Single injection of CD8+ T lymphocytes derived from hematopoietic stem cells – Mathematical and numerical insights

BioSystems 144 (2016) 46–54

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Single injection of CD8+ T lymphocytes derived from hematopoietic stem cells – Mathematical and numerical insights Adam Korpusik a,∗ , Mikhail Kolev b a b

Faculty of Technical Sciences, University of Warmia and Mazury, ul. Oczapowskiego 11, 10-719 Olsztyn, Poland Faculty of Mathematics and Computer Science, University of Warmia and Mazury, ul. Słoneczna 54, 10-710 Olsztyn, Poland

a r t i c l e

i n f o

Article history: Received 28 December 2014 Received in revised form 1 April 2016 Accepted 14 April 2016 Available online 16 April 2016 Keywords: Mathematical model Stem cell based therapy Hematopoietic stem cells (HSC) Cytotoxic T lymphocytes (CTL) HIV

a b s t r a c t Recently, hematopoietic stem cell (HSC) based therapy is being discussed as a possible treatment for HIV infection. The main advantage of this approach is that it limits the immune impairing effect of infection by introducing an independent influx of antigen-specific cytotoxic T lymphocytes (CTL). In this paper, we present a mathematical approach to predict the dynamics of HSC based therapy. We use a modification of a basic mathematical model for virus induced impairment of help to study how virus – immune system dynamics can be influenced by a single injection of CD8+ T lymphocytes derived from hematopoietic stem cells. Our mathematical and numerical results indicate that a single, large enough dose of genetically derived CTL may lead to restoration of the cellular immune response and result in long-term control of infection. © 2016 Elsevier Ireland Ltd. All rights reserved.

1. Introduction Many viruses (e.g. HIV, LCMV, HCV) have the ability to escape the immune response (Wodarz, 2007; Thomsen et al., 1996; Yao et al., 2001). Infection by a virus of this kind is very hard to cure (or incurable), even with proper medical intervention. Pharmaceutical treatment greatly reduces the numbers of free virus particles in the system, but does not eradicate them completely. Virus still persists in the patient’s body and can reproduce with very little resistance from the immune system. As a result, the infection is never cleared from the body, as opposed to a normal, healthy immune reaction (Anderson and May, 1991; Kolev, 2011). There are several mechanisms by which viruses can evade the defense system of the host, e.g. antigenic escape or virus latency (Lydyard et al., 2004; McMichael and Phillips, 1997; Farci et al., 2000; Valyi-Nagy et al., 2000). Some viral infections can also impair the immune response by targeting cells of the immune system specifically. One way to achieve immune impairment is to infect the CD4+ T lymphocytes (Th cells). These cells play an important role in the regulation of the adaptive immune response (Pinchuk, 2002; Janssen et al., 2003). They are necessary for the secondary

Abbreviations: HSC, hematopoietic stem cell; CTL, cytotoxic T lymphocyte; APC, antigen presenting cell; Th cell, CD4+ T lymphocyte. ∗ Corresponding author. E-mail addresses: [email protected] (A. Korpusik), [email protected] (M. Kolev). http://dx.doi.org/10.1016/j.biosystems.2016.04.010 0303-2647/© 2016 Elsevier Ireland Ltd. All rights reserved.

expansion of antigen specific cytotoxic T lymphocytes (CTLs, CD8+ T lymphocytes), whose main role is to recognize and kill infected, cancerous and foreign cells. The lack of functional Th cells results in CTL population decline and reduces the cellular immune response to a weak helper-independent response (Rosenberg et al., 1997; Christensen et al., 2001). Another way to impair the immune response is to target the antigen presenting cells (APCs), which take part in CTL activation and expansion as well (Abbas and Lichtman, 2004; Borrow et al., 1995). Recent advantages in genetic engineering create new possibilities for personalized treatment of a vast variety of diseases. Gene based therapies are currently being developed to fight cancer (Clay et al., 1999; Kalos et al., 2011; Davila et al., 2014; Distler et al., 2016) as well as viral infections (Rossi et al., 2007; van Lunzen et al., 2011; Hoxie and June, 2012; Dey and Pillai, 2015). Many of these studies concentrate on modifications of mature CTLs in order to redirect them toward a selected antigen (Cooper et al., 2000; Hughes et al., 2005; Joseph et al., 2008). However, last decades have also been a time of enormous progress in the stem cell research, e.g. pluripotent stem cells have been obtained from blood and somatic cells (Yu et al., 2007; Brown et al., 2010; Zhou et al., 2011; Park et al., 2012). These advances in genetic engineering and stem cell research make it possible to consider stem cell based gene approaches to treat various diseases (Kitchen and Zack, 2011; Lai, 2012; Bexell and Svensson, 2012; Leventhal et al., 2013; Bigger and Wynn, 2014; Garcia et al., 2015; Aronovich and Hackett, 2015). Over the past few years, hematopoietic stem cells (HSCs) have been discussed as a basis for a gene therapy aiming to cure immune

