Solution of a mathematical model for the treatment of rheumatoid arthritis

Rheumatoid arthritis is an autoimmune disease of unknown etiology that manifests as a persistent inflammatory synovitis and eventually destroys the joints. The immune system recognizes synovial cells as not self and consequently causes lymphocyte and antibody proliferation that is promoted by the pro-inflammatory cytokines, the most significant being tumor necrosis factor TNF-α. In the treatment of rheumatoid arthritis either monoclonal antibodies or soluble receptors are used to neutralize the TNF-α bioactivity, such as sTNFR2, Etanercept and Infliximab. In [M. Jit et al. Rheumatology 2005;44:323-331] a mathematical model that represents the TNF-α dynamics in the inflamed synovial joint within which locally produced TNF-α can bind to cell-surface receptors was proposed. It consists of four coupled ordinary differential equations, that were integrated numerically assuming a range of estimates of the key parameters. In this paper we complement the previous work by determining the general solution of those equations for specific conditions on the parameters. Then we characterize the behavior of TNF-α in the presence of different inhibitors and also evaluate the inhibitors effectiveness in the treatment of rheumatoid arthritis.