2018, 15(3): 717-738. doi: 10.3934/mbe.2018032

Mathematical insights on psoriasis regulation: Role of Th1 and Th2 cells

1. 

Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata-700032, India

2. 

Department of Mathematics and Computer Sciences, Texas Womans University, Denton, TX 76204, USA

Received  April 10, 2017 Revised  July 6, 2017 Published  December 2017

Fund Project: The research is supported by Department of Mathematics, Jadavpur University, PURSE-DST, Government of India

Psoriasis is an autoimmune disorder, characterized by hyper-proli-feration of Keratinocytes for the abnormal activation of T Cells, Dendritic Cells (DCs) and cytokine signaling. Interaction of DCs and T Cells enable T Cell to differentiate into Type 1 (Th1), Type 2 (Th2) helper T Cell depending on cytokine release. Hyper-proliferation of Keratinocytes may occur due to over expression of pro-inflammatory cytokines secreted by Th1-Cells viz. Interferon gamma ($\mbox{IFN}-{γ}$), Transforming growth factor beta ($\mbox{TGF}-β$) and Tumor necrosis factor alpha ($\mbox{TNF}-α$) etc. Deregulation of epidermal happens due to signaling of anti-inflammatory cytokines like Interleukin 10 ($\mbox{IL}-{10}$), Interleukin 4 ($\mbox{IL}-{4}$) etc., released by Th2-Cells. In this article, we have constructed a set of nonlinear differential equations involving the above cell population for better understanding the impact of cytokines on Psoriasis. System is analyzed introducing therapeutic agent (Biologic / $\mbox{IL}-{10}$) for reducing the hyper-proliferation of Keratinocytes. Effect of Biologic is used as a surrogate of control parameter to reduce the psoriatic lesions. We also studied its effect both in continuous and impulsive dosing method. Our study reveals that impulsive dosing is more applicable compare with continuous dosing to prevent Psoriasis.

Citation: Amit Kumar Roy, Priti Kumar Roy, Ellina Grigorieva. Mathematical insights on psoriasis regulation: Role of Th1 and Th2 cells. Mathematical Biosciences & Engineering, 2018, 15 (3) : 717-738. doi: 10.3934/mbe.2018032
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show all references

References:
[1]

F. O. NestleP. D. MeglioJ. Z. Qin and B. J. Meglio, Skin immune sentinels in health and disease, Nature Reviews Immunology, 9 (2009), 679-691. doi: 10.1038/nri2622.

[2]

World Health Organization, Global report on Psoriasis, WHO Library Cataloguing-in-Publication Data, 2016.

[3]

M. L. FordB. H. KoehnM. E. WagenerW. JiangS. GangappaT. C. Pearson and C. P. Larsen, Antigen-specific precursor frequency impacts T cell proliferation, differentiation, and requirement for costimulation, Journal of Experimental Medicine, 204 (2007), 299-309. doi: 10.1084/jem.20062319.

[4]

J. SongF. T. LeiX. Xiong and R. Haque, Intracellular signals of T cell costimulation, Cellular & Molecular Immunology, 5 (2008), 239-247. doi: 10.1038/cmi.2008.30.

[5]

T. J. Kindt, R. A. Goldsby, B. A. Osborne and J. Kuby, Kuby Immunology, Macmillan, 2007.

[6]

A. YatesC. BergmannJ. L. Van HemmenJ. Stark and R. Callard, Cytokine-modulated regulation of helper T cell populations, Journal of Theoretical Biology, 206 (2000), 539-560. doi: 10.1006/jtbi.2000.2147.

[7]

P. Bousso, T-cell activation by dendritic cells in the lymph node: Lessons from the movies, Nature Reviews Immunology, 8 (2008), 675-684. doi: 10.1038/nri2379.

[8]

Y. CaiC. Fleming and J. Yan, New insights of T cells in the pathogenesis of psoriasis, Cellular and Molecular Immunology, 9 (2012), 302-309. doi: 10.1038/cmi.2012.15.

