Coupling CPM and PDE: Vascular patterning

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.

Model 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.

Model

In Morpheus GUI: Examples 🠒 Multiscale 🠒 VascularPatterning.xml.

<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_on*u*s + k_off*b - delta*u</Expression>
            </DiffEqn>
            <DiffEqn symbol-ref="s">
                <Expression>gamma*cell - k_on*u*s+k_off*b</Expression>
            </DiffEqn>
            <DiffEqn symbol-ref="b">
                <Expression>k_on*u*s - k_off*b</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 + 0.0*t</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>

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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