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In this work we derive a hierarchy of new mathematical models for describing the motion of phototactic bacteria, i.e., bacteria that move towards light. These models are based on recent experiments suggesting that the motion of such bacteria depends on the individual bacteria, on group dynamics, and on the interaction between bacteria and their environment. Our first model is a collisionless interacting particle system in which we follow the location of the bacteria, their velocity, and their internal excitation (a parameter whose role is assumed to be related to communication between bacteria). In this model, the light source acts as an external force. The resulting particle system is an extension of the Cucker-Smale flocking model. We prove that when all particles are fully excited, their asymptotic velocity tends to an identical (pre-determined) terminal velocity. Our second model is a kinetic model for the one-particle distribution function that includes an internal variable representing the excitation level. The kinetic model is a Vlasov-type equation that is derived from the particle system using the BBGKY hierarchy and molecular chaos assumption. Since bacteria tend to move in areas that were previously traveled by other bacteria, a surface memory effect is added to the kinetic model as a turning operator that accounts for the collisions between bacteria and the environment. The third and final model is derived as a formal macroscopic limit of the kinetic model. It is shown to be the Vlasov-McKean equation coupled with a reaction-diffusion equation.
The purpose of this study is to develop automatic algorithms for the segmentation phase of radiotherapy treatment planning. We develop new image processing techniques that are based on solving a partial differential equation for the evolution of the curve that identifies the segmented organ. The velocity function is based on the piecewise Mumford-Shah functional. Our method incorporates information about the target organ into classical segmentation algorithms. This information, which is given in terms of a three-dimensional wireframe representation of the organ, serves as an initial guess for the segmentation algorithm. We check the performance of the new algorithm on eight data sets of three different organs: rectum, bladder, and kidney. The results of the automatic segmentation were compared with a manual segmentation of each data set by radiation oncology faculty and residents. The quality of the automatic segmentation was measured with the ''$\kappa$-statistics'', and with a count of over- and undersegmented frames, and was shown in most cases to be very close to the manual segmentation of the same data. A typical segmentation of an organ with sixty slices takes less than ten seconds on a Pentium IV laptop.
A multiscale image registration technique is presented for the registration of medical images that contain significant levels of noise. An overview of the medical image registration problem is presented, and various registration techniques are discussed. Experiments using mean squares, normalized correlation, and mutual information optimal linear registration are presented that determine the noise levels at which registration using these techniques fails. Further experiments in which classical denoising algorithms are applied prior to registration are presented, and it is shown that registration fails in this case for significantly high levels of noise, as well. The hierarchical multiscale image decomposition of E. Tadmor, S. Nezzar, and L. Vese  is presented, and accurate registration of noisy images is achieved by obtaining a hierarchical multiscale decomposition of the images and registering the resulting components. This approach enables successful registration of images that contain noise levels well beyond the level at which ordinary optimal linear registration fails. Image registration experiments demonstrate the accuracy and efficiency of the multiscale registration technique, and for all noise levels, the multiscale technique is as accurate as or more accurate than ordinary registration techniques.
Resistance to drugs has been an ongoing obstacle to a successful treatment of many diseases. In this work we consider the problem of drug resistance in cancer, focusing on random genetic point mutations. Most previous works on mathematical models of such drug resistance have been based on stochastic methods. In contrast, our approach is based on an elementary, compartmental system of ordinary differential equations. We use our very simple approach to derive results on drug resistance that are comparable to those that were previously obtained using much more complex mathematical techniques. The simplicity of our model allows us to obtain analytic results for resistance to any number of drugs. In particular, we show that the amount of resistance generated before the start of the treatment, and present at some given time afterward, always depends on the turnover rate, no matter how many drugs are simultaneously used in the treatment.