[9]

H. ValdimarssonB. S. BakeI. Jónsdótdr and L. Fry, Psoriasis: A disease of abnormal keratinocyte proliferation induced by T lymphocytes, Immunology Today, 7 (1986), 256-259. doi: 10.1016/0167-5699(86)90005-8.

[10]

J. T. ChangV. R. PalanivelI. KinjyoF. SchambachA. M. IntlekoferA. BanerjeeS. A. LongworthK. E. VinupP. MrassJ. Oliaro and N. Killeen, Asymmetric T lymphocyte division in the initiation of adaptive immune responses, Science, 315 (2007), 1687-1691. doi: 10.1126/science.1139393.

[11]

J. RengarajanS. J. Szabo and L. H. Glimcher, Transcriptional regulation of Th1/Th2 polarization, Immunology Today, 21 (2000), 479-483. doi: 10.1016/S0167-5699(00)01712-6.

[12]

J. H. MaoE. F. SaunierJ. P. de KoningM. M. McKinnonM. N. HigginsK. NicklasH. T. YangA. Balmain and R. J. Akhurst, Genetic variants of Tgfb1 act as context-dependent modifiers of mouse skin tumor susceptibility, Proceedings of the National Academy of Sciences, 103 (2006), 8125-8130. doi: 10.1073/pnas.0602581103.

[13]

A. Balato, F. Ayala, M. Schiattarella, M. Megna, N. Balato and S. Lembo, Pathogenesis of Psoriasis: the role of pro-inflammatory cytokines produced by keratinocytes, J. Soung (Ed. ), Pathogenesis of Psoriasis: The Role of Pro-Inflammatory Cytokines Produced by Keratinocytes, InTech, Shanghai, (2012), p372. doi: 10.5772/26163.

[14]

J. BaliwagD. H. Barnes and A. Johnston, Cytokines in Psoriasis, Cytokine, 73 (2015), 342-350. doi: 10.1016/j.cyto.2014.12.014.

[15]

Y. KoganZ. Agur and M. Elishmereni, A mathematical model for the immunotherapeutic control of the Th1/Th2 imbalance in melanoma, Discr Cont Dyn Syst Ser B, 18 (2013), 1017-1030. doi: 10.3934/dcdsb.2013.18.1017.

[16]

A. T. PietrzakA. ZalewskaG. ChodorowskaD. KrasowskaA. Michalak-StomaP. NockowskiP. OsemlakT. Paszkowski and J. M. Roliński, Cytokines and anticytokines in Psoriasis, Clinica Chimica Acta, 394 (2008), 7-21. doi: 10.1016/j.cca.2008.04.005.

[17]

K. GhoreschiP. ThomasS. BreitM. DugasR. MailhammerW. van EdenR. van der ZeeT. BiedermannJ. PrinzM. Mack and U. Mrowietz, Interleukin-4 therapy of Psoriasis induces Th2 responses and improves human autoimmune disease, Nature Medicine, 9 (2002), 40-46. doi: 10.1038/nm804.

[18]

K. AsadullahW. Sterry and U. Trefzer, Cytokines: Interleukin and interferon therapy in dermatology, Clinical and Experimental Dermatology, 27 (2002), 578-584. doi: 10.1046/j.1365-2230.2002.01144.x.

[19]

A. D'andreaM. Aste-AmezagaN. M. ValianteX. MaM. Kubin and G. Trinchieri, Interleukin 10 (IL-10) inhibits human lymphocyte interferon gamma-production by suppressing natural killer cell stimulatory factor/IL-12 synthesis in accessory cells, The Journal of Experimental Medicine, 178 (1993), 1041-1048.

[20]

S. JainI. R. KaurS. DasS. N. Bhattacharya and A. Singh, T helper 1 to T helper 2 shift in cytokine expression: An autoregulatory process in superantigen-associated Psoriasis progression?, Journal of Medical Microbiology, 58 (2009), 180-184. doi: 10.1099/jmm.0.003939-0.