A. Korpusik, M. Kolev / BioSystems 144 (2016) 46–54

impairing infections, most of all HIV (Kitchen et al., 2011; Kiem et al., 2012). Hematopoietic stem cells are the cells of the bone marrow that give rise to all blood cells (Pinchuk, 2002; Abbas and Lichtman, 2004). T cell precursors (a fraction of HSC population) are translocated to the thymus, where they mature into T lymphocytes. These new cells are specific for a vast repertoire of antigenic epitopes. In order to avoid autoimmune reaction, they undergo a strict selection process within the thymus. Mature CD4+ and CD8+ T lymphocytes that survive the selection process are then transferred to the lymph nodes. Upon contact with specific antigen, they are activated, they proliferate, carry out their effector functions and establish immunological memory (Lydyard et al., 2004; Abbas and Lichtman, 2004; Flossdorf et al., 2015). The HSC based therapy (Kitchen et al., 2011; Kiem et al., 2012; Zhen et al., 2015) assumes that the HSCs of the patient are obtained and modified to bear a T cell receptor that reacts specifically to the given antigenic epitope. Next, these genetically modified cells are introduced into the patient’s thymus. This should result in the production of fully functional CTLs specific to the selected virus strain, capable of killing infected cells. Apart from the antiviral effect of the therapy, there are two other major advantages of this approach. Firstly, since the production of new antigen specific CTLs is independent of viral stimulation and Th cell help, the effects of immune impairment are limited. Secondly, the engineered T cells still have to undergo the thymic selection process, which eliminates the possible defects of the cells, such as autoimmune reactions. There are still very little experimental data on the HSC based therapy of viral infections. However, CTLs derived from genetically modified HSCs have already been shown to suppress the virus population, both ex vivo (Kitchen et al., 2009) and in vivo (Kitchen et al., 2012). In the latter experiment, the HSC based therapy resulted in significant reduction of plasma viremia, the number of cells infected with HIV was much lower as well. Moreover, the following results were reported: multilineage hematopoiesis, creation of effector CTLs, and reduced Th cell depletion. The antigenic escape of the virus was not observed during the experiment. Mathematical and numerical approaches have also been applied to predict the dynamics of HSC based therapy of viral infections, see e.g. (Murray et al., 2009). In our previous work (Korpusik, 2014; Korpusik and Kolev, 2013), we have studied the influence of a constant influx of new CTLs on the virus – immune system interactions, using a modification of the basic mathematical model of virus induced impairment of help (Wodarz, 2007). Our results were qualitatively the same as those obtained in the in vivo experiment (Kitchen et al., 2012): restored cellular response, limited Th cell depletion, suppression of plasma virus and lower numbers of infected Th cells. However, the therapy does not result in a stable influx of cells. Therefore our simplifying assumption can only give sensible results in the short-term dynamics. Moreover, the strengthening of the immune response with engineered T cells can have effects exceeding the time of the actual cell influx. It is possible that the resulting higher number of memory CTL can lead to the immune control of the virus, or even viral clearance (Kitchen et al., 2011; Murray et al., 2009). To investigate this possibility, we take a look at the virus – immune system interactions in an APC/Th cell targeting infection after a single injection with a limited dose of CD8+ T lymphocytes derived from HSCs. 2. Mathematical model In order to predict the course and outcome of the therapy, we use a mathematical model based on a basic model for virus induced impairment of help (Wodarz, 2007; Wodarz et al., 1998; Wodarz and Nowak, 1999). We introduce an additional equation (Eq. (5)) describing the dynamics of a limited dose of CD8+ T lymphocytes

47

derived from HSCs. This equation is introduced similarly as in Kronik et al. (2010). Moreover, an additional term, illustrating the influx of new CTLs, is added in the equation for CTL precursor cells (Eq. (3)). The proposed model of HSC based therapy consists of ordinary differential equations describing the dynamics of the following five populations: CD8+ T lymphocytes derived from HSCs (h), uninfected Th cells/APCs (x), infected Th cells/APCs (y), precursor (w) and effector CTL (z). The model is given as follows: x˙ =  − dx − ˇxy,

(1)

y˙ = ˇxy − ay − pyz,

(2)

w˙ = mh + cwxy − qwy − b1 w,

(3)

z˙ = qwy − b2 z,

(4)

h˙ = −h.

(5)

