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Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39300232
71251
10.1038/s41598-024-71251-3
Article
A mathematical insight to control the disease psoriasis using mesenchymal stem cell transplantation with a biologic inhibitor
Kushary Subhankar 1
Cao Xianbing 2
Ghosh Tushar 1
Roy Priti Kumar pritiju@gmail.com

1
1 https://ror.org/02af4h012 grid.216499.1 0000 0001 0722 3459 Department of Mathematics, Jadavpur University, Kolkata, West Bengal 700032 India
2 https://ror.org/013e0zm98 grid.411615.6 0000 0000 9938 1755 School of Mathematics and Statistics, Beijing Technology and Business University, Beijing, 100048 China
19 9 2024
19 9 2024
2024
14 2189718 1 2024
26 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Psoriasis is a chronic, non-contagious, immune-mediated skin disorder. Inflammation of the skin’s surface is characterised by scaly white, red, or silvery spots that occur due to the hyper-proliferation of keratinocytes in the epidermal layer. Primarily, pharmaceutical drugs or immune therapy are used to treat psoriasis. We are all aware that, certain therapeutic strategies can have some adverse effects, and over time, that hidden inflammation may manifest. This article introduces a mathematical model for psoriasis, formulated by employing a set of nonlinear ordinary differential equations (ODEs) that describe the densities of T-cells, dendritic cells (DCs), keratinocytes, and mesenchymal stromal cells (MSCs) as basic cell populations. A tumor necrosis factor-α (TNF-α) inhibitor has been imposed from the initial stage of the treatment regime, using the optimal control theoretic approach, and the numerical results have been observed. After 80 days of monitoring using only biologic TNF-α inhibitors, if this approach did not provide the intended outcomes (when severity arises), stem cells are administered a few times in a pulsed manner as a cell replacement technique in addition to this anti TNF-α medicine. We have observed the combined therapeutic benefit of stem cell replacement with a TNF-α inhibitor from a mathematical point of view. The theoretical analysis and the numerical results revealed that stem cell transplantation, along with a TNF-α inhibitor, is a promising psoriasis treatment option moving forward.

Keywords

Mesenchymal stromal cells
TNF-α inhibitor
Optimal control
Pulsed manner
Subject terms

Cell biology
Stem cells
Diseases
Mathematics and computing
Project of National Statistical Science Foundation of China2021LD01 Cao Xianbing issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

Human skin is the outer covering of the body that protects the underlying tissues and organs. It is composed mainly of two layers: the epidermal layer (The outermost layer) and the dermal layer (The innermost layer). Various skin diseases may occur due to many factors, including environmental, genetic, infectious, or immune system disorders. Psoriasis is considered one of the chronic, non-contagious, immune-mediated skin disorders1. Both genetic and environmental factors play a significant role in the occurrence of psoriasis. There are several types of psoriasis, each of which has different signs and symptoms. Psoriasis has a number of distinct symptoms, including skin flaking, irritation, and thick, white, silvery, or red plaques2,3. Along with its skin-related symptoms, psoriasis can also affect a number of joints, including the knees, elbows, and, most frequently, the scalp. Geographical differences influence the frequency of psoriasis. Psoriasis affects fewer than 1% of children worldwide, and its frequency in adults ranges from 0.17% in East Asia to 2.5% in Western Europe4. Patients who misuse alcohol or who are obese may get more widespread psoriasis5. Psoriasis can also have a considerable psychological or emotional impact on an individual. Currently, severe psoriasis can be treated with the proper medical approaches and stem cell transplantation6,7.

Since psoriasis is an autoimmune disorder, the development of the disease is mainly characterized by the hyper-proliferation and abnormal differentiation of keratinocytes and the infiltration of multiple inflammatory cells. The human immune system, especially helper T-cells and dendritic cells, is responsible for this excessive growth. T-cells are an important component of the human immune system and are the kind of leukocyte (White blood cell) that develops in the thymus after originating in the bone marrow8. In a person with psoriasis, these T-cells are stimulated by the antigens and release excessive amounts of inflammatory cytokines, which causes overgrowth of keratinocytes in the skin’s epidermal layer. Dendritic cells (DCs) are the most potent antigen-presenting cells (APCs), a unique sub-type of immune cell that are found in the tissues of the skin. They develop through hematopoiesis from progenitors in the bone marrow and are trained to handle antigens8. Keratinocytes are essential for keeping the body safe from environmental hazards and are also involved in the healing of wounds under normal homeostatic conditions. Keratins, which are naturally present in keratinocytes, are crucial in cytokine synthesis in the epidermis9. This keratinocyte population becomes the passive target of immunological assault by T-cells in the psoriatic condition, while dendritic cells actively manage epidermal immune responses10.

Stem cells are a special kind of cell in the body that has the potential to renew itself as well as replace or develop different kinds of other cells under the proper circumstances in the body or in the laboratory. Embryonic stem cells and adult stem cells are the two primary sub-types of stem cells. Adult stem cells of a special kind known as mesenchymal stromal cells (MSCs) can be found in a variety of organs and tissues, including the umbilical cord, bone marrow, and others11. These are multi-potent, have the ability to regenerate themselves, and can differentiate into a wide range of cell types. They have anti-inflammatory properties as they have the ability to control the cytokine secretion from the lymphocytes12.