This paper focuses on the characterization of delay effects on the asymptotic stability of some continuous-time delay systems encountered in modeling the post-transplantation dynamics of the immune response to chronic myelogenous leukemia. Such models include multiple delays in some large range, from one minute to several days. The main objective of the paper is to study the stability of the crossing boundaries of the corresponding linearized models in the delay-parameter space by taking into account the interactions between small and large delays. Weak, and strong cell interactions are discussed, and analytic characterizations are proposed. An illustrative example together with related discussions completes the presentation.
This work presents a hierarchy of mathematical models for describing the motion of phototactic bacteria, i.e., bacteria that move towards light. Based on experimental observations, we conjecture that the motion of the colony towards light depends on certain group dynamics. This group dynamics is assumed to be encoded as an individual property of each bacterium, which we refer to as ’excitation’. The excitation of each individual bacterium changes based on the excitation of the neighboring bacteria. Under these assumptions, we derive a stochastic model for describing the evolution in time of the location of bacteria, the excitation of individual bacteria, and a surface memory effect. A discretization of this model results in an interacting stochastic many-particle system. The third, and last model is a system of partial differential equations that is obtained as the continuum limit of the stochastic particle system. The main theoretical results establish the validity of the new system of PDEs as the limit dynamics of the multi-particle system.
We present a new algorithm for segmenting organs in CT scans for radiotherapy treatment planning. Given a contour of an organ that is segmented in one image, our algorithm proceeds to segment contours that identify the same organ in the consecutive images. Our technique combines partial differential equations-based morphing active contours with algorithms for joint segmentation and registration. The coupling between these different techniques is done in order to deal with the complexity of segmenting ''real'' images, where boundaries are not always well defined, and the initial contour is not an isophote of the image.
Multiscale image registration techniques are presented for the reg- istration of medical images using deformable registration models. The tech- niques are particularly effective for registration problems in which one or both of the images to be registered contains significant levels of noise. A brief overview of existing deformable registration techniques is presented, and exper- iments using B-spline free-form deformation registration models demonstrate that ordinary deformable registration techniques fail to produce accurate re- sults in the presence of significant levels of noise. The hierarchical multiscale image decomposition described in E. Tadmor, S. Nezzar, and L. Vese's, ''A multiscale image representation using hierarchical (BV,L2) decompositions'' (Multiscale Modeling and Simulations, 2 (2004): 4, pp. 554-579) is reviewed, and multiscale image registration algorithms are developed based on the mul- tiscale decomposition. Accurate registration of noisy images is achieved by obtaining a hierarchical multiscale decomposition of the images and iteratively registering the resulting components. This approach enables a successful reg- istration of images that contain noise levels well beyond the level at which ordinary deformable registration fails. Numerous image registration experi- ments demonstrate the accuracy and efficiency of the multiscale registration techniques.
An image registration technique is presented for the registration of medical images using a hybrid combination of coarse-scale landmark and B-splines deformable registration techniques. The technique is particularly effective for registration problems in which the images to be registered contain large localized deformations. A brief overview of landmark and deformable registration techniques is presented. The hierarchical multiscale image decomposition of E. Tadmor, S. Nezzar, and L. Vese, A multiscale image representation using hierarchical $(BV,L^2)$ decompositions, Multiscale Modeling and Simulations, vol. 2, no. 4, pp. 554--579, 2004, is reviewed, and an image registration algorithm is developed based on combining the multiscale decomposition with landmark and deformable techniques. Successful registration of medical images is achieved by first obtaining a hierarchical multiscale decomposition of the images and then using landmark-based registration to register the resulting coarse scales. Corresponding bony structure landmarks are easily identified in the coarse scales, which contain only the large shapes and main features of the image. This registration is then fine tuned by using the resulting transformation as the starting point to deformably register the original images with each other using an iterated multiscale B-splines deformable registration technique. The accuracy and efficiency of the hybrid technique is demonstrated with several image registration case studies in two and three dimensions. Additionally, the hybrid technique is shown to be very robust with respect to the location of landmarks and presence of noise.
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