NON-EQUILIBRIUM THERMODYNAMICS OF HETEROGENEOUS GROWING BIOSYSTEMS - - PowerPoint PPT Presentation

NON-EQUILIBRIUM THERMODYNAMICS OF HETEROGENEOUS GROWING BIOSYSTEMS

Natalya Kizilova Department of Theoretical and Applied Mechanics Kharkov National University Ukraine

Outline

1. Biological growth: definition, types, properties 2. Experiments with growing plant materials (leaves) 3. Experiment-based mathematical model of growing

• continuum. Parameter identification.

4. Biological growth in tissue engineering. Experimental technologies and models. 5. A mixture model of the inhomogeneous growing tissue. Application to the tissue growth in the degradable scaffold 6. Conclusions

Growth = irreversible changes in the mass (volume, size) of an object provided by new mass accumulation

Tissues=cells + extracellular solid matter + interstitial liquid Plant cells = immovable cells + rigid cellular walls Animal cells = movable (migrating) cells + extracellular solids and liquids I: cell growth and divisions II: extracellular matter production and self-assembling

Biosystems are

• open TD systems with are in permanent mass and energy

exchange with environment (circulatory, respiratory, excretory systems; outer and internal surfaces)

• in permanent non-equilibrium (NE) state working against

equilibrium; supporting non-zero gradients and corresponding fluxes; exhibiting complex cross-related phenomena

• non-uniform systems (cell types, gradient fields) at permanent

dynamical loading (gravity, muscle contractions, flow

• scillations, electric impulses)
• active systems (parameter-dependent properties; local chemical

and mechanical + central nervous and humoral systems)

• optimal systems possessing maximal performance at given

conditions (minimal energy expenses/entropy production)

Growth types:

Surface growth

• Mass accumulation/resorbtion at

external surfaces

• Coupling of dissolution-crystallization
• Driven by
• Features: growth anisotropy; non-

uniformity

• TD consideration: solidification fronts
• Examples: bones, skull, tree trunks,

branches, shoots

a e

c , ,...   

Inner growth (remodeling)

• Mass increase/decrease in each point
• Non-zero stress field
• Examples: plant leaves and roots, inner
• rgans, tumors
• Features: anisotropic growth; residual

stresses

Volume growth

• Mass increase/decrease in each point
• Non-zero stress field
• Examples: plant leaves and roots, inner organs,

tumors

• Features: anisotropic growth; residual stresses

Experimental study of plant leaf growth at zero stress conditions

) v , v ( v

r 

 

    



v r ) t ( a v

r r

     

 

r ) t ( a v r ) t ( a v

r r

Experimental study of plant leaf growth at mechanical restrictions

Leaf blade deflection and boundary angle measurements

Experiment-based conclusions:

• Extraction/compression stimulates/oppresses growth in

the corresponding direction

• Growth rate at zero-stress conditions is a function of

time and concentrations of growth factors/regulators

• Growth rate at nonzero-stress conditions is a function of

stress tensor components

• New material accumulates according to principals of the

stress tensor providing the lightweight design

• Stress-induced elongation of cells (endothelial cells in

vessel wall, skeletal muscle cells, conducting vessels)

Mathematical modeling of growing continua

div( v) q t ˆ div         

1 i k k i

ˆ ˆ ˆ ˆ ˆ ˆ e A(t) B (E) d / dt 1 v v ˆ e 2 x x  



              

1 2

n

0, v

 

    



3 , 2 , 1 j , i , x x A 2 x A x A

j i ij 2 2 i jj 2 2 j ii 2

         

2 2 ii ij 2 i j i

A , A x x x         

y x A 2 x A y A

xy 2 2 yy 2 2 xx 2

        

xy y x yy y xx x

A 2 x v y v A y v A x v            

Growth viscosity tensor, Beltrami-Michell equations, growth problem formulation

                    

66 55 44 33 32 31 23 22 21 13 12 11

B B B B B B B B B B B B B

lm iklm ik ik

B A e   

F ˆ div    

* n

  



 

v 





3 j m i mm ij i m j pp j i m 1 i k ik i k qq k i

v 1 v 1 v A A x b x x 2B x x 1 v v A F x 2B x x i,j,k 1 ,2,3, q 9 i k, p 9 i j



                                                                          



