Smooth intensity maps and the Bortfeld-Boyer sequencer - Ph. Süss, K.-H. Küfer - Berichte des Fraunhofer ITWM, Nr. 109 (2007)
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Ph. Süss, K.-H. Küfer Smooth intensity maps and the Bortfeld-Boyer sequencer Berichte des Fraunhofer ITWM, Nr. 109 (2007)
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Smooth intensity maps and the Bortfeld-Boyer
sequencer
Philipp Süss∗, Karl-Heinz Küfer
Department of Optimization
Fraunhofer Institute for Industrial Mathematics
Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany
Abstract
It has been empirically verified that smoother intensity maps can be expected
to produce shorter sequences when step-and-shoot collimation is the method
of choice. This work studies the length of sequences obtained by the se-
quencing algorithm by Bortfeld and Boyer using a probabilistic approach.
The results of this work build a theoretical foundation for the up to now only
empirically validated fact that if smoothness of intensity maps is considered
during their calculation, the solutions can be expected to be more easily ap-
plied.
∗
Corresponding author: philipp.suess@itwm.fraunhofer.de
11 INTRODUCTION 1
1 Introduction
A treatment plan in intensity modulated radiotherapy treatment (IMRT) is applied
by blocking parts of the beam surface for predetermined times. This is realized
by multileaf collimators (MLCs). If the beam is switched off while the leaves
configure a “shape” through which some radiation is emitted, then this is referred
to as “step-and-shoot” delivery. The collection of all shapes and the time the beam
is switched on for a shape (its “monitor unit”) is referred to as the “sequence” of a
given treatment plan.
It has been empirically verified that smoother maps can be expected to produce
shorter sequences. However, the absolute statement that smoother maps always
result in sequences with fewer shapes is incorrect. Counterexamples exist where
a higher variation does not lead to an increase in the number of shapes. There-
fore, any statement about the relationship between the smoothness of a map and
the length of its sequence in static collimation can only be made in terms of their
expected values when a probability distribution of the intensity maps is assumed.
Thus, the appropriate mathematical equivalent statement to the observed phenom-
ena can be given by
Proposition 1.1
If the total variation of an intensity map increases, the number of shapes required
to sequence it can be expected to increase as well.
The total variation of an intensity map is taken as the inverse to the measure of
smoothness and given by
n
m X
(xi,j − xi,j−1 )2 ,
X
T V := (1)
i=1 j=1
where m is the number of rows in the map, and n is the number of columns. xi,j
denotes the intensity with which beamlet (i, j) contributes to the dose. In this work,
we will study the number of shapes produced by the sequencing algorithm written
by Bortfeld and Boyer [1] when a certain type of intensity map is sequenced. These
special intensity maps are created using a specified probabilistic method. Then, a
result similar to Proposition 1.1 for the case of sequences produced by the Bortfeld
algorithm is proven. In particular, we will show that by varying a parameter of
the probabilistic method used to produce the intensity maps both, the expected
total variation and the expected number of shapes will increase. In the discussion
we point to an obvious generalization of the probabilistic method to create the
intensity maps which will not alter the results of the analysis. We then conclude
that the stated Proposition is true for a rather large class of intensity maps.
This work is, to our knowledge, the first attempt to construct a theoretical founda-
tion to the afore-mentioned empirical knowledge about the tendency of correlation
between smoothness and sequences.2 RANDOM INTENSITY MAPS 2
The implications are not limited to the mathematical insight: the message that
smoothing maps is beneficial to the treatment is carried implicitly and explicitly
in many publications [2, 3, 4, 5, 6, 7, 8, 9]. This work supplies some theoretical
justification to these undertakings.
2 Random intensity maps
To construct a random intensity map, we let each beamlet intensity xi,j be a random
variable with value depending on its left neighbor xi,j−1 and an added stochastic
term. To this end, let a random variable ∆i,j ∈ {−1, 0, 1} be distributed as follows.
p is a parameter between 0 and 1.
