MBE
Evolution of uncontrolled proliferation and the angiogenic switch in cancer
John D. Nagy Dieter Armbruster
Mathematical Biosciences & Engineering 2012, 9(4): 843-876 doi: 10.3934/mbe.2012.9.843
The major goal of evolutionary oncology is to explain how malignant traits evolve to become cancer "hallmarks." One such hallmark---the angiogenic switch---is difficult to explain for the same reason altruism is difficult to explain. An angiogenic clone is vulnerable to "cheater" lineages that shunt energy from angiogenesis to proliferation, allowing the cheater to outcompete cooperative phenotypes in the environment built by the cooperators. Here we show that cell- or clone-level selection is sufficient to explain the angiogenic switch, but not because of direct selection on angiogenesis factor secretion---angiogenic potential evolves only as a pleiotropic afterthought. We study a multiscale mathematical model that includes an energy management system in an evolving angiogenic tumor. The energy management model makes the counterintuitive prediction that ATP concentration in resting cells increases with increasing ATP hydrolysis, as seen in other theoretical and empirical studies. As a result, increasing ATP hydrolysis for angiogenesis can increase proliferative potential, which is the trait directly under selection. Intriguingly, this energy dynamic allows an evolutionary stable angiogenesis strategy, but this strategy is an evolutionary repeller, leading to runaway selection for extreme vascular hypo- or hyperplasia. The former case yields a tumor-on-a-tumor, or hypertumor, as predicted in other studies, and the latter case may explain vascular hyperplasia evident in certain tumor types.
keywords: evolutionary suicide adaptive dynamics energy charge multiscale models perturbation expansion. Cancer ATP
NHM
Integrating release and dispatch policies in production models
Dieter Armbruster Michael Herty Xinping Wang Lindu Zhao
Networks & Heterogeneous Media 2015, 10(3): 511-526 doi: 10.3934/nhm.2015.10.511
Aggregate production planning for highly re--entrant production processes is typically generated by finding optimal release rates based on clearing function models. For production processes with very long cycle times, like in semiconductor production, dispatch policies are used to cover short term fluctuations. We extend the concept of a clearing function to allow control over both, the release rates and priority allocations in re-entrant production. This approach is used to improve the production planning problem using combined release and the allocation dispatch policy. The control parameter for priority allocation, called the push-pull point (PPP), separates the beginning of the factory which employs a push policy from the end of the factory, which uses a pull policy. The extended clearing function model describes the output of the factory as a function of the work in progress (wip) and the position of the PPP. The model's qualitative behavior is analyzed. Numerical optimization results are compared to production planning based only on releases. It is found that controlling the PPP significantly reduces the average wip in the system and hence leads to much shorter cycle times.
keywords: partial differential equations Production planning dispatch control re-entrant production.
KRM
Kinetic models and intrinsic timescales: Simulation comparison for a 2nd order queueing model
Dieter Armbruster Matthew Wienke
Kinetic & Related Models 2019, 12(1): 177-193 doi: 10.3934/krm.2019008

Kinetic models of stochastic production flows can be expanded into deterministic moment equations and thus approximated with appropriate closures. A second order model for the product density and the product speed has previously been proposed. A systematic analysis comparing simulations of the partial differential equations (PDE) with discrete event simulations (DES) is performed. Specifically, factory production is modeled as an M/M/1 queue where the arrival process is a non-homogeneous Poisson process. Three fundamental scenarios for such a time dependent influx are studied: An instant step up/step down of the arrival rate, an exponential step up/step down and periodic variation of the average arrival rate. It is shown that the second order model in general yields significant improvements over the first order model. Adding diffusion into the PDE further improves the agreement in particular for queues with low utilization. The analysis also points to fundamental open issues regarding kinetic models of time dependent agent based simulations. Memory effects and the possibility of resonance in deterministic models are caused by intrinsic timescales of the PDE that are not present in the original stochastic processes.

keywords: Kinetic models production systems simulations 2nd order models transients
NHM
A kinetic model for an agent based market simulation
Dieter Armbruster Christian Ringhofer Andrea Thatcher
Networks & Heterogeneous Media 2015, 10(3): 527-542 doi: 10.3934/nhm.2015.10.527
A kinetic model for a specific agent based simulation to generate the sales curves of successive generations of high-end computer chips is developed. The resulting continuum market model consists of transport equations in two variables, representing the availability of money and the desire to buy a new chip. In lieu of typical collision terms in the kinetic equations that discontinuously change the attributes of an agent, discontinuous changes are initiated via boundary conditions between sets of partial differential equations. A scaling analysis of the transport equations determines the different time scales that constitute the market forces, characterizing different sales scenarios. It is argued that the resulting model can be adjusted to generic markets of multi-generational technology products where the innovation time scale is an important driver of the market.
keywords: agent based simulations Kinetic theory multi-generational technology products. market models
PROC
A hyperbolic relaxation model for product flow in complex production networks
Ali Unver Christian Ringhofer Dieter Armbruster
Conference Publications 2009, 2009(Special): 790-799 doi: 10.3934/proc.2009.2009.790
This paper presents a continuum - traffic flow like - model for the flow of products through complex production networks, based on statistical information obtained from extensive observations of the system. The resulting model consists of a system of hyperbolic conservation laws, which, in a relaxation limit, exhibit the correct diffusive properties given by the variance of the observed data.
keywords: hyperbolic systems relaxation methods Supply chains

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