We assume nonnegative initial conditions and positive values of all the parameters. Target cells (Eq. (1)), namely uninfected Th cells or APCs, are assumed to be produced at a rate  and die at a rate dx. The more cells are infected, the more free virus particles are present in the system. Therefore, the rate of infection (ˇxy) is assumed to be proportional to the concentrations of both uninfected and infected cells and to the rate constant ˇ describing the ability of the virus strain to infect the target cells. Excessive virus production inside the infected cells (Eq. (2)) can result in shortening their lifespan. This is denoted by the death rate constant a. Infected cells can also be killed by the immune system of the host. Note that the rate of effector CTL mediated killing of infected cells (pyz) is proportional to the parameter p denoting the strength of the cellular immune response and to the numbers of infected cells and effector CTL. The cellular immune response is described by Eqs. (3) and (4). We assume that precursor CTLs proliferate at the rate cwxy. It is important to note that this rate does not depend solely on the numbers of precursor CTLs (w) and their ability to proliferate (denoted by c). The more infected cells (y) there are, the stronger the stimulus for the immune response is. Moreover, the numbers of uninfected target cells (x) play a major role in the cellular immune response. This is because the functional Th cells and APCs are necessary for expansion of memory CTL (Rosenberg et al., 1997; Borrow et al., 1995). Lack of these cells results in impairment of the immune response. A fraction (denoted by q ∈ [0, 1]) of precursor CTL population turns into the effector CTLs that identify and kill the infected cells. Parameters b1 and b2 denote the death rate constants of precursor and effector CTLs, respectively. The last equation of the model (Eq. (5)) describes the dynamics of a single dose of genetically modified CD8+ T lymphocytes derived from HSCs. In order to cure the infection, these cells are introduced into the patient’s thymus. They leave the thymus at a rate h. The depletion rate constant  is determined by the average time between the introduction of engineered T cells and the end of the selection process. A fraction of cells, given by the parameter m ∈ [0, 1], successfully passes the thymic selection and become functional CTL. The model is an autonomous system of first order ordinary differential equations, which right-hand sides are of the C1 class. Therefore, the existence and uniqueness of the solutions is guaranteed for all t ≥ 0. Moreover, it can be shown that these solutions are nonnegative. Let us start from Eq. (1) and assume that there exist some values of t > 0 for which x(t) < 0. From the initial condition x(0) ≥ 0 and the continuity of x(t), we obtain that there exists a certain moment t1 > 0, such that x(t) > 0 for t < t1 , x(t1 ) = 0 and x(t) < 0 for ˙ 1 ) < 0. On the other hand, from Eq. (1) we obtain t > t1 . This yields x(t ˙ 1 ) =  − dx(t1 ) − ˇx(t1 )y(t1 ) =  > 0, which is contradictory to x(t ˙ 1 ) < 0. Therefore, the function x(t) has to be nonnegative for all x(t

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t ≥ 0. Applying the same reasoning to the other equations of the model (in the following order: Eqs. (2), (5), (3) and (4)) guarantees the nonnegativity of the solutions. Finally, it can be shown that the solutions of the model are bounded on any finite time interval [0, T], which excludes the pos˙ sibility of a finite time blow-up. From Eq. (5), we have h(t) < 0, ˙ ≤ , which yields therefore h(t) ≤ h(0). Next, from Eq. (1) we get x(t) ˙ ˙ x(t) ≤ T + x(0). From Eqs. (1) and (2), we obtain x(t) + y(t) ≤ . Since the solutions of the model are nonnegative, the function y(t) has to be bounded as well. Applying these results to Eqs. (3)–(4) yields the boundedness of w(t) and z(t) on any arbitrary time interval [0, T]. 3. Results 3.1. Equilibria of the model Our modification of the model is made under the assumption that the dose of engineered CD8+ T lymphocytes is eventually depleted. For every equilibrium, we have h = 0 (see Eq. (5)) and, consequently, mh = 0 in Eq. (3). Therefore, the equilibria of our mathematical model are analogous to those of the original model for virus induced impairment of help. Following the results from Wodarz and Nowak (1999), we obtain 3 biologically significant equilibria. The first outcome is observed when the viral replication is too low to establish a long-term infection (y = 0 in Eq. (2)). It is given by the following condition: R0 = ˇ/ad < 1, where R0 is the basic reproductive ratio of the virus. If this condition is satisfied, our system of Eqs. (1)–(5) converges to the following virus-free equilibrium: xvf =

 , d

yvf = 0,

wvf = 0,

zvf = 0,

hvf = 0.

(6)

The condition for R0 can be obtained from a standard analysis of the eigenvalues of the Jacobian matrix. The virus-free equilibrium is globally stable in the case of R0 < 1. From Eq. (1), we obtain: x˙ =  − dx − ˇxy <  − dx, therefore x < (/d) + , where  → 0 as t → ∞. Next, Eq. (2) yields: y˙ = ˇxy − ay − pyz < ˇxy − ay < ˇ((/d) + )y − ay = y(ˇ(/d) − a) + ˇy. Since R0 < 1, we have y(ˇ(/d) − a) < 0. It can be shown that y has to be bounded, i.e. there exists ymax such that y ≤ ymax . This implies that ˇy ≤ ˇymax → 0 as t → ∞. Therefore, after a finite time y can only decrease (y˙ < 0 for t > tvf ), which results in y → 0 as t → ∞. Since h, y → 0 as t → ∞, w and z have to decrease to 0 as well (see Eqs. (3)–(4)). Finally, Eq. (1) produces: x˙ →  − dx, which results in x → /d. If the condition is not satisfied (i.e. R0 = ˇ/ad > 1), the virusfree equilibrium becomes unstable and the infection is successfully established. In this case, due to the immune impairing ability of the virus, two outcomes can be observed, depending on the rate of viral replication and the strength of immune response (Wodarz, 2007). Provided that a large fraction of Th cells/APCs get infected, the cellular immune response is successfully impaired and the CTL population eventually goes extinct (y = / 0 in Eq. (2) and w = 0 in Eq. (3)). The resulting CTL exhaustion equilibrium is given by: xex =

a , ˇ

yex =

 d − , a ˇ

wex = 0,

zex = 0,

hex = 0.