1−p
−1 with probability 2
∆i,j = 0 with probability p (2)
1−p
1 with probability 2
We will then let each beamlet intensity be given by xi,j = xi,j−1 + c ∆i,j , where
c is a positive constant. RandomIntM ap defined in Algorithm 2.1 formalizes
this procedure. To ensure non-negativity, xi,0 := nc for every row i. Taking the
Algorithm 2.1 Method to create random intensity maps
Procedure: RandomIntM ap
Input: Constant c, a stream of random variables ∆1,1 , . . . , ∆m,n identically and
independently distributed with probability mass function (2)
Output: Intensity map x
1: for row i = 1 to m do
Let xi,0 := nc
for column j = 1 to n do
Let xi,j := xi,j−1 + c ∆i,j
5: next j
next i
expectation of the total variation T V (1), we have
X m X n X n
m X
2
E (T V ) = E (xi,j − xi,j−1 ) = E (xi,j − xi,j−1 )2 .
i=1 j=1 i=1 j=1
The expected value of the squared level jump is given by
E (xi,j − xi,j−1 )2 = E (xi,j−1 + c ∆i,j − xi,j−1 )2 = E (c ∆i,j )2 = c2 E ∆2i,j ,
and the expected value of ∆2i,j is given by
1−p 1−p
E ∆2i,j = (−1)2 + (0)2 p + (1)2
= 1 − p.
2 23 NUMBER OF SHAPES IN BORTFELD SEQUENCES 3
The expected total variation is then given by
E (T V ) = mnc2 (1 − p),
and is linear in the probability that the level jump is not zero. This parameter
q := 1 − p can be thought of as our “control” for the smoothness of the maps we
produce.
3 Number of shapes in Bortfeld sequences
The sequencing algorithm given by Bortfeld and Boyer [1] was the first that re-
sulted in sequences with provably optimal beam-on time (total number of monitor
units). The number of shapes resulting from an application of this sequencer can
approximately be given by
n
X
N S(Bortfeld) ≈ max SP Gi = max max (0, xi,j − xi,j−1 ) ,
i=1,...,m i=1,...,m
j=1
where SP Gi stands for the “sum of positive gradients” in row i. This is the number
of iterations the algorithm will perform, and in each iteration a shape with monitor
unit 1 is created. The actual number of shapes will be slightly less since different
iterations may produce the same shapes. However, this is very unlikely if the in-
tensities are continuos variables and we neglect this fact for the remainder of this
work.
Next we determine the expected value of maximum sum of positive gradients in
terms of q = 1 − p. Denote by SP G(m) the maximum sum of positive gradients
over m rows of the intensity map:
SP G(m) := max SP Gi .
i=1,...,m
Further denote by L the maximum intensity value in x: L := maxi,j xi,j . Note
that the maximum sum of positive gradients in one row is bounded by n̂ := nL
2 .
Then
n̂
X
E (N S(Bortfeld)) ≈ E SP G(m) = t Pr SP G(m) = t
t=0
n̂
X
= t Pr SP G(m) ≤ t − Pr SP G(m) ≤ t − 1
t=1
n̂−1
X
= n̂ Pr SP G(m) ≤ n̂ − Pr SP G(m) ≤ t
t=0
n̂−1
X m
= n̂ − Pr (SP Gi ≤ t) ,
t=03 NUMBER OF SHAPES IN BORTFELD SEQUENCES 4
where SP Gi is treated as a random variable.
Now we are interested in the rate of change of E SP G(m) with respect to q :=
1−p. If it can be shown to be positive for all 0 ≤ q ≤ 1, then a positive relationship
between the variation and the length of Bortfeld sequences would be established.
Let us first study the distribution of SP Gi . That is, we would like to calculate
Xn
Pr (SP Gi ≤ t) = Pr max(0, xi,j − xi,j−1 ) ≤ t .
j=1
Expressing SP Gi a little differently, we obtain
n
X
SP Gi = χ (xi,j > xi,j−1 ) (xi,j − xi,j−1 )
j=1
n
X
= χ (xi,j > xi,j−1 ) c ∆ij ,
j=1
where χ (A) represents the characteristic function of event A. Thus, we obtain
n
X t
Pr (SP Gi ≤ t) = Pr ∆ij χ (xi,j > xi,j−1 ) ≤ =: Pr Pi ≤ t̂ ,
c
j=1
where Pi is the number of positive level jumps in row i, and t̂ := ct .