3 i m ; 3 , 2 , 1 k , j , i x x ) B ( 2 x ) B B B ( x ) B B B (

j i ij mm 2 2 i kk jk jj jj ii ji 2 2 j kk ik jj ij ii ii 2

                       

ik

b det b 

,

33 32 31 23 22 21 13 12 11 ik

B B B B B B B B B b 

Conclusions

• In spite of different shape, size, physiology, evolutionary

age, etc… the narrow limits for growth parameters have been found

* * * 1

~ 0.03 – 0.05 MPa A ~ 0.5 3 mm / day B ~ 0.1 1(Pa s) 



  

• Transportation systems have the same principles of

design (dependences between the lengths, diameters, branching angles, drianage areas) which corresponds to the model of optimal pipeline providing homogenous flow delivery at minimum energy expences.

Biological growth in tissue engineering

Successful laboratory and clinical reports on tissue engineering of:

blood and lymphatic vessels [Shin’oka T., et al, 2001] heart valves [Sodian R., et al, 2000] cardiac tissue [Carrier R.L., et al, 1999] bone and cartilage [Vacanti C.A., et al, 1994] tendon [Cao D., et al, 2006] skin [Parenteau N.L., et al, 1991] liver [Kim T.H., et al, 2000] stomach [Maemura T., et al, 2003] intestine [Choi R.S., et al, 1998] bladder [Oberpenning F., et al, 1998] skeletal muscle [Geris L., et al, 2001] nerves [Fansa H., et al, 2003]

3D tissue and organ printing

Polymer and metal scaffolds with regular structure

• Role of geometry (strength, lightweight design,

porosity, shape of pores, adequate pore sizes for easy penetration of the growing cells/structures)

• Role of material (biocompatibility and non-toxicity;

controlled degradation kinetics corresponding to the new tissue formation).

Diffusion models of growth

C b b b C C C C

C J C t b J Cb t J D b J D C f(b) b b b D D ( ,b, ,R, , (T)) F R 6 R                                           

Particle dynamic growth models

( j) 2 ( j) ( j) ( j) (k) ( j) ( j) rm a S r 2 n ( j) (k) k j

dr d r dr r r m k (r r ) 6 R k dt dt dt r r adhesion drag repulsion random walks  



      



        

A multi-phase model of the growing inhomogeneous tissue

Solid phases: 1 – cells of different types 2 - vessel walls, connective tissues, airways 3 – extracellular matrix Liquid phases: 4 – intracellular liquids 5 – extracellular (tissue) liquid 6 – delivering liquid Components: 1 - nutrition (glucose, O2, …) 2 – growth factors, …

Mass balance equations

 

div v t dC divJ M k dt J C (v v)          



    

               

      

Momentum balance equations

k k j kj k k k j j k k k k

v ( v v ) p R M f t M v (R M )

                 

   



           

 

Energy balance equations

 

k j k k j j k k k

E ( v E ) Q v R f N W t N E (v R N W )

                     

   



             

 

Additional equations (active movement, structure formation, aggregation, …)

1

(v ) (C ,v ,...) (C ,v ,...) t n div(nv ) G t

 

                

        

Internal energy

kj kj kj

U (S ,C ), 4 6 (liquid phases) U U (S ,C , ), 1 3 (solid phases) U U U T , , S C

                

                        

Entropy balance equation

j j S n n S n kj kj kjpq p q kj kj kj kjpq p q k k k k k k k k k

S 1 G , S S t X Y p pC g v , 4 6 p p pC g v , 1 3 R p C k (v v ) J D (v v )

                    

                                        

 

Cell proliferation, migration, adhesion, interaction, vascularization in a biodegradable scaffold can be studied as slow flow through a porous media with increasing porosity

Conclusions

1. Biological growth is a complex phenomena that can be described and understood on the concepts of NET of

• pen systems

2. Mixture models based on Onsager theory are useful for slow normal growth and tissue engineering problems while they are failed in some special occasions 3. Tissue engineering technologies need development of the TD theory enable to describe non-uniform multicellular growth at mechanical load and chemical regulation conditions; control over tissue anisotropy, vascularisation and innervation