Psoriasis is primarily brought on by the malfunctioning of T-cell subsets, especially Th1 and Th1713. DCs are able to secrete multiple types of cytokines, including TNF-α, interleukin-12 (IL-12), and interleukin-23 (IL-23), which allows helper T-cells to be activated in the proper dermal region14,15. Among these, IL-23 stimulates Th-17 cells and enables them to release TNF-α and interleukin-17 (IL-17), which are the most essential components in the pathogenesis of psoriasis. Besides that, interferon-gamma (IFN-γ) promotes DC cell maturation and TNF-α activation through specific cell biology16. According to recent studies, IL-23 is one of the most important cytokines and directly induces interleukin-22 (IL-22), which controls keratinocyte differentiation and migration as well as the production of antimicrobial proteins from keratinocytes17. A transitional state between keratinocyte hyperplasia and IL-23 is indicated by this cytokine, IL-2213. The cytokines TNF-α and IL-17/22 axis, together with IFN-γ, form a synergistic effect that increases the cytokine storm and triggers up the keratinocyte growth factor (KGF). Finally, anti-genetic molecules are to be created in the dermal blood vessels as a result of this excessive growth of keratinocytes16. All of these things contribute to the hyper-proliferation of keratinocytes and psoriasis development.

Several mathematicians have drawn attention to psoriasis in this way by creating mathematical models of the disease. Roy et. al. studied the role of cytokine signalling in the disease’s cell-biological network to develop mathematical models for the dynamics of psoriasis based on immune cell kinetics. They have proposed a mathematical model to describe the dynamics of psoriasis in a cytokine-interacting environment and shown the results of drug holidays during induction treatment18. In recent research, the modified impulse theory was used to assess the safety and effectiveness of two drugs used as combined biologics19.

In this research work, we have introduced stem cell therapy along with a biologic TNF-α inhibitor to completely cure the disease psoriasis from a mathematical point of view. The approach to treatment using stem cells has enlightened a new way to cell-based therapies in medical science. Various cytokines are secreted from the MSCs that can contribute to the suppression of psoriatic inflammation and inhibit the overproduction of keratinocytes20. Transforming Growth Factor-β (TGF-β) and Interleukin-10 (IL-10) are some immunoregulatory cytokines that are secreted from the MSCs21. TGF-β inhibits the activation of T-cells and DC cells, can promote the differentiation of Th17 cells, and suppresses the production of IL-17, which is a key cytokine in psoriasis pathogenesis. On the other hand, IL-10 restrains the production of IFN-γ, mainly through the suppression of IL-12 in peripheral cells22. MSCs also control the cytokine profile of DCs by secreting IL-10, which blocks the synthesis of TNF-α and IL-1223. Figure 1 illustrates the interactions between immune cells and the stem cell replacement procedure using the cytokine network. Therefore, the aim of this work is to mathematically and numerically investigate the potential impacts of stem cell treatment on psoriasis in combination with a TNF-α inhibitor24.

This manuscript is organized as follows: In Sect. "Mathematical model formulation with suitable assumptions", we formulate a mathematical model with suitable assumptions, incorporating MSC as a population. Section "Mathematical properties of the model" examines the mathematical properties of the formulated model. In Sect. "Optimal control strategy using TNF-α inhibitor", we introduce an optimal control treatment strategy for the system, utilizing a biologic TNF-α inhibitor in a cost-effective manner. Section "Impulsive MSC replacement on the system with optimal control by TNF-α inhibitor" demonstrates an impulsive stem cell replacement approach, in conjunction with a biologic TNF-α inhibitor, to suppress keratinocyte density in severe psoriasis. Section "Numerical simulation results" provides several numerical simulations for both treated and non-treated systems to validate our analytical results. In Sect. "Discussion", we discussed the theoretical implications of our study. Finally, Sect. "Conclusion" concludes with a summary of our findings, their potential applications, and the novelty of our mathematical model, as revealed through our analytical and numerical results.Fig. 1 Schematic representation of model populations and their interrelationships mediated by cytokine signaling.

Mathematical model formulation with suitable assumptions

A variety of cells represent the cell-biological interactions contributing to the development of psoriasis. To simplify the mathematical model, we divided it into three key populations to illustrate the main epidermal transmission pathway. The densities of helper T-cell, dendritic cell, and keratinocyte cell populations are denoted by T(t), D(t), and K(t), respectively, at any time ‘t’. The following hypotheses are considered when formulating the mathematical model for the disease psoriasis :Here, we have used αT and αD as the constant accumulation rates of T-cells and dendritic cells, respectively, to construct the growth equations.

Psoriasis is a chronic, inflammatory disorder associated with T-cells, but the total number of T-cells cannot increase endlessly; thus, we take the logistical expansion of T-cells. The term ηT(1-TTmax) serves to highlight the growth of T-cells, where η is the proliferation rate and Tmax is the maximum carrying capacity.

In the human immune system, T-cells and dendritic cells are two distinct kinds of immune cells with diverse characteristics. Psoriasis begins as a result of the differentiation of Th1 and Th17 cells by activated dermal DC-derived cytokines (mainly IL-12 and IL-23)25. Here, we have considered β1 as the rate at which DCs activate T-cells (via TNF-α, IL-12, and IL-23) and β2 as the rate at which T-cells activate DCs (through the cytokines TNF-α and IL-17). T-cells and DCs engage in mutual interaction (denoted by the terms β1TD and β2TD) to generate a cytokine storm that eventually infiltrates and contributes to the growth of the keratinocyte population.

The hyper-reactivity of T-cells is initially inhibited by keratinocytes (via IL-4 and TGF-β) at a rate of ξ, which is considered in the first population by the term (-ξTK). Throughout the typical sequence, the per-capita clearance rates of T-cells and DCs are considered to be μT and μD, respectively.