(7)

From the stability analysis (using the eigenvalues of the Jacobian matrix), we obtain the following stability condition: (/a − d/ˇ) (ca/ˇ − q) < b1 . Moreover, we have 0 < yex = /a − d/ˇ ⇔ 1 < ˇ/ad = R0 , which means that virus-free and CTL exhaustion equilibria can not exist simultaneously. Finally, the virus can establish a long-term infection, without / 0 in Eq. (2) and w = / 0 in Eq. (3)). It this causing CTL exhaustion (y = case, the cellular immune response manages to control the virus population. The solutions to our model converge to the following

immune control equilibrium: √ c + dq − ˇb1 +  , xic = 2cd wic =

b2 zic , qyic

zic =

yic =

ˇxic − a , p

 − dxic b1 = , cxic − q ˇxic

hic = 0,

(8)

where  = (c + dq − ˇb1 )2 − 4cdq (xic is a root of the quadratic polynomial P2 (x) =− cdx2 + (c + dq − ˇb1 )x − q). This outcome is stable if  > 0. Moreover, since yic and zic are both positive, we obtain /d > xic > a/ˇ ⇔ 1 < ˇ/ad = R0 . This result excludes the simultaneous existence of the immune control and virus-free equilibria. It can be shown that if the CTL exhaustion equilibrium is locally stable and the immune control equilibrium is not locally stable, then the CTL exhaustion is globally stable. Firstly, we will show that the trajectory has to enter a region given by the inequality cxy − qy − b1 < 0. Let us assume that cx0 y0 − qy0 − b1 > 0. Then, from Eq. (3), we have w˙ 0 > 0. Moreover, it can be shown (from Eqs. (3), (4) and (2)) that w is bounded. Therefore, one of the following cases has to occur: (i) w˙ = 0 in a finite time (trajectory intersects the curve cxy − qy − b1 = 0), or (ii) 0 < w˙ → 0 as t → ∞ (therefore cxy − qy − b1 → 0, because h → 0). However, as the trajectory approaches the curve cxy − qy − b1 = 0, Eq. (1) yields x˙ → P2 (x)/(cx − q). Since the immune control equilibrium is not stable, we have x˙ → P2 (x)/(cx − q) ≤ 0 and there is no locally stable equilibrium on the curve cxy − qy − b1 = 0. Because cxy − qy − b1 → 0, while x is decreasing and y is bounded, the trajectory cannot stay in the region given by cxy − qy − b1 ≥ 0 for an infinite amount of time. Therefore, the trajectory has to enter the region given by cxy − qy − b1 < 0. This property is also true in the case of cx0 y0 − qy0 − b1 = 0 (the proof is analogous). Next, we investigate the stability of the CTL exhaustion equilibrium using a Lyapunov function whose domain is given by the inequality cxy − qy − b1 < 0. A similar approach can be found in the proof of Theorem 2.3.2. in (Chan, 2011). We choose the Lyapunov function as follows:  = x − xex − xex ln + mh + w +

 x  xex

+ y − yex − yex ln

 py  ex

b2

 y  yex

z,

(9)

where  is a positive coefficient yet to be determined. Clearly  ≥ 0 and (P) = 0 only if P = (xex , yex , wex , zex , hex ). The derivative of Eq. (9) with respect to time has the following form: ˙ =



1−

xex x





x˙ + 1 −

yex y



y˙ + mh˙ + w˙ +

 py  ex

b2

˙ z.

(10)

After the substitution of Eqs. (1)–(5) into Eq. (10), we obtain:



x xex ˙ =  2 − − x xex





− pyz

+ w (cxy − qy − b1 ) +

pyex qy b2

 (11)

It can be shown that (2 − xex /x − x/xex ) ≤ 0, and (2 − xex /x − x/xex ) = 0 only if x = xex (see e.g. Chan, 2011; Korobeinikov, 2004; Ansari and Hesaaraki, 2013). Since cxy − qy − b1 < 0, there exists a positive constant  such that ˙ < 0. On the other hand, if the stability condition for CTL exhaustion equilibrium is not satisfied, the immune control equilibrium is locally stable. The CTL exhaustion is not stable if (/a − d/ˇ)(ca/ˇ − q) ≥ b1 . After multiplying both sides of the inequality by ˇ and some transformations, we obtain: c + dq − ˇb1 ≥ cad/ˇ + ˇq/a. Since the right-hand side of the inequality is non-negative, the left-hand side has to be non-negative as well. Therefore, we have: (c + dq − ˇb1 )2 ≥ (cad/ˇ + ˇq/a)2 = (cad/ˇ − ˇq/a)2 + 4cdq.

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49

Fig. 1. The idea of our numerical simulations.

