Coupling CPM and PDE: Vascular Patterning

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Introduction

This example shows a model of vascular network formation by paracrine signaling (Köhn-Luque et al., 2011) and employs a coupled CPM and reaction-diffusion model.

Cells organize into networks due to matrix-anchorage of chemoattractant.
Cells organize into networks due to matrix-anchorage of chemoattractant.

Description

The model defines a CPM as well as a PDE. These models are coupled by two processes:

  1. Cells, specified in CellTypes, respond chemotactically to a Layer (or species) in the PDE.
  2. Conversely, the production term of one PDE Layer is coupled to the presence/absence of cell.

Reference

A. Köhn-Luque, W. de Back, J. Starruß, A. Mattiotti, A. Deutsch, J. M. Pérez-Pomares: Early Embryonic Vascular Patterning by Matrix-Mediated Paracrine Signalling: A Mathematical Model Study. PLoS ONE 6 (9): e24175, 2011.

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<MorpheusModel version="3">
    <Description>
        <Title>Example-VascularPatterning</Title>
        <Details>Reference:
- A Köhn-Luque, W de Back, Y Yamaguchi, K Yoshimura, M A Herrero, T Miura (2013) Dynamics of VEGF matrix-retention in vascular network patterning, Physical Biology, 10(6) : 066007
http://dx.doi.org/10.1088/1478-3975/10/6/066007

Related to:

  • Köhn-Luque A, de Back W, Starruß J, Mattiotti A, Deutsch A, Perez-Pomares JM, Herrero MA (2011) Early Embryonic Vascular Patterning by Matrix-Mediated Paracrine Signalling: A Mathematical Model Study. PLoS ONE 6(9): e24175. http://dx.doi.org/10.1371/journal.pone.0024175

</Details> </Description> <Global> <Field symbol="u" value="1.5e-6" name="VEGF"> <Diffusion rate="58.7"/> </Field> <Field symbol="s" value="0" name="Free ECM"> <Diffusion rate="0.001"/> </Field> <Field symbol="b" value="0" name="VEGF_b"> <Diffusion rate="0"/> </Field> <Field symbol="VEGF_all" value="0" name="VEGF_s + VEGF_b"> <Diffusion rate="0"/> </Field> <System solver="heun" time-step="5.0"> <Constant symbol="gamma" value="5e-3" name="Production ECM"/> <Constant symbol="k_on" value="8.5e-4" name="Binding rate VEGF/ECM"/> <Constant symbol="k_off" value="3.6e-3" name="Unbinding rate VEGF/ECM"/> <Constant symbol="delta" value="2.6e-6" name="Decay VEGF "/> <DiffEqn symbol-ref="u"> <Expression>- k_onus + k_offb - deltau</Expression> </DiffEqn> <DiffEqn symbol-ref="s"> <Expression>gammacell - k_onus+k_offb</Expression> </DiffEqn> <DiffEqn symbol-ref="b"> <Expression>k_onus - k_offb</Expression> </DiffEqn> <Rule symbol-ref="VEGF_all"> <Expression>u+b</Expression> </Rule> </System> <Constant symbol="cell" value="0.0"/> <Constant symbol="cell_density" value="0.0045"/> </Global> <Space> <Lattice class="square"> <Size symbol="size" value="200 200 0"/> <BoundaryConditions> <Condition boundary="x" type="periodic"/> <Condition boundary="y" type="periodic"/> </BoundaryConditions> <NodeLength value="2"/> <Neighborhood> <Order>2</Order> </Neighborhood> </Lattice> <SpaceSymbol symbol="l"/> </Space> <Time> <StartTime value="0"/> <StopTime value="3600"/> <SaveInterval value="0"/> <RandomSeed value="1"/> <TimeSymbol symbol="t"/> </Time> <CellTypes> <CellType class="biological" name="HUVEC"> <Property symbol="cell" value="1.0" name="cell"/> <Property symbol="str" value="3e7" name="chemotactic strength"/> <VolumeConstraint target="90" strength="1"/> <Chemotaxis field="b" contact-inhibition="false" strength="str" retraction="false"/> <!– <Disabled> <AddCell mode="exclude"> <Condition>rand_uni(0,1) &lt; 0.24 + 0.0t</Condition> <Triggers/> <Distribution>l.x / size.x</Distribution> </AddCell> </Disabled> –> </CellType> <CellType class="medium" name="medium"/> </CellTypes> <CPM> <Interaction default="0"> <Contact type1="medium" type2="HUVEC" value="3.2"/> <Contact type1="HUVEC" type2="HUVEC" value="6.4"/> </Interaction> <MonteCarloSampler stepper="edgelist"> <MCSDuration value="1.0"/> <Neighborhood> <Order>2</Order> </Neighborhood> <MetropolisKinetics temperature="1"/> </MonteCarloSampler> <ShapeSurface scaling="norm"> <Neighborhood> <Order>2</Order> </Neighborhood> </ShapeSurface> </CPM> <CellPopulations> <Population size="0" type="HUVEC"> <InitRectangle mode="regular" number-of-cells="cell_density * size.x * size.y"> <Dimensions size="200,200,0" origin="0,0,0"/> </InitRectangle> </Population> </CellPopulations> <Analysis> <Gnuplotter time-step="250" decorate="false"> <Terminal opacity="0.65" name="png"/> <Plot> <Field symbol-ref="b"/> <Cells value="cell"> <ColorMap> <Color value="1" color="gray"/> <Color value="0.0" color="gray"/> </ColorMap> </Cells> </Plot> </Gnuplotter> <!– <Disabled> <HistogramLogger normalized="false" number-of-bins="10" time-step="100"> <Plot minimum="0" maximum="1.0" terminal="png"/> <Column symbol-ref="cell.id" celltype="Angioblasts"/> </HistogramLogger> </Disabled> –> <DependencyGraph format="svg" exclude-plugins="Gnuplotter, CPM" exclude-symbols="cell, cell.center, str_2"/> </Analysis> </MorpheusModel>

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