[21]

A. Coondoo, The role of cytokines in the pathomechanism of cutaneous disorders, Indian Journal of Dermatology, 57 (2012), 90-96. doi: 10.4103/0019-5154.94272.

[22]

G. D. WeinsteinJ. L. McCullough and P. A. Ross, Cell kinetic basis for pathophysiology of Psoriasis, Journal of Investigative Dermatology, 85 (1985), 579-583. doi: 10.1111/1523-1747.ep12283594.

[23]

B. S. BakerA. F. SwainL. Fry and H. Valdimarsson, Epidermal T lymphocytes and HLA-DR expression in Psoriasis, British Journal of Dermatology, 110 (1984), 555-564.

[24]

P. NockowskiJ. C. SzepietowskiM. Ziarkiewicz and E. Baran, Serum concentrations of transforming growth factor beta 1 in patients with Psoriasis vulgaris, Acta Dermatovenerologica Croatica: ADC, 12 (2003), 2-6.

[25]

M. A. LowesA. M. Bowcock and J. G. Krueger, Pathogenesis and therapy of Psoriasis, Nature, 445 (2007), 866-873. doi: 10.1038/nature05663.

[26]

K. AsadullahW. Sterry and H. D. Volk, Interleukin-10 therapy--review of a new approach, Pharmacological Reviews, 55 (2003), 241-269. doi: 10.1124/pr.55.2.4.

[27]

J. TzuA. J. Mamelak and D. N. Sauder, Current advancements in the treatment of Psoriasis: Immunobiologic agents, Clinical and Applied Immunology Reviews, 6 (2006), 99-130. doi: 10.1016/j.cair.2006.06.003.

[28]

K. ReichM. BruckA. GrafeC. VenteC. Neumann and C. Garbe, Treatment of Psoriasis with interleukin-10, J Invest Dermatol, 111 (1998), 1235-1236. doi: 10.1046/j.1523-1747.1998.00444.x.

[29]

K. AsadullahW. D. DöckeM. EbelingM. FriedrichG. BelbeH. AudringH. D. Volk and W. Sterry, Interleukin 10 treatment of Psoriasis: Clinical results of a phase 2 trial, Archives of Dermatology, 135 (1999), 187-192. doi: 10.1001/archderm.135.2.187.

[30]

K. ReichV. BlaschkeC. MaurerU. LippertC. NeumannC. GarbeP. Middel and G. Westphal, Response of Psoriasis to interleukin-10 is associated with suppression of cutaneous type 1 inflammation, downregulation of the epidermal interleukin-8/CXCR2 pathway and normalization of keratinocyte maturation, Journal of Investigative Dermatology, 116 (2001), 319-329. doi: 10.1046/j.1523-1747.2001.01248.x.

[31]

M. FriedrichW. D. DöckeA. KleinS. PhilippH. D. VolkW. Sterry and K. Asadullah, Immunomodulation by interleukin-10 therapy decreases the incidence of relapse and prolongs the relapse-free interval in Psoriasis, Journal of Investigative Dermatology, 118 (2002), 672-677. doi: 10.1046/j.1523-1747.2002.01731.x.

[32]

I. B. McInnesG. G. IlleiC. L. DanningC. H. YarboroM. CraneT. KuroiwaR. SchlimgenE. LeeB. FosterD. Flemming and C. Prussin, IL-10 improves skin disease and modulates endothelial activation and leukocyte effector function in patients with psoriatic arthritis, The Journal of Immunology, 167 (2001), 4075-4082. doi: 10.4049/jimmunol.167.7.4075.

[33]

P. K. Roy and A. Datta, Negative Feedback Control may Regulate Cytokines Effect during Growth of Keratinocytes in the Chronic Plaque of Psoriasis: A Mathematical Study, International Journal of Applied Mathematics, 25 (2012), 233-254.