Apart from the border case (cad/ˇ − ˇq/a) = (/a − d/ˇ)(ca/ˇ − q) − b1 = 0, which is irrelevant for practical purposes, we obtain:  = (c + dq − ˇb1 )2 − 4cdq > 0. Therefore, the immune control equilibrium is locally stable. It is important to note that there exists a parameter region, in which the stability conditions for both equilibria are satisfied (there are two locally stable equilibria). In this case, the outcome of the infection depends on the initial conditions, including the dose of modified HSCs introduced into the patient’s thymus (h0 ). In primary infection of untreated patients (h0 = 0), the system naturally favors the CTL exhaustion equilibrium. This is because the initial physiological level of antigen specific CTLs is very low and it takes time to establish a potent cellular immune response. The virus quickly manages to infect a significant fraction of Th cells or APCs, which results in immune impairment and exhaustion of CTL population. HSC based therapy provides an independent influx of new CTLs to the system. Therefore, it is possible that a single treatment can strengthen the cellular immune response enough to cause the shift of equilibria in some patients, resulting in restoration of cellular response and immune control of the infection. In Section 3.2, we present a numerical simulation illustrating how a single injection of CD8+ T lymphocytes derived from HSCs can change the outcome of infection. On the other hand, if the cellular immune response is capable of the immune control of the virus on its own, applying the HSC based therapy should also be beneficial for the patient. The introduction of engineered CD8+ T lymphocytes causes a boost of the cellular immune response, lowering the numbers of infected cells and suppressing the plasma viremia (Korpusik, 2014; Korpusik and Kolev, 2013). Even if the therapy will not result in viral clearance (see Section 3.5), it will have a temporary positive influence on the virus immune system dynamics.

For the first part of our simulations (infection), we assume the initial presence of the virus and a low number of antigen specific precursor CTL in a healthy individual (x0∗ = /d as in Eq. (6)):

3.2. Numerical simulations – shifting equilibria

x0∗ = 1,

In this section, we present numerical simulations illustrating the mathematical results obtained in Section 3.1. The idea of our simulations is depicted in Fig. 1. We assume that a healthy (virusfree) individual is infected with an immune impairing virus. The infection results in successful suppression of the cellular immune response and CTL exhaustion is observed. Next, HSC based therapy is applied, resulting in restoration of the immune response and immune control of the virus population. The choice of initial conditions and parameters of the model (Eqs. (1)–(5)) is arbitrary. This is because there are very little experimental data documenting the real course of the therapy. Moreover, we do not consider any specific infection, but rather a class of viruses having the ability to impair the immune response. Of course, we need to have c > q (only a fraction of precursor CTLs become effector cells) and b1 < b2 (effector cells have a shorter lifespan than precursor CTLs). We have decided to simulate a nonpathogenic infection (a = d), so that any additional destruction of infected cells comes only from the cellular immune response. For now, we assume that every CD8+ T cell engineered from HSC is able to pass the process of thymic selection (m = 1). The role of parameter m ∈ [0, 1] is studied in Section 3.3. Parameters are set as follows:  = 5, c = 1.5,

 = 1,

d = 1,

q = 0.2,

ˇ = 5,

b1 = 0.1,

a = 1, b2 = 1.

p = 5,

m = 1,

Fig. 2. Depletion (1) and restoration (2) of the Th cell/APC population.

y0∗ = 0.01,

w0∗ = 0.01,

z0∗ = 0,

h∗0 = 0.

With this choice of parameters and initial conditions, the virus manages to successfully impair the immune response. Therefore, the initial conditions for the second part of our simulations (therapy) can be calculated from the CTL exhaustion equilibrium expressions (see Eq. (7)). Now, the HSC based therapy is applied and new CD8+ T lymphocytes are introduced into the patient’s thymus (h0 > 0). The new initial conditions are set as follows: x0 = 0.2,

y0 = 0.8,

w0 = 0,

z0 = 0,

h0 = 2.

The simulated dynamics of infection and therapy are illustrated in Figs. 2 and 3. Our numerical simulations are consistent with the mathematical results from Section 3.1. Even without the initial presence of CTLs, the HSC based therapy manages to achieve long-term control of the infection. The number of infected cells is being significantly reduced, while the population of uninfected Th cells/APCs starts to recover, as seen in Fig. 2. Moreover, the treatment results in restoration of the cellular immune response (see Fig. 3). The restored CTL population is now capable of immune control of the infection without any additional intervention. Please note that in Figs. 2 and 3 the equilibrium for therapy dynamics has not been reached yet. The time frame of these figures is selected to best illustrate the shift of equilibria. The interacting populations eventually tend to numbers given by the immune control equilibrium expressions (Eq. (8)).

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Fig. 4. Relationship between the dose depletion rate (), the thymic selection success rate (m) and the minimal initial dose size (h0 ) required to cause the change in equilibria.

Fig. 3. Suppression (1) and restoration (2) of the cellular immune response.

3.3. Numerical simulations – treatment parameters Next, we have studied the parameters of HSC based therapy: the size of the dose (h0 ), the rate constant of cell depletion in the thymus () and the thymic selection success rate constant (m). In Section 3.2, we have demonstrated that a large enough dose of new engineered CTLs can result in shifting the equilibria from CTL exhaustion to immune control of infection. Even with maximal and fastest possible assimilation of injected cells (m = 1 and  → ∞), there exist a threshold dose size hmin required to cause this effect (for our set of parameters hmin = 0.2953). If the size of the dose is lower than this threshold (h0 < hmin ), there will be no shift in equilibria. The effect of the therapy, although beneficial, will only be temporary. However, if the condition for the dose size is satisfied (h0 ≥ hmin ), the outcome of the therapy depends on other parameters. Higher value of the depletion rate constant  results in a faster influx of new CTLs, thus promoting the shift in equilibria. Lower value of  requires higher dose size (h0 ) to successfully change the outcome of the infection. The influence of the depletion rate constant  on the minimal initial dose size h0 required to cause the change in equilibria is illustrated in Fig. 4. The results presented in Section 3.2 have been obtained under the assumption of maximal assimilation of the dose (m = 1). In the opposite case, only a fraction of cells derived from HSCs will successfully become functional CTLs. Parameter values m ∈ (0, 1) result in reducing the influx of new precursor CTLs by k = 1/m times. Therefore, for a fixed value of , the dose size has to be increased k times in order to ensure the same outcome. Increasing the depletion rate constant  can also compensate for the reduced influx of new CTLs, but only to some extent. This is because the value of the minimal dose threshold is increased as well. The case of h0 < khmin will result in CTL exhaustion independently of . The comparison

