Dendritic Cell Based Immunotherapy for Melanoma

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1 Denritic Cell Base Immunotherapy for Melanoma Kun Dong University of California, Los Angeles Binan Gu University of Southern California Lucy Oom California State University, Fullerton Aparna Sarkar Pomona College Avisor: Cymra Haskell University of Southern California August 10, 2012 Abstract Denritic cells DC) are important immunostimulatory cells that facilitate antigen transport to lymphoi tissues an provie stimulation of cytotoxic T lymphocyte CTL) cells. In this paper, we attempt to unerstan the etails of the regulation an kinetics of the DC-CTL interaction in the DC-base immunotherapeutic treatment of human melanoma cancer. We stuy a previously efine moel, which integrates enritic cell populations in the bloo, spleen, an the tumor. Ultimately, we are intereste in applying analysis of the moel towars higher rates of efficacy of DC treatment. 1 Introuction Denritic cell treatment is a recently evelope immunotherapy for cancer. Immunotherapies boost a patient s immune response to a pathogen by aministering vaccinations or introucing antiboies. In the case of enritic cell therapy, the vaccination consists of enritic cells. Denritic cells DCs) act as messengers between a specific pathogen an cytotoxic T- lymphocytes CTLs), which are the fighter cells of the immune system. After encountering a tumor, DCs travel to the lymphoi organs an present the tumor-specific antigen to naive CTLs. The naive CTLs then either evelop into activate CTLs or memory CTLs. The activate CTLs travel to the tumor site an fight the pathogen, killing tumor cells via apoptosis. Memory CTLs circulate in the bloo an lymphoi organs, prepare for a future attack on the boy by the same pathogen. Previous biological stuies have emonstrate the efficacy of enritic cell treatment on fiel mice. In these clinical trials, researchers cultivate a subject s immature DCs ex 1

2 vivo an loa them with tumor-specific antigens. This culture is use as a patient-specific vaccine, injecte to boost the native response against tumor cells. Success of these stuies has le to FDA approval of the first enritic cell vaccine for prostate cancer. However, it has proven ifficult to preict how ifferent patients will react to DC treatment. In this paper we are concerne with analyzing a mathematical moel of the immune response with DC treatment in orer to etermine an optimal treatment regimen an to unerstan how this might vary from patient to patient. Luewig et al. presente results from murine experiments of DC treatment in the form of ata an a mathematical moel of DC-CTL interactions. In their experiments, mice receive injections of melanoma-specific DCs. No tumor was present in their boies. So, the resulting moel is one of DC trafficking in the boy specifically in the lung, liver, spleen, an bloo. DePillis et al. moifie an extene Luewig et al. s moel by combining all the lymphoi organs into a single compartment which they call the spleen an by incluing a tumor compartment. The result is a moel of the immune response to DC treatment in the presence of a tumor, allowing for analysis of varying DC treatments in terms of tumor growth an patients particular parameter values. The values we have use in our analysis are those estimate by DePillis et al. to fit experimental ata from Luewig et al. 2 The Moel We are stuying DC trafficking an immune activity between the spleen an the tumor via the bloo. Denritic cells an active CTLs move between both the spleen an the tumor, while memory CTLs o not travel to the tumor remaining in the bloo an spleen in case of future attack). The following iagrams explain the boy s native immune response to a tumor. 2

