08 - basics constitutive equations - volume growth...

108 - basic constitutive equations

08 - basics constitutive equations -volume growth - growing tumors

2where are we???

3homework 02 4homework 02

9kinematic equations

kinematics of finite growth

• incompatible growth configuration & growth tensor

rodriguez, hoger & mc culloch [1994]

10balance equations

balance of mass of open systems

biological equilbrium

mass flux• cell movement (migration)

mass source

• cell enlargement (hypertrophy)• cell division (hyperplasia)• cell growth (proliferation)

cowin & hegedus [1976], beaupré, orr & carter [1990], harrigan & hamilton [1992],jacobs, levenston, beaupré, simo & carter [1995], huiskes [2000], carter & beaupré [2001]

11balance equations

generic balance equation of open systems

… balance quantity… flux… source… production

general format

12constitutive equations

constitutive equations instructural analysis, constitutive rela-tions connect applied stresses or forces tostrains or deformations. the constitutiverelations for linear materials are linear.more generally, in physics, a constitutiveequation is a relation between two physicalquantities (often tensors) that is specificto a material, and does not follow directlyfrom physical law. some constitutiveequations are simply phenomenological;others are derived from first principles.

constitutive equations


constitutive equationsor equations of state bring in the charac-terization of particular materials withincontinuum mechanics. mathematically, thepurpose of these relations is to supplyconnections between kinematic, mechanicaland thermal fields. physically, constitu-tive equations represent the various formsof idealized material response which serveas models of the behavior of actualsubstances.

constitutive equations

chadwick “continuum mechanics“ [1976]


tensor analysis - these derivatives might be handy


tensor analysis - these derivatives might be handy


neo hooke‘ian elasticity

• definition of stress

• free energy

• definition of tangent operator




• free energy

• definition of tangent operator




• free energy

undeformedpotato

deformedpotato




• free energy

undeformedpotato

mashedpotatoes

• remember! mashing potatoes is not an elastic process!



• free energy

undeformedpotato

deformedpotato

• large strain - lamé parameters and bulk modulus

• small strain - young‘s modulus and poisson‘s ratio


neo hooke‘ian elasticity - cellular tissues

carter & hayes [1977]

free energy


additional constitutive equations for volume growth

mass source

mass flux

growth tensor

multiplicative decomposition - but what is ?lee [1969], simo [1992], rodriguez, hoger & mc culloch [1994], epstein & maugin [2000],

humphrey [2002], ambrosi & mollica [2002], himpel, kuhl, menzel & steinmann [2005]


volume growth at constant density

kinematic coupling of growth and deformation

• free energy

• mass source

• stress

rodriguez, hoger & mc culloch [1994], epstein & maugin [2000], humphrey [2002]

• growth tensorpressuregrowth function

increase in mass

24example - breast cancer

volume growth - cylindrical tumor

ductal carcinoma is a very common type of breast cancer inwomen. infiltrating ductal carcinoma refers to the development ofinvasive cancer cells within the milk ducts of the breast. it accountsfor 80% of all types of breast cancer. on a mammography, it isusually visualized as a mass with fine spikes radiating from theedges, and small microcalcification may be seen as well. on physicalexamination, the lump usually feels hard or firm. on microscopicexamination, the cancerous cells invade and replace the surroundingnormal tissue inside the breast. the subtypes with histology ofmucinous, papillary, cribriform, and tubular carcinomas have abetter prognosis, longer survival, and lower recurrence ratesThan those with histology like signet-ring cell or inflammatorycarcinoma or carcinoma with sarcomatoid metaplasia.


volume growth - cylindrical tumornormal cells grow andmultiply at a specific rate.cells that grow andmultiply without stoppingare called cancerous ormalignant. however, theyare not detectable whenthey first start growing.

ductal carcinoma in situ is a non-invasive change in the cells that line

the milk tubes that bring milk from themilk lobules to the nipple

invasive or infiltrating ductalcarcinoma is the most

common type of breastcancer.it occurs when thecells that line the milk duct

become abnormal and spreadinto the surrounding breast

tissue.




normal ductal epithelialcells (negative)

tumor cells of ductalcarcinoma (positive)

invasive ductal carcinomacells (strongly positive)

model assumptions• ductal cardinoma - tumor grows in breast duct for up to 10 cm• model - homogeneous growth inside a rigid cylinder• assumption - rotational symmetry• strategy - solve for deformation that satisfies equi‘m and boundary cond‘s




• homogeneous deformation inside a rigid cylinder

• deformation gradient




• free energy

• deformation

• stress

normal ductal epithelialcells (negative)

• growth tensor

tumor cells of ductalcarcinoma (positive)

invasive ductal carcinomacells (strongly positive)



ambrosi & mollica [2002]



• axial displacement as a function of growth

• stress


• bc‘s

• growth induced stress on tumor wall

cancer cells duct wall




tumor pressure on duct walls increases with growth

cancer cells duct wall

growth multiplier

rela

tive

axia

l elo

ngat

ion

growth multiplier

gro

wth

-induc

ed w

all s

tress

32example - tumor growth

volume growth - spherical tumor

model assumptions• tumor model - inhomogeneous growth of a sphere• inhomogeneity - residual stress even in the absense of applied loads• assumption - spherical symmetry• strategy - solve for deformation that satisfies equi‘m and boundary cond‘s


kinematic coupling of growth and deformationambrosi & mollica [2002]




• free energy

• deformation

• stress


• growth tensor






• balance of momentum

• stress


• bc‘s




ordinary differential eqn of 2nd order in 2B solved numerically





hoop stress tensile inside / compressive outside

inhomogeneous deformation hoop stress

radial stress

radial position radial position

gro

wth

and

rad

ial d

ispla

cem

ent

radia

l stre

ss a

nd h

oop s

tressincreased growth in outer layers

08 - basics constitutive equations - volume growth...

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