# Mathematical Modeling to Study Multistage Stem Cell Transplantation in HIV-1 Patients

Authors: Manar A. Al Qudah; Sana’a A. Zarea; Saoussan A. Kallel-Jallouli
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

Stem cells as a therapeutic measure for the treatment of different diseases have a great potential to give rise to different mature cells as they could be used to treat HIV-1 patients when provided with the convenient factors. Thus, this paper proposes a new mathematical model, represented by a system of ODEs, to study the effect of stem cell transplantation for HIV-1 patients. Since stem cells lineage passes through many stages to become more specialized cell types, investigating (theorizing) the best stage for these cells to be engrafted was needed. The proposed system of ODEs can help medicine make the right decision about the proposed therapy.


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## Body

## 1. Introduction

A virus called HIV (human immunodeficiency virus) causes AIDS (acquired immunodeficiency syndrome). HIV is a retrovirus that attacks the vital organs and the cells of the human immune system, making people more vulnerable to diseases and infections. Without treatment, HIV can develop into AIDS in many cases. In the absence of therapy the virus progresses. The developing rate of the virus differs between individuals and changes according to many factors. These factors include ability of the body to fight against HIV, the age of the individual, access to healthcare, the individual’s genetic inheritance, the presence of other infections, and the resistance to certain strains of HIV.According to the World Health Organization’s guidelines issued in June 2018, there is no viable cure or vaccine for HIV, but treatments can improve patients’ quality of life.When given the right treatment, some patients can live with HIV and reduce their viral load to such a degree that it is not discoverable. Studies by the CDC (Centers for Disease Control and Prevention) proved that in individuals who have no discoverable viral load “through sex there is effectively no risk of transmitting HIV to an HIV-negative partner.” Generally, people who live with HIV take a composition of medications called HAART (highly active antiretroviral therapy) or CART (combination antiretroviral therapy) [1]. This antiretroviral therapy (ART) prevents or slows the virus from developing.We can find two types of HIV: HIV-1 and HIV-2. HIV-1 is more infective and more dangerous than HIV-2 as mentioned in [2]. Also HIV-1 concretes both lymphadenopathy associated virus (LAV) and human T-lymphotropic virus-3 (HTLV-III). HIV-2 is largely found in West Africa [3], according to its relatively poor capacity for transmission.Understanding the dynamics of HIV-1 by mathematical modeling plays a basic role in medicine. All mathematical models of HIV-1 are based on the model in [4] where they considered the interaction between four components: uninfected CD4 + T-cells, latent infected CD4 + T-cells, infected CD4 + T-cells, and the free virus. Then, many researchers modified this model such as in [5] where they presented the same model without the latent infected CD4 + T-cells. The models that are studied in [6–10] focused on models interested in the dynamics of HIV and a hope for a cure for the infection.Stem Cell Therapy. Nowadays, stem cell studies are a field of much interest, and within this area there is a need for mathematical model representation of the dynamics growth for stem cells. Inside any organism the new cells need a source, which requires a continuous role to replace the older cells, to keep the balance of homeostasis for tissue, and to react to the external ecological stresses. In each organ adult stem cells [11, 12] are planning this important role to replace the older cells. Through its niche in the organism and its special nature, the stem cell is able to renew itself. The daughter cells are generated from the stem cells through cell divisions and will differentiate into one of any number of organ specific cell types; in this process of differentiation, they lose their initial capacity for self-renewal. Lie and Xie [13] discussed an accurate equilibrium between stem cell self-renewal and differentiation; also they studied the way stem cells regulate the body and treat human disease. A lot of interest has also been focused on embryonic stem cells [14, 15], a kind of stem cell from earlier in human development, and it is a smart cell that can be differentiated into any type of cells within the body.Adult stem and embryonic cells have the same properties of proliferation, differentiation, and self-renewal. Hence, thinking about their transplantation in the patient can be a good idea to replace injured cells and help in restoring the function of damaged cells. Recent studies have shown the efficacy of this technique to regenerate some body organs [16]. All of the abovementioned properties are a critical field within the research of stem cells. According to the recent researches that studied proliferation of stem cells, this paper introduces a mathematical model. As discussed in Henrigou et al. [17] and Bonab et al. [18], many therapies of stem cells depend on the availability of the number of the stem cells to transfer. Therefore, the representation of the rate of growth for a population of stem cells as a mathematical model will be very useful. Several models applied to stem cell proliferation as in Loeffler and Wichman [19] and as in Cowan and Morris [20] have been reviewed and accordingly this paper’s mathematical model was constructed.Depending on the ability of stem cells to differentiate into some of the specialized phenotypes in response to appropriate signals [21], this paper proposes to treat HIV-1 patients by engraftment of the type of cells able to transform to T-cells (CD4+T). Other varieties of cells were tested for restoring the heart [22–26], the cornea [27], and pancreatic functions [28, 29].This paper begins by recalling a known mathematical model for multilineage stem cells, which was used to formulate a new mathematical model of the viral dynamic of HIV-1 treatment by stem cell therapy. The study of the model enables us to choose the best stage in the lineage to engraft these cells and help medicine make the good decision.


