Allosteric control model of bone remodelling containing periodical modes

Abstract

To help to understand the modelling process that occurs when a scaffold is implanted it is vital to understand the rather complex bone remodelling process prevalent in native bone.We have formulated a mathematical model that predicts osteoactivity both in scaffolds, as well as in bone in vivo and could set a basis for the more detailed allosteric models. The model is extended towards a bio-cybernetic vision of basic multicellular unit (BMU) action, when some of the regulation loops have been modified to reflect the allosteric control mechanisms, developed by Michaels-Menten, Hill, Koshland-Nemethy-Filmer, Monod-Wyman-Changeux.By implementation of this approach a four-dimensional system was obtained that shows steady cyclic behaviour using a wide range of constants with clear biological meaning. We have observed that a local steady-state appears as a limiting cycle in multidimensional phase space and this is discussed in this paper. Physiological interpretation of this limiting four-dimension cycle possibly related to a conservative-like value has been proposed. Analysis and simulation of the model has shown an analogy between this conservative value, as a kind of substrateenergy regenerative potential of the bone remodelling system with a molecular nature, and to the classical physical value -energy. This dynamic recovery potential is directed against both mechanical and biomechanical damage to the bone. Furthermore, the current model has credibility when compared to the normal bone remodelling process. In the framework of widely recognised Hill mechanisms of allosteric regulation the cyclic attractor, described formerly for a pure cellular model, prevails for different forms of feedback control. This result indicates the viability of the proposed existence of a conservative value (analogous to energy) that characterises the recovery potential of the bone remodelling cycle. Linear stability analysis has been performed in order to determine the robustness of the basic state, however, additional work is required to study a wider range of constants.

A review of mathematical modeling of bone remodeling from a systems biology perspective

“…These distinct approaches form the basis of many temporal and spatiotemporal models of bone remodeling (Figure 5, which includes all publications from Tables 1-3 and Supplementary Tables S1-S2 but not all labels are shown because of space constraints). Models that do not explicitly follow either approach (Kroll, 2000;Rattanakul et al, 2003;Martin and Buckland-Wright, 2004;Moroz et al, 2006;Moroz and Wimpenny, 2007;Akchurin et al, 2008;Ji et al, 2012;Proctor and Gartland, 2016;Chaiya and Rattanakul, 2017;Javed et al, 2018;Zhao and Zhang, 2019;Javed et al, 2020;Nutini et al, 2021;Jorg et al, 2022) (Table 4) are not included in Figure 5; however, most of those detailed in Supplementary Table S3 show citation connections for the field's literature.…”

“…The power law models cannot be easily extended for situations like diseases or treatments when these signals are perturbed outside the conditions used to fit the power law parameters. The lack of direct mechanistic interpretation is a common criticism of the power law approach (Moroz and Wimpenny, 2007).…”

“…For example, Moroz et al (2006) is the earliest model with osteocytes, and the model includes mechanical stress. Moroz and Wimpenny (2007) introduces osteocyte regulation and, similar to Pivonka et al (2008), defines autocrine and paracrine regulation mechanisms with more biologically accurate formulas, exploring four different receptor-ligand binding equations (Michaelis-Menten, Hill, Koshland-Nemethy-Filmer, and Monod-Wyman- Quantitative comparison of cells (top row) and chemical signals (bottom row) commonly included in the 88 mathematical models of bone remodeling detailed in Tables 1-4. Abbreviations: pOCL, preosteoclasts; OCL, osteoclasts; pOBL, preosteoblasts; OBL, osteoblasts; OCY, osteocytes; Immune, immune cells; A&P, general autocrine and paracrine signaling; RANK, receptor activator of nuclear factor kappa-B; RANKL, receptor activator of nuclear factor kappa-B ligand; OPG, osteoprotegerin; TGF-β, transforming growth factor beta; SCL, sclerostin; Wnt, wingless-related integration site; PTH, parathyroid hormone; Cyt, cytokines other than RANK, RANKL, OPG, and TGF-β; E, estrogen.…”

Fractional Order Nonlinear Bone Remodeling Dynamics Using the Supervised Neural Network

“…Many studies of bone metabolism based on the signal transduction models have been presented using the OC and osteoblasts (OB) [7]. The behavior of OC and OB in a cellular environment is recognized as a basic multicellular unit [8][9][10][11][12].…”

Multiscale investigation of the functional properties of the human femur

“…If this model is coupled with the higher scale (the aforementioned tissue-level model), and with the lower scale (with models of cellular networks that predict how osteoclastic and osteoblastic activities are modulated by metabolic, pharmacological or environmental determinants), it could be possible to predict how alteration in such determinants affects the strength of the skeleton. In a recent interesting example of these cell-level models, it is suggested that Hill mechanisms of allosteric regulation could also be used in this context (Moroz & Wimpenny 2007).…”