Psoriasis is mainly characterized by the hyper-proliferation and abnormal differentiation of keratinocytes and the infiltration of multiple inflammatory cells. The immune system also plays a crucial role in regulation. However, incorporating all these factors into a mathematical model is complex due to the numerous parameters involved, which complicates computations. For the sake of simplicity, we focus on the significant causes: hyper-proliferation and the infiltration of cells via interaction between immune cells (specifically T-cells and DCs) through inflammatory cytokines, which leads to excessive growth in the keratinocyte population. Excessive production of pro-inflammatory cytokines by T-cells and dendritic cells (DCs) significantly contributes to the cytokine storm. IFN-γ is a pivotal cytokine in this process, its secretion being stimulated by IL-12. Additionally, the synthesis of IL-17 is promoted by IL-23. Keratinocytes over-proliferate in conjunction with TNF-α and increase IFN-γ and IL-17/22 axis production, which promotes inflammation and maintains hyper-proliferation over the course of the disease26. We have taken, αK as the constant accumulation of keratinocyte population, and μK stands for the keratinocytes inherent mortality rate. The cytokine network in Fig. 1 illustrates how immune cells interact with one another to promote disease progression.

Combining the aforementioned hypotheses, we have formulated the following mathematical model:1 dT(t)dt=αT+ηT(t)(1-T(t)Tmax)-β1T(t)D(t)-ξT(t)K(t)-μTT(t),dD(t)dt=αD-β2T(t)D(t)-μDD(t),dK(t)dt=αK+β1T(t)D(t)+β2T(t)D(t)-μKK(t).

Adult stem cells are essential for rejuvenating ageing cells in every organ. The ability of MSCs to suppress immune cell function and prevent the entry of infiltrating cells into inflamed areas indicates that this approach could obstruct the migration of T-cells27. MSCs secrete a substantial amount of tolerogenic anti-inflammatory cytokines, including TGF-β, IL-10, and others, when sufficiently high amounts of pro-inflammatory cytokines are exposed to inhibit the overproduction of T-cells and keratinocytes28. Moreover, it contributes to increasing the number of immature DCs, resulting in elevated IL-10 expression. Simultaneously, it diminishes the population of mature inflammatory DCs, along with a decrease in the expression of TNF-α and IL-1229,30. Therefore, transplanting them into the inflamed area of the patient will replace damaged cells and promote the functional recovery of those cells.

In this direction, we have introduced an additional population of matured stem cells (MSCs) to the previously considered cell populations reflected in system (1) to investigate how this cell therapy inhibits the excessive production of inflammatory cytokines. To express the effect of MSCs on the core populations of our model, we have considered the interactions between T-cells and DCs that are inhibited by the MSCs at the rates of γ1 and γ2, so that the interaction terms are multiplied by (1+γ1SM)-1 and (1+γ2SM)-1. MSCs do not affect the interaction terms explicitly if γ1 and γ2 fade away. The coefficient αSM denotes the constant production or the average of all mature stem cells, and μSM is the sum of natural mortality and absorption rate in the inflamed region. From the above-mentioned cell biological discussion, the complete mathematical model that includes MSC as a population takes the form:2 dT(t)dt=αT+ηT(t)(1-T(t)Tmax)-β1T(t)D(t)1+γ1SM(t)-ξT(t)K(t)-μTT(t),dD(t)dt=αD-β2T(t)D(t)1+γ2SM(t)-μDD(t),dK(t)dt=αK+β1T(t)D(t)1+γ1SM(t)+β2T(t)D(t)1+γ2SM(t)-μKK(t),dSM(t)dt=αSM-μSMSM(t).

subject to the initial conditions: T(0)>0,D(0)>0,K(0)>0, and SM(0)>0.

Mathematical properties of the model

Positivity of the invariant region and boundedness of the model

In this subsection, we have looked at the non-negativity and boundedness properties of the solutions for our formulated mathematical system (2) to prove that the model is biologically feasible and well-posed. Now we provide the following theorem, which guarantees the non-negativity of the solutions of system (2).

Theorem 1

All solutions of the system (2) together with the initial conditions are positive for all t>0.

Proof

To prove this theorem, let us take that z1(t)=T(t), z2(t)=D(t), z3(t)=K(t), and z4(t)=SM(t). We now rewrite the system (2) in the following manner:3 dZ(t)dt=Π(Z(t)),whereZ(t)=(z1(t),z2(t),z3(t),z4(t))TandΠ=(Π1,Π2,Π3,Π4)T.

Here T denotes the transpose and Π denote the right hand sides of the system (2).We have,Π1(0,z2,z3,z4)=αT≥0ifz2≥0,z3≥0,z4≥0,Π2(z1,0,z3,z4)=αD≥0ifz1≥0,z3≥0,z4≥0,Π3(z1,z2,0,z4)=αK+β1z1z21+γ1z4+β2z1z21+γ2z4≥0ifz1≥0,z2≥0,z4≥0,Π4(z1,z2,z3,0)=αSM≥0ifz1≥0,z2≥0,z3≥0.

It is easy to verify that for the system (2),   Πi(Z)zi=0,Z∈R+4≥0. The condition clearly ensures the non-negativity of the solutions of a system according to the well-known Nagumo finding34. It means that all the solutions of the system (2) starting with a point Z0=(z1(0),z2(0),z3(0),z3(0))T∈R+4, such as Z(t)=Z(t;Z0), and Z(t)∈R+4,∀t>0. This implies that the non-negative octant R+4 becomes invariant for system (2). □

It is also very essential to show that all the state variables of system (2) are bounded for all time t>0. This will ensure that the system (2) is epidemiologically meaningful and mathematically well-posed. The following theorem demonstrates the invariant region Γ where the solutions of system (2) are bounded.

Theorem 2

All non-negative solutions of the system (2) enter into the region Γ⊆R+4 and are ultimately bounded, where Γ is defined by Γ={(T,D,K,SM)T∈R+4:0<T≤B1,0<D≤B2,0<K≤B3,SMmin≤SM≤B4}, where Bi's(i=1,2,3,4) are defined by,B1=αTμT+ηTmax4μT,B2=αDμD,B3=αKμK+β1B1B2μK(1+γ1SMmin)+β2B1B2μK(1+γ2SMmin),andB4=αSMμSM.

