Coupling CPM and PDE: Vascular Patterning

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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.


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.


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">
    - 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
    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.
            <Field symbol="u" value="1.5e-6" name="VEGF">
                <Diffusion rate="58.7"/>
            <Field symbol="s" value="0" name="Free ECM">
                <Diffusion rate="0.001"/>
            <Field symbol="b" value="0" name="VEGF_b">
                <Diffusion rate="0"/>
            <Field symbol="VEGF_all" value="0" name="VEGF_s + VEGF_b">
                <Diffusion rate="0"/>
            <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 symbol-ref="s">
                    <Expression>gamma*cell - k_on*u*s+k_off*b</Expression>
                <DiffEqn symbol-ref="b">
                    <Expression>k_on*u*s - k_off*b</Expression>
                <Rule symbol-ref="VEGF_all">
            <Constant symbol="cell" value="0.0"/>
            <Constant symbol="cell_density" value="0.0045"/>
            <Lattice class="square">
                <Size symbol="size" value="200 200 0"/>
                    <Condition boundary="x" type="periodic"/>
                    <Condition boundary="y" type="periodic"/>
                <NodeLength value="2"/>
            <SpaceSymbol symbol="l"/>
            <StartTime value="0"/>
            <StopTime value="3600"/>
            <SaveInterval value="0"/>
            <RandomSeed value="1"/>
            <TimeSymbol symbol="t"/>
            <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>
                <Distribution>l.x / size.x</Distribution>
            <CellType class="medium" name="medium"/>
            <Interaction default="0">
                <Contact type1="medium" type2="HUVEC" value="3.2"/>
                <Contact type1="HUVEC" type2="HUVEC" value="6.4"/>
            <MonteCarloSampler stepper="edgelist">
                <MCSDuration value="1.0"/>
                <MetropolisKinetics temperature="1"/>
            <ShapeSurface scaling="norm">
            <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"/>
            <Gnuplotter time-step="250" decorate="false">
                <Terminal opacity="0.65" name="png"/>
                    <Field symbol-ref="b"/>
                    <Cells value="cell">
                            <Color value="1" color="gray"/>
                            <Color value="0.0" color="gray"/>
            <!--    <Disabled>
            <HistogramLogger normalized="false" number-of-bins="10" time-step="100">
                <Plot minimum="0" maximum="1.0" terminal="png"/>
                <Column symbol-ref="" celltype="Angioblasts"/>
            <DependencyGraph format="svg" exclude-plugins="Gnuplotter, CPM" exclude-symbols="cell,, str_2"/>


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