Fig. 5. The minimal initial dose size (h0 ) required to cause the shift in equilibria (for m = 1) at different times of therapy initiation: at the beginning of infection (t0 ), at the time of maximal precursor CTL concentration (tw ) and after the CTL exhaustion equilibrium has already been reached (tex ).

between the minimal dose size h0 required to change the outcome of the infection for m = 1, m = 1/2 and m = 1/3 is depicted in Fig. 4. Note that the dose required to achieve the immune control for m = 1/2 is two times larger than in the case of m = 1 (and three times larger for m = 1/3). 3.4. Time of treatment initiation In the CTL exhaustion dynamics, the number of precursor CTLs decreases asymptotically to zero (w(t) → 0, where w(t) > 0). Therefore, the sooner the therapy starts, the more CTL precursors are still present in the system (see Fig. 3) and less engineered CTLs are needed to cause the shift of equilibria. Since hmin was calculated assuming that CTL exhaustion has already been reached (w0 = 0), it can be possible to achieve immune control with a dose size less than hmin in early treatment. These results are illustrated in Fig. 5. After the initial expansion, the CTL population reaches a maximal level and then starts to decline. It may seem that this point of maximal CTL concentration is the best time for treatment initiation. However, our previous results indicate that it is not true (Korpusik, 2014). The same dose of engineered cells applied in the initial expansion phase should result in a stronger cellular response, even though there is less CTLs in the system. This is because the virus has not infected enough target cells yet and the effect of immune impairment is limited. Our numerical simulations (see Fig. 5) confirm these theoretical results.

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If the cellular immune system is already capable of immune control of the infection, applying the therapy will not influence the established equilibrium. It will, however, lead to a temporary reduction in the concentration of infected cells and virus particles, thus promoting viral clearance (see Section 3.5). Therefore, in this case, it is also better to initiate the treatment as early as possible. The therapy of HIV infected mice in experiment (Kitchen et al., 2012) was started prior to infection. This kind of treatment results in higher initial precursor CTL count, which is one of the major factors influencing the outcome of the virus - immune system dynamics (see Section 3.1). Injecting the genetically derived CD8+ T lymphocytes prior to infection may cause the shift of equilibria as well. Even if the HSC based vaccine will fail to change the outcome of the infection, it will at least slow down the progression of the disease. In conclusion, our results indicate that the HSC based therapy of immune impairing infection should be initiated as early as possible. Pre-infection vaccines can also prove to be very beneficial to achieve long-term immune control of the virus. 3.5. Viral clearance

51

w˙ = G + mh + cwxy − qwy − b1 w,

(12)

z˙ = qwy − b2 z,

(4)

h˙ = −h.

(5)

The remaining equations, as well as all the parameters and initial conditions remain the same as in the original model (see Section 2). 4.2. Equilibria Similarly to Section 3.1, we obtain h = 0 (see Eq. (5)) and mh = 0 (Eq. (12)) in every equilibrium. The outcome of the infection depends on the basic reproductive ratio of the virus R0 . The condition for R0 remains unchanged (R0 = ˇ/ad), according to our previous work (Korpusik and Kolev, 2013). If R0 < 1, the virus fails to establish the infection and the system converges to the following virus-free equilibrium: ∗ xvf =

 , d

∗ yvf = 0,

∗ wvf =

G , b1

∗ zvf = 0,

h∗vf = 0.

(13)

Note that the numbers of precursor CTL remain positive even in the absence of infection, as opposed to Eq. (6). If R0 > 1, the infection is successfully established. The resulting virus persistence equilibrium is given by:

From a mathematical point of view, the viral clearance (y = 0) is not possible in our model. For both cases with R0 > 1 (see Section 3.1), we obtain a positive equilibrium number of infected cells, i.e. yex > yic > 0. However, depending on parameter values, the numbers of infected cells in the immune control equilibrium can actually reach a very low level. From Eq. (8), we obtain that low equilibrium concentration of infected cells is promoted by the long lifespan of precursor CTLs (low values of b1 ) and high CTL responsiveness (high values of c). Moreover, during the course of therapy, variable y can temporarily reach levels lower than its equilibrium value. Numbers of infected cells can be reduced even more by introducing additional pharmaceutical treatment or applying the HSC based therapy multiple times, especially after the immune control is already achieved. In reality, the number of infected cells is not continuous, as opposed to variable y. Sufficiently strong reduction of y can, in practical terms, result in virus extinction. Therefore, from theoretical perspective, the HSC based therapy can lead to viral clearance, as long as there is no antigenic escape.