Proof

The maximum value of the quadratic term ηT(1-TTmax) of the first equation of (2) is ηTmax4. Using this fact, from the first equation of system (2) we have:dTdt<αT+ηTmax4-μTT=μTB1-μTT.

By comparison principle35, we obtain from the above inequality:4 0<T(t)<B1(1-e-μTt)+T(0)e-μTt(t>0),which implies thatT(t)≤B1ifT(0)≤B1.

In similar manner from the second equation of system (2) we get:5 0<D(t)<B2(1-e-μDt)+D(0)e-μDt(t>0),which implies thatD(t)≤B2ifD(0)≤B2.

From the fourth equation of system (2) we get:dSMdt≤αSM-μSMSM.

By solving this inequality for sufficiently large t we get:6 SMmin≤SM(t)<B4,t>0.

Now, using T(t)≤B1,D(t)≤B2andSM(t)≥SMmin, from the third equation of the system (2) we have:dKdt<αK+β1B1B21+γ1SMmin+β2B1B21+γ2SMmin-μKK=μKB3-μKK.

Utilizing the same comparison principle, we have obtained:7 0<K(t)<B3(1-e-μKt)+K(0)e-μKt(t>0),which implies thatK(t)≤B3ifK(0)≤B3.

Hence, all the solutions (T(t),D(t),K(t),SM(t))T of system (2) that start in the region Γ, remain within it ∀t>0. This evidently implies, Γ is invariant set for the system (2). Furthermore, the set Γ is bounded and therefore all solutions of the system (2) are eventually bounded.

It is noteworthy that in this case, any solution of system (2) with non-negative initial conditions eventually enters and remains inside the set Γ. This property is justified by the definition of the set Γ and the subsequent relationships:8 dT(t)dt∂Γ<0,dD(t)dt∂Γ<0,dK(t)dt∂Γ<0,dSM(t)dt∂Γ<0.

which are actually performed at the points of the boundary ∂Γ of Γ. Also, note that, these relationships in (8) hold outside the set Γ, which guarantees the boundedness of all solutions (T(t),D(t),K(t),SM(t))T of system (2) along with the positive initial conditions. □

Feasibility of positive equilibrium

To find the interior equilibrium point E∗(T∗,D∗,K∗,SM∗) we set all the equations of the system (2) to zero and solve for the state variables when its co-ordinates satisfy the conditions: T∗>0,D∗>0,K∗>0andSM∗>0.Fig. 2 Simulation of the system (2) showing system boundedness and convergence of solutions with various initial conditions to the endemic equilibrium point E∗ for (a) T-cells (b) Dendritic cells (c) Keratinocytes and (d) Mesenchymal stromal cells using the parameter values given in Table 1.

Table 1 In numerical simulations, the values of the system parameter are employed.

Parameter	Definition	Assigned value (day-1)	Range (day-1)	Reference	
αT	Constant accumulation of T-cells	25 mm-3	9–30 mm-3	19	
αD	Constant accumulation of dendritic-cells	18 mm-3	12–24 mm-3	16	
αK	Constant accumulation of keratinocytes	30 mm-3	10–40 mm-3	19	
β1	T-cell activation rate in response to dendritic cells	0.025	0.003–0.1	31	
β2	Dendritic cell activation rate in response to T-cells	0.03	0.00004–0.4	31	
γ1	Rate of inhibition of the response of dendritic cells to T-cells via MSCs	0.035	–	Chosen	
γ2	Rate of inhibition of the response of T-cells to dendritic cells via MSCs	0.25	–	Chosen	
η	Proliferation rate of T-cells	0.03	0.03–0.7	16,31	
ξ	Rate of inhibition of T cells by keratinocytes acting as an initial barrier	0.003	0.001–0.8	18	
Tmax	Maximum carrying capacity of the T-cell population	300 mm-3	300–1500 mm-3	2,32	
μT	Mortality rate of T-cells	0.06	0.01–0.05	16,19	
μD	Mortality rate of dendritic cells	0.15	0.002–0.05	16,19	
μK	Mortality rate of keratinocytes	0.26	0.01–0.08	2,16	
αSM	The constant production or the average of all MSCs in the inflamed area	0.0033	–	33	
μSM	Sum of natural mortality and absorption rate of MSCs	0.3	0.01–1.1	19	
We have sourced a few parameter ranges from the literature to help us choose the parameter values. Numerous model parameters were determined from various literature, which allowed for the biological viability of the model behaviour.

From the fourth equation of the system (2), we get:SM∗=αSMμSM.

Using the value of SM∗, we have chosen two constants P1 and P2 in order to simplify the computation, which are given by,P1=β11+γ1SM∗andP2=β21+γ2SM∗.

Following that, we have determined the values of D∗ and K∗ in terms of T∗ by using the aforementioned values and setting the second equation of the system (2) to zero, which are provided by,9 D∗=αDμD+P2T∗,K∗=αKμK+αD(P1+P2)T∗μK(μD+P2T∗).

Setting the first equation of the system (2) to zero and substituting these values of state variables into the equation, we have obtained a cubic equation of T∗.10 T∗3+A1T∗2+A2T∗+A3=0.

whereA1=1ημKP2[ημKμD+{(ξαK+μTμK-ημK)P2+αDξ(P1+P2)}Tmax],A2=1ημKP2[ξαKμD-ημDμK+μTμDμK+αDμKP1-αTμKP2]Tmax,A3=-αTμDTmaxηP2.

Using the parameter values listed in Table 1, we have observed that A1>0 is positive and A2<0 and A3<0 are negative. By Descartes’ rule of sign the system (10) has exactly one positive root. By substituting the value of T∗ in (9), we can determine the interior equilibrium point E∗(T∗,D∗,K∗,SM∗).