## 2. Mathematical Modeling of n-Multistage Stem Cell Maturation

Letsi be the density of stem cells at the ith stage for i=1,2,…,n-1, of differentiation and the density of mature cells for i=n. Let h(t) be the concentration of signaling molecules which depends only on the density of mature cells sn; thus the dynamics of a cytokine (a protein, polypeptide, or glycoprotein that are used in signaling and intracellular signaling and also aid in cell-to-cell communication in immune responses and stimulate the movement of cells towards sites of inflammation and infection) can be represented as a first-order differential equation:dh/dt=μ(1-h-γhsn(t)), where μ is the death rate of a mature stem cell population and γ is a +ve constant; solving this equation using the approximation of the quasi-steady state we get h=hγt=1/(1+γsn(t)). The stem cell lineage can be expressed as an ODEs system [30]:(1)ds1dt=f1ht,s1t,ds2dt=f2ht,s2t+g1ht,s1t,⋮dsndt=fnht,snt+gn-1ht,sn-1t.Herefiht,sit is the change of sit that is caused by processes at the ith stage of maturation. If gain cells caused by the loss of differentiation or death are more weakened than the self-renewal and proliferation, then fiht,sit>0; otherwise fiht,sit<0. But if we suppose that mature cells are postmitotic (they cannot proliferate), then since fn is accounted only for cell death, we have fnht,snt<0. We denote giht,sit≥0 to be the flux of cells from the maturation stage i to maturation stage i+1 due to differentiation.Letpit denote the proliferation rate of the subpopulation of type i at time t, ai(t) denote the fraction of self-renewal, and μi(t) denote the death rate. Then, the flux to mitosis at time t will be pi(t)sit. The fraction ai(t) of daughter cells dose not differentiate. Thus, the influx to cell population i after cell division is 2ai(t)pi(t)sit and the flux to the next cell compartment is 2(1-ait)pi(t)sit. The flux to death at time t is given by μi(t)sit. Hence we obtain(2)fit=2ait-1pitsit-μitsit,fori<n,git=21-ai-1tpi-1tsi-1t,for1<i<n,fnt=-μntsnt.Assume that the proliferation rates and the death rates are constant with time, and self-renewal fractions are controlled by the signal feedback,ait=aih(sit=ai,maxh(sit, where ai,max=ai is the maximal fraction of self-renewal. Therefore, system (1) can be written as (3)ds1dt=2a1h-1p1s1-μ1s1,ds2dt=2a2h-1p2s2+21-a1hp1s1-μ2s2,⋮dsidt=2aih-1pisi+21-ai-1hpi-1si-1-μisi,⋮dsndt=21-an-1hpn-1sn-1-μnsn.Using thath=1/(1+γsn), then system (3) becomes(4)ds1dt=2a11+γsn-1p1s1-μ1s1,ds2dt=2a21+γsn-1p2s2+21-a11+γsnp1s1-μ2s2,⋮dsidt=2ai1+γsn-1pisi+21-ai-11+γsnpi-1si-1-μisi,⋮dsndt=21-an-11+γsnpn-1sn-1-μnsn.For the initial data and the model parameters, we add the following biologically relevant assumptions as in [30]:(5)tϵ0,∞.si0≥0,fori=1,…,n,μi≥0,fori=1,…,n-1,μn>0,pi>0,fori=1,…,n-1,ai,maxϵ0,1,fori=1,…,n-1,γ>0.


