Mathematical Modeling of Systemic Sclerosis and Its Treatment

Abstract Systemic sclerosis (SSc) is an autoimmune disease marked by excessive extracellular matrix (ECM) deposited by myofibroblasts. The disease carries the risk of pathologically progressing to internal organs, particularly the lung. Since abnormally large densities of myofibroblasts are associated with SSc, clinical studies consider trials that reduce the density of myofibroblasts: imatinib treatment ( $$N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> ), which promotes apoptosis in myofibroblasts, and SSc by fresolimumab ( $$S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> ), which inhibits TGF- $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> , a key growth factor of myofibroblasts. In this paper, we develop a mathematical model of SSc by a system of partial differential equations, and use it to explore a range of treatment strategies with $$N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> and $$S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> . For example, we considered administering $$S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> in fractions three weeks apart. In this case, we determined the smallest amount of $$S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> such that $$\rho (t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , the density of the ECM at time $$t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> </mml:math> , will continuously and oscillatingly decrease from a disease level $$2\rho ^0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:msup> <mml:mi>ρ</mml:mi> <mml:mn>0</mml:mn> </mml:msup> </mml:mrow> </mml:math> , where $$\rho ^0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ρ</mml:mi> <mml:mn>0</mml:mn> </mml:msup> </mml:math> is the density $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> in health. Since SSc has no cure, $$\rho (t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> cannot decrease below $$\rho ^0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ρ</mml:mi> <mml:mn>0</mml:mn> </mml:msup> </mml:math> . We found that with the smallest amount of $$S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> , $$\rho (t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> decreases over a few months to $$1.19\rho ^0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1.19</mml:mn> <mml:msup> <mml:mi>ρ</mml:mi> <mml:mn>0</mml:mn> </mml:msup> </mml:mrow> </mml:math> and remains nearly stable thereafter. The results of the paper could be useful in the design of future clinical trials aimed to decrease the excessive extracellular matrix in SSc patients.