Mariia Soloviova, Juan Carlos Beltran Vargas, Luis Fernandez de Castro, Juan Belmonte-Beitia, Victor M. Perez-Garcia, Magdalena Caballero AbstractFibrous 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. 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.
4.3.2 Mathematical model for dysplastic bone shows that even if the departure percentage of mutant cells is small, mutant and WT phenocopying osteoprogenitors become the main populations
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For the initial conditions of the FD model, we haven chosen the values given by the equilibrium point of the healthy bone model, and we have added Pm(0) = 104 and Pp(0) = 0. Let us remark that no estimation can be found in the literature of the initial mutational load of a bone region affected by FD. It is thought to depend on the moment at which the initial mutation took place, which is different in each patient, and impossible to be determined. The mutation that gives rise to FD happens in a single cell during embryonic development, , either after or before gastrulation. Even if it is not known if the mutation represents an evolutionary advantage or disadvantage for the mutant initial cell in comparison with the WT ones, , the percentage of mutant cells before lesions become visible is expected to be very small.
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Figure 4(b), shows an example of the typical dynamics of the disease burden in a dysplastic bone ruled by Eqs. (1), where mutant osteoprogenitors are present. We can observe that mutant and WT phenocopying osteoprogenitors expanded, while osteocytes experience a continuous decrease, as it was pointed out in [42, 44]. Their numbers stabilized after approximately 600 days. Even for very small values of Pm(0), in the equilibrium, most cells are immature progenitors and either bear the mutation or are WT phenocopying cells, in agreement with experimental observations . 4.3.3 Parameters influencing the severity of the disease
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The parameters that characterize the disease include: m, indicating the efficiency of mutant osteoprogenitors in converting WT osteoprogenitors into phenocopying WT ones; m, representing the actual proliferation rate of mutant osteoprogenitors; m, denoting the flow of cells from mutant SSCs; , reflecting the efficiency of bone dissolution by osteoclasts, known to be higher in dysplastic bone compared to healthy bone; and c, signifying the formation and activation of osteoclasts due to signaling from osteoprogenitors and osteocytes, also observed to be higher in dysplastic bone than in healthy bone. 17
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