## 3. Mathematical Modeling of the n-Multistage Stem Cell Therapy for HIV-1 Patients

In our previous work [31], we proposed for the first time in literature a model that explains the effect of stem cell transplantation on HIV-1 viral dynamics. The basic model structure that describes the therapy uses four state variables: stem cells (S), healthy T-cells (T), infected T-cells (Ti), and the virus (V): (6)dSdt=bαS-αD-δSSdTdt=λT-dTT-kTTV+2αD+αAbASdTidt=kTTV-ρTiTidVdt=πTiTi-cV,where the parameters b,αS,αD are nonnegative and bαS-αD-δS are nonpositive. We know there are three possibilities for a stem cell to divide [32]:(i) Symmetric self-renewal, where a stem cell can divide to become two stem cells, with probabilityαs(ii) Asymmetric self-renewal, where one daughter cell remains a stem cell while the other does not inherit this characteristic, with probabilityαA(iii) Symmetric commitment differentiation, where a stem cell can divide to become two committed cells, with probabilityαDThereforeαA+αS+αD=1. We suppose that the stem cells S divide at rate b and die at rate δS. We need to introduce an amplification factor A to finally get the simplified ODE: dS/dt=[bαS-αD-δS]S, such that bαS-αD-δS represents the net per-capita growth rate of stem cells [33]; for the uninfected susceptible T cells the production rate is λT, and the death rate is dT and they become infected cells (Ti) at rate kT; the infected cells are produced and die at the rates kT and ρTi, respectively; free virions (V) are produced from infected cells at rate πTi and decline at rate c.Since stem cellsS=S1 need many stages to transform to mature cells Sn=T, in this paper we shall perform our previous work [31] by including the stages of the lineage of stem cells given by system (4). Our system (6) can then transform to a new system that describes the n-multistage stem cells transformation in the therapy of HIV-1 patients. This new model describes the dynamics of stem cell transplantation on the virus using (n+2) state variables ((n+2) ODEs): (7)dS1dt=2a11+γT-1p1S1-μ1S1,dS2dt=2a21+γT-1p2S2+21-a11+γTp1S1-μ2S2,⋮dSidt=2ai1+γT-1piSi+21-ai-11+γTpi-1Si-1-μiSi,⋮dSn-1dt=2an-11+γT-1pn-1Sn-1+21-an-21+γTpn-2Sn-2-μn-1Sn-1,dTdt=λT-dTT-kTTV+21-an-11+γTpn-1Sn-1-μnT,dTidt=kTTV-ρTiTi,dVdt=πTiTi-cV.Different numbers of compartment n were chosen in previous researches, as mentioned in [34]. By n=8 we mean that the direct reflection of the biological case of differentiation of white blood stem cells into neutrophilic granulocytes [35]; in this case we treat all immature and mature cells as separated populations so they might be understood as stem cells and their direct progeny. The compartment n=6 corresponds to short- and long-term repopulating stem cells, multipotent progenitor cells, committed progenitor cells, precursors, and mature cells, so it is related to the properties of division cells. For simplicity we consider n=2. The nth compartment is the population of mature cells, which we suppose, for simplicity, do not divide any more. In the next section, we will study the following 5-dimensional version extracted from model (7):(8)dS1dt=2a11+γT-1p1S1-μ1S1,dS2dt=2a21+γT-1p2S2+21-a11+γTp1S1-μ2S2,dTdt=λT-dT+μ3T-kTTV+21-a21+γTp2S2,dTidt=kTTV-ρTiTi,dVdt=πTiTi-cV.Here μ3 is the death rate of T-cells that come out from the final stage of the stem cell maturation, so for simplicity let dT+μ3=β.


