A mathematical model for fibrous dysplasia: The role of the flow of mutant cells

TL;DR

An analytical study of the model is developed and the existence and stability of steady states are studied, an analysis of sensitivity on the model parameters is done, and different numerical simulations provide findings in agreement with the analytical results.

Abstract

Fibrous dysplasia (FD) is a mosaic non-inheritable genetic disorder of the skeleton in which normal bone is replaced by structurally unsound fibro-osseous tissue. There is no curative treatment for FD, partly because its pathophysiology is not yet fully known. We present a simple mathematical model of the disease incorporating its basic known biology, to gain insight on the dynamics of the involved bone-cell populations, and shed light on its pathophysiology. Our mathematical models account for the dynamic evolution over time of several interacting populations of bone cells averaged over a volume of bone of sufficient size in order to obtain consistent results. We develop an analytical study of the model and study its basic properties. The existence and stability of steady states are studied, an analysis of sensitivity on the model parameters is done, and different numerical simulations provide findings in agreement with the analytical results. We discuss the model dynamics match with known facts on the disease, and how some open questions could be addressed using the model.

Figures (8)

• Figure 1: Bone-cell populations relevant in bone remodeling, for both healthy bone and FD tissue. In (a), bone-cell populations involved in bone remodeling are shown. Bone remodeling consists of three coupled processes: resorption, deposition, and mineralization. The first of these processes is carried out by osteoclasts, whereas the last two are performed by cells of the osteoblastic lineage. Osteoclasts are large multinucleated bone-resorbing cells formed by the fusion of monocyte/macrophage-derived precursors that reside in the bone marrow. Osteoclastogenesis is driven by signals coming from osteocytes and osteoprogenitors. In healthy bone, fusion can only occur at the bone surface to be resorbed. Mature activated osteoclasts have a finite lifetime and after resorption, osteoclasts either undergo apoptosis (a rare event with a high-energy cost due to the removal of the apoptotic debris) or disassembly into mononucleated cells unable to resorb bone, osteomorphs, that remain in the adjacent bone marrow, Bolamperti2022McDonald2021. The osteoblastic lineage derives from the differentiation of skeletal stem cells (SSCs) in the bone marrow. SSCs are self-renewing multipotent progenitors that can give rise to osteoblasts (bone-depositing cells), chondrocytes, hematopoietic-supportive stromal cells, and marrow adipocytes, Bianco2015. When the remodeling process finishes, osteoblasts destiny is any of the following three: undergoing apoptosis (60-80%), becoming lining cells (cells that cover quiescent bone surfaces that are not undergoing remodeling, which can also be osteblasts precursors), or being trapped in the bone matrix (mainly consisting of collagen), mineralizing it and differentiating into osteocytes, Manolagas2010Bonewald2011Zhao2000. Osteocytes are terminally differentiated cells that represent more than 90% of bone cells in the adult skeleton, and they are considered to be the orchestrators of the bone remodeling, Bolamperti2022Bonewald2011. In (b), bone-cell populations appearing on FD tissue are added. Mutant SSCs differentiate into mutant osteoprogenitors which overproduce cAMP. Surrounding WT osteoprogenitors adopt a cAMP related reversible cell phenotype, being called WT phenocopying osteoprogenitors. Both mutant and WT phenocopying osteoprogenitors overproliferate and differentiate abnormally, giving rise to altered osteoblasts and osteocytes, and producing aberrant abnormal woven bone and fibrous matrix Riminucci1997Marie1997Xiao2019Zhao2018Hartley2019. They also release factors to induce osteoclastogenesis and bone resorption, causing the formation of ectopic, numerous, large (and so, more active) osteoclasts, Riminucci2003Whitlock2023.
• Figure 2: Schematic representation of the bone-cell populations and interactions between them considered in the FD mathematical model given by Eqs. \ref{['FD_simple_model']}. Bone remodeling consists of three coupled processes: resorption, deposition, and mineralization. For each of those processes, we have chosen a bone-cell population that accomplishes the process: osteoclasts ($C_L$), WT osteoprogenitors ($P$), and mature osteocytes ($C_I$). For the sake of simplification, both osteoprogenitors and osteoblasts (as well as linning and reversal cells) are clustered into a unique population, which by abuse of notation will also be referred to as osteoprogenitors. Since we are considering FD tissue, we have added two more populations, mutant and WT phenocopying osteoprogenitors ($P_m$ and $P_p$, respectively). They give rise to fibrous tissue rich in fibroblast-like cells that express markers of early stages of osteogenic maturation. For this reason, in our FD model, the mutant osteoprogenitor population also comprehends mutant osteoprogenitors progeny, and the same applies for the WT phenocopying osteoprogenitor population.
• Figure 3: Drawings of the quadratic function $y(P) =-\rho P^2+b P + c$ for (a) $b<0$, (b) $b\geq 0$ and $c>0$, and (c) $b>0$ and $c\leq 0$.
• Figure 4: (a) Time dynamics of osteoprogenitor cells (blue), osteocytes (purple), and osteoclasts (green) compartments according to Eqs. \ref{['HB_simple_model']} for parameters $\rho = 0.15$ day$^{-1}$, $\alpha = 0.2$ day$^{-1}$, $\mu = 0.99$, $\tau_s = 0.2$ day$^{-1}$,$\tau_c = 1/350000$ day$^{-1}$, $\gamma_c = 1/35$ day$^{-1}$, and $\delta=90$ day$^{-1}$. (b) Dynamics of osteoprogenitor cells (blue), mutant osteoprogenitors (red), WT phenocopying osteoprogenitors (yellow), osteocytes (violet), and osteoclasts (green) compartments according to Eqs. \ref{['FD_simple_model']} for parameters $\rho = 0.15$ day$^{-1}$, $\rho_m = 0.3$ day$^{-1}$, $\alpha = 0.2$ day$^{-1}$, $\mu = 0.99$, $\tau_s = 0.2$ day$^{-1}$,$\tau_c = 1/350000$ day$^{-1}$, $\gamma_c = 1/35$ day$^{-1}$, $\delta=100$ day$^{-1}$, $\delta_m=1$ day$^{-1}$, $\tau_m=0.2$ day$^{-1}$.
• Figure 5: Dependence of the equilibrium point on the model parameter values, fixing all parameters except for one. The fixed parameters take the values $\rho = 0.15$ day$^{-1}$, $\rho_m = 0.3$ day$^{-1}$, $\alpha = 0.2$ day$^{-1}$, $\mu = 0.99$, $\tau_s = 0.2$ day$^{-1}$,$\tau_c = 1/350000$ day$^{-1}$, $\gamma_c = 1/35$ day$^{-1}$, $\delta=100$ day$^{-1}$, $\delta_m=1$ day$^{-1}$, and $\tau_m=0.2$ day$^{-1}$. Whereas the value of the varying parameter is specified in each case. (a) Varying parameter $\tau_m \in [0.05, 0.22]$, (b) Varying parameter $\tau_c \in [2 \cdot 10^{-6}, 8 \cdot 10^{-6}]$, (c) Varying parameter $\mu \in [0.9, 99]$, (d) Parameter values $\delta \in [90, 129]$, (e) Varying parameter $\rho_m \in [0,27, 0,495]$, (f) Varying parameter $\delta_m \in [1, 3]$.

Theorems & Definitions (12)

• Proposition 1
• proof
• Proposition 2
• proof
• Remark 1
• Proposition 3
• proof
• Proposition 4
• proof
• Remark 2