A computational analysis of bone formation in the cranial vault in the mouse

Abstract

Bones of the cranial vault are formed by the differentiation of mesenchymal cells into osteoblasts on a surface that surrounds the brain, eventually forming mineralized bone. Signaling pathways causative for cell differentiation include the actions of extracellular proteins driven by information from genes. We assume that the interaction of cells and extracellular molecules, which are associated with cell differentiation, can be modeled using Turing's reaction-diffusion model, a mathematical model for pattern formation controlled by two interacting molecules (activator and inhibitor). In this study, we hypothesize that regions of high concentration of an activator develop into primary centers of ossification, the earliest sites of cranial vault bone. In addition to the Turing model, we use another diffusion equation to model a morphogen (potentially the same as the morphogen associated with formation of ossification centers) associated with bone growth. These mathematical models were solved using the finite volume method. The computational domain and model parameters are determined using a large collection of experimental data showing skull bone formation in mouse at different embryonic days in mice carrying disease causing mutations and their unaffected littermates. The results show that the relative locations of the five ossification centers that form in our model occur at the same position as those identified in experimental data. As bone grows from these ossification centers, sutures form between the bones.

Keywords: computational morphogenesis; developmental biology; finite volume method; skull growth; skull sutures.

Figures

Figure 1

Components associated with generalized mouse cranial vault bone development ranging from molecules to tissue according to time and length scale. Data are generated from observations of C57Bl6 mice.

Figure 2

Formation of the mouse skull. (A–C) Histological images of developing mouse embryos showing formation of bone (magenta stained by alizarin red) and cartilage (stained by alcian blue) of the embryonic skull. (A) lateral view of embryonic mouse head at embryonic day 14.5 (E14.5) showing site of initial ossification of the frontal bone of the cranial vault (red arrow) and skull cartilage; (B) at E15.5, the forming frontal and parietal bones are clearly visible and the ossification of the interparietal bone is beginning (arrow); (C) By E17.5, the interparietal is well formed and other bones of the skull are clearly visible. (D) 3D reconstruction of micro computed tomography image of the mouse skull at birth (P0) colored to indicate the placement and level of maturity of all skull bones. (E) Adult mouse skull colored to show relative position of skull bones.

Figure 3

Schematic diagram of extracellular and cellular process associated with differentiation of mesenchymal cells to osteoblast cells. Undifferentiated mesenchymal cells surrounding the brain express diffusible extracellular molecules, which play a key role in cell differentiation (① and ②). One of the molecules (activator) activates signaling pathways to initiate cell differentiation of mesenchymal cells into osteoblasts (③). In an extracellular process, the activator simultaneously enhances itself (④) and the other key molecule (inhibitor) (⑤), while the inhibitor inhibits the activator (⑥). These two proteins eventually establish a regulatory loop and diffuse in space with different speed (⑦) to form an inhomogeneous spatial pattern of concentration.

Figure 4

Stages of osteoblast lineage cell (OLC) differentiation. Mesenchymal progenitors give rise to osteoblasts and chondrocytes and are usually marked by SOX9. If SOX9+ cells do not differentiate into chondrocytes, they progress along an osteogenic path and as they mature, they are marked progressively by the expression of RUNX2, followed by OSX. Additional evidence suggests that SOX9+ cells can switch fates under certain conditions (indicated by dotted lines) [adapted from Long (2011)].

Figure 5

Lateral view of a 3D domain. The relative distance from inferior surface of the domain (xrel) and reference position (xref) are indicated. Our model limits differentiation to those cells below the reference position.

Figure 6

Schematic diagram of bone growth from primary ossification centers. Osteoblasts in the ossification centers begin to secret a morphogen, which diffuses into the neighboring space and allows adjacent mesenchymal cells to differentiate into osteoblasts. Suture formation between bone elements is modeled by assuming that differentiation of mesenchymal cells adjacent to bone is restricted by the morphogen secreted from the adjacent bone elements.

Figure 7

Procedure for constructing a computational domain. (A) An isosurface of the cranial vault bones from a three-dimensional reconstruction of micro CT images acquired from a mouse at E17.5 was used for making the domain and generating a mesh. Rostral end (nose) is toward the top while the caudal end of the skull is at bottom. (B) Only parts of cranial vault are considered. (C) Surface surrounding the cranial vault is made. (D) Shell-shaped domain around the surface and mesh are generated.

Figure 8

Initial condition. A small perturbation of activator on two points on the frontal side of domain was given as an initial condition. Elsewhere the domain has homogeneous concentration of activator and inhibitor. This condition is based on the biological observation showing that mesenchymal condensations initiate at the supra orbital region just above the globe of the eye and develop into right and left frontal bones.

Figure 9

Concentrations of activator and inhibitor at embryonic day 0 and 15.4 from simulation result. Small perturbation of activator at initial time increases through regulatory reaction between activator and inhibitor, and finally makes a specific pattern of concentration of activator and inhibitor after 15.4 days. Activator and inhibitor are in phase. Six regions of high concentration of the molecules appear, two on the front, two on the side, one on the rear, and one on the top of the domain.

Figure 10

Concentration of osteoblast at embryonic day 0 and 15.4 from simulation result. Osteoblasts are differentiated in the regions where concentration of activator is high. And these regions of osteoblasts can develop into mineralized bone that is primary centers of ossification. Osteoblasts are not differentiated on the top of the domain although concentration of activator is high there, due to the spatial effect on cell differentiation in our model. Five primary centers of ossification and this agree well with experimental observation showing two frontal, two parietal, and a single interparietal bones.

Figure 11

Change of region of high concentration of osteoblast over time. The regions originally marked by the differentiation of osteoblasts expand from the primary centers of ossification over time. Because these osteoblasts differentiate into osteocytes and eventually become trapped within mineralized bone, it can be said to show pattern of bone growth. The results agree well with experimental observation showing two frontal bones, two parietal bones, and one interparietal bone. Sutures form between bones as bones grow according to repulsive effect between bones in our model.