Project PlacentaProject Placenta

Ethan JewettEthan Jewett

&&Megan LewisMegan Lewis

OutlineOutline

MotivationMotivation Biological BackgroundBiological Background GoalGoal Factors Factors First model (arterial dilation)First model (arterial dilation) Expanded model (trophoblast Expanded model (trophoblast

invasion)invasion) Corrected arterial modelCorrected arterial model ConclusionConclusion

MotivationMotivation

Maternal blood flow to a growing Maternal blood flow to a growing foetus affects foetal developmentfoetus affects foetal development

Too little blood flow can cause Too little blood flow can cause miscarriages or pre-eclampsia (a miscarriages or pre-eclampsia (a condition that causes hypertension condition that causes hypertension in the mother)in the mother)

Question: Can something be done to Question: Can something be done to alleviate these problems?alleviate these problems?

GoalGoal

To model the effect of trophoblast To model the effect of trophoblast movement, apoptosis and movement, apoptosis and invasiveness on maternal arterial invasiveness on maternal arterial dilation in order to determine their dilation in order to determine their effect on foetal developmenteffect on foetal development

Background: The Background: The PlacentaPlacenta

During pregnancy, During pregnancy, blood flows from blood flows from maternal spiral maternal spiral arteries into the arteries into the placenta, allowing placenta, allowing nutrient/oxygen nutrient/oxygen and waste and waste exchange between exchange between the mother and the mother and the foetusthe foetus

Maternal spiral arteries Maternal spiral arteries provide oxygen and provide oxygen and nutrients to a growing nutrients to a growing foetus, as well as foetus, as well as removing waste removing waste productsproducts

During the first During the first trimester, these trimester, these arteries are widened by arteries are widened by trophoblast cellstrophoblast cells

Trophoblast cells are Trophoblast cells are produced by the foetus, produced by the foetus, and invade the uterine and invade the uterine tissue by random tissue by random motion and chemotaxismotion and chemotaxis

Once a trophoblast cell Once a trophoblast cell reaches an artery, it reaches an artery, it proceeds to degrade proceeds to degrade the artery wallthe artery wall

In addition, In addition, trophoblasts migrate up trophoblasts migrate up the spiral arteriesthe spiral arteries

The trophoblasts The trophoblasts replace the smooth replace the smooth muscle inside the muscle inside the arteryartery

The arteries can The arteries can then deliver the then deliver the blood required by blood required by the developing the developing foetusfoetus

FactorsFactors Density of trophoblast cells in the arteryDensity of trophoblast cells in the artery Rate at which trophoblast cells degrade Rate at which trophoblast cells degrade

the arterythe artery Chemotaxis and random motion causing Chemotaxis and random motion causing

trophoblast cells to arrive at an arterytrophoblast cells to arrive at an artery Oxygen and temperature gradients Oxygen and temperature gradients

which provide stimulus to direct which provide stimulus to direct trophoblast cells to an arterytrophoblast cells to an artery

Amount of muscular material Amount of muscular material trophoblasts can absorb before trophoblasts can absorb before maturationmaturation

First model: arterial First model: arterial degredation by trophoblastsdegredation by trophoblasts

Artery smooth muscle

Artery

Ror

Artery wall

Trophoblasts

Assumptions for first Assumptions for first modelmodel Trophoblasts are at the artery (not Trophoblasts are at the artery (not

worrying about invasion process)worrying about invasion process) Rate of change of trophoblast density is Rate of change of trophoblast density is

dependent on the radius of the arterydependent on the radius of the artery When an artery reaches maximum When an artery reaches maximum

radius, no more dilation occursradius, no more dilation occurs Rate of change of the radius is Rate of change of the radius is

dependent on density of trophoblastsdependent on density of trophoblasts As the radius reaches its maximum As the radius reaches its maximum

value, the rate of change of value, the rate of change of trophoblasts decreases to zerotrophoblasts decreases to zero

Artery modelArtery model

)()]([)(

)]([)(

)()(

tbTtRRkdt

tdT

tRRtTs

tTw

dt

tdR

o

o

R(t) – Radius of artery at time tT(t) – Density of trophoblast cells in artery at time tRo – Maximum radius of the arterys – Density of trophoblasts at which the rate of increase of the radius reaches half its value (assuming Ro-R(t) fixed)k – Parameter determining rate T(t) increases w.r.t. the radiusb – Parameter determining the mortality rate of trophoblast cellsw – Parameter affecting rate of increase of radius rate w.r.t. time

