MOSAIC: a multiscale model of osteogenesis and sprouting angiogenesis with lateral inhibition of endothelial cells

Abstract

The healing of a fracture depends largely on the development of a new blood vessel network (angiogenesis) in the callus. During angiogenesis tip cells lead the developing sprout in response to extracellular signals, amongst which vascular endothelial growth factor (VEGF) is critical. In order to ensure a correct development of the vasculature, the balance between stalk and tip cell phenotypes must be tightly controlled, which is primarily achieved by the Dll4-Notch1 signaling pathway. This study presents a novel multiscale model of osteogenesis and sprouting angiogenesis, incorporating lateral inhibition of endothelial cells (further denoted MOSAIC model) through Dll4-Notch1 signaling, and applies it to fracture healing. The MOSAIC model correctly predicted the bone regeneration process and recapitulated many experimentally observed aspects of tip cell selection: the salt and pepper pattern seen for cell fates, an increased tip cell density due to the loss of Dll4 and an excessive number of tip cells in high VEGF environments. When VEGF concentration was even further increased, the MOSAIC model predicted the absence of a vascular network and fracture healing, thereby leading to a non-union, which is a direct consequence of the mutual inhibition of neighboring cells through Dll4-Notch1 signaling. This result was not retrieved for a more phenomenological model that only considers extracellular signals for tip cell migration, which illustrates the importance of implementing the actual signaling pathway rather than phenomenological rules. Finally, the MOSAIC model demonstrated the importance of a proper criterion for tip cell selection and the need for experimental data to further explore this. In conclusion, this study demonstrates that the MOSAIC model creates enhanced capabilities for investigating the influence of molecular mechanisms on angiogenesis and its relation to bone formation in a more mechanistic way and across different time and spatial scales.

Figure 1. Schematic overview of the multiscale model and a flowchart of its numerical implementation.

(A) Scale separation map indicating schematically the modeled processes at different spatial and temporal scales. Intracellular variables are further explained in Table 1 and govern endothelial cell (EC) behavior. Cell types that are considered at the tissue scale (MSCs: mesenchymal stem cells, CHs: chondrocytes, OBs: osteoblasts, FBs: fibroblasts) can migrate (only MSCs and FBs), proliferate (circular arrows), differentiate (vertical arrows) and produce growth factors (gb: osteogenic growth factor concentration, gc: chondrogenic growth factor concentration, gv: angiogenic growth factor concentration) and extracellular matrix (mf: fibrous tissue density, mb: bone density, mc: cartilage density, m: total tissue density). Blood vessels serve as an oxygen source (n: oxygen concentration). Variables next to an arrow indicate their mediating role for a certain tissue level process. (B) Flowchart of the numerical implementation of the MOSAIC model (: vector of the continuous variables, t: time, δt: time step of the inner loop, Δt: time step of the outer loop, c_v: endothelial cells).

Figure 2. Conceptual representation of the Dll4 distribution across the cell membranes of endothelial cells.

The dark grey ECs represent the original branch whereas Ti and Si are part of a newly formed sprout (T = tip cell, S = stalk cell). The amount of Dll4 was arbitrarily chosen, a maximal amount of 25 000 ligands per EC (for an EC with a length of 25 µm) was calculated by Bentley et al. , estimated from Liu et al. .

Figure 3. Geometrical domain and boundary conditions.

(left) Geometrical domain deduced from the real callus geometry at postfracture week 3 ; 1 periosteal callus; 2 intercortical callus; 3 endosteal callus; 4 cortical bone. (right) Boundary conditions. The mesenchymal stem cells (c_m) and fibroblasts (c_f) are released from the periosteum, the surrounding soft tissues and the bone marrow . The chondrogenic growth factors (g_c) are released from the degrading bone ends whereas the cortex is modeled to be the source of osteogenic growth factors (g_b) . c_v indicates the initial position of the ECs.

Figure 4. In silico and in vivo evolution of normal fracture healing.

Temporal evolution of the bone, cartilage and fibrous tissue fractions (%) in the periosteal, intercortical and endosteal callus as predicted using the hybrid model of Peiffer et al. and the newly developed multiscale model and as measured by Harrison et al. .

Figure 5. Image of the amount of VEGFR-2 per EC for different Dll4 expression and Notch activity levels at post fracture day 19.

The figure is focused on the periosteal callus (area 1 in Figure 3). Remark that the tip cells have a lot of VEGFR-2 (dark brown) whereas the following stalk cells are inhibited, giving rise to a low number of VEGFR-2 (dark blue). (A) Influence of Dll4 levels. (B) Influence of Notch1 activity levels. Left: heterozygous knockout, middle: standard condition, right: overexpression.

Figure 6. Temporal evolution of the average amount of VEGFR-2 on ECs in the callus for different levels of VEGF addition (corresponding to Figure 7B ).

Evolution of the average VEGFR-2 concentration (day 0 corresponds to the time of fracture). The threshold line represents the first requirement that needs to be fulfilled to obtain the tip cell phenotype, i.e. V>V_max/2. Remark that in very high VEGF concentrations (+10%) the average receptor concentration is far below the threshold.

Figure 7. Vasculature at 35 days post fracture for different conditions.

(A) Variation in the amount of the decoy receptor VEGFR-1 (i: V_sink = 100% (standard condition), ii: V_sink = 45%, iii: V_sink = 9%, iv: V_sink = 0.9%); (B) Variation in the amount of VEGF addition in the multiscale model with the standard tip cell selection criterion (based on V; Equation 6) (i: 0% (standard condition), ii: 0.1%, iii: 2%, iv: 10%); (C) Variation in the amount of VEGF addition in the multiscale model with an altered tip cell selection criterion (based on V′) (i: 0%, ii: 0.1%, iii: 2%, iv: 10%); (D) Variation in the amount of VEGF addition in the hybrid model (i: 0%, ii: 0.1%, iii: 2%, iv: 10%).

Figure 8. In silico evolution of fracture healing for various conditions.

Temporal evolution of the bone, cartilage and fibrous tissue fractions (%) in the periosteal, intercortical and endosteal callus as predicted by the MOSAIC model. (full: normal fracture healing; dashed: Dll4 overexpression (δ = 12.64, Figure 5A); dotted: VEGF-addition (+ 0.1%, Figure 7Bii).

Figure 9. Amount of VEGFR-2 receptors on every endothelial cell.

(A) standard condition; (B) addition of 2% VEGF.

Figure 10. Schematic representation of the Dll4-Notch pathway in high VEGF environments for two neighboring ECs.

Remark that although the VEGFR-2 receptor is down-regulated, the VEGF concentration is high enough to compensate for this effect resulting in high V′ levels (Equation 1).