TemporalTissueGraph implementation proposal

Biological observations

The typical object is a discrete time-series of multicellular tissue observations (from microscopy). Cell-based segmentation, manual or algorithmic, is required to access the spatial information hidden in the intensity image. Cell lineage, is required to access the temporal information.

Goal

The goal of this data structure is to provide a flexible graph object dedicated to the exploration of discrete dynamic multicellular tissue properties.

• It leverages the TissueGraph class.

• Biological semantic should be used when possible.

Structure

A lineage graph define the temporal relationships between the -spatial- TissueGraph objects. A dictionary of TissueGraph gather the discrete observations of multicellular tissues.

The lineage graph is a networkx.DiGraph().

The time intervals between each observation is required to perform temporal differentiation of spatial properties!

Elements ids

The topological element ids are now also indexed by time.

Primal graph

• temporal cell ids as nodes: tcid=(t, cid) an integer t for time and an integer cid for the cell id, e.g. (0, 5) for cell 5 at time 0

• temporal cell-wall ids as edges:

  1. twid=(t, wid)=(t, (cid_i, cid_j)) an integer t for time and a len-2 tuple of integers wid=(cid_i, cid_j) for the cell-wall id, e.g. (0, (5, 6)) for cell-wall (5, 6) at time 0

  2. twid=(tcid_i, tcid_j)=((t, cid_i), (t, cid_j)) two len-2 tuple of integers tcid_i and tcid_j for the two temporal cell ids defining the cell-wall, e.g. ((0, 5), (0, 6)) for cells 5 and 6 at time 0.

Dual graph

• temporal cell-vertex ids as nodes:

  1. tvid=(t, vid)=(t, (cid_i, cid_j, cid_k, cid_l)) an integer t for time and a len-4 tuple of integers vid=(cid_i, cid_j, cid_k, cid_l) for the cell-vertex id, e.g. (0, (5, 6, 7, 8)) for cell-vertex (5, 6, 7, 8) at time 0

  2. tvid=(tcid_i, tcid_j, tcid_k, tcid_l)=((t, cid_i), (t, cid_j), (t, cid_k), (t, cid_l)) four len-2 tuple of integers tcid_i, tcid_j, tcid_k and tcid_l for the four temporal cell ids defining the cell-vertex, e.g. ((0, 5), (0, 6), (0, 7), (0, 8)) for cells 5, 6, 7 and 8 at time 0.

• temporal cell-edge ids as edges:

  1. teid=(t, eid)=(t, (vid_1, vid_2))=(t, ((cid_i1, cid_j1, cid_k1, cid_l1), (cid_i2, cid_j2, cid_k2, cid_l2))) an integer t for time and two len-4 tuple of integers vid_1=(cid_i1, cid_j1, cid_k1, cid_l1) and vid_2=(cid_i2, cid_j2, cid_k2, cid_l2) for the two cell-vertex id, e.g. (0, ((5, 6, 7, 8), (5, 6, 7, 9))) for cell-vertices (5, 6, 7, 8) and (5, 6, 7, 9) at time 0

  2. teid=(tvid_1, tvid_2)=((tcid_i1, tcid_j1, tcid_k1, tcid_l1), (tcid_i2, tcid_j2, tcid_k2, tcid_l2))=(((t, cid_i1), (t, cid_j1), (t, cid_k1), (t, cid_l1)), ((t, cid_i2), (t, cid_j2), (t, cid_k2), (t, cid_l2))) two len-4 tuple of integers tvid_1 and tvid_2 for the two temporal vertex ids defining the cell-edge, e.g. (((0, 5), (0, 6), (0, 7), (0, 8)), ((0, 5), (0, 6), (0, 7), (0, 9))) for the two cell-vertices (5, 6, 7, 8) and (5, 6, 7, 9) at time 0.

Methods

Creation methods

The first step is to populate the data structure with the spatial and temporal data.

Add topological tissue graph

The following methods allow to add topological tissue graph:

ttg = TemporalTissueGraph()
ttg.add_tissue_graph(tissue, elapsed_time, unit='h')

Add cell lineage

The following methods allow to add cell lineages as temporal relationships between the tissue graphs:

ttg.add_lineage(ancestor_tp, lineage)

Exploration methods

We want to be able to “explore” the data structure

Temporal exploration

ttg.child(tp, cell_id)
ttg.parent(tp, cell_id)
ttg.siblings(tp, cell_id)

Spatial exploration

Warning

Should we create methods ttg.neighbors(cid, tp) or access from tissueGraph with ttg._tissue_graph[tp].neighbors(cid) ?

Properties methods

Add temporal properties