Quantized Transport in Biological Systems

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
Soil
Spinal Cord
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)

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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)

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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)

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3.Expansion of Variables

• Transported entity, u
• Where

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3.Expansion of Variables

• Nonlinear parameter, D 1st order Taylor series
• Redefine variables to get

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3.Expansion of Variables

• Diffusive flux
• Where

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3.Expansion of Variables

• Reactive term

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4.Extract Equations

• Upscaled equations in correlation-based form

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4.Extract Equations

• Simplification
• The gradient of ?u is small
• k is correlated to p1 only
• D is correlated to the derivative of p1 only
• G is constant

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5.Transformations

• 1 - Stationary approximation
• Starting point
• Assume minor temporal variations of ?u and solve
• Substitute

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5.Transformations

• 2 Nonlocal (memory, Integro-PD) form
• Starting point
• Assume k and D are linear and solve for ?u
• Substitute

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5.Transformations

• 3a Quantized form
• Define characteristic variables
• Substitute

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5.Transformations

• 3b Simplified Quantized form (bi-continuum)
• Assume D has only small spatial variations

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Application Example

• Water Infiltration in a heterogeneous soil

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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

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Future Research

• Verify accuracy in Fisher-Kolmogoroff and other
  biotransport processes
• Investigate higher-order approximations
• Investigate equivalence with iterated Greens
  functions techniques
• Investigate relationship with Quantum Mechanics
  (Heisenberg/Schrödinger)

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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