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Mathematical modeling of liver fibrosis

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  • Fibrosis is the formation of excessive fibrous connective tissue in an organ or tissue, which occurs in reparative process or in response to inflammation. Fibrotic diseases are characterized by abnormal excessive deposition of fibrous proteins, such as collagen, and the disease is most commonly progressive, leading to organ disfunction and failure. Although fibrotic diseases evolve in a similar way in all organs, differences may occur as a result of structure and function of the specific organ. In liver fibrosis, the gold standard for diagnosis and monitoring the progression of the disease is biopsy, which is invasive and cannot be repeated frequently. For this reason there is currently a great interest in identifying non-invasive biomarkers for liver fibrosis. In this paper, we develop for the first time a mathematical model of liver fibrosis by a system of partial differential equations. We use the model to explore the efficacy of potential and currently used drugs aimed at blocking the progression of liver fibrosis. We also use the model to develop a diagnostic tool based on a combination of two biomarkers.

    Mathematics Subject Classification: 62P10, 35Q92, 92C42, 92C50.


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  • Figure 1.  Functions of the liver.

    Figure 2.  Network of the fibrosis.

    Figure 3.  The average concentrations of cells, cytokines and ECM

    Figure 4.  Treatment studies

    Figure 5.  Biomarkers HA and TIMP with respect to scar density; the color column scales the scar density in $g/cm^3$.

    Figure 6.  The sensitivity analysis for the cytokine production rates. The figure shows the partial rank correlation (PRCC) between the cytokine production rate and the scar concentration at day 200.

    Table 1.  The variables of the model; concentration and densities are in units of $g/cm^3$

    $M_1$:  density of M1 macrophages $M_2$:  density of M2 macrophages
    $T_1$:  Th1 cell density $T_2$:  Th2 cell density
    $E_0$:  density of tissue epithelial cells (TECs) $E$:   density of activated TECs
    $H$:  density of HSCs $f$:  density of fibroblasts
    $m$:  density of myofibroblasts $\rho$:  density of ECM
    $G$:  concentration of PDGF $T_\beta$:  concentration of activated TGF-$\beta$
    $Q$:  concentration of MMP $Q_r$:  concentration of TIMP
    $T_\alpha$:  concentration of TNF-$\alpha$ $I_{\gamma}$:  concentration of IFN-$\gamma$
    $I_{2}$:  IL-2 concentration $I_{4}$:  IL-4 concentration
    $I_{10}$:  IL-10 concentration $I_{13}$:  IL-13 concentration
    $P$:  concentration of MCP-1 $H_A$:  Hyaluronic acid concentration
    $S$  scar density
     | Show Table
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    Table 2.  Parameters' description and value

