Reaction-diffusion simulation model (draft)
Reaction-diffusion systems are mathematical models, derived from the work of Alan Turing in 1952. Many extensions have been proposed to the model, but originally the model consisted of two substances in certain concentrations inside a certain space, with four variables per substance: the rates of production and degradation, rate of diffusion, and the activating or inhibiting property on the reaction rates of the other substance. The model can explain formation of gradients (patterns) within the space, and has found some applications not only in chemistry but in physics, biology, ecology, epidemiology, oncology, and even in positron emission tomography.
Reaction-diffusion models are too complex (overdetermined) to be used in the analysis of PET data, but may be used to simulate data.
Andasari V, Gerisch A, Lolas G, South AP, Chaplain MAJ. Mathematical modeling of cancer cell invasion in tissue: biological insight from mathematical analysis and computational simulation. J Math Biol. 2011; 63: 141-171.
Capasso V, Lachowicz M (eds). Multiscale Problems in the Life Sciences: From Microscopic to Macroscopic., Springer, 2008.
Dalah E, Bradley D, Nisbet A. Simulation of tissue activity curves of 64Cu-ATSM for sub-target volume delineation in radiotherapy. Phys Med Biol. 2010; 55: 681-694.
Jennings M, Marcu LG, Bezak E. PET-specific parameters and radiotracers in theoretical tumour modelling. Comput Math Methods Med. 2015: 415923.
Kelly CJ, Brady M. A model to simulate tumour oxygenation and dynamic [18F]-Fmiso PET data. Phys Med Biol. 2006; 51: 5859-5873.
Kondo S, Miura T. Reaction-diffusion model as a framework for understanding biological pattern formation. Science 2010; 329(5999): 1616-1620.
Li F, Joergensen JT, Hansen AE, Kjaer A. Kinetic modeling in PET imaging of hypoxia. Am J Nucl Med Mol Imaging 2014; 4(5): 490-506.
Liu D, Chalkidou A, Landau DB, Marsden PK, Fenwick JD. Interstitial diffusion and the relationship between compartment modelling and multi-scale spatial-temporal modelling of 18F-FLT tumour uptake dynamics. Phys Med Biol. 2014; 59(17): 5175-5202.
Maini PK, Woolley TE, Baker RE, Gaffney EA, Lee SS. Turing’s model for biological pattern formation and the robustness problem. Interface Focus 2012; 2: 487-496.
Munk OL, Bass L, Feng H, Keiding S. Determination of regional flow by use of intravascular PET tracers: microvascular theory and experimental validation for pig livers. J Nucl Med. 2003; 44: 1862-1870.
Mönnich D, Troost EGC, Kaanders JHAM, Oyen WJG, Alber M, Thorwarth D. Modelling and simulation of [18F]fluoromisonidazole dynamics based on histology-derived microvessel maps. Phys Med Biol. 2011; 56: 2045-2057.
Mönnich D, Troost EGC, Kaanders JHAM, Oyen WJG, Alber M, Thorwarth D. Modelling and simulation of the influence of acute and chronic hypoxia on [18F]fluoromisonidazole PET imaging. Phys Med Biol. 2012; 57: 1675-1684.
Reps L, Engbers R, Burger M. Dynamic PET reconstruction based on a reaction-diffusion model. Proc Appl Math Mech. 2012; 12: 683-684.
Soltani M, Sefidgar M, Casey ME, Wahl RL, Subramaniam RM, Rahmim A. Comprehensive modeling of the spatiotemporal distribution of PET tracer uptake in solid tumors based on the convection-diffusion-reaction equation. IEEE Nucl Sci Symp Conf Record 2014. DOI: 10.1109/NSSMIC.2014.7430813
Soltani M, Sefidgar M, Bazmara H, Rahmim A. Enhanced modeling of spatiotemporal distribution of PET tracers in solid tumors and estimation of transport parameters. J Nucl Med. 2015; 56(Suppl 3): 1221.
Tang S, Qin S, Weber RO. Numerical studies on 2-dimensional reaction-diffusion equations. J Austral Math Soc Ser B 1993; 35: 223-243.
Vijayendran RA, Ligler FS, Leckband DE. A computational reaction-diffusion model for analysis of transport-limited kinetics. Anal Chem. 1999; 71: 5405-5412.
Wang Q, Liu Z, Ziegler SI, Shi K. A Reaction-Diffusion Simulation Model of [18F]FDG PET imaging for the Quantitative Interpretation of Tumor Glucose Metabolism. In: Gao F, Shi K, Li S (eds.). Computational Methods for Molecular Metabolism. Springer, 2015, pp 123-137.
Wang Q, Vaupel P, Ziegler SI, Shi K. Exploring the quantitative relationship between metabolism and enzymatic phenotype by physiological modeling of glucose metabolism and lactate oxidation in solid tumors. Phys Med Biol. 2015; 60: 2547-2571.
Wilks MQ, Knowles SM, Wu AM, Huang S-C. Improved modeling of in vivo kinetics of slowly diffusing radiotracers for tumor imaging. J Nucl Med. 2014; 55: 1539-1544.
Zierler K. A critique of compartmental analysis. Annu Rev Biophys Bioeng. 1981; 10:531-562.
Updated at: 2016-04-05
Created at: 2016-03-31
Written by: Vesa Oikonen