Cell-based, mathematical models helpmake sense of morphogenesis—i.e. cells organizing intoshape and pattern—by capturing cell behavior in simple, purelydescriptive models. Cell-based models then predict thetissue-level patterns the cells produce collectively. The firststep in a cell-based modeling approach is to isolatesub-processes, e.g. the patterning capabilities of one or afew cell types in cell cultures. Cell-based models can thenidentify the mechanisms responsible for patterning in vitro.This review discusses two cell culture models of morphogenesisthat have been studied using this combinedexperimental-mathematical approach: chondrogenesis (cartilagepatterning) and vasculogenesis (de novo blood vessel growth). Inboth these systems, radically different models can equallyplausibly explain the in vitro patterns. Quantitativedescriptions of cell behavior would help choose betweenalternative models. We will briefly review the experimentalmethodology (microfluidics technology and traction forcemicroscopy) used to measure responses of individual cells to theirmicro-environment, including chemical gradients, physical forcesand neighboring cells. We conclude by discussing how to includequantitative cell descriptions into a cell-based model: theCellular Potts model.

Vasculogenesis and angiogenesis are two different mechanisms for bloodvessel formation. Angiogenesis occurs when new vessels sprout from pre-existing vasculature in response to external chemical stimuli. Vasculogenesis occurs via the reorganization of randomly distributed cells into a blood vessel network. Experimental modelsof vasculogenesis have suggested that the cells exert traction forcesonto the extracellular matrix and that these forces may playan important role in the network forming process.In order to study the role of the mechanical and chemical forcesin both of these stages of blood vessel formation, we present amathematical model which assumes that (i) cells exert traction forcesonto the extracellular matrix, (ii) the matrix behaves as a linearviscoelastic material, (iii) the cells move along gradients ofexogenously supplied chemical stimuli (chemotaxis) and (iv) these stimuli diffuse or are uptaken by the cells.We study the equations numerically, present an appropriate finite difference scheme and simulate the formation of vascular networks in a plane. Our results compare very well with experimental observations and suggest that spontaneous formation of networks can be explained via a purely mechanical interaction betweencells and the extracellular matrix. We find that chemotaxis alone is not a sufficient force to stimulate formation of pattern. Moreover, duringvessel sprouting, we find that mechanical forces can help in the formationof well defined vascular structures.

The effects of lithium on vascular development were examined using the chick embryo area vasculosa in shell-less culture as an experimental model. Embryos were explanted after 48 h in ovo and LiCl (50, 100, 150 and 200 μg in 10 μl water) was applied to the centre of the blastodisc. Controls were untreated or given equimolar amounts of NaCl. At 24 h and 48 h after treatment, untreated and NaCl controls were identical, having well developed extraembryonic vessels. At doses of 100 μg and greater, LiCl significantly inhibited normal vascular development and expansion of the area vasculosa in the majority of explants. In many specimens blood islands continued to form but their assembly into primitive vessels was prevented, indicating that lithium affects the mechanism regulating the assembly of vascular endothelium. At the same time the embryos were alive but retarded in development compared with controls. When LiCl (150 μg) was applied to cultures explanted after 72 h in ovo (when the primary vascular network had already formed through vasculogenesis) no adverse effects were seen. This suggests that lithium affects vasculogenesis but not angiogenesis. Treatment with myo-inositol completely reversed the effects of lithium in a time dependent manner indicating that the phosphatidylinositol second messenger cycle may be involved in the cellular events of vasculogenesis. Finally the results of this study show that the yolk sac vasculature is particularly vulnerable to lithium and the consequent effects of this interference on embryonic development are discussed.