Korpusik and Kolev (2013). In other words, there exists at least one stable and biologically significant equilibrium for the case of R0 > 1. Moreover, there exists a parameter region in which we have two stable and biologically significant equilibria given by Eq. (14). The obtained equilibria correspond to CTL exhaustion and immune control equilibria from Section 3.1. Therefore, similarly as in Section 3, high enough HSC-based injection can shift the outcome of the infection. The conditions for such parameter region can easily be derived. However, we will not present them due to the large and impractical form of the corresponding equations.

4. Additional modification of the mathematical model

4.3. Numerical simulations

4.1. Modification

In order to illustrate the results from Section 4.2, we conduct numerical simulations analogously to those from Section 3.2 (as depicted in Fig. 1). The parameters of the modified model are chosen identically as in the previous simulations and the additional influx of precursor CTL is set to G = 0.001. The initial conditions for the first part of our simulations (infection) are also set identically as in Section 3.2. Note that the presence of precursor CTLs is now consistent with the new virus-free equilibrium (Eq. (13)), as opposed to the non-modified one (Eq. (6)). In the second part of our simulations (therapy), the dose size is set identically as in Section 3.2. The rest of the initial conditions are calculated from the new virus persistence equilibrium (Eq. (14)):

In this section, we present an additional modification of the mathematical model from Section 2. The main cause for this modification is that in the virus-free equilibrium, the number of precursor CTLs is equal to zero (wvf = 0 in Eq. (6)), while virus specific CD8+ T lymphocytes are continuously produced in the thymus, independently of antigen stimulation. A small physiological concentration of these cells is always present in the system. Moreover, infection of Th cells/APCs impairs the expansion of memory CTLs (see e.g. Janssen et al., 2003; Borrow et al., 1995). However, the CD8+ T cells still undergo the initial expansion and memory cells, though impaired, are still present. Therefore, it should be impossible for the entire CTL population to go extinct, as opposed to wex = 0 in Eq. (7). In order to deal with these issues, a new term G > 0, denoting the virus-independent production of CTLs in the thymus, is introduced into Eq. (3). We get the following modified model: x˙ =  − dx − ˇxy,

(1)

y˙ = ˇxy − ay − pyz,

(2)

y∗ =

( − dx∗ ) , ˇx∗

w∗ =

b2 z ∗ , qy∗

z∗ =

(ˇx∗ − a) , p

h∗ = 0,

(14)

where x* is the root of a third degree polynomial f(x) with a negative leading coefficient. It can be shown that there exists at least one x0 ∈ (0, /d) satisfying the following conditions: (1) f(x0 ) = 0, (2) lim f (x) > 0 > lim f (x), (3)  − dx0 > 0 and (4) ˇx0 − a > 0, see x→x− 0

x→x+ 0

x0 = 0.2122,

y0 = 0.7424,

z0 = 0.0122,

h0 = 2.

w0 = 0.0823,

The obtained results correspond to those described in Sections 3.2–3.5. High enough dose of CD8+ T lymphocytes derived from HSCs can shift the equilibria in the modified model as well. This is illustrated in Fig. 6. There also exists a minimal dose threshold hmin and the influence of parameters  and m on the therapy

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5. Discussion

Fig. 6. Suppression (1) and restoration (2) of the cellular immune response in the modified model.

dynamics is qualitatively identical. The conclusions regarding the time of treatment initiation and viral clearance hold as well. Apart from influencing the population sizes in equilibria, the additional influx of precursor CTLs (parameter G) also affects the minimal dose size (h0 ) required to change the outcome of infection. Higher values of G require lower minimal initial dose sizes h0 , as illustrated in Fig. 7. Moreover, for high enough values of G (in our case G ≥ 0.0012), the system converges to the immune control equilibrium, even without the application of treatment (h0 = 0). Since the additional parameter G is present only in Eq. (12), the case of G = 0 corresponds to the non-modified model from Section 2.

Fig. 7. Influence of the virus-independent production of CTLs in the thymus (G) on the minimal initial dose size (h0 ) required to cause the change in equilibria. The case of G = 0 corresponds to the non-modified model from Section 2.