[34]

P. K. Roy and A. Datta, Impact of perfect drug adherence on immunopathogenic mechanism for dynamical system of Psoriasis, Biomath, 2 (2013), 1212101, 6 pp.

[35]

A. Datta and P. K. Roy, T-cell proliferation on immunopathogenic mechanism of Psoriasis: A control based theoretical approach, Control and Cybernetics, 42 (2013), 365-386.

[36]

H. Zhang, W. Hou, L. Henrot, S. Schnebert, M. Dumas, C. Heuséle and J. Yang, Modelling epidermis homoeostasis and Psoriasis pathogenesis, Journal of The Royal Society Interface, 12 (2015), Article ID : 20141071.

[37]

H. B. OzaR. PandeyD. RoperY. Al-NuaimiS. K. Spurgeon and M. Goodfellow, Modelling and finite time stability analysis of Psoriasis pathogenesis, International Journal of Control, 90 (2017), 1664-1677.

[38]

J. C. Maxwell, On governors, Proceedings of the Royal Society of London, 16 (1867), 270-283.

[39]

E. J. Routh, A treatise on the stability of a given state of motion: Particularly steady motion, Macmillan and Company, 1877.

[40]

A. Hurwitz, On the conditions under which an equation has only roots with negative real parts, Selected papers on mathematical trends in control theory, 65 (1964), 273-284.

[41]

E. V. GrigorievaE. N. KhailovN. V. Bondarenko and A. Korobeinikov, Modeling and optimal control for antiviral treatment, Special issue on Analytic Modeling in Biology and Medicine of Journal of Biological Systems, 22 (2014), 199-217.

[42]

E. V. Grigorieva, E. N. Khailov and A. Korobeinikov, Optimal control problem in HIV treatment Discrete and Continuous Dynamical Systems, supplement volume, (2011), 311-322.

[43]