## 4. Study of the Mathematical Model of the 2-Multistage Stem Cell Therapy for HIV-1 Patients

In this section, we are going to discuss the stability of the model of viral dynamics corresponding to the stem cell therapy given by system (8). We shall try to find out equilibrium points of system (8) in the form (S1,S2,T,Ti,V). As in [36] we take γ to be very small; therefore we replace ai/(1+γT) by ai, for i=1,2,…,n-1, which is equivalent to choosing h=1/(1+γT)ϵ(0,1) as in [37].The following theorem gives us the equilibrium points for the following system:(9)dS1dt=2a1-1p1S1-μ1S1,dS2dt=2a2-1p2S2+21-a1p1S1-μ2S2,dTdt=λT-βT-kTTV+21-a2p2s2,dTidt=kTTV-ρTiTi,dVdt=πTiTi-cV.If the basic reproduction ratio of the viruses is given by R0=kTλTπTi/cρTiβ, then we get the following theorem.Theorem 1.
System (9) has two equilibrium points given byP1S1,S2,T,Ti,V=(0,0,λT/β,0,0) corresponding to the free disease case and P2S1,S2,T,Ti,V = 0,0,(λT/β)(1/R0),(βc/kTπTi)R0-1,(β/kT)R0-1 corresponding to the endemic case.Proof.
To find the equilibrium points, we solve the following system forS1,S2,T,Ti,V:(10)2a1-1p1S1-μ1S1=02a2-1p2S2+21-a1p1S1-μ2S2=0λT-βT-kTTV+21-a2p2s2=0kTTV-ρTiTi=0πTiTi-cV=0.By introducing R0=kTλTπTi/cρTiβ, we get the result.In the next theorems we study the stability at the equilibrium pointsP1 and P2.Theorem 2.
The free disease caseP1 is asymptotically stable if R0<1 and unstable if R0>1.Proof.
The Jacobian matrix corresponding to system (9) is given by(11)J=2a1-1p1-μ1000021-a1p12a2-1p2-μ2000021-a2p2-β-kTV0-kTT00kTV-ρTikTT000πTi-c.At the health point P1 the Jacobian matrix becomes(12)J1=2a1-1p1-μ1000021-a1p12a2-1p2-μ2000021-a2p2-β0-kTλTβ000-ρTikTλTβ000πTi-c.So the characteristic equation det⁡J1-rI=0 has the eigenvalues(13)r1=2a1-1p1-μ1,r2=2a2-1p2-μ2,r3=-β,r4,5=-βc+ρTi±βc+ρTi2-4ββcρTi-kTλTπTi2β=-βc+ρTi±ββc2-2βcρTi+4kTλTπTi+βρ2Ti2β.To get the asymptotically stable case for the free disease equilibrium point we shall have for the first and the second roots2a1-1p1<μ1 and 2a2-1p2<μ2 which are true and agree with [30, 31, 36]; since the population of mature cells S3 reaches some value, then the term (2ai(t)-1)pi(t)Si(t)<0, for i<n, and the number of cells at stage i decreases, while the lowness of the density for the mature cells causes (2ai(t)-1)pi(t)si(t)>0, for i<n, and if the death rates are not too high, then the number of cells at stage i increases. These show how the level of mature cells affects the dynamics of each cell subpopulation. Also it is clear that r3=-β<0, and if R0=kTλTπTi/cρTiβ<1, then cρTiβ2>kTλTπTiβ, so βc+ρTi2>ββc2-2βcρTi+4kTλTπTi+βρ2Ti; then r4,5<0.IfR0=kTλTπTi/cρTiβ>1, then cρTiβ2<kTλTπTiβ, so βc+ρTi2<ββc2-2βcρTi+4kTλTπTi+βρ2Ti; then r4,5>0. Thus, we get the result.Remark 3.
(1) IfR0=1, then P1=P2
(2) IfR0<1, then P2 is impossibleTo study the endemic pointP2, we need the following lemma.Lemma 4 (see [37]).
The polynomialfr=r3+a1r2+a2r+a3 has the following results:
(i) Ifa3<0, then f(r) has at least one positive root
(ii) Ifa1>0,a3≥0, and a2≥0, then f(r) has no positive root
(iii) Ifa1>0,a3≥0, and a2<0, then f(r) has a positive root: (14)r=13-a1+a12-3a2Theorem 5.
IfR0≥1, then the endemic point P2 is asymptotically stable.Proof.
At the point0,0,(λT/β)(1/R0),(βc/kTπTi)R0-1,(β/kT)R0-1 the Jacobian matrix becomes(15)J2=2a1-1p1-μ1000021-a1p12a2-1p2-μ2000021-a2p2-βR00-kTλTβR000βR0-1-ρTikTλTβR0000πTi-c.So the characteristic equation det⁡J2-rI=0 has the eigenvalues r1=2a1-1p1-μ1, r2=2a2-1p2-μ2, and the other eigenvalues are roots of the polynomial fr=r3+a1r2+a2r+a3=0, where a1=c+ρTi+βR0,a2=βR0c+ρTi, and a3=cρTiβ(R0-1).SinceR0≥1, then βR0c+ρTi≥0, and cρTiβ(R0-1)≥0. Then, according to Lemma 4, we deduce that P2 is asymptotically stable.


