1. INTRODUCTION: Biological systems are characterized by significant heterogeneity at multiple scales. The fine scale (local scale) heterogeneity often has significant effects, even on large scale transport of drugs, fertilizers, pesticides, invasive species and bioagents. Epithelium Spinal Cord Soil Landscape COMPLEX SIMPLIFICATION OF QUANTIZED TRANSPORT EQUATIONS IN BIOLOGICAL SYSTEMS Hubert J. Montas, Ph.D. Biological Resources Engineering, University of Maryland at College Park To be effective, the design of diagnosis and treatment strategies (targeted at biological systems) needs to consider the effects of local scale heterogeneity. However, considering such heterogeneity in detail, at the scale at which it arises, is computationally inefficient or altogether impossible in many situations. Commonly used upscaled formulations are much more efficient but are overwhelmingly focused on statistical means only, which is inaccurate. OBJECTIVE: Develop and evaluate upscaled transport equations that are applicable at problem scales and that incorporate the effects of local scale heterogeneity on the process A reaction-diffusion equation with spatially-varying coefficients is assumed to apply at the local scale: Example 1: Richards’ equation (soils) Example 2: Fischer-Kolmogoroff (tissues/ecosys.) MATERIALS: METHODS: Stochastic-Perturbation Volume Averaging (inspired by research related to Yucca Mountain) 1.Develop a statistical description of the local scale heterogeneity of the material 2.Define a system of orthogonal fields from 1 3.Expand (project) local scale variables in terms of 2 and correlations with the fields in 2 4.Extract individual correlation equations (simplify) 5.Perform canonical transformation (and others) 1. Heterogeneity Statistics It is assumed that spatial fluctuations of one of the parameters of the governing PDE (e.g. p 1 ) dominate The mean and variance of p 1 are known The standard deviation of the spectral density function of p 1 is known (characteristic spatial frequency) 2.Orthogonal Fields P 1 is normalized: Normalized complex orthogonal fields that combine p 1 with its spatial derivative are defined: (complex treatment of deriv. is analog. to Fourier) 3.Expansion of Variables Transported entity, u: Where: Nonlinear parameter, D: 1 st order Taylor series: Redefine variables to get: Diffusive flux: Where: Reactive term: 4.Extract Equations Upscaled equations in correlation-based form: 5.Transformations and Approximations a) Stationary approximation: Starting point: Assume minor temporal variations of  u and solve: Substitute: Simplifications: 1.The gradient of  u is small 2.k is correlated to p 1 only 3.D is correlated to the derivative of p 1 only 4.G is constant b) Nonlocal (memory, Integro-PDE) form: Starting point: Assume k and D are linear and solve for  u : Substitute: c) Quantized form (full, canonical): Define characteristic variables: Substitute: Where: d) Simplified Quantized form (bi-continuum): Assume ( D 1 – D 2 ) is relatively small: APPLICATION EXAMPLE: Water Infiltration in heterogeneous soils: ACKNOWLEDGEMENTS: This study was supported in part by the Maryland Agricultural Experiment Station (MAES) and by the National Science Foundation (NSF) under Grant No. 0134424. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the MAES or NSF. The funding given by these agencies is gratefully acknowledged. e) Complex Simplification (bi-cont. assumptions): Define complex variables: Substitute: SUMMARY: Derived problem scale transport equations that incorporate the effects of local scale heterogeneity Asymptote: harmonic reactions and geometric diffusion Nonlocal form obtained in linear case Quantized form obtained in general case Complex notation simplified the quantized form Equations were found accurate for soils REFERENCES: Montas H.J., 2003. Quantized Transport in Biological Systems. IBE Conference Proceedings, Jan 15-17 2003, University of Georgia, Athens, GA, Institute of Biological Engineering (IBE), www.ibeweb.org. Montas, H.J., 2003. An Analytical Solution of the Three-Component Transport Equation with Application to Third-Order Transport. Water Resources Research, 39:(2), 1036, doi:10.1029/2002WR001288. Montas H.J. and A. Shirmohammadi, 2000. Equivalence of Bicontinuum and Second-Order Transport in Heterogeneous Soils and Aquifers. Water Resources Research, 36(12):3427-3446.