On kinematics of a parallel robot used for the minimally invasive surgery

One of the pioneer fields for robots is their assimilation in surgery, especially in minimally invasive procedures. The paper presents the kinematics of an innovative parallel structure for the manipulation of surgical instruments in minimally invasive surgery. The parallel architecture has been chosen for its superiority in precision, repeatability, stiffness, higher speeds and occupied volume. The equations, which model the kinematics are pointed out for this robot based on its mathematically determined functional parameters. The results of the kinematical modeling are systematically presented and lead to the facile solving of the parallel structure. Some simulation results have been presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Circular-imperfection of triangle-free graphs

Circular-perfect graphs form a natural superclass of the well-known perfect graphs by means of a more general coloring concept.For perfect graphs, a characterization by means of forbidden subgraphs was recently settled by Chudnovsky et al. [Chudnovsky, M., N. Robertson, P. Seymour, and R. Thomas, The Strong Perfect Graph Theorem, Annals of Mathematics 164 (2006) 51–229]. It is, therefore, natural to ask for an analogous characterization for circular-perfect graphs or, equivalently, for a characterization of all minimally circular-imperfect graphs.Our focus is the circular-(im)perfection of triangle-free graphs. We exhibit several different new infinite families of minimally circular-imperfect triangle-free graphs. This shows that a characterization of circular-perfect graphs by means of forbidden subgraphs is a difficult task, even if restricted to the class of triangle-free graphs. This is in contrary to the perfect case where it is long-time known that the only minimally imperfect triangle-free graphs are the odd holes [Tucker, A., Critical Perfect Graphs and Perfect 3-chromatic Graphs, J. Combin. Theory (B) 23 (1977) 143–149].     

Identification of Temperature Dependent Parameters in Radiative Heat Transfer

Laser-induced thermotherapy (LITT) is an established minimally invasive percutaneous technique of tumor ablation. Nevertheless, there is a need to predict the effect of laser applications and optimizing irradiation planning in LITT. Optical attributes (absorption, scattering) change due to thermal denaturation. The work presents the possibility to identify these temperature dependent parameters from given temperature measurements via an optimal control problem. The solvability of the optimal control problem is analyzed and results of successful implementations are shown. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linearly implicit time discretization for free surface problems

Deeper investigation of time discretization for free surface problems is a widely neglected problem. Many existing approaches use an explicit decoupling which is only conditionally stable. Only few unconditionally stable methods are known, and known methods may suffer from too strong numerical dissipativity. They are also usually of first rder only [1, 9]. We are therefore looking for unconditionally stable, minimally dissipative methods of higher order. Linearly implicit Runge-Kutta (LIRK) methods are a class of one-step methods that require the solution of linear systems in each time step of a nonlinear system. They are well suited for discretized PDEs, e.g. parabolic problems [7]. They have been used successfully to solve the incompressible Navier-Stokes equations [5]. We suggest an adaption of these methods for free surface problems and compare different approximations to the Jacobian matrix needed for such methods. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new method of construction of resonances that applies to critical models

We introduce a new method of multi-scale analysis that can be used to study the spectral properties of operators in non-relativistic quantum electrodynamics with critical coupling functions. We utilize our method to prove the existence of resonances of nonrelativistic atoms which are minimally coupled to the quantized (ultraviolet-regularized) radiation field and construct them together with the corresponding resonance eigenvector in case of critical coupling, i.e., without any infrared regularization. This result was first proved in [19] with the ingredient of a suitable Pauli-Fierz transformation. The purpose of the present paper, however, is to demonstrate the power of our new method for the estimation of resolvents that is based on the isospectral Feshbach-Schur map [8]. The reconstruction formula for the resolvent of an operator in terms of the resolvent of its image under the Feshbach-Schur map allows us to use a fixed projection and to compare two resolvents without actually decimating the degrees of freedom. This is in contrast to the renormalization group based on Feshbach-Schur map, developed in [8], [5], that uses a decreasing sequence of ever-smaller projections and successively decimates the degrees of freedom. It is this new method that allows us to treat the critical and physically relevant Standard model of non-relativistic quantum electrodynamics [7] which is intractable by standard methods.     

Toroidal Dehn surgery on hyperbolic knots and hitting number

In this paper, we prove that for any positive even integer m, there exists a hyperbolic knot such that its longitudinal Dehn surgery yields a 3-manifold containing a unique separating, incompressible torus, which meets the core of the attached solid torus in m points minimally.     

On a certain class of nonideal clutters

In this paper we define the class of near-ideal clutters following a similar concept due to Shepherd [Near perfect matrices, Math. Programming 64 (1994) 295-323] for near-perfect graphs. We prove that near-ideal clutters give a polyhedral characterization for minimally nonideal clutters as near-perfect graphs did for minimally imperfect graphs. We characterize near-ideal blockers of graphs as blockers of near-bipartite graphs. We find necessary conditions for a clutter to be near-ideal and sufficient conditions for the clutters satisfying that every minimal vertex cover is minimum.     