    Parameter Description Value
    $D_M$ dispersion coefficient of macrophages $8.64\times10^{-7}$ $cm^2$ day$^{-1}$[32]
    $D_T$ diffusion coefficient of T cell $8.64\times10^{-7}$ $cm^2$ day$^{-1}$[33]
    $D_{I_\gamma}$ diffusion coefficient of IFN-$\gamma$ $1.08\times10^{-2} $ $cm^2$ day$^{-1}$[29]
    $D_{I_{2}}$ diffusion coefficient of IL-2 $1.08\times10^{-2} $ $cm^2$ day$^{-1}$[29]
    $D_{I_{4}}$ diffusion coefficient of IL-4 $1.08\times10^{-2} $ $cm^2$ day$^{-1}$[29]
    $D_{I_{12}}$ diffusion coefficient of IL-12 $1.08\times10^{-2} $ $cm^2$ day$^{-1}$[29]
    $D_{I_{13}}$ diffusion coefficient of IL-13 $1.08\times10^{-2} $ $cm^2$ day$^{-1}$[29]
    $D_{P}$ diffusion coefficient of MCP-1 $1.728\times10^{-1}$ $cm^2$ day$^{-1}$[32]
    $D_{G}$ diffusion coefficient of PDGF $8.64\times10^{-2} $ $cm^2$ day$^{-1}$[32]
    $D_{Q}$ diffusion coefficient of MMP $4.32\times10^{-2} $ $cm^2$ day$^{-1}$[32]
    $D_{Q_r}$ diffusion coefficient for TIMP $4.32\times 10^{-2}$ $cm^2$ day$^{-1}$ [32]
    $D_{T_\beta}$ diffusion coefficient for TGF-$\beta$ $4.32\times 10^{-2}$ $cm^2$ day$^{-1}$ [32]
    $D_{T_\alpha}$ diffusion coefficient for TNF-$\alpha$ $1.29\times10^{-2} $ $cm^2$ day$^{-1}$[31]
    $D_{f}$ dispersion coefficient of fibroblasts $1.47\times10^{-6} $ $cm^2$ day$^{-1}$ [32]
    $D_M$ dispersion coefficient of myofibroblasts $1.47\times10^{-5} $ $cm^2$ day$^{-1}$ [32]
    $\lambda_{M_2}$ Differentiation rate of M1 to M2 $0.3$ day$^{-1}$ [33]
    $\lambda_{M_1}$ Maximal rate at which M2 is activated to become M1 $0.6$/day [33]
    $\lambda_{M T}$ transition rate of M2 to M1 macrophages by TNF-$\alpha$ $5\times10^{-3}$ day$^{-1}$ [15]
    $\lambda_{MI_r}$ Production rate by IFN-$\gamma$ $1$/day [33]
    $\lambda_{MI_4}$ Production rate by IL-4 $1$/day [33]
    $\lambda_{MI_{13}}$ Production rate by IL-13 $1$/day [33]
    $\lambda_{T1M1}$ Production rate of Th1 cells by M1 macrophages $10$/day [33]
    $\lambda_{T1I2}$ Production rate of Th1 cells by IL-12 $1$/day [29]
    $\lambda_{T2}$ Production rate of Th2 cells by M1 $0.75$/day estimated
    $\lambda_{E_0}$ production rate of AEC 0.25 day$^{-1}$ [31]
    $\lambda_{1}$ repair rate of AEC $10^{-3}$ $g/cm^3$ day$^{-1}$ [31]
    $\lambda_{EM}$ EMT rate of AEC $1.65\times10^{-3}$ day$^{-1}$ [31]
    $\lambda_{H G}$ production rate of HSCs by PDGF $1.5\times10^{-3}$ day$^{-1}$ estimated
    $\lambda_{HT_\beta}$ production rate of HSCs by TGF-$\beta$ $3.32\times10^{-3}$ day$^{-1}$ estimated
    $\lambda_{H_A H}$ production rate of HA by HSCs $2.9\times10^{-2}$ day$^{-1}$ estimated
    $\lambda_{T_\beta M}$ production rate of TGF-$\beta$ by macrophages $1.5\times10^{-2}$ day$^{-1}$ [32]
    $\lambda_{T_\beta f}$ production rate of TGF-$\beta$ by fibroblast $7.5\times10^{-3}$ day$^{-1}$ [31]
    $\lambda_{T_\beta I_{13}}$ production rate of TGF-$\beta$ by IL-13 $2$ [31]
    $\lambda_{GM}$ production rate of PDGF by macrophages $2.4\times10^{-5}$ day$^{-1}$ [32]
    $\lambda_{QM}$ production rate of MMP by macrophages $2\times 10^{-3}$ day$^{-1}$ estimated
    $\lambda_{Q_rM}$ production rate of TIMP by macrophages $4\times 10^{-4}$ day$^{-1}$ estimated
    $\lambda_{P E}$ activation rate of MCP-1 due to AECs $1\times10^{-8}$ day$^{-1}$ [32]
    $\lambda_{\rho f}$ activation rate of ECM due to fibroblasts $3\times10^{-3}$ day$^{-1}$ [32]
    $\lambda_{\rho m}$ activation rate of ECM due to myofibroblasts $6\times10^{-3}$ day$^{-1}$ [32]
    $\lambda_{\rho T_\beta}$ activation rate of ECM due to TGF-$\beta$ 2 [32]
    $\lambda_{Ef}$ activation rate of fibroblasts due to bFGF and TGF-$\beta$ $2.5\times10^{-1}$ day$^{-1}$ [31]
    $\lambda_{fH_A}$ production rate of fibroblasts by HA $2.5\times10^{-3}$ day$^{-1}$ estimated
    $\lambda_{fE}$ production rate of fibroblasts $5\times10^{-4}$ day$^{-1}$ [31]
    $\lambda_{mfT}$ activation rate of myofibroblasts due to TGF-$\beta$ $0.3$ day$^{-1}$ estimated
    $\lambda_{mfG}$ activation rate of myofibroblasts due to PDGF $0.3$ day$^{-1}$ estimated
     | Show Table
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    Table 3.  Parameters' description and value