Stability analysis of interior equilibrium

The local stability analysis of the interior equilibrium point E∗=(T∗,D∗,K∗,SM∗) has been examined here. The Jacobian matrix calculated at the interior equilibrium point E∗ is provided by,

J(E∗)=-(αTT∗+ηT∗Tmax)-P1T∗-ξT∗P12γ1T∗D∗β1-P2D∗-αDD∗0P22γ2T∗D∗β2(P1+P2)D∗(P1+P2)T∗-μK-(P12γ1T∗D∗β1+P22γ2T∗D∗β2)000-μM Fig. 3 Identifying the endemic equilibrium E∗(T∗,D∗,K∗,SM∗) of system (2) by stability analysis of the system employing two groups of cell populations: (a) Considering three types of cells (T-cell, Dendritic cell and Keratinocyte); (b) Considering three sorts of cells (Keratinocyte, Mesenchymal stromal cell and T-cell).

The eigen values of the above Jacobian matrix determine the stability criterion of the dynamical system at the equilibrium point E∗. The characteristic equation of the matrix J is given by,11 (λ+μM)(λ3+C1λ2+C2λ+C3)=0.

whereC1=(αTT∗+ηT∗Tmax)+αDD∗+μK,C2=(αDD∗+μK)(αTT∗+ηT∗Tmax)+αDμKD∗-P1P2T∗D∗+ξ(P1+P2)T∗D∗,C3=ξT∗(P1+P2)(αD-P2T∗D∗)+αDμKD∗(αTT∗+ηT∗Tmax)-μKP1P2T∗D∗.

Here C1>0 and C1C2-C3>0, applying the Routh Hurwitz criterion36, we deduce that all the roots of the equation (11) is either negative or have negative real parts. We construct the following lemma based on the aforementioned considerations.

Lemma 3

The interior equilibrium point E∗(T∗,D∗,K∗,SM∗) of system (2) is locally asymptotically stable if the following inequality holds:max{(αTT∗+ηT∗Tmax),(αDD∗-P2T∗)}<C1<1.

Optimal control strategy using TNF-α inhibitor

The optimal control approach is a powerful mathematical tool for developing and managing treatment policies for various diseases through modeling and system analysis. It is crucial for addressing drug resistance in long-term disease treatment by enabling strategic planning, improving therapies, and evaluating cost-effectiveness. This technique facilitates the creation of precise, individualized treatment regimens by determining time-dependent control functions over specific intervals.

In the context of psoriasis, the interaction rate between T-cells and DCs is amplified by the cytokine TNF-α, resulting in the abnormal proliferation of keratinocytes. Effective control of psoriasis requires the regulation of keratinocyte overgrowth, which is the primary pathological feature of this condition.Fig. 4 Dynamical behaviour of T-cells, dendritic cells, and keratinocytes in System (12) for several fixed values of the control variable.

In this section, we have examined our formulated mathematical model by introducing one control function, u(t), where u(t) is a permissible control representing the effect of a TNF-α inhibitor that can restrict the reaction rates between T-cells and dendritic cells. Therefore, the model (2) equipped with optimum control is governed by the following set of equations for the specified time interval [ts,tf],12 dT(t)dt=αT+ηT(t)(1-T(t)Tmax)-β1(1-u(t))T(t)D(t)1+γ1SM(t)-ξT(t)K(t)-μTT(t),dD(t)dt=αD-β2(1-u(t))T(t)D(t)1+γ2SM(t)-μDD(t),dK(t)dt=αK+β1(1-u(t))T(t)D(t)1+γ1SM(t)+β2(1-u(t))T(t)D(t)1+γ2SM(t)-μKK(t),dSM(t)dt=αSM-μSMSM(t).

with T(0)=T0>0,D(0)=D0>0,K(0)=K0>0  and  SM(0)=SM0>0.Fig. 5 Dynamical behavior of all model populations (T-cell, Dendritic cell, Keratinocyte and Mesenchymal stem cell) for psoriatic people (blue solid line) and after using optimal control (brown dotted line).

The description of objective functional

The problem is in minimizing the objective cost functionally expressed as follows:13 J(u(t))=∫tsf[K(t)+Wu2(t)]dt.

subject to the optimal control induced system (12). Here, our aim is to suppress the keratinocytes using TNF-α inhibitor with minimum cost. The control function u(t) in system (13) represents our objective, where it represents the effect of the TNF-α inhibitor and W is the positive weight constant on the benefit of the cost of the TNF-α inhibitor.

The control set is defined on the interval [ts,tf], where ts and tf stand for the treatment’s starting and finishing times of control. Let us define the control set,U={u(t):u(t)is Lebesgue measurable function on[ts,tf],0≤u(t)≤umax<1,t∈[ts,tf]}.

The aim of the optimum control problem is to determine the optimal control function for the system (12), represented as u∗(t).14 J(u∗(t))=min{J(u(t)):u(t)∈U}.

We applied Pontryagin’s Minimum Principle to determine the prerequisites for solving this optimum control problem37. There are non-negative bounded solutions to the optimal control-induced system (12) for the bounded Lebesgue-measurable control function and the non-negative initial conditions.

Optimal control’s characteristics

For characterization of optimal control, we apply Pontryagin’s Minimum Principle37. In order to do this, let us define the Hamiltonian H for the control problem (12) as:15 H=K(t)+Wu2(t)+Θ1(t)[αT+ηT(t)(1-T(t)Tmax)-β1(1-u(t))T(t)D(t)1+γ1SM(t)-ξT(t)K(t)-μTT(t)]+Θ2(t)[αD-β2(1-u(t))T(t)D(t)1+γ2SM(t)-μDD(t)]+Θ3(t)[αK+β1(1-u(t))T(t)D(t)1+γ1SM(t)+β2(1-u(t))T(t)D(t)1+γ2SM(t)-μKK(t)]+Θ4(t)[αSM-μSMSM(t)].

where, Θi(t)’s (i=1,2,3,4) are the adjoint variables to be determined suitably.