On the set covering polyhedron of circulant matrices

A well known family of minimally nonideal matrices is the family of the incidence matrices of chordless odd cycles. A natural generalization of these matrices is given by the family of circulant matrices. Ideal and minimally nonideal circulant matrices have been completely identified by Cornuéjols and Novick [G. Cornuéjols, B. Novick, Ideal 0 - 1 matrices, Journal of Combinatorial Theory B 60 (1994) 145–157]. In this work we classify circulant matrices and their blockers in terms of the inequalities involved in their set covering polyhedra. We exploit the results due to Cornuéjols and Novick in the above-cited reference for describing the set covering polyhedron of blockers of circulant matrices. Finally, we point out that the results found on circulant matrices and their blockers present a remarkable analogy with a similar analysis of webs and antiwebs due to Pêcher and Wagler [A. Pêcher, A. Wagler, A construction for non-rank facets of stable set polytopes of webs, European Journal of Combinatorics 27 (2006) 1172–1185; A. Pêcher, A. Wagler, Almost all webs are not rank-perfect, Mathematical Programming Series B 105 (2006) 311–328] and Wagler [A. Wagler, Relaxing perfectness: Which graphs are ‘Almost’ perfect?, in: M. Groetschel (Ed.), The Sharpest Cut, Impact of Manfred Padberg and his work, in: SIAM/MPS Series on Optimization, vol. 4, Philadelphia, 2004; A. Wagler, Antiwebs are rank-perfect, 4OR 2 (2004) 149–152].     

On a certain class of nonideal clutters

In this paper we define near-ideal clutters following a similar concept due to Shepherd [Mathematical Programming 64 (1994) 295–323] for near-perfect graphs. We find necessary conditions for a clutter to be near-ideal. From these conditions, we prove that near-ideal clutters give a polyhedral characterization for minimally nonideal clutters as well as near-perfect graphs did for minimally imperfect graphs. We characterize near-ideal blockers of graph-clutters as blockers of near-bipartite graphs. We find that two of the necessary conditions of near-ideal clutters become sufficient for clutters such that every minimal vertex cover is minimum.     

On a mathematical model for laser-induced thermotherapy

We study a mathematical model for laser-induced thermotherapy, a minimally invasive cancer treatment. The model consists of a diffusion approximation of the radiation transport equation coupled to a bio-heat equation and a model to describe the evolution of the coagulated zone. Special emphasis is laid on a refined model of the applicator device, accounting for the effect of coolant flow inside. Comparisons between experiment and simulations show that the model is able to predict the experimentally achieved temperatures reasonably well.     

Identification of the thermal growth characteristics of coagulated tumor tissue in laser‐induced thermotherapy

We consider an inverse problem arising in laser‐induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues are destroyed by coagulation. For the dosage planning, numerical simulations play an important role. To this end, a crucial problem is to identify the thermal growth kinetics of the coagulated zone. Mathematically, this problem is a nonlinear and nonlocal parabolic heat source inverse problem. The solution to this inverse problem is defined as the minimizer of a nonconvex cost functional in this paper. The existence of the minimizer is proven. We derive the Gateaux derivative of the cost functional, which is based on the adjoint system, and use it for a numerical approximation of the optimal coefficient. Numerical implementations are presented to show the validity of the optimization schemes. Copyright © 2012 John Wiley & Sons, Ltd.     

Decomposition of 3-connected graphs

Cunningham and Edmonds [4[ have proved that a 2-connected graphG has a unique minimal decomposition into graphs, each of which is either 3-connected, a bond or a polygon. They define the notion of a good split, and first prove thatG has a unique minimal decomposition into graphs, none of which has a good split, and second prove that the graphs that do not have a good split are precisely 3-connected graphs, bonds and polygons. This paper provides an analogue of the first result above for 3-connected graphs, and an analogue of the second for minimally 3-connected graphs. Following the basic strategy of Cunningham and Edmonds, an appropriate notion of good split is defined. The first main result is that ifG is a 3-connected graph, thenG has a unique minimal decomposition into graphs, none of which has a good split. The second main result is that the minimally 3-connected graphs that do not have a good split are precisely cyclically 4-connected graphs, twirls (K _3,n for somen3) and wheels. From this it is shown that ifG is a minimally 3-connected graph, thenG has a unique minimal decomposition into graphs, each of which is either cyclically 4-connected, a twirl or a wheel.Research partially supported by Office of Naval Research Grant N00014-86-K-0689 at Purdue University.     

Modeling the heating of biological tissue based on the hyperbolic heat transfer equation

In modern surgery, a multitude of minimally intrusive operational techniques are used which are based on the point heating of target zones of human tissue via laser or radiofrequency currents. Traditionally, these processes are modeled by the bioheat equation introduced by Pennes, who considers Fourier’s theory of heat conduction. We present an alternative and more realistic model established using the hyperbolic equation of heat transfer. To demonstrate some features and advantages of our proposed method, we apply the results obtained to different types of tissue heating with high energy fluxes, in particular radiofrequency heating and pulsed laser treatment of the cornea to correct refractive errors. We hope that the results from our approach will help with refining surgical interventions in this novel field of medical treatment.     