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4 The moel evelope by DePillis et al. [2] is a compartment moel involving three compartments: the bloo, spleen, an tumor. It involves nine state variables an boils own to a system of nine first orer non-linear ifferential equations. The variables are: D bloo D spleen Ebloo a Espleen a Ebloo m Espleen m Etumor a T D tumor Number of enritic cells in the bloo; Number of enritic cells in the spleen; Number of activate CTLs in the bloo; Number of activate CTLs in the spleen; Number of memory CTLs in the bloo; Number of memory CTLs in the spleen; Number of activate CTLs in the tumor; Number of tumor cells; Number of melanoma infiltrating enritic cells MIDCs). Notice that the moel tracks four ifferent types of cells: enritic cells, activate CTLs, memory CTLs, an tumor cells. Most of the terms in the equations moel eath of the cells or their migration from one organ to another. These terms all follow the pattern of a rate coefficient multiplying a variable; the first two terms in Equation 1) are two simple examples. Some of the percentage rates of change are not constant but epen on other variables, as in the first term in Equation 2). Other terms represent the growth or prouction of cells an the injection of DCs through vaccination. We will examine each compartment an equation separately see Table 1 in the Appenix for a complete list of parameters). Bloo Compartment t D bloo = µ B D bloo + µ T B D tumor + v bloo t) 1) t Ea bloo = µ D spleen )Espleen a µ BBEbloo a 2) t Em bloo = µ D spleen )Espleen m µ BBEbloo m 3) Equation 1) provies a goo example of the general pattern of the moel. The first term moels the number of DCs leaving the bloo an traveling to all lymphoi organs in the boy. This inclues the spleen, so jumping ahea to 4), we see how the DC transfer rate from bloo to spleen µ BS ) is a small portion of the emigration rate of DCs out of the bloo µ B ). Our secon term is a straight trafficking term, moeling the transfer of DCs from tumor to bloo. This term correspons to the secon term in 9). The last term of 1) inicates the therapeutic injection of enritic cells into the bloo. This term is typically piecewise constant. The first term in Equation 2) is a trafficking term inicating the migration of active CTLs from spleen to bloo. It correspons to the secon term in Equation 5). However, we see that the percentage rate at which these cells move from the spleen is a function that 4

5 epens on D spleen : µ D spleen ) = µ µ D spleen θ shut where µ = µ Normal µ This term comes from Luewig et al. an moels the empirical observation that active CTLs are hel back in the spleen when there are a large number of DCs there. The graph of µ D spleen ) is shown in Figure 1. Notice that it rops very rapily an is essentially equal to µ for any value of D spleen more than about 50. The secon term moels the Figure 1: The percentage rate at which active CTLS migrate from the spleen to the bloo is close to µ Normal when D spleen is very small but ecays rapily to µ as D spleen increases. migration of active CTLs out of the bloo, traveling to all other parts of the boy among which are the spleen an tumor. This term correspons to the first terms in Equation 5) an 7) where we notice that µ BT E T ) < µ BB for all T, so µ BSE + µ BT E T ) < µ BB. Equation 3) follows equation 2), with memory CTLs substitute for active CTLs; biologically, in fluctuation between spleen an bloo, active an memory cells are treate the same. Spleen Compartment 5

6 ) t D spleen = MaxD1 e µ BS D bloo ) MaxD a D D spleen b DE Espleen a D spleen 4) t Ea spleen = µ BSEEbloo a µ D spleen )Espleen a + b ad spleen Espleen m a EaSEspleen a + a E asdc on D spleen )E naive r am Espleen a D spleen t τ D )Espleen a + b t τ D) p 5) θ D + D spleen t τ D ) t Em spleen = r amespleen a a E m Espleen m b ad spleen Espleen m µ D spleen )Espleen m + µ BSE Ebloo m 6) The first term of Equation 4) moels the migration of enritic cells in the bloo to the spleen. It stans out as the only migration term in the moel that is written as a the transfer rate of DCs from the bloo to the spleen, instea of as a percentage rate multiplie by D bloo. The graph of this term is shown in Figure 2. The term is a moification of the Figure 2: When D bloo is close to 0, the transfer rate of DCs from bloo to spleen is close to µ BS. When D bloo is large, the actual rate an not the percentage rate) is close to MaxD this means the percentage rate tens to 0 as D bloo ). Luewig et al. moel by DePillis et al., accounting for the existence of a maximum rate at which DCs enter the spleen [2]. Through moel simulation an fit to experimental ata, DePillis et al. foun this rate to be limite to about 400 cells per ay: MaxD = 400 [2]. Notice that the percentage rate at which DCs enter the spleen from the bloo is MaxD1 e µ BS D bloo MaxD ) ) D bloo < µ BS. 6