figure Pi is a binomial random variable with distribu-
Note that by (2),the random
1−p
tion parameters n, 2 . Thus,
t̂
1 − p u 1 + p n−u
X n
Pr (SP Gi ≤ t) = Pr Pi ≤ t̂ =
u 2 2
u=0
Differentiating now E SP G(m) with respect to q gives
n̂−1 m−1
∂E SP G(m) X
c(q) := = −m Pr (SP Gi ≤ t) d(t, q),
∂q
t=0
where
t̂
u q u−1 2 − q n−u
n−u−1
X n n − u q u 2−q
d(t, q) = −
u 2 2 2 2 2 2
u=0
t̂
q u 2 − q n−u u
X n n−u
= −
u 2 2 q 2−q
u=0
t̂
1 X
= (2u − nq) Pr (Pi = u) .
q(2 − q)
u=03 NUMBER OF SHAPES IN BORTFELD SEQUENCES 5
Let now
q
v := , 0 < v < 1. (3)
2−q
Then
n̂−1
" t̂ n !m−1
m (1 + v)2 X
X n 1
c(q) = − vu ·
2v u 1+v
t=0 u=0
t̂ n t̂ n !#
X n 1 nv X n u 1
uv u − v . (4)
u 1+v 1 + v u=0 u 1+v
u=0
Taking the terms (1 + v)−1 out of the summations, we get
n̂−1
" t̂ !m−1
m X X n
c(q) = − mn−2 vu ·
2v (1 + v) u
t=0 u=0
t̂ t̂
!#
X n u nv X n u
uv − v . (5)
u 1 + v u
u=0 u=0
Simplifying once by taking out the v in the denominator of the first fraction, we
obtain the following difference in the last bracket
t̂ t̂
X n n X n
uv u−1 − vu (6)
u 1+v u
u=0 u=0
Now note that
t̂ n
X n u
X n u n
v = (1 + v) − v ,
u u
u=0 u=t̂+1
and
t̂ t̂
X n u−1 ∂ X n u
uv = v .
u ∂v u
u=0 u=0
This implies
n̂−1
" t̂ !m−1
m X X n
c(q) = − mn−2 vu ·
2 (1 + v) u
t=0 u=0
n n !#
n−1
X n u−1 n−1
X n u
n(1 + v) − uv − n(1 + v) + v . (7)
u u
u=t̂+1 u=t̂+1
And this finally simplifies to
n̂−1
" t̂
!m−1 n
#
m X X n u
X n u−1
c(q) = − mn−2 v v (v − u) .
2 (1 + v) u u
t=0 u=0 u=t̂+1
(8)4 DISCUSSION 6
Now let us look at the terms involved:
t̂
!m−1
X n u
v >0 (9)
u
u=0
n
X n u−1
v (v − u) < 0 (10)
u
u=t̂+1
n̂−1
" t̂ !m−1 n #
X X n X n
thus vu v u−1 (v − u) < 0 (11)
u u
t=0 u=0 u=t̂+1
m
for all possible choices of t and q. And, because − 2(1+v) mn−2 < 0, we arrive at
the conclusion that c(q) > 0.
In other words, the Bortfeld sequencing algorithm produces sequences in proba-
bly increasing lengths as the intensity maps from the random method become less
smooth.
4 Discussion
The main result from this work is that for the type of intensity map created by
RandomIntM ap, the Bortfeld sequences increase with total variation. Notice that
the crucial point in the argumentation was the fact that RandomIntM ap produces
level jumps that are binomially distributed - just like a series of coin flips to decide
whether the jump goes up or not. That is, the magnitude of the jumps, c remains in
the calculations, but has less impact on the validity of the result.
Imagine a modified randomized method that creates level jumps according to (2)
but distributes the magnitudes of the jumps randomly. It is easy to see, that if
the jumps are not too small, the variation (1) of the intensity maps increases. It
is also relatively easy to imagine that SPG will increase as a result of sufficient
variation introduced by differing level jump magnitudes cij . In other words, the
main result that sequences will increase with increasing variation will hold even
for the generalized random intensity map creation.
This argument shows that Propsition 1.1 holds for a large class of intensity maps.
