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. 2014 Sep 30;111(39):14193-8.
doi: 10.1073/pnas.1413970111. Epub 2014 Sep 15.

Mathematical model of renal interstitial fibrosis

Wenrui Hao  1 , Brad H Rovin  2 , Avner Friedman  3
Affiliations

Mathematical model of renal interstitial fibrosis

Wenrui Hao et al. Proc Natl Acad Sci U S A. .

Abstract

Lupus nephritis (LN) is an autoimmune disease that occurs when autoantibodies complex with self-antigen and form immune complexes that accumulate in the glomeruli. These immune complexes initiate an inflammatory response resulting in glomerular injury. LN often concomitantly affects the tubulointerstitial compartment of the kidney, leading first to interstitial inflammation and subsequently to interstitial fibrosis and atrophy of the renal tubules if not appropriately treated. Presently the only way to assess interstitial inflammation and fibrosis is through kidney biopsy, which is invasive and cannot be repeated frequently. Hence, monitoring of disease progression and response to therapy is suboptimal. In this paper we describe a mathematical model of the progress from tubulointerstitial inflammation to fibrosis. We demonstrate how the model can be used to monitor treatments for interstitial fibrosis in LN with drugs currently being developed or used for nonrenal fibrosis.

Keywords: math modeling; renal fibrosis; tubulointerstitial inflammation.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Network of the renal interstitial fibrosis. Arrows indicate activation or induction; hammerheads indicate inhibition or killing.
Fig. 2.
Fig. 2.
Domain Ω with a damaged area D.
Fig. 3.
Fig. 3.
(A) Blood vessels depicted as circles in ω periodic structure and (B) 1/ω magnification of one period. R is a unit square, and B is a circle of area 0.05 with its center at the square’s center.
Fig. 4.
Fig. 4.
Simulation of all variables over a period of 60 d with λPE = 5 × 10−9 and D = 0.2 × 0.2.
Fig. 5.
Fig. 5.
Comparison of patient data to model simulations for different levels of interstitial fibrosis based on kidney biopsy. (A) Low fibrosis; the size of damaged area is 0.4 × 0.4, λPE = 1 × 10−8 d−1. (B) Intermediate fibrosis; the size of damaged area is 0.5 × 0.5, λPE = 4 × 10−8 d−1. (C) High fibrosis; the size of damaged area is 0.6 × 0.6, λPE = 8 × 10−8 d−1. Most of the data points fall near the simulated curves, but there are several outliers expected in real patients.
Fig. 6.
Fig. 6.
Treatment with a TGF-β inhibitor (K1 = 20 in Eq. 15).
Fig. 7.
Fig. 7.
Treatment with anti-C5 to decrease MCP-1 concentration (K2 = 50 in Eq. 16).
Fig. 8.
Fig. 8.
Treatment with MCP-1 blockade by a CCR2 antagonist to decrease macrophage density (K3 = 10 in Eq. 17).
Fig. 9.
Fig. 9.
Anti-PDGF treatment to decrease PDGF concentration (K4 = 1,000 in Eq. 18).
Fig. 10.
Fig. 10.
MMP injection (K5 = 5 in Eq. 19).
See this image and copyright information in PMC

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