    Parameter Description Value
    $\lambda_{T_\alpha M} $ activation rate of TNF-$\alpha$ due to macrophage $1.39\times10^{-2}$ day$^{-1}$ [31]
    $\lambda_{T_\alpha E} $ activation rate of TNF-$\alpha$ due to macrophage $6.9\times10^{-4}$ day$^{-1}$ [31]
    $\lambda_{I_2 T_1} $ production rate of IL-2 by Th1 cells $4.2\times10^{-4}$ day$^{-1}$ [33]
    $\lambda_{I_{4}T_2}$ production rate of IL-10 by Th2 cells $5.96\times10^{-4}$ day$^{-1}$ [33]
    $\lambda_{I_{4}M_2}$ production rate of IL-10 by M2 macrophages $2.38\times10^{-3}$ day$^{-1}$ [33]
    $\lambda_{I_{10}M2}$ production rate of IL-10 by M2 macrophages $6.67\times10^{-4}$ day$^{-1}$ [33]
    $\lambda_{I_{12}M_1}$ production rate of IL-12 by M1 macrophages $9.64\times10^{-2}$ day$^{-1}$ [33]
    $\lambda_{I_{13}T_2}$ production rate of IL-13 by Th2 cells $2.24\times10^{-4}$ day$^{-1}$ [33]
    $\lambda_{I_{13}M_2}$ production rate of IL-13 by macrophages $5.94\times10^{-4}$ day$^{-1}$ [33]
    $\lambda_{I_{r}T_1}$ production rate of IFN-$\gamma$ by Th1 cells $2.87\times10^{-5}$ day$^{-1}$ [33]
    $d_{M_2}$ death rate of macrophages 0.015 day$^{-1}$ [32]
    $d_{M_1}$ death rate of macrophages 0.02 day$^{-1}$ [31]
    $d_{T_1}$ death rate of Th1 cell $1.97\times10^{-1}$ day$^{-1}$ [33]
    $d_{T_{2}}$ death rate of Th2 cell $1.97\times10^{-1}$ day$^{-1}$ [33]
    $d_E$ death rate of activated AECs $1.65\times10^{-2}$ day$^{-1}$ [32]
    $d_H$ death rate of HSCs $1.66\times10^{-2}$ day$^{-1}$ estimated
    $d_{E_0}$ death rate of inactivated AECs $1.65\times10^{-2}$ day$^{-1}$ [32]
    $d_{E_0T}$ death rate of AECs $1.65\times10^{-3}$ day$^{-1}$ [32]
    $\delta$ increased death rate of AECs by injury $1\times10^{-3}$ day$^{-1}$ [31]
    $d_\rho$ degradation rate of ECM $0.37$ day$^{-1}$ [32]
    $d_{H_A}$ degradation rate of HA $0.1$ day$^{-1}$ estimated
    $d_P$ degradation rate of MCP-1 $1.73$ day$^{-1}$[32]
    $d_{PM}$ internalization rate of MCP-1 by M1 macrophages $2.08\times10^{-4}$ day$^{-1}$[32]
    $d_G$ degradation rate of PDGF $3.84$ day$^{-1}$ [32]
    $d_{QQ_r}$ binding rate of MMP to TIMP $4.98\times10^{5}$ $cm^3g^{-1}$ day$^{-1}$ [32]
    $d_{Q_rQ}$ binding rate of TIMP to MMP $1.04\times10^{6}$ $cm^3g^{-1}$ day$^{-1}$ [32]
    $d_Q$ degradation rate of MMP $4.32$ day$^{-1}$[32]
    $d_{Q_r}$ degradation rate of TIMP $21.6$ day$^{-1}$ [32]
    $d_{\rho Q}$ degradation rate of ECM due to MMP $2.59\times 10^{5}$ $cm^3g^{-1}$ day$^{-1}$ [32]
    $d_{T_\beta}$ degradation rate of TGF-$\beta$ $3.33\times10^2$ day$^{-1}$ [32]
    $d_f$ death rate of fibroblasts $1.66\times10^{-2}$ day$^{-1}$ [32]
    $d_m$ death rate of myofibroblasts $1.66\times10^{-2}$ day$^{-1}$ [32]
    $d_{T_\alpha}$ degradation rate of TNF-$\alpha$ 55.45 day$^{-1}$ [33]
    $d_{I_{2}}$ degradation rate of IL-2 2.376 day$^{-1}$ [33]
    $d_{I_{4}}$ degradation rate of IL-4 50 day$^{-1}$ [33]
    $d_{I_{10}}$ degradation rate of IL-10 8.32 day$^{-1}$ [33]
    $d_{I_{12}}$ degradation rate of IL-12 1.38 day$^{-1}$ [33]
    $d_{I_{13}}$ degradation rate of IL-13 12.47 day$^{-1}$ [33]
    $d_{I_{\gamma}}$ degradation rate of IFN-$\gamma$ $2.16$ day$^{-1}$ [33]
     | Show Table
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    Table 4.  Parameters' description and value