## 5. Study of the Mathematical Model of the n-Multistage Stem Cell Therapy for HIV-1 Patients

In this section, we will consider previously given(n+2)-dimensional system (7). As mentioned in the previous section, we can replace ai/(1+γT) by ai for i=1,2,…,n-1. Then (7) can be written as(16)dS1dt=2a1-1p1S1-μ1S1,dS2dt=2a2-1p2S2+21-a1p1S1-μ2S2,⋮dSn-1dt=2an-1-1pn-1Sn-1+21-an-2pn-2Sn-2-μn-1Sn-1,dTdt=λT-dT+μnT-kTTV+21-an-1pn-1Sn-1,dTidt=kTTV-ρTiTi,dVdt=πTiTi-cV.Assume thatdT+μn=β′. If the basic reproduction ratio of the viruses (that depends on the patient) is given by R0=kTλTπTi/cρTiβ′, then, using mathematical induction, we can deduce the following theorem.Theorem 6.
System (16) has two equilibrium points given by the following:
P 1 S 1 , S 2 , … , S n - 1 T , T i , V = 0,0 , … , 0 , λ T / β ′ , 0,0, corresponding to the free disease case, and (17)P2S1,S2,…,Sn-1T,Ti,V=0,0,…,0,λTβ′1R0,β′ckTπTiR0-1,β′kTR0-1,corresponding to the endemic case.Theorem 7.
The free disease caseP1 is asymptotically stable if R0<1 and unstable if R0>1.Remark 8.
(1) IfR0=1, then P1=P2
(2) IfR0<1, then P2 is impossibleTheorem 9.
IfR0≥1, then the endemic point P2 is asymptotically stable.


## 6. Conclusion

We have formulated an n-multistage system of ODEs model to study the influence of the treatment of HIV-1 infection with stem cell transplantation. Evidently the effect of the therapy on the viral dynamics depends on the patient (depends onR0).(i) IfR0≤1, we have only one equilibrium point corresponding to the free disease case P1S1,S2,…,Sn-1,T,Ti,V=0,0,…,0,λT/dT,0,0, which is asymptotically stable. We can then conclude that the infected T-cells and virus particles will be cleared and the stem cells exhausted. In this case, the patient is healedHere we can also add a comment on the best stage we can choose for the mutant cells to be transplanted on the patient.Since all the componentsS1,S2,…,Sn-1 of the point P1 are equal to zero, then it is better to transplant the stem cells when they are nearly transformed to mature T-cells. This result is known in the cases of corneal transplantation, skin transplantation, and cardiac disease.(ii) IfR0>1, the free disease case P1 is unstable, and the infected state P2S1,S2,…,Sn-1,T,Ti,V=0,0,…,0,(λT/dT)(1/R0),(dTcV/kTπTi)(R0-1),(dT/kT)(R0-1) is asymptotically stable, which means that any HIV-1 infection for that patient will progress to chronic infection after stem cell exhaustionTherefore, stem cell therapy will not offer a cure to that infected person (R0>1) but simply will help delay progression to the chronic stage. If we repeat the transplantation of stem cells in a manner to prevent its exhaustion from the patient, we can prevent AIDS.We advise medicine to try to replace the chemical drug therapy with the apparently safe stem cell therapy. But will clinical experiments meet the expectations?


## 7. Future Works

For more interest we will study a fractional order of mathematical modeling to study multistage stem cell transplantation on HIV-1 patients based on [38, 39]. The presence of a fractional differential order in differential equations can lead to more consistency and more accuracy to real life than the integer-order models because the fractional derivatives enable the description of the memory and hereditary properties inherent in various processes and materials. The most important property of these models is their nonlocal property which does not exist in the integer-order differential operators. We mean by this property that the next state of a model depends not only on its current state but also on all of its historical states.



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