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J. Puerto (21 pages, 2003)
Keywords: lattice Boltzmann equation, boudary condis-
tions, bounce-back rule, Navier-Stokes equation Exact Procedures for Solving the Discrete
(72 pages, 2002) Ordered Median Problem 57. H. Knaf, P. Lang, S. Zeiser
Keywords: discrete location, Integer programming
(41 pages, 2003) Diagnosis aiding in Regulation
39. R. Korn Thermography using Fuzzy Logic
Elementare Finanzmathematik Keywords: fuzzy logic,knowledge representation, ex-
Keywords: Finanzmathematik, Aktien, Optionen, Port 48. S. Feldmann, P. Lang pert system
folio-Optimierung, Börse, Lehrerweiterbildung, Mathe- Padé-like reduction of stable discrete linear (22 pages, 2003)
matikunterricht systems preserving their stability
(98 pages, 2002) Keywords: Discrete linear systems, model reduction, 58. M. T. Melo, S. Nickel, F. Saldanha da Gama
stability, Hankel matrix, Stein equation
(16 pages, 2003) Largescale models for dynamic multi
40. J. Kallrath, M. C. Müller, S. Nickel
commodity capacitated facility location
Batch Presorting Problems: Keywords: supply chain management, strategic
Models and Complexity Results 49. J. Kallrath, S. Nickel planning, dynamic location, modeling
Keywords: Complexity theory, Integer programming, A Polynomial Case of the Batch Presorting (40 pages, 2003)
Assigment, Logistics Problem
(19 pages, 2002) Keywords: batch presorting problem, online optimiza- 59. J. Orlik
tion, competetive analysis, polynomial algorithms, lo-
gistics Homogenization for contact problems with
41. J. Linn
(17 pages, 2003) periodically rough surfaces
On the frame-invariant description of the Keywords: asymptotic homogenization, contact
phase space of the Folgar-Tucker equation problems
Key words: fiber orientation, Folgar-Tucker equation, in- 50. T. Hanne, H. L. Trinkaus (28 pages, 2004)
jection molding knowCube for MCDM –
(5 pages, 2003) Visual and Interactive Support for 60. A. Scherrer, K.-H. Küfer, M. Monz, F. Alonso,
Multicriteria Decision Making T. Bortfeld
42. T. Hanne, S. Nickel Key words: Multicriteria decision making, knowledge
management, decision support systems, visual interfac- IMRT planning on adaptive volume struc-
A Multi-Objective Evolutionary Algorithm tures – a significant advance of computa-
es, interactive navigation, real-life applications.
for Scheduling and Inspection Planning in (26 pages, 2003) tional complexity
Software Development Projects Keywords: Intensity-modulated radiation therapy
Key words: multiple objective programming, project (IMRT), inverse treatment planning, adaptive volume
management and scheduling, software development, 51. O. Iliev, V. Laptev
structures, hierarchical clustering, local refinement,
evolutionary algorithms, efficient set On Numerical Simulation of Flow Through adaptive clustering, convex programming, mesh gen-
(29 pages, 2003) Oil Filters eration, multi-grid methods
Keywords: oil filters, coupled flow in plain and porous (24 pages, 2004)
media, Navier-Stokes, Brinkman, numerical simulation
(8 pages, 2003)61. D. Kehrwald 70. W. Dörfler, O. Iliev, D. Stoyanov, D. Vassileva 79. N. Ettrich
Parallel lattice Boltzmann simulation On Efficient Simulation of Non-Newto- Generation of surface elevation models for
of complex flows nian Flow in Saturated Porous Media with a urban drainage simulation
Keywords: Lattice Boltzmann methods, parallel com- Multigrid Adaptive Refinement Solver Keywords: Flooding, simulation, urban elevation
puting, microstructure simulation, virtual material de- Keywords: Nonlinear multigrid, adaptive renement, models, laser scanning
sign, pseudo-plastic fluids, liquid composite moulding non-Newtonian in porous media (22 pages, 2005)
(12 pages, 2004) (25 pages, 2004)
80. H. Andrä, J. Linn, I. Matei, I. Shklyar,
62. O. Iliev, J. Linn, M. Moog, D. Niedziela, 71. J. Kalcsics, S. Nickel, M. Schröder K. Steiner, E. Teichmann
V. Starikovicius Towards a Unified Territory Design Ap- OPTCAST – Entwicklung adäquater Struk-
