quantized transport in biological systems hubert j. montas, ph.d. biological resources engineering...
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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