1. Quantized Transport in Biological Systems Hubert J. Montas, Ph.D. Biological Resources Engineering University of Maryland at College Park

2. Introduction: Biological systems are characterized by significant heterogeneity at multiple scales Fine scale (local scale) heterogeneity often has significant effects on large scale transport Epithelium Spinal Cord Soil Landscape

3. Introduction Engineering design and analysis of diagnosis and treatment strategies needs to incorporate local scale heterogeneity effects (using mean values is not accurate) Accuracy is needed to maximize efficiency with minimal side-effects Drug/pesticide encapsulation Drug/fertilizer application strategies Control of invasive species/epidemics/bioagents

4. Objective Develop and evaluate transport equations applicable at problem scales that incorporate the effects of local scale heterogeneity on the process

5. Materials 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)

6. Methods Stochastic-Perturbation Volume Averaging (inspired by research related to Yucca Mountain) Develop a statistical description of the local scale heterogeneity of the material Define a system of orthogonal fields from 1 Expand (project) local scale variables in terms of 2 and correlations with the fields in 2 (entails averaging over REAs) Extract individual correlation equations (simplify) Perform canonical transformation (and others)

7. 1. Heterogeneity Statistics It is assumed that spatial fluctuations of one of the parameters of the governing PDE (e.g. p1) dominate The mean and variance of p1 are determined The standard deviation of the spectral density function of p1 is determined (characteristic spatial frequency)

8. 2.Orthogonal Fields P1 is normalized: Normalized complex orthogonal fields that combine p1 with its spatial derivative are defined: (treatment of the derivative is analogous to Fourier)

9. 3.Expansion of Variables Transported entity, u: Where:

10. 3.Expansion of Variables Nonlinear parameter, D: 1 st order Taylor series: Redefine variables to get:

11. 3.Expansion of Variables Diffusive flux: Where:

12. 3.Expansion of Variables Reactive term:

13. 4.Extract Equations Upscaled equations in correlation-based form:

14. 4.Extract Equations Simplification: 1.The gradient of  u is small 2.k is correlated to p1 only 3.D is correlated to the derivative of p1 only 4.G is constant

15. 5.Transformations 1 - Stationary approximation: Starting point: Assume minor temporal variations of  u and solve: Substitute:

16. 5.Transformations 2 – Nonlocal (memory, Integro-PD) form: Starting point: Assume k and D are linear and solve for  u : Substitute:

17. 5.Transformations 3a – Quantized form: Define characteristic variables: Substitute:

18. 5.Transformations 3b – Simplified Quantized form (bi-continuum): Assume D has only small spatial variations:

19. Application Example Water Infiltration in a heterogeneous soil

20. Summary Derived problem scale transport equations that incorporate the effects of local scale heterogeneity Asymptotic behavior corresponds to harmonic reactions and geometric diffusion Nonlocal form obtained in linear case Quantized form obtained in general case Equations are accurate for soils

21. Future Research Verify accuracy in Fisher-Kolmogoroff and other biotransport processes Investigate higher-order approximations Investigate equivalence with iterated Green’s functions techniques Investigate relationship with Quantum Mechanics (Heisenberg/Schrödinger)

22. Conclusion The developed approach has significant prospect for improving the engineering design and analysis of diagnosis and treatment strategies applicable to heterogeneous bioenvironments in areas such as: Drug/pesticide encapsulation Drug/fertilizer application strategies Control of invasive species / epidemics / bioagents