On the Performance of Certain Iterative proach – Applications, Algorithms and GIS turoptimierungsverfahren für Gießereien
Solvers for Coupled Systems Arising in Integration Technischer Bericht (KURZFASSUNG)
Discretization of Non-Newtonian Flow Keywords: territory desgin, political districting, sales Keywords: Topologieoptimierung, Level-Set-Methode,
Equations territory alignment, optimization algorithms, Geo- Gießprozesssimulation, Gießtechnische Restriktionen,
Keywords: Performance of iterative solvers, Precondi- graphical Information Systems CAE-Kette zur Strukturoptimierung
tioners, Non-Newtonian flow (40 pages, 2005) (77 pages, 2005)
(17 pages, 2004)
72. K. Schladitz, S. Peters, D. Reinel-Bitzer, 81. N. Marheineke, R. Wegener
63. R. Ciegis, O. Iliev, S. Rief, K. Steiner A. Wiegmann, J. Ohser Fiber Dynamics in Turbulent Flows
On Modelling and Simulation of Different Design of acoustic trim based on geometric Part I: General Modeling Framework
Regimes for Liquid Polymer Moulding modeling and flow simulation for non-woven Keywords: fiber-fluid interaction; Cosserat rod; turbu-
Keywords: Liquid Polymer Moulding, Modelling, Simu- Keywords: random system of fibers, Poisson line lence modeling; Kolmogorov’s energy spectrum; dou-
lation, Infiltration, Front Propagation, non-Newtonian process, flow resistivity, acoustic absorption, Lattice- ble-velocity correlations; differentiable Gaussian fields
flow in porous media Boltzmann method, non-woven (20 pages, 2005)
(43 pages, 2004) (21 pages, 2005) Part II: Specific Taylor Drag
Keywords: flexible fibers; k- e turbulence model; fi-
ber-turbulence interaction scales; air drag; random
64. T. Hanne, H. Neu 73. V. Rutka, A. Wiegmann Gaussian aerodynamic force; white noise; stochastic
Simulating Human Resources in Explicit Jump Immersed Interface Method differential equations; ARMA process
Software Development Processes for virtual material design of the effective (18 pages, 2005)
Keywords: Human resource modeling, software pro- elastic moduli of composite materials
cess, productivity, human factors, learning curve Keywords: virtual material design, explicit jump im- 82. C. H. Lampert, O. Wirjadi
(14 pages, 2004) mersed interface method, effective elastic moduli,
composite materials
An Optimal Non-Orthogonal Separation of
(22 pages, 2005) the Anisotropic Gaussian Convolution Filter
65. O. Iliev, A. Mikelic, P. Popov Keywords: Anisotropic Gaussian filter, linear filtering, ori-
Fluid structure interaction problems in de- entation space, nD image processing, separable filters
formable porous media: Toward permeabil- 74. T. Hanne (25 pages, 2005)
ity of deformable porous media Eine Übersicht zum Scheduling von Baustellen
Keywords: fluid-structure interaction, deformable po- Keywords: Projektplanung, Scheduling, Bauplanung, 83. H. Andrä, D. Stoyanov
rous media, upscaling, linear elasticity, stokes, finite Bauindustrie
elements (32 pages, 2005) Error indicators in the parallel finite ele-
(28 pages, 2004) ment solver for linear elasticity DDFEM
Keywords: linear elasticity, finite element method, hier-
75. J. Linn archical shape functions, domain decom-position, par-
66. F. Gaspar, O. Iliev, F. Lisbona, A. Naumovich, The Folgar-Tucker Model as a Differetial allel implementation, a posteriori error estimates
P. Vabishchevich Algebraic System for Fiber Orientation (21 pages, 2006)
On numerical solution of 1-D poroelasticity Calculation
equations in a multilayered domain Keywords: fiber orientation, Folgar–Tucker model, in- 84. M. Schröder, I. Solchenbach
Keywords: poroelasticity, multilayered material, finite variants, algebraic constraints, phase space, trace sta-
bility
Optimization of Transfer Quality in
volume discretization, MAC type grid
(41 pages, 2004) (15 pages, 2005) Regional Public Transit
Keywords: public transit, transfer quality, quadratic
assignment problem
67. J. Ohser, K. Schladitz, K. Koch, M. Nöthe 76. M. Speckert, K. Dreßler, H. Mauch, (16 pages, 2006)
Diffraction by image processing and its ap- A. Lion, G. J. Wierda