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Figure 1.  Schematic of the interactions between the components of the model.
Figure 2.  Time series plot of cell population and contour plot (A) Qualitative nature of all cells (T Cell, Dendritic Cell, $\mbox{Th}_{1}$-Cell, $\mbox{Th}_{2}$-Cell and Keratinocyte) during the disease progression. (B) Contour plot of $R_E$ as a function of $\mu_5$ and $c$.
Figure 3.  Stability analysis using different cell population and finding the endemic equilibrium of system dynamics (A) Considering three cells Keratinocyte, T Cell, Dendritic Cell. (B) Considering three cells $\mbox{Th}_{1}$-Cell, $\mbox{Th}_{2}$-Cell and Keratinocyte.
Figure 4.  Qualitative behaviour of $\mbox{Th}_{1}$, $\mbox{Th}_{2}$ and Keratinocyte with and with out control. Dotted line represents the cell dynamics without control and solid line represents the cell dynamics with control.
Figure 5.  Control parameter with respect to time for better impact on psoriatic plaque.
Figure 6.  Comparative system behaviour of continuous therapy (blue line) and impulse therapy (green line) using Biologic ($\mbox{IL}-{10}$).
Figure 7.  System behaviour with perfect dose of Biologic ($\mbox{IL}-{10}$) in impulsive way. Different cells population are denoted by different colour line i.e Keratinocyte (red), $\mbox{Th}_{1}$-Cells (green) and $\mbox{Th}_{2}$-Cells (blue).
Table 1.  Parameters value using for numerical simulation.
ParameterAssigned ValueRangeReferences
$ a $12 $\mbox{mm}^{-3}\mbox Day^{-1}$9 -15 $\mbox{mm}^{-3}\mbox Day^{-1}$[33,34,54]
$ b $14 $\mbox{mm}^{-3}\mbox Day^{-1}$12 -14 $\mbox{mm}^{-3}\mbox Day^{-1}$[33,35,55]
$ \delta_1 $0.07 $\mbox Day^{-1}$0.005 -0.15 $\mbox Day^{-1}$[34,35,56]
$ \delta_2 $0.08 $\mbox Day^{-1}$0.00004 -0.4 $\mbox Day^{-1}$[34,35,55]
$ \mu_1 $0.02 $\mbox Day^{-1}$0.007 -0.1 $\mbox Day^{-1}$[33,34,54]
$ \eta_1 $0.05 $\mbox Day^{-1}$Estimated[57]
$ \eta_2 $0.0025 $\mbox Day^{-1}$Estimated[57]
$ \alpha $0.002 $\mbox Day^{-1}$-Assumed
$ \mu_2 $0.05 $\mbox Day^{-1}$0.002-0.05 $\mbox Day^{-1}$[33,35]
$ \beta_1 $0.02 $\mbox Day^{-1}$Estimated[15,58]
$ \beta_2 $0.0001 $\mbox Day^{-1}$Estimated[15,58]
$ \mu_3 $0.12 $\mbox Day^{-1}$0.012 -0.12 $\mbox Day^{-1}$[37]
$ \mu_4 $0.24 $\mbox Day^{-1}$0.24 $\mbox Day^{-1}$[58]
$ \gamma_1 $0.51 $\mbox Day^{-1}$Estimated[15,58]
$ \gamma_2 $0.035 $\mbox Day^{-1}$Estimated[15,58]
$ \xi_1 $0.90 $\mbox Day^{-1}$-Assumed
$ \xi_2 $0.15 $\mbox Day^{-1}$-Assumed
$ \mu_5 $0.65 $\mbox Day^{-1}$0.04-0.9 $\mbox Day^{-1}$[33]
$ c $0.50 $\mbox Day^{-1}$Estimated[22,60]
ParameterAssigned ValueRangeReferences
$ a $12 $\mbox{mm}^{-3}\mbox Day^{-1}$9 -15 $\mbox{mm}^{-3}\mbox Day^{-1}$[33,34,54]
$ b $14 $\mbox{mm}^{-3}\mbox Day^{-1}$12 -14 $\mbox{mm}^{-3}\mbox Day^{-1}$[33,35,55]
$ \delta_1 $0.07 $\mbox Day^{-1}$0.005 -0.15 $\mbox Day^{-1}$[34,35,56]
$ \delta_2 $0.08 $\mbox Day^{-1}$0.00004 -0.4 $\mbox Day^{-1}$[34,35,55]
$ \mu_1 $0.02 $\mbox Day^{-1}$0.007 -0.1 $\mbox Day^{-1}$[33,34,54]
$ \eta_1 $0.05 $\mbox Day^{-1}$Estimated[57]
$ \eta_2 $0.0025 $\mbox Day^{-1}$Estimated[57]
$ \alpha $0.002 $\mbox Day^{-1}$-Assumed
$ \mu_2 $0.05 $\mbox Day^{-1}$0.002-0.05 $\mbox Day^{-1}$[33,35]
$ \beta_1 $0.02 $\mbox Day^{-1}$Estimated[15,58]
$ \beta_2 $0.0001 $\mbox Day^{-1}$Estimated[15,58]
$ \mu_3 $0.12 $\mbox Day^{-1}$0.012 -0.12 $\mbox Day^{-1}$[37]
$ \mu_4 $0.24 $\mbox Day^{-1}$0.24 $\mbox Day^{-1}$[58]
$ \gamma_1 $0.51 $\mbox Day^{-1}$Estimated[15,58]
$ \gamma_2 $0.035 $\mbox Day^{-1}$Estimated[15,58]
$ \xi_1 $0.90 $\mbox Day^{-1}$-Assumed
$ \xi_2 $0.15 $\mbox Day^{-1}$-Assumed
$ \mu_5 $0.65 $\mbox Day^{-1}$0.04-0.9 $\mbox Day^{-1}$[33]
$ c $0.50 $\mbox Day^{-1}$Estimated[22,60]
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