The adjoint system with transversality criteria Θi(tf)=0 for i=1,2,3,4 can be obtained by using Pontryagin’s Minimum Principle as:dΘ1dt=-∂H∂T,dΘ2dt=-∂H∂D,dΘ3dt=-∂H∂K,dΘ4dt=-∂H∂SM.

The optimality of the system (12) consists of the optimal control and corresponding states, thus the adjoint system takes the form16 dΘ1dt=-Θ1[η(1-2TTmax)-β1(1-u∗)D1+γ1SM-ξK-μT]+Θ2[β2(1-u∗)D1+γ2SM]-Θ3[β1(1-u∗)D1+γ1SM+β2(1-u∗)D1+γ2SM],dΘ2dt=Θ1[β1(1-u∗)T1+γ1SM]+Θ2[β2(1-u∗)T1+γ2SM+μD]-Θ3[β1(1-u∗)T1+γ1SM+β2(1-u∗)T1+γ2SM],dΘ3dt=-1+Θ1ξT+Θ3μK,dΘ4dt=-Θ1[γ1β1(1-u∗)TD(1+γ1SM)2]-Θ2[γ2β2(1-u∗)TD(1+γ2SM)2]+Θ3[γ1β1(1-u∗)TD(1+γ1SM)2+γ2β2(1-u∗)TD(1+γ2SM)2]+Θ4μSM.

with transversality conditions Θi(tf)=0 for all i=1,2,3,4. We now determine the optimal control u∗(t), using the optimality condition provided by:17 ∂H∂u|u=u∗(t)=0.

Differentiating the equation (15), partially with respect to ‘u’ and using the condition (17) we get:u∗(t)=12W{Θ3(β1TD1+γ1SM+β2TD1+γ2SM)-Θ1(β1TD1+γ1SM)-Θ2(β2TD1+γ2SM)}=Φ(t)(say).

Now, using the conventional control’s boundedness criteria and following the characteristics of the control set U that the admissible control takes the values such that 0≤u(t)≤umax<1, we can have:18 u∗(t)=0,whenΦ(t)≤0Φ(t),when0≤Φ(t)≤umaxumax,whenΦ(t)≥umax.

Therefore we can derive the following theorem from this result.

Theorem 4

If the objective cost functional J(u) attains its minimum value for the optimal control u∗(t) and (T,D,K,SM) is the corresponding optimal state for the optimal control problem (12), then there exist adjoint functions Θi(t)(i=1,2,3,4) satisfying the transversality conditions Θi(tf)=0, for i=1,2,3,4. Furthermore, the optimal control solution is given by,u∗(t)=max{0,min{1,Θ3(β1TD1+γ1SM+β2TD1+γ2SM)-Θ1(β1TD1+γ1SM)-Θ2(β2TD1+γ2SM)2W}}.

Fig. 6 Control function u∗(t) with respect to time regulated by TNF-α inhibitor.

Impulsive MSC replacement on the system with optimal control by TNF-α inhibitor

MSC is considered as a population in our proposed mathematical model stated in system (2). We have already used the optimal control strategy mathematically with a biologic TNF-α inhibitor from the initial phase of the treatment of the disease psoriasis by employing the system (12). In cases where the disease severity is high and the TNF-α inhibitor is not effective in eradicating the disease completely, stem cell replacement therapy will be used in conjunction with the biologic TNF-α inhibitor, under the guidance of specialists and with strict adherence to safety protocols. A patient with severe psoriasis requires replacement MSC, possibly for complete eradication of the disease, because long-term psoriasis causes a deficit of mature stem cells in the affected part. In stem cell therapies, an intravenous infusion of MSC from the outside of the body is often administered to the patient’s affected region to fill this deficiency. Due to this reason, we rely on an impulsive approach38. This approach involves administering a certain amount of MSC periodically, at a certain time interval, into the patient’s affected region during the course of treatment. The model presented below is almost equivalent to system (2). The dynamics of cell therapy can be modelled using impulsive differential equations:19 dT(t)dt=αT+ηT(t)(1-T(t)Tmax)-β1(1-u∗)T(t)D(t)1+γ1SM(t)-ξT(t)K(t)-μTT(t),dD(t)dt=αD-β2(1-u∗)T(t)D(t)1+γ2SM(t)-μDD(t),dK(t)dt=αK+β1(1-u∗)T(t)D(t)1+γ1SM(t)+β2(1-u∗)T(t)D(t)1+γ2SM(t)-μKK(t),dSM(t)dt=αSM-μSMSM(t),[SM(nτ)=δ+SM(nτ)-].

where ‘δ’ represents the average quantity of MSCs administered, and ‘τ’ denotes the time interval between successive infusions. Each dose, containing a fixed amount δ, is infused into the inflamed region at intervals of τ, for a total of ‘n’ (nτ≤t<(n+1)τ,n∈N∪{0}) impulsive infusions. SM(nτ)- represents the population value just before receiving the next input of MSC, i.e. just before t=nτ.

The production of MSC in the affected region is at an extremely low constant rate. We investigate the effects of pulsed treatment strategy with MSC in this model both mathematically and numerically. However, in clinical settings, a restricted number of MSC infusions are feasible for use in clinical trials. These infusions typically involve dosage amounts ranging from (1–6) ×106/kg of body weight. The infusions are administered five to six times, with intervals of 7 to 15 days between each administration throughout the course of treatment11. The main advantage of this strategy is its ability to enhance the immune system’s response against psoriasis.Fig. 7 Diagrammatic representation of the model populations with optimum control and after 80 days of treatment with TNF-α inhibitor, we perform the numerical simulation of the systemic stem cell therapy simultaneously with the same biologic (i.e. TNF-α inhibitor).