plication in materials science Simulation eines neuartigen Prüfsystems 85. A. Naumovich, F. J. Gaspar
Keywords: porous microstructure, image analysis, ran- für Achserprobungen durch MKS-Model-
On a multigrid solver for the three-dimen-
dom set, fast Fourier transform, power spectrum, lierung einschließlich Regelung
Bartlett spectrum
sional Biot poroelasticity system in multi-
Keywords: virtual test rig, suspension testing, multi-
(13 pages, 2004) body simulation, modeling hexapod test rig, optimiza-
layered domains
tion of test rig configuration Keywords: poroelasticity, interface problem, multigrid,
(20 pages, 2005) operator-dependent prolongation
68. H. Neunzert (11 pages, 2006)
Mathematics as a Technology: Challenges
for the next 10 Years 77. K.-H. Küfer, M. Monz, A. Scherrer, P. Süss,
F. Alonso, A. S. A. Sultan, Th. Bortfeld, 86. S. Panda, R. Wegener, N. Marheineke
Keywords: applied mathematics, technology, modelling,
simulation, visualization, optimization, glass processing, D. Craft, Chr. Thieke Slender Body Theory for the Dynamics of
spinning processes, fiber-fluid interaction, trubulence Multicriteria optimization in intensity mod- Curved Viscous Fibers
effects, topological optimization, multicriteria optimiza- Keywords: curved viscous fibers; fluid dynamics; Navier-
ulated radiotherapy planning
tion, Uncertainty and Risk, financial mathematics, Mal- Stokes equations; free boundary value problem; asymp-
Keywords: multicriteria optimization, extreme solutions,
liavin calculus, Monte-Carlo methods, virtual material totic expansions; slender body theory
real-time decision making, adaptive approximation
design, filtration, bio-informatics, system biology (14 pages, 2006)
schemes, clustering methods, IMRT planning, reverse
(29 pages, 2004)
engineering
(51 pages, 2005) 87. E. Ivanov, H. Andrä, A. Kudryavtsev
69. R. Ewing, O. Iliev, R. Lazarov, Domain Decomposition Approach for Auto-
A. Naumovich 78. S. Amstutz, H. Andrä matic Parallel Generation of Tetrahedral Grids
On convergence of certain finite difference A new algorithm for topology optimization Key words: Grid Generation, Unstructured Grid, Delau-
discretizations for 1D poroelasticity inter- nay Triangulation, Parallel Programming, Domain De-
using a level-set method
face problems composition, Load Balancing
Keywords: shape optimization, topology optimization,
(18 pages, 2006)
Keywords: poroelasticity, multilayered material, finite topological sensitivity, level-set
volume discretizations, MAC type grid, error estimates (22 pages, 2005)
(26 pages,2004)88. S. Tiwari, S. Antonov, D. Hietel, J. Kuhnert, 97. A. Dreyer 107. Z. Drezner, S. Nickel
R. Wegener Interval Methods for Analog Circiuts Solving the ordered one-median problem in
A Meshfree Method for Simulations of Inter- Keywords: interval arithmetic, analog circuits, tolerance the plane
actions between Fluids and Flexible Structures analysis, parametric linear systems, frequency response, Keywords: planar location, global optimization, ordered
Key words: Meshfree Method, FPM, Fluid Structure In- symbolic analysis, CAD, computer algebra median, big triangle small triangle method, bounds,
teraction, Sheet of Paper, Dynamical Coupling (36 pages, 2006) numerical experiments
(16 pages, 2006) (21 pages, 2007)
98. N. Weigel, S. Weihe, G. Bitsch, K. Dreßler
89. R. Ciegis , O. Iliev, V. Starikovicius, K. Steiner Usage of Simulation for Design and Optimi- 108. Th. Götz, A. Klar, A. Unterreiter,
Numerical Algorithms for Solving Problems zation of Testing R. We gener
of Multiphase Flows in Porous Media Keywords: Vehicle test rigs, MBS, control, hydraulics, Numerical evidance for the non-existing of
Keywords: nonlinear algorithms, finite-volume method, testing philosophy solutions of the equations desribing rota-
software tools, porous media, flows (14 pages, 2006)
tional fiber spinning
(16 pages, 2006) Keywords: rotational fiber spinning, viscous fibers,
99. H. Lang, G. Bitsch, K. Dreßler, M. Speckert boundary value problem, existence of solutions