The main problem with our previous mathematical approach is that we have assumed a constant influx of new CD8+ T lymphocytes (Korpusik, 2014; Korpusik and Kolev, 2013). This is a big simplification of the treatment process, which makes our results valid only in the short-term dynamics. In reality, the applied dose is finite and the strength of the therapy decreases in time, as engineered cells are depleted in the thymus. Although the analysis of a mathematical model with a constant influx of cells may give some insights on the treatment, it cannot forecast the infection dynamics after therapy termination. Some studies suggest that strengthening the immune system with engineered CD8+ T lymphocytes can lead to long-term control of the virus population (Kitchen et al., 2011; Murray et al., 2009). In order to deal with these issues, in the present paper, we have analyzed a new mathematical model for HSC based therapy of an immune impairing infection. We have assumed a finite and decreasing dose of CD8+ T lymphocytes engineered from HSCs. Our results show that during the course of therapy, introduction of engineered CD8+ T lymphocytes should lead to restoration of the cellular immune response and significant virus suppression, without any additional pharmaceutical treatment. This is compatible with the course of infection in relevant in vivo experiment (Kitchen et al., 2012). We have also shown that it is best for the patient to initiate the therapy as quickly as possible, similarly to our previous results (Korpusik, 2014). The most important result of our studies is that a large enough dose of new CTLs may cause a shift in the outcome of infection and lead to long-term immune control of virus population. This result is consistent with biological intuition and previous theoretical studies on the HSC based therapy (Kitchen et al., 2011; Murray et al., 2009). The main advantage of the HSC based therapy is the introduction of a new, independent influx of CD8+ T lymphocytes derived from hematopoietic stem cells. These engineered virus specific CTLs are capable of killing the infected cells. Therefore, this kind of treatment presents a great potential for a wide variety of diseases. One of the most important applications for the HSC based therapy is the HIV infection, in which introducing engineered CD8+ T lymphocytes can be a way of bypassing the immune impairing properties of the virus (Kiem et al., 2012). HSC based therapy is still in the early development phase. Many experimental studies need to be done, first on laboratory mice and then in human organism, to assess its real potential. Theoretical studies indicate that such a treatment can result in long-term control of the virus population. The viral clearance is theoretically possible as well, especially with the aid of additional pharmaceutical treatment. Applying the HSC based therapy multiple times can also lead to viral clearance. Furthermore, the analyses of mathematical models suggest that even if the HSC based therapy fails to establish long-term control of the virus, it can prove to be a very useful as a supplementary treatment. Combined with drug therapy, it can significantly slow down the disease progress. On the other hand, there are some drawbacks in the HSC based approach to fight viral infections. First of all, it is a personalized treatment. The therapy has to be tailored to a patient, so that new cells can survive the thymic selection process (Kitchen et al., 2011, 2012). Moreover, the engineered CTLs are specific to a selected antigenic epitope, while viral infections are rarely homogeneous. Suppressing a given virus strain lessens the chances for viral evolution, but the antigenic escape still remains a viable option for escaping the immune response, both natural and engineered. This has negative implications for treating the advanced HIV infection (McMichael and Phillips, 1997; Nowak et al., 1995). In addition, strengthening the cellular immune response against a given antigen also affects other immune responses (Selin et al., 1999; Liu

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et al., 2003). The influence of these mechanisms on the efficacy of the multiple-strain infection therapy remains unknown. There are also some features that are not taken into account in our mathematical approach. Most of all, we have only considered one virus strain, assuming no viral evolution. Next, the role of humoral immune response was assumed to be constant, while, in reality, it is an adaptive mechanism. The B lymphocyte population expands in response to viral stimulation and produces antibodies that can greatly influence the dynamics of the infection (Marchuk, 1997; Wodarz, 2003). What is more, we did not distinguish between classes of target cells, given e.g. by their activation state (Douek et al., 2002). There was no distinction between the primary (helper independent) and memory expansion of CTL population as well. Many of the limitations for the description of biological phenomena, such as the heterogeneity among cells of the same population, their progression in performing biological functions, their selection and mutations, follow directly from our modeling approach (Wodarz, 2007). By using a system of coupled differential equations, we actually consider the corresponding populations as homogeneous, averaging over the space and the biological specificity of the cells belonging to the same population (Bellomo et al., 2008). This problem can be solved, for example, by introducing additional internal variables describing the specific biological functions of each cell, following the ideas of the kinetic theory of active particles (Bellomo and Carbonaro, 2011; De Angelis, 2014). Apart from laboratory experiments, there is still much theoretical analysis to be done before the therapy is tested on humans. Future analyzes of mathematical models can give important insights on many aspects of the HSC based therapy of viral infections. First of all, they can help to propose vaccination/multiple injection strategies aimed to achieve long-term control or viral clearance, similarly to epidemiological models (see e.g. Qiao et al., 2013). Moreover, the analysis of the models describing a multiplestrain infection can result in designing approaches to treat AIDS or evolving viruses. It is also important to note that the HSC based approach can be used against other mechanisms of viral infections, e.g. virus latency (Valyi-Nagy et al., 2000) or CTL-induced pathology (Thomsen et al., 2000). The theoretical analysis of these cases can result in fast and relatively inexpensive prediction of the course of infection and therapy effectiveness. In conclusion, the hematopoietic stem cell based therapy is a very promising approach for treating a vast repertoire of diseases. It introduces an independent influx of CD8+ T lymphocytes, which are able to recognize and kill infected cells. This way the immune impairing ability of viruses (e.g. HIV) can be omitted. The theoretical results indicate that the therapy can lead to long-term immune control of infection, or even viral clearance. Nevertheless, there is still much experimental and theoretical work before the therapy can be introduced into practice. Acknowledgments The authors would like to express their gratitude to Shingo Iwami (Kyushu University, Fukuoka, Japan) and to anonymous reviewers for their useful comments and suggestions that led to improvement of this paper. This research is supported by the internal Grant for Young Scientists, ID number: 0620-0883 (Faculty of Technical Sciences, University of Warmia and Mazury in Olsztyn, Poland). References Abbas, A., Lichtman, A., 2004. Basic Immunology: Functions and Disorders of the Immune System. Saunders, Philadelphia. Anderson, R.M., May, R.M., 1991. Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford.

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