The solution of the last equation of the system (19) is of the following form,SM(t)=αSMμSM+Ce-μSMt.

where C is a constant to be determined using the following initial condition.

When 0≤t<τdSMdt=αSM-μSMSM,SM(0)=δ⇒C=(δ-αSMμSM).

20 Therefore,SM(t)=αSMμSM+(δ-αSMμSM)e-μSMt.

Now,SM(τ)=SM(τ)-+δ=αSMμSM+δ(1+e-μSMτ)-αSMμSMe-μSMτ.

We examine the system at each time interval because the MSC population’s initial condition changes over time.

When τ≤t<2τdSMdt=αSM-μSMSM,SM(τ)=αSMμSM+δ(1+e-μSMτ)-αSMμSMe-μSMτ⇒C=[δ(1+e-μSMτ)-αSMμSMe-μSMτ]eμSMτ.

21 Therefore,SM(t)=αSMμSM+[δ(1+e-μSMτ)-αSMμSMe-μSMτ]e-μSM(t-τ).

Now,SM(2τ)=SM(2τ)-+δ=αSMμSM+δ(1+e-μSMτ+e-2μSMτ)-αSMμSMe-2μSMτ.

Continuing this process up to the nth iteration, we have :

When nτ≤t<(n+1)τdSMdt=αSM-μSMSM,SM(nτ)=αSMμSM+δ(1+e-μSMτ+e-2μSMτ+……+e-nμSMτ)-αSMμSMe-nμSMτ.⇒C=[δ(1+e-μSMτ+e-2μSMτ+……+e-nμSMτ)-αSMμSMe-nμSMτ]eμSMτ.

Therefore, the concentration of MSC after n-th infusion is given by:22 SM(t)=αSMμSM+[δ(1+e-μSMτ+e-2μSMτ+……+e-nμSMτ)-αSMμSMe-nμSMτ]e-μSM(t-nτ)=αSMμSM+[δ(1-e-(n+1)μSMτ1-e-μSMτ)-αSMμSMe-nμSMτ]e-μSM(t-nτ)→αSMμSM+[δe-μSM(t-nτ)1-e-μSMτ]asn→∞.

Our analytical results and numerical findings have demonstrated that pulsed cell therapy can effectively treat severe forms of the disease in combination with a TNF-α inhibitor under optimal control approach. We have also demonstrated how the entire system dynamics will change with this combined control approach numerically.

Numerical simulation results

In this section, we have performed several types of numerical simulations of our proposed mathematical model of psoriasis, namely, systems (2), (12), and (19) to validate our theoretic analytical results. It should be noted that the initial values we have chosen for the system populations in order to perform the numerical simulations must satisfy the biological hypothesis based on the theoretical analysis discovered during the mathematical study. The values of the parameters that we have utilized here are depicted in the Table 1. All of our Numerical simulations are performed using MATLAB.

In Fig. 2, we have plotted the time series solution of the proposed system (2) for various initial conditions of the cell populations. We have observed the rising trend of keratinocyte populations while MSC counts have been sharply declining, and T-cell and dendritic cell counts will eventually reach saturation to some extent. This figure demonstrated the stability of the endemic equilibrium point E∗, which justified the analytic outcome of the asymptotic stability of the psoriatic state. This simulation reflects the behaviour of disease progression in the absence of any control or pulsed treatment.

In Fig. 3, we have plotted two different 3-D phase portraits of system (2) by taking three populations at a time for different initial values of model populations. The trajectories in Fig. 3a converge to a unique point E1∗(26.43,17.44,224.29) for various initial values of T-cells, dendritic cells, and keratinocytes. Similarly, in Fig. 3b, keratinocytes, MSCs, and T-cells converge to a certain equilibrium state, E2∗(26.43,224.29,0.34). Since it is quite difficult to evaluate the endemic equilibrium analytically, this figure allows us to estimate the endemic equilibrium point E∗(T∗,D∗,K∗,SM∗)=(26.43,17.44,224.29,0.34) numerically.

To observe the effect of the optimum control u∗(t) with respect to a TNF-α inhibitor, we have solved the optimal control problem for the mathematical system (12) numerically by taking the objective functional (13) and system (14). In a healthy state, the normal concentration of keratinocytes is considered to be K∼=200 mm-332.

In Fig. 4, we have plotted T-cell, dendritic cell and keratinocyte populations with respect to time without optimal control and considered three fixed values (u=0.55,0.64,0.7) of u for the control-induced mathematical model (12). This figure demonstrates that, although T-cell and dendritic cell concentrations rose with the higher values of the control parameter ‘u’, the keratinocyte concentrations approached a state that is considerably closer to the healthy state. Even if this option would probably make psoriatic symptoms easier to manage, the disease is not in remission since the concentration of keratinocytes is still above the threshold value of 200mm-3. As a result, we need to provide the optimal control strategy for this situation.

In Fig. 5, we have plotted both the systems of ODEs, without control and with optimal control, where the optimal value of the control function u(t) is umax=0.86. This allows us to see how treatment with a biologic TNF-α inhibitor reduces keratinocyte density. According to clinical investigations, keratinocyte hyper-proliferation can be effectively reduced by TNF-α inhibitor treatment39. However, in the severe form of the disease, this approach will not be enough to reach the keratinocyte density within the threshold value K∼=200mm-3.

