Role of mathematical modeling in bone fracture healing

Abstract

Bone fracture healing is a complex physiological process commonly described by a four-phase model consisting of an inflammatory phase, two repair phases with soft callus formation followed by hard callus formation, and a remodeling phase, or more recently by an anabolic/catabolic model. Data from humans and animal models have demonstrated crucial environmental conditions for optimal fracture healing, including the mechanical environment, blood supply and availability of mesenchymal stem cells. Fracture healing spans multiple length and time scales, making it difficult to know precisely which factors and/or phases to manipulate in order to obtain optimal fracture-repair outcomes. Deformations resulting from physiological loading or fracture fixation at the organ scale are sensed at the cellular scale by cells inside the fracture callus. These deformations together with autocrine and paracrine signals determine cellular differentiation, proliferation and migration. The local repair activities lead to new bone formation and stabilization of the fracture. Although experimental data are available at different spatial and temporal scales, it is not clear how these data can be linked to provide a holistic view of fracture healing. Mathematical modeling is a powerful tool to quantify conceptual models and to establish the missing links between experimental data obtained at different scales. The objective of this review is to introduce mathematical modeling to readers who are not familiar with this methodology and to demonstrate that once validated, such models can be used for hypothesis testing and to assist in clinical treatment as will be shown for the example of atrophic nonunions.

Figure 1

Conceptual models of bone fracture healing: (a) four-phase model based on histology providing a static picture and dynamic overlap of phases (yellow=granulation tissue; dark gray, fibrous tissue, light gray, cartilaginous tissue, dashed, bone); (b) extended four-phase model according to Claes et al. superposing IFM, blood flow and tissue volume fractions; (c) anabolic/catabolic model according to Little and co-workers.

Figure 2

Mathematical models of bone fracture healing: (a) Organ-scale model of human tibia including external fixator device using finite element analysis (FEA) modified from Bryne et al.; (b) tissue-scale model of the biomechanical stimulus based on fluid flow and tissue shear strain according to Prendergast et al. to direct cell response; (c) cell-scale model including biochemical and biomechanical regulation of cell migration, proliferation and differentiation; (d) multiscale model (coupling organ, tissue and cell scale) including biochemical and biomechanical regulation (adapted from Geris et al.6).

Figure 3

Mathematical model of the regeneration process in healing and nonunion groups: (a) different domains of the simulations (1, periosteal callus, 2, intercortical gap, 3, endosteal callus, 4, cortical bone); (b) boundary conditions for healing and nonunion model (FB, fibroblast, EC, endothelial cell, CGGF, chondrogenic growth factor, OGGF, osteogenic growth factor); (c) comparison of experimentally measured (Exp) and numerically calculated (Sim) tissue constituents present within the interfragmentary gap of healing and nonunion groups; (d) temporal evolution of the numerically calculated tissue fraction for the healing group (spatially averaged over the interfragmentary region—see insert); (e) temporal evolution of the numerically calculated tissue fraction for the healing group with 10-fold reduced cartilage and bone production rates leading to better correspondence between experimental and simulation results (adapted from Geris et al.57).

Figure 4

Effect of MSC transplantation on atrophic nonunion: (a_I) simulation results for the treatment with the cell transplant injected at the center of the callus; (a_II) comparison of experimentally measured (Exp) and numerically calculated (Sim) tissue constituents present within the interfragmentary gap of control (carrier solution injected) and treatment (MSC transplant) groups (*P<0.005, Student's t-test); (b_I) Simulation results for the treatment with cell transplant injected externally in the callus; (b_II) comparison of experimentally measured (Exp) and numerically calculated (Sim) tissue constituents present within the interfragmentary gap of control (carrier solution injected) and treatment (MSC transplant) groups (*P<0.005, Student's t-test); (c) simulation results for the treatment with the cell transplant injected outside the callus (adapted from Geris et al.57).