Initial conditions:R(0)=r

T(0)=0

Phase portraitPhase portrait

Phase portrait Phase portrait of T(t) vs. R(t)of T(t) vs. R(t)

(0,R(0,Roo) is a ) is a steady statesteady state

Initial radius

Maximum radius Ro

Parameters set to:Ro – 20 (length)s – 10 k – 3 (1/length)b – 4w – 1 (1/time)

r – 5 (length)

Arterial degredation model Arterial degredation model (non-dimensionalized)(non-dimensionalized)

)())(1()(

)](1[)(1

)()(

tTtRqdt

tdT

tRtT

tTa

dt

tdR

sb

kRq

b

wawhere o ,:

Changing the value of a:Changing the value of a:Each line represents a different Each line represents a different

value of ‘a’value of ‘a’• blue: a= 1blue: a= 1• black: a= 2black: a= 2• red: a= 3red: a= 3• green: a=4green: a=4• pink: a=5pink: a=5

Shows as ‘a’ increases, the rate at Shows as ‘a’ increases, the rate at which the radius reaches its which the radius reaches its maximum also increases and the maximum also increases and the density of trophoblasts decreases. density of trophoblasts decreases.

R(t) and T(t) versus time

b

wa

b – Parameter determining the mortality rate of trophoblast cellsw – Parameter affecting rate of increase of radius rate w.r.t. time

Changing the value of q:Changing the value of q:Each line represents a different Each line represents a different

value of ‘q’value of ‘q’• blue: q= 1blue: q= 1• black: q= 2black: q= 2• red: q= 3red: q= 3• green: q=4green: q=4• pink: q=5pink: q=5

Shows as ‘q’ increases, the rate Shows as ‘q’ increases, the rate at which the radius grows at which the radius grows increases and the density of increases and the density of trophoblasts increases.trophoblasts increases.

R(t) and T(t) versus time

sb

kRq o

ProblemsProblems

Model isn’t very realisticModel isn’t very realistic Trophoblast density goes to zero as radius Trophoblast density goes to zero as radius

goes to zero... It might happen earlier!goes to zero... It might happen earlier! With this model, the radius always With this model, the radius always

reaches its maximumreaches its maximum Trophoblast density should be Trophoblast density should be

independent of the artery radiusindependent of the artery radius Also would like to model flow of Also would like to model flow of

trophoblasts from developing foetus to an trophoblasts from developing foetus to an arteryartery

How to fix the problem?How to fix the problem?

Make trophoblast density dependent Make trophoblast density dependent on time, instead of on artery radiuson time, instead of on artery radius

Develop a model which takes into Develop a model which takes into account more details, specifically the account more details, specifically the flow of trophoblasts from the foetus flow of trophoblasts from the foetus to the maternal arteriesto the maternal arteries

Diffusion model sketchDiffusion model sketch

Artery smooth muscle

Artery

Ror

Artery wall

Trophoblasts

Uterine lining

Trophoblast movement (diffusion)

L

Diffusion/chemotaxis Diffusion/chemotaxis equation coupled with equation coupled with

artery model artery model

x

vtxu

xx

txuD

t

txu

tbTt

tLu

dt

tdT

tRRtTs

tTk

dt

tdRo

),(),(),(

)(),()(

)]([)(

)()(

2

2

0)0,(

),(),(

1

1),0(

:_

xu

tLutLux

ttu

ConditionsBoundary

)10()(),( xexvtxv

R(t) :=Radius of artery at time t

T(t) :=Density of trophoblasts in artery at time t

u(x,t) :=Density of trophoblasts at position x at time t

Diffusion with Diffusion with chemotaxis plotchemotaxis plotu(x,t) vs. x at different

times

t values range approx. from 0 to 20

Fitting a curve at x=LFitting a curve at x=L

1)(),(

2

t

ttLu

Putting estimate into Putting estimate into ODESODES

R(t) and T(t) versus time

Each line represents a Each line represents a different value of ‘different value of ‘ψψ’’

• blue: blue: ψ ψ = 1= 1• black: black: ψψ = 2 = 2• red: red: ψψ = 3 = 3• green: green: ψψ =4 =4• pink: pink: ψψ =5 =5

Shows as ‘Shows as ‘ψψ’ increases, the ’ increases, the rate at which the radius rate at which the radius grows decreases and the grows decreases and the rate of change of density rate of change of density of trophoblasts also of trophoblasts also decreases.decreases.