In Fig. 6, we have plotted the dynamics of the control function, u(t) described in system (18), with respect to time. This figure demonstrates that at the initial stage of treatment, it required high doses of a TNF-α inhibitor to control the disease. After that, the dose decreases over time until 74 days. Later, we noticed a significant rise in the control function between 75 and 78 days, depending on the severity of the disease. Then the trajectory gradually decreases, even though the keratinocyte concentration has not reached the desired threshold value K∼.

In a severe psoriatic scenario, when treatment with a TNF-α inhibitor was unable to yield the desired outcomes, we performed stem cell replacement in combination with TNF-α inhibitor (MSCs derived from the umbilical cord is the best). We have started this approach mathematically and numerically after 80 days of observation of the disease treated with only a biologic TNF-α inhibitor.Fig. 8 The sensitivity indices of the parameters are shown in relation to the excessive growth of the keratinocyte cell population during psoriasis formation. Green trajectories represent parameters with positive sensitivity, while red trajectories represent those with negative sensitivity.

In Fig. 7, we have depicted MSC replacement with six pulsed infusions at a single dose of 3 cells/mm3 (i.e. δ=3) once every 10 days simultaneously with optimum control via a TNF-α inhibitor. In this figure, the blue trajectories for the cell populations indicate the uncontrolled psoriatic state. The brown colour trajectories represent the dynamics of the cell densities, with the optimum control strategy alone using a TNF-α inhibitor. Following eighty days of treatment with a TNF-α inhibitor, stem cell therapy combined with a TNF-α inhibitor is started. The dynamics of this combined effect of MSC infusion simultaneously with TNF-α inhibitor are shown by the green colour trajectories, and it is also clearly visible via our numerical simulation that the keratinocytes rapidly decrease to the desired density levels (≤200 mm-3) in a short period. Since the differential equation for the MSC population does not depend on the TNF-α inhibitor, the trajectories of MSC with and without TNF-α inhibitor control coincide, leading to a decline in MSC levels in inflamed regions. However, with this combined therapy, T-cell and dendritic cell concentrations are raised, and the MSC levels in the inflamed regions return to a normal state. It is important to note that in an impulsive approach, the population must be in an equilibrium state. Consequently, for this approximation, the implementation of treatment strategies will not be impeded.

In Fig. 8, we have shown the sensitivity index of a few model parameters as a function of time in accordance with Saltelli et al.40. We have found that certain parameters are positively and a few are negatively responsive with respect to the excessive growth of the keratinocyte population. The green lines indicate those parameters that are positively sensitive, and the red lines indicate those parameters that are negatively sensitive.

Discussion

The prediction of psoriasis progression using mathematical models is an effective approach to forecasting disease progression, optimizing treatment applications, and anticipating future manifestations. Mathematical models can be developed to devise treatment strategies for various ailments. Previously, no cell-dynamic mathematical model of psoriasis had incorporated stem cell replacement as a therapeutic approach. In this study, we introduce MSC replacement combined with a biologic TNF-α inhibitor through mathematical and numerical analysis for severe forms of the disease, particularly when biologic drugs alone are insufficient for complete eradication.

Initially, we focus on cell-to-cell interactions between T-cells, dendritic cells, and keratinocytes and consider an additional population of MSCs, which can modulate immune system responses. Although various factors, like abnormal differentiation and a complex network, contribute to disease progression, which we have not considered here in order to simplify the model. We have considered only keratinocyte hyper-proliferation as the key factor. MSCs have multiple aspects influencing disease progress, but we emphasize their positive role in inhibiting keratinocyte hyper-proliferation through the production of tolerogenic anti-inflammatory cytokines and reestablishing normal autoimmune responses.

In severe psoriasis, local MSC counts decline drastically and become dysfunctional, so we infuse healthy MSCs from outside (umbilical cord-derived MSC will be one of the best options). According to Fig. 7, when TNF-α inhibitor treatment alone fails to produce the desired results, we introduce a combined stem cell replacement strategy in an impulsive manner alongside the same TNF-α inhibitor. Our study demonstrates an impulsive therapeutic approach using MSCs in conjunction with the biologic TNF-α inhibitor, following 80 days of monitoring with the same biologic alone.

Conclusion

Analytical and numerical results revealed that within two months of combined therapy (replacement of MSC with a TNF-α inhibitor), keratinocyte density dropped to desired levels, and other cell concentrations returned to normal in the affected region, with increased local MSC counts. Consequently, the hyper-proliferation of keratinocytes was effectively controlled. This suggests that revising parameters in the carefully constructed disease model to maintain a proper balance between immune cells, keratinocytes, and local MSCs could potentially lead to a complete cure for psoriasis. The proposed mathematical model provides valuable insights for mathematicians and physicians in developing appropriate diagnostic and therapeutic tools for psoriasis. Additionally, this study suggests that stem cell treatments could be a viable future alternative to biologic drugs for the complete eradication of severe cases of the disease.

Acknowledgements

This research work was supported by the National Social Science fund of China under Grant No. 22VSZ032, School of Mathematics and Statistics at Beijing Technology and Business University, Beijing, P.R. China, and the DST FIST Programme, Government of India [no.: SR/FST/MS-II/2021/101(C)], Department of Mathematics, Jadavpur University, Kolkata-700032, India. The authors would like to thank sincerely the anonymous reviewers for their valuable suggestions, which enhanced the manuscript in various aspects.

Author contributions

Conceptualization, S.K. and P.K.R.; methodology, S.K.; software,S.K.; validation, S.K. and T.G.; formal analysis, S.K. and T.G.; investigation, S.K. and X.C.; writing-original draft preparation, S.K. and T.G.; supervision, P.K.R. and X.C. All authors reviewed the manuscript.

Data availibility

No data associated in the manuscript. However, the parameters that were provided for numerical simulation to validate the theoretical findings are included in this article.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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