    Parameter Description Value
    $\chi_{P}$ chemotactic sensitivity parameter by MCP-1 10 $cm^5g^{-1}$ day$^{-1}$[32]
    $A_{H}$ HSC proliferation $3.32\times10^{-5}~g/cm^3\hbox{ day}^{-1}$ estimated
    $A_{E0}$ intrinsic AEC proliferation $1.65\times10^{-3}~g/cm^3\hbox{ day}^{-1}$ estimated
    $K_{G}$ PDGF saturation for activation of myofibroblasts $1.5 \times 10^{-8}$ $gcm^{-3}$ [32]
    $K_{T_\beta}$ TGF-$\beta$ saturation for apoptosis of AECs $1\times 10^{-10}$ $gcm^{-3}$ [32]
    $K_{P}$ MCP-1 saturation for influx of macrophages $5\times 10^{-9}$ $gcm^{-3}$ [32]
    $K_{T_\alpha}$ TNF-$\alpha$ saturation $5\times 10^{-7}$ $gcm^{-3}$ [29]
    $K_{I_{13}}$ IL-13 saturation $2\times10^{-7}$ g/$cm^3$ [29]
    $K_{H_A}$ HA saturation $2\times10^{-3}$ g/ml estimated
    $K_{T_1}$ Th1 cell saturation $1\times10^{-1}$ g/ml [33]
    $K_{I_\gamma}$ IFN-$\gamma$ saturation $2\times10^{-7} ~gcm^{-3}$ [33]
    $K_{I_{2}}$ IL-2 saturation $5\times10^{-7}$ g/$cm^3$ [33]
    $K_{I_{4}}$ IL-4 saturation $2\times10^{-7}$ g/$cm^3$ [33]
    $K_{I_{10}}$ IL-10 saturation $2\times10^{-7}$ g/$cm^3$ [29]
    $K_{I_{12}}$ IL-12 saturation $1.5\times10^{-5}$ g/$cm^3$ [29]
    $K_{I_{13}}$ IL-13 saturation $2\times10^{-7}$ g/$cm^3$ [29]
    $K_{E}$ AEC saturation $0.1$ g/$cm^3$ [31]
    $\rho_0$ ECM saturation $10^{-2}$ $gcm^{-3}$ [32]
    $\rho^*$ ECM density in health $3.26\times10^{-3}$ $gcm^{-3}$ [31]
    $E^*$ TEC density in health $0.799$ $gcm^{-3}$ [31]
    $f^*$ fibroblast density in health $4.75\times10^{-3}$ $gcm^{-3}$ [31]
    $M_0$ source/influx of macrophages from blood $5\times 10^{-5}$ $gcm^{-3}$ [32]
    $\beta$ influx rate of macrophages into interstitium $0.2 ~cm^{-1}$ [32]
    $A_{M_2}$ Source term of M2 $0.05$ [29]
    $K_{M_1}$ M1 saturation $0.5$ [15]
    $K_{M_2}$ M2 saturation $1$ [15]
    $K_{p}$ MCP-1 saturation $5\times10^{-9}$ [32]
    $E_0$ TEC saturation $0.1$ g/ml estimated
    $\rho_0$ ECM saturation $10^{-2}$ g/ml [29]
    $T_0$ T cells saturation $3\times10^{-5}$ g/ml [30,30]
     | Show Table
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