7 As mentione in the explanation of the first term in Equation 1), µ BS < µ B because that term also inclues migration of DCs into other organs of the boy. The secon term of Equation 4) moels the eath of enritic cells in the spleen. When enritic cells meet active CTLs in the spleen they are consume enabling the proliferation of more active CTLs. The thir term in Equation 4) moels this an it correspons to the last term in Equation 5). Equation 5) has many components, incluing two new an exciting terms. It begins with the migration of active CTLs from bloo to spleen, which we previously iscusse in Equation 2). Similarly, the secon term is a trafficking term moeling the migration of active CTLs from spleen to bloo an correspons to the secon term in Equation 2. The thir term is another trafficking term, moeling the conversion of memory CTLs to active CTLs in the presence of DCs. It correspons to the thir term in Equation 6). Moving to the secon line, the fourth term moels the eath of active CTLs in the spleen. The fifth term moels the conversion of naive CTLs to active CTLs in the presence of enritic cells. Note that E naive is the number of naive CTLs in the spleen a constant in this moel an not a variable. The function DC on is a step function: { 0 if Dspleen = 0 DC on D spleen ) = 1 if D spleen > 0 DePillis et al. introuce DC on D spleen ) to prevent the creation of tumor-specific CTLs before there are any DCs to present the antigen to the naive T-cells. We fin it a little strange that the coefficient in this term, a E a S, is the same as the percentage rate at which active CTLs ie; this assumption seems tenuous at the moment, meriting further exploration an potential moification. However, we on t expect any moification to yiel substantive ifferences in our analysis, so for now we have accepte that the rates are the same. Just as DCs cause some memory CTLs to activate, some activate CTLs revert to memory CTLs. This reversion from active to memory CTLs is moele by the sixth term in the equation. It is a trafficking term, so we see it again as the first term of Equation 6). The final term of 5) incorporates a time elay, which presents consierable challenges in the analysis. Biologically, the elay represents the time that DCs an naive CTLs in the spleen nee to be in contact before proliferation of active CTLs begins [2]. Accoringly, this last term moels the proliferation of more active CTLs from some active CTLs. Recall from Equation 4) that active CTLs kill some DCs in the spleen in orer to proliferate The percentage rate of profileration is equal to b p D spleen t τ D ) θ D + D spleen t τ D ). When D spleen t τ D ) = 0 there is no proliferation. As D spleen t τ D ) increases the percentage rate of proliferation increases with a limiting value of b p. Moving on to equation 6), we first observe the reversion of active CTLs to memory CTLs. The secon term moels the natural eath of memory CTLs in the spleen. With its counterpart back in Equation 5), the thir term moels the activation of memory CTLs by DCs. The fourth term is familiar memory cells are transferre from spleen to bloo in the same way that active cells are as seen in Equation 5)) an its counterpart is the first 7

8 term in Equation 3. The last terms moels the migration of memory CTLs from the bloo to the spleen an its counterpart is the secon term in Equation 3. Tumor Compartment t Ea tumor = µ BT E T )Ebloo a a E at Etumor a cetumort a 7) t T = rt 1 T ) DE a k tumor, T )T 8) t D tumor = mt q + T µ T BD tumor a D D tumor + v tumor t) 9) where an T µ BT E T ) = µ BB α + T DE a tumor, T ) = E a tumor T ) l s + E a tumor T ) l 10) We jump right into 7). The first term moels the migration of active CTLs in the bloo to the tumor. The rate function µ BT E T ) can be unerstoo as the a fraction of the total elimination of CTLs from the bloo, which increases as the number of tumor cells increases. So when there are more tumor cells, a greater proportion of the active CTLs leaving the bloo will go to the tumor instea of traveling to other organs.as previously note in our iscussion of the bloo compartment, this term correspons to a portion of emigration of active CTLs from the bloo in equation 2): µ BB T α + T < µ BB. The secon term moels the natural eath of active CTLs in the tumor. The thir term moels the inactivation of active CTLs by tumor cells; think of this as the tumor s combat of the immune system s fighter cells. The two terms of equation 8), the rate of change in number of tumor cells, are a growth term an a ecay term respectively. The growth term represents logistic growth, as use in previous moels an fit to experimental ata [2]. The percentage rate at which tumor cells grow is equal to the parameter r when T is small an ecreases to 0 as T increases. The percentage rate at which active CTLs kill tumor cells, D, epens on the kill ratio of active CTLs in the tumor to tumor cells. Its graph is shown in Figure 3. This rate is equal to 0 when there are no active CTLs an increases up to a maximum of as the kill ratio increases. Notice that D is not continuous when E a tumor = T = 0, so this term is not ifferentiable in this 7-imensional subspace of phase space. 8

9 Figure 3: D is the percentage rate at which tumor cells are kille off; it ranges between 0 when Ea tumor T is small to when Ea tumor T is large. The first term in Equation 9) moels the migration of DCs to the tumor ue to the native immune response. This rate increases as a saturation-limite function of the number of tumor cells; the moