Quantized Transport in Biological Systems

Hubert J. Montas, Ph.D.

Biological Resources Engineering

University of Maryland at College Park

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

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

Objective

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

Materials• A reaction-diffusion equation with spatially-

varying coefficients is assumed to apply at the local scale:

upuDupukt

upuG

;;;

• Example 1: Richards’ equation (soils)

upuKt

upuC

;;

• Example 2: Fischer-Kolmogoroff (tissues/ecosys)

uDuupukrt

u

;

MethodsStochastic-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)

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)

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)

1

11 ),(),(

p

pyxpyx

2

' i

3.Expansion of Variables

• Transported entity, u:

Where:

),(),,(

),(),,(

),,(),,(

** dyydxxtyx

dyydxxtyx

tyxutdxydxxu

u

u

*),,(

1),,(

utyx

utyxu

u

3.Expansion of Variables

• Nonlinear parameter, D: 1st order Taylor series:

• Redefine variables to get:

pdyydxxpp

D

yxudyydxxuu

DpyxuDpuD

pyxu

pyxudyydxx

),(

),(),());,(();(

));,((

));,((),(

**);( DDDpuD

3.Expansion of Variables• Diffusive flux:

Where:

*

**

*

**

*

uD

uD

uuD

uuD

Du

Du

ei

eiuDJ

u

ue u

3.Expansion of Variables

• Reactive term:

****

**

uk

uk

ukuk

ku

ku

ukuk

4.Extract Equations

• Upscaled equations in correlation-based form:

**

***

*2****

2

**

**

****

2

2

uD

uuuuDu

uukGu

uD

uuuuDu

uukGu

uDuD

uuDuuD

ukuku

Gu

G

Du

DeiDeiuei

Dkut

u

tG

Du

DeiDeiuei

Dkut

u

tG

uD

eiei

ukttt

uG

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

uDkut

uDeukt

u

Duuku

uuDuk

22

5.Transformations1 - Stationary approximation:

• Starting point:

• Assume minor temporal variations of u and solve:

• Substitute:

uDkut

uDeukt

u

Duuku

uuDuk

22

uDk

uDk

Dku

22 22

uDkD

DuDkk

kt

u Dk

22

22

22

2

21

21

5.Transformations2 – Nonlocal (memory, Integro-PD) form:

• Starting point:

• Assume k and D are linear and solve for u:

• Substitute:

uDkut

uDeukt

u

Duuku

uuDuk

22

duut

t

Dk

tkDu e

0

2 2

duuuDuk

t

uDk

ttkD

e 2222

0

2 2

5.Transformations3a – Quantized form:

• Define characteristic variables:

• Substitute:

ppu

ppu

puDDpukkuu

puDDpukkuu

;;;;

;;;;

22222

11111

22221122122212

11112211211121

uDuvuuLukt

u

uDuvuuLukt

u

21

21212112

212211221

212112

2;

2;

2 uu

uuDDvv

DDLLuu

DD

5.Transformations

3b – Simplified Quantized form (bi-continuum):

• Assume D has only small spatial variations:

221221222

112112111

uDuuLukt

u

uDuuLukt

u

Application Example• Water Infiltration in a heterogeneous soil

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

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)

ConclusionThe 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