Zariski density in lie groups

In [7] Furstenberg gave a proof of Borel’s density theorem [1], which depended not on complete reducibility but rather on properties of the action of a minimally almost periodic group on projective space. In [9] and [10] the basic idea of this proof was extended in various ways to deal with other particular classes of Lie groupsG and closed subgroupsH of cofinite volume. In [5] Dani gives a more general form of the density theorem in whichH need only be non-wandering. In the present paper we define the condition ofk-minimal quasiboundedness, and prove that this condition is necessary and sufficient for the density theorem to hold ((2.4) and (2.6)). Here we replace the arguments of [9] and [10] simply by proofs that the groups considered there satisfy this condition (2.10). We extend the results of [9] and [10] by considering groups which are analytic rather than algebraic, and in the solvable case we completely characterize thek-minimally quasibounded groups (2.9). In the last section we give two applications of the density theorem.     

Uniqueness for an inverse problem for a nonlinear parabolic system with an integral term by one-point Dirichlet data

We consider an inverse problem arising in laser-induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues is destroyed by coagulation. For the dosage planning quantitatively reliable numerical simulation are indispensable. To this end the identification of the thermal growth kinetics of the coagulated zone is of crucial importance. Mathematically, this problem is a nonlinear and nonlocal parabolic inverse heat source problem. We show in this paper that the temperature dependent thermal growth parameter can be identified uniquely from a one-point measurement.     

Computational modeling of reconstructive surgery: The effects of the natural tension on skin wrinkling

A computational model is presented for the simulation of procedures of reconstructive surgery characterized by the excision of a cutaneous defect and the closure and suture of the wound edges. The skin is modeled as a plane membrane with zero flexural stiffness. The membrane undergoes large deformations and is characterized by a Fung type constitutive response in biaxial tension. Skin wrinkling, which is a typical outcome of the surgery in the form of extrusion of the wound edges and dog-ears, is considered through a modification of the elastic potential as originally proposed by Pipkin's Relaxed Energy Density theory [A.C. Pipkin, The relaxed energy density for isotropic elastic membranes, IMA J. Appl. Math. 36 (1986) 85–99; A.C. Pipkin, Relaxed energy density for large deformations of membranes, IMA J. Appl. Math. 52 (1994) 197–308]. The post-buckling analysis of a stretched annular membrane performed by Geminard et al. [Wrinkle formations in axi-symmetrically stretched membranes, Eur. Phys. J. E 15 (2004) 117–126] is used to validate the model under conditions similar to those of the surgery and to discuss the influence of a pre-existing tension in the membrane on the extension of the wrinkled regions. The model is applied to simulate different surgical procedures and investigate the effects of the natural state of the skin and the shape and size of the excisions. The results explain and validate current practice.     

Globally Linked Pairs of Vertices in Equivalent Realizations of Graphs

A two-dimensional framework (G,p) is a graph G = (V,E) together with a map p: V → ℝ². We view (G,p) as a straight line realization of G in ℝ². Two realizations of G are equivalent if the corresponding edges in the two frameworks have the same length. A pair of vertices {u,v} is globally linked in G if %and for all equivalent frameworks (G,q), the distance between the points corresponding to u and v is the same in all pairs of equivalent generic realizations of G. The graph G is globally rigid if all of its pairs of vertices are globally linked. We extend the characterization of globally rigid graphs given by the first two authors [13] by characterizing globally linked pairs in M-connected graphs, an important family of rigid graphs. As a byproduct we simplify the proof of a result of Connelly [6] which is a key step in the characterization of globally rigid graphs. We also determine the number of distinct realizations of an M-connected graph, each of which is equivalent to a given generic realization. Bounds on this number for minimally rigid graphs were obtained by Borcea and Streinu in [3].     

Combinatorial pseudo-triangulations

We prove that a planar graph is generically rigid in the plane if and only if it can be embedded as a pseudo-triangulation. This generalizes the main result of [Haas et al. Planar minimally rigid graphs and pseudo-triangulations, Comput. Geom. 31(1–2) (2005) 31–61] which treats the minimally generically rigid case.The proof uses the concept of combinatorial pseudo-triangulation, CPT, in the plane and has two main steps: showing that a certain “generalized Laman property” is a necessary and sufficient condition for a CPT to be “stretchable”, and showing that all generically rigid plane graphs admit a CPT assignment with that property.Additionally, we propose the study of CPTs on closed surfaces.     

A new infinite family of minimally nonideal matrices

Minimally nonideal matrices are a key to understanding when the set covering problem can be solved using linear programming. The complete classification of minimally nonideal matrices is an open problem. One of the most important results on these matrices comes from a theorem of Lehman, which gives a property of the core of a minimally nonideal matrix. Cornuéjols and Novick gave a conjecture on the possible cores of minimally nonideal matrices. This paper disproves their conjecture by constructing a new infinite family of square minimally nonideal matrices. In particular, we show that there exists a minimally nonideal matrix with r ones in each row and column for any r?3.