The relative biologic effectiveness versus linear energy transfer
curve as an outputinput relation for linear cellular systems

1.
Department of Radiation Oncology, Stanford University, Stanford, CA 94305

2.
Department of Mathematics, Colorado State University, Fort Collins, CO 80523

Received:
01 September 2008
Accepted:
29 June 2018
Published:
01 June 2009


MSC :
Primary: 92C99, 93A30; Secondary: 92B05.


Experiments have established that different radiation types have different magnitudes of biological response. When biological response is defined in terms of the Relative Biologic Effectiveness (RBE) and different radiation type is characterized by Linear Energy Transfer (LET), the plot of the RBE versus LET (RBELET) curve shows RBE to increase with increasing LET, to reach a maximum, and to decrease with further increasing LET. Perhaps due to the descriptive nature of biology, most quantitative models for the RBELET curve ignore the reality of the underlying molecular biology. On the other hand, the molecular basis for the RBELET curve is not completely known despite recent efforts.
Here we introduce a differential equation formulation for a signalandsystem model that sees cells as systems, different radiation types as input, and cellular responses as output. Because of scant knowledge of the underlying biochemical network, the current version is necessarily a work in progress. It explains the RBELET curve using not just input parameters but also systems internal state parameters. These systems internal state parameters represent parts of a biochemical network within a cell. Although multiple biochemical parts may well be involved, the shape of the RBELET curve is reproduced when only three system parameters are related to three biochemical parts: the molecular machinery for DNA double strand break repair; the molecular pathways for handling oxidative stress; and the radiolytic products of the cellular water.
Despite being a simplified ''toy model,'' changes in the systems state parameters lead to model curves that are refutable in a modern molecular biology laboratory. As the parts in the biochemical network of the radiation response are being further elucidated, this model can incorporate new systems state parameters to allow a more accurate fit.
Citation: Quoc T. Luu, Paul DuChateau. The relative biologic effectiveness versus linear energy transfercurve as an outputinput relation for linear cellular systems[J]. Mathematical Biosciences and Engineering, 2009, 6(3): 591602. doi: 10.3934/mbe.2009.6.591

Abstract
Experiments have established that different radiation types have different magnitudes of biological response. When biological response is defined in terms of the Relative Biologic Effectiveness (RBE) and different radiation type is characterized by Linear Energy Transfer (LET), the plot of the RBE versus LET (RBELET) curve shows RBE to increase with increasing LET, to reach a maximum, and to decrease with further increasing LET. Perhaps due to the descriptive nature of biology, most quantitative models for the RBELET curve ignore the reality of the underlying molecular biology. On the other hand, the molecular basis for the RBELET curve is not completely known despite recent efforts.
Here we introduce a differential equation formulation for a signalandsystem model that sees cells as systems, different radiation types as input, and cellular responses as output. Because of scant knowledge of the underlying biochemical network, the current version is necessarily a work in progress. It explains the RBELET curve using not just input parameters but also systems internal state parameters. These systems internal state parameters represent parts of a biochemical network within a cell. Although multiple biochemical parts may well be involved, the shape of the RBELET curve is reproduced when only three system parameters are related to three biochemical parts: the molecular machinery for DNA double strand break repair; the molecular pathways for handling oxidative stress; and the radiolytic products of the cellular water.
Despite being a simplified ''toy model,'' changes in the systems state parameters lead to model curves that are refutable in a modern molecular biology laboratory. As the parts in the biochemical network of the radiation response are being further elucidated, this model can incorporate new systems state parameters to allow a more accurate fit.
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