90. D. Niedziela, O. Iliev, A. Latz Comparison of the solutions of the elastic (11 pages, 2007)
On 3D Numerical Simulations of Viscoelastic and elastoplastic boundary value problems
Fluids Keywords: Elastic BVP, elastoplastic BVP, variational 109. Ph. Süss, K.-H. Küfer
Keywords: non-Newtonian fluids, anisotropic viscosity, inequalities, rate-independency, hysteresis, linear kine- Smooth intensity maps and the Bortfeld-
integral constitutive equation matic hardening, stop- and play-operator
Boyer sequencer
(18 pages, 2006) (21 pages, 2006)
Keywords: probabilistic analysis, intensity modulated
radiotherapy treatment (IMRT), IMRT plan application,
91. A. Winterfeld 100. M. Speckert, K. Dreßler, H. Mauch step-and-shoot sequencing
MBS Simulation of a hexapod based sus- (8 pages, 2007)
Application of general semi-infinite Pro-
gramming to Lapidary Cutting Problems pension test rig
Keywords: large scale optimization, nonlinear program- Keywords: Test rig, MBS simulation, suspension,
ming, general semi-infinite optimization, design center- hydraulics, controlling, design optimization
ing, clustering (12 pages, 2006) Status quo: March 2007
(26 pages, 2006)
101. S. Azizi Sultan, K.-H. Küfer
92. J. Orlik, A. Ostrovska A dynamic algorithm for beam orientations
Space-Time Finite Element Approximation in multicriteria IMRT planning
and Numerical Solution of Hereditary Lin- Keywords: radiotherapy planning, beam orientation
ear Viscoelasticity Problems optimization, dynamic approach, evolutionary algo-
rithm, global optimization
Keywords: hereditary viscoelasticity; kern approxima-
(14 pages, 2006)
tion by interpolation; space-time finite element approx-
imation, stability and a priori estimate
(24 pages, 2006) 102. T. Götz, A. Klar, N. Marheineke, R. Wegener
A Stochastic Model for the Fiber Lay-down
93. V. Rutka, A. Wiegmann, H. Andrä Process in the Nonwoven Production
EJIIM for Calculation of effective Elastic Keywords: fiber dynamics, stochastic Hamiltonian sys-
Moduli in 3D Linear Elasticity tem, stochastic averaging
Keywords: Elliptic PDE, linear elasticity, irregular do- (17 pages, 2006)
main, finite differences, fast solvers, effective elas-
tic moduli 103. Ph. Süss, K.-H. Küfer
(24 pages, 2006)
Balancing control and simplicity: a variable
aggregation method in intensity modulated
94. A. Wiegmann, A. Zemitis radiation therapy planning
EJ-HEAT: A Fast Explicit Jump Harmonic Keywords: IMRT planning, variable aggregation, clus-
veraging Solver for the Effective Heat
A tering methods
(22 pages, 2006)
Conductivity of Composite Materials
Keywords: Stationary heat equation, effective thermal
conductivity, explicit jump, discontinuous coefficients, 104. A. Beaudry, G. Laporte, T. Melo, S. Nickel
virtual material design, microstructure simulation, EJ- Dynamic transportation of patients in hos-
HEAT
pitals
(21 pages, 2006)
Keywords: in-house hospital transportation, dial-a-ride,
dynamic mode, tabu search
95. A. Naumovich (37 pages, 2006)
On a finite volume discretization of the
three-dimensional Biot poroelasticity sys- 105. Th. Hanne
tem in multilayered domains Applying multiobjective evolutionary algo-
Keywords: Biot poroelasticity system, interface prob- rithms in industrial projects
lems, finite volume discretization, finite difference Keywords: multiobjective evolutionary algorithms, dis-
method. crete optimization, continuous optimization, electronic
(21 pages, 2006) circuit design, semi-infinite programming, scheduling
(18 pages, 2006)
96. M. Krekel, J. Wenzel
A unified approach to Credit Default 106. J. Franke, S. Halim
Swaption and Constant Maturity Credit De- Wild bootstrap tests for comparing signals
fault Swap valuation and images
Keywords: LIBOR market model, credit risk, Credit De- Keywords: wild bootstrap test, texture classification,
fault Swaption, Constant Maturity Credit Default Swap- textile quality control, defect detection, kernel estimate,
method. nonparametric regression
(43 pages, 2006)
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