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Thread: Cell Growth: Single Cell Growth

  1. #1
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    Default Cell Growth: Single Cell Growth

    In FEBio, it is possible to model the growth of cells in biological tissues using the material type "cell growth". This model describes a mechanism of interstitial (volumetric) growth, where growth is driven by the exchange of mass between the cell and its surrounding environment. The mass exchanged may consist of: (1) membrane-impermeant solutes that remain in soluble form inside the cell; (2) solutes that bind to (or get released from) the intracellular solid matrix (e.g., cytoskeleton or other osmotically inactive constituents of the intracellular environment); and (3) water that enters or leaves the intracellular environment in response to osmotic gradients relative to the extracellular environment. It is assumed that all these constituents are intrinsically incompressible. This model does not account for the electric charges of the solutes or intracellular solid matrix, i.e., solutes are assumed neutral. Membrane-permeant solutes are thus assumed to maintain the same concentration between the intracellular and extracellular environments, having no influence on the osmotic gradients.

    The material parameters associated with this cell growth model are:
    • cr: number of moles of membrane-impermeant intracellular solutes, per volume of the cell in the reference configuration.
    • phir: volume of intracellular solid matrix, normalized by the volume of the cell in the reference configuration.
    • ce: osmolarity of the extracellular environment.


    The reference configuration is the stress-free configuration of the intracellular solid matrix. Typically, it is the initial configuration of the model. Since cr and phir are normalized by the cell volume in the reference configuration (an invariant quantity), any changes to these parameters respectively represent changes in the number of intracellular membrane-impermeant solutes and volume of solid matrix. Therefore, cell growth may be modeled by associating these parameters with load curves in FEBio, so that the user can determine how these parameters should vary over time. For example, if we want to model doubling of the cell volume, cr and phir may be set to double in value over the desired time span. Biological cells typically double in size [the interphase, consisting of the synthesis (S) and gap phases (G1 & G2)] either prior to, or after dividing into two daughter cells (mitosis, or M phase).
    (click on image to see movie)
    SingleCellGrowth.feb
    For example, consider the model of a spherical cell. Assume that the extracellular osmolarity is ce=300 mM, which remains fixed throughout the analysis. Assume that phir=0.3 initially, implying that the intracellular solid matrix occupies 30 percent of the cell volume. In order to ensure that our initial configuration is also the reference configuration, we must pick the value of cr such that there is no initial osmotic gradient between the intracellular and extracellular environments. In other words, we would like to set the intracellular osmolarity to 300 mM as well. Since intracellular osmolarity is evaluated as the number of solutes per volume of intracellular fluid, this means that our initial value of cr should be cr = (1-phir)*ce = (1-0.3)*300 = 210 mM. Now, to model doubling of the cell volume, we can create a load curve for cr that varies from 210 to 420 mM, and another for phir that varies from 0.3 to 0.6 (e.g., from time t=0 to t=1). The FEBio file for this example is SingleCellGrowth.feb. The resulting response is displayed in the movie.

    Keep in mind that this volume growth represents a combination of the three mass additions described above: Increasing mass of intracellular solute and solid matrix (as controlled by the user-defined load curves), and obligatory uptake of water as a result of osmotic effects. Since the intracellular volume of water has also doubled in this example, the intracellular osmolarity of membrane-impermeant solutes remains constant (300 mM) throughout the growth process.

    Also note that there was no constitutive relation provided for the elasticity (or viscoelasticity) of the intracellular solid matrix. Indeed, the cell growth model assumes that the intracellular solid matrix stiffness is zero. Therefore, as the cell grows, the solid matrix stress remains zero, implying that the intracellular solid matrix offers no resistance to the growth. If we would like to include some measure of stiffness to the solid matrix, we may combine the "cell growth" material with an elastic material (e.g., "neo-Hookean") inside a "solid mixture". This approach is illustrated in the example titled Internal Constraints to Growth on the forum.

    The theoretical framework behind this cell growth model is described in the following publications:

    Ateshian, G.A., Costa, K.D., Azeloglu, E.U., Morrison, B., 3rd, and Hung, C.T., 2009. Continuum modeling of biological tissue growth by cell division, and alteration of intracellular osmolytes and extracellular fixed charge density. J Biomech Eng 131(10), 101001 PMCID: 2860886.

    Ateshian, G.A., Morrison, B., 3rd, Holmes, J.W., and Hung, C.T., 2012. Mechanics of cell growth. Mechanics Research Communications, In Press.
    Last edited by ateshian; 03-25-2012 at 06:20 PM.

  2. #2

    Default

    Hi,
    Great material. Just wondering whether it is possible to have two of these materials in the same contiguous mesh (object)? Thinking about a slower growing population of cells (part1) within, or adjacent to, a faster growth population (part2) by altering phir and cr. Is there any reason why this wouldn't work?
    best
    nathan

  3. #3
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    Default

    Hi Nathan,

    In principle, a heterogeneous model should work. If you run into any problems please let me know.

    Best,

    Gerard

  4. #4

    Default

    Thanks Gerard,
    First attempt is attached. Just about works but not very stable. Any suggestions for improving it - in particular to sustain bigger differences of phir and cr and more complex geometries?

    Also wondering if these should be separate geometries and then use a biphasic or facet-facet sliding interface?

    Any advice greatly appreciated

    best
    nathan
    Attached Files Attached Files

  5. #5
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    Default

    Hi Nathan,

    This model only has one internal node that is fixed (all three components). Due to the asymmetric nature of the response, there is nothing that prevents the model from rotating around that fixed node, so this could be the principal reason for the poor stability (even though the model converges in the end). Unfortunately, there are no natural planes of symmetry for this analysis, so you may have to pick another arbitrary set of constraints (e.g., fixing all three components of another node, or the X, Y, Z components of three separate nodes, respectively). This should hopefully help.

    Also wondering if these should be separate geometries and then use a biphasic or facet-facet sliding interface?
    I don't think that should be necessary.

    Best,

    Gerard

  6. #6
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    Default

    Hi Gerard,

    Is it possible to convert this .feb file to .prv file in linux. I would like to play around with this model in preview.

    Beat Regards,
    Ishan Gupta

  7. #7
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    Default

    Hi Ishan,

    In principle all you have to do is launch PreView and File->Import FE Model... to read the .feb file, then save the resulting .prv file. I was able to do that successfully with the latest version of PreView on the Mac, even though the file available in this post is in 1.2 format. Nevertheless, in case you are having trouble importing that format in linux, I have attached a .feb file in 2.5 format (SingleCellGrowth.feb). Let me know if that doesn't work for you.

    Best,

    Gerard

  8. #8
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    Apr 2019
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    Default

    Hi Gerard,

    The 1.2 format works perfectlz fine, though there is some missing boundary condition in 2.5 version. Hence, it's not importing.

    Best Regards,
    Ishan

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