongrel dogs under piritramid-N2O anesthesia were examined under base-line conditions, in orthostasis to alter the distribution of pulmonary blood flow (9 dogs), and in oleic acid edema (8 dogs). Using the deconvolution approach, thermal recovery was 0.97 +/- 0.04 under base-line conditions, 0.96 +/- 0.03 in orthostasis, and 0.96 +/- 0.05 in pulmonary edema. Thermal recovery determined from extrapolated dilution curves was greater than 100% in all groups, a physically impossible finding. It is concluded that thermal recovery is incomplete but insensitive with respect to the distribution of blood flow and to the size of the extravascular compartment. Monoexponential extrapolation is unsuited for the determination of thermal recovery.

Evaluation of monoexponential extrapolation of transpulmonary thermal-dye kinetics by use of a new model-free deconvolution algorithm.

This study evaluates the routine mathematic approach (monoexponential extrapolation) for analysis of transpulmonary thermal-dye dilution curves and estimates the effects of systemic-indicator recirculation by use of a deconvolution technique. Fifteen dogs anesthetized with N2O-piritramid were studied before and after induction of pulmonary edema by oleic acid. After introduction of central venous indicator (10 ml of a mixture of cold blood and indocyanine green dye), dilution data were recorded from the pulmonary artery and the ascending aorta. The conclusions were: (1) monoexponential extrapolation yields reasonably good estimates of the mean transit times of dye; (2) mean transit times of heat are usually overestimated by monoexponential extrapolation; (3) extravascular lung thermal volume assessed by monoexponential extrapolation is overestimated by 2.03 ml/kg of body mass under baseline conditions; and (4) the prepulmonary volume of distribution of heat exceeds that of dye by 1.4 ml/kg of body mass, thus increasing the overestimation of pulmonary extravascular heat-accessible space by the conventional technique.