)(),()(

)](1[)(1

)()(

tTt

tLu

dt

tdT

tRtT

tT

dt

tdR

Diffusion/logistic growth Diffusion/logistic growth equation coupled with equation coupled with

artery modelartery model

})],(1){[,(),(),(

)(),()(

)]([)(

)()(

2

2

btxutxux

txuD

t

txu

tbTt

tLu

dt

tdT

tRRtTs

tTk

dt

tdRo

R(t) :=Radius of artery at time t

T(t) :=Density of trophoblasts in artery at time t

u(x,t) :=Density of trophoblasts at position x at time t

)()0,(

0),0(

),(),(

:_

xlHeavisidexu

tux

tLcutLux

ConditionsBoundary

Diffusion with logistic Diffusion with logistic growth plotgrowth plot

u(x,t) vs. x at different times

Fitting a curve at x=LFitting a curve at x=L

)(),( 2)(terf

e

ttLu

t

Problems/SolutionsProblems/Solutions

Problem: The PDEs are too difficult to Problem: The PDEs are too difficult to solve analytically using Maple (can be solve analytically using Maple (can be solved numerically), but since the ODES solved numerically), but since the ODES require a solution at the boundary, this require a solution at the boundary, this is a problemis a problem

Solution: find a function which Solution: find a function which resembles the graph of the numerical resembles the graph of the numerical solution of the PDE at the endpoint, and solution of the PDE at the endpoint, and use it in the original model of the arteryuse it in the original model of the artery

Corrected Artery model Corrected Artery model (DL)(DL)

)()()(

)]([)(

)()(

2)(

2

tbTe

tk

dt

tdT

tRRtTs

tTc

dt

tdR

t

o

Corrected term, no longer dependent on radius

Initial conditions:R(0)=r

T(0)=0

Non-dimensionalized Non-dimensionalized corrected model (DL)corrected model (DL)

)()(

)](1[)(1

)()(

2

2

tTe

t

dt

tdT

tRtT

tT

dt

tdR

t

Corrected model:Corrected model:changing the value of alphachanging the value of alpha

Each line represents a Each line represents a different value of ‘different value of ‘αα’’

• blue: blue: αα= 1= 1• black: black: αα = 2 = 2• red: red: αα = 3 = 3• green: green: αα =4 =4• pink: pink: αα =5 =5

Shows as ‘Shows as ‘αα’ increases, the ’ increases, the rate at which the radius rate at which the radius grows increases and the grows increases and the rate of change of density rate of change of density of trophoblasts does not of trophoblasts does not change.change.

R(t) and T(t) versus time

Corrected model:Corrected model:changing the value of changing the value of

omegaomega

Each line represents a Each line represents a different value of ‘different value of ‘ωω’’

• blue: blue: ω ω = 1= 1• black: black: ωω = 2 = 2• red: red: ωω = 3 = 3• green: green: ωω =4 =4• pink: pink: ωω =5 =5

Shows as ‘Shows as ‘ωω’ increases, the ’ increases, the rate at which the radius rate at which the radius grows increases and the grows increases and the rate of change of density rate of change of density of trophoblasts also of trophoblasts also increases.increases.

R(t) and T(t) versus time

Corrected model:Corrected model:changing the value of psichanging the value of psi

Each line represents a Each line represents a different value of ‘different value of ‘ψψ’’

• blue: blue: ψ ψ = 1= 1• black: black: ψψ = 2 = 2• red: red: ψψ = 3 = 3• green: green: ψψ =4 =4• pink: pink: ψψ =5 =5

Shows as ‘Shows as ‘ψψ’ increases, the ’ increases, the rate at which the radius rate at which the radius grows decreases and the grows decreases and the rate of change of density rate of change of density of trophoblasts also of trophoblasts also decreases.decreases.

R(t) and T(t) versus time

ConclusionConclusion

No data, making it very difficult to No data, making it very difficult to determine if our models are determine if our models are biologically correctbiologically correct

The models show what was The models show what was expected, although with data it expected, although with data it would be possible to determine how would be possible to determine how different parameters affect the total different parameters affect the total blood flow to the foetusblood flow to the foetus

Further workFurther work

Model the movement down the Model the movement down the arteryartery

Solve the combined PDE and ODE Solve the combined PDE and ODE system numerically (coding in system numerically (coding in numerics)numerics)

Find data in order to fit the results Find data in order to fit the results and potentially make predictionsand potentially make predictions

Thank YouThank You

Gerda de VriesGerda de Vries Jim MuirheadJim Muirhead Gustavo CarreroGustavo Carrero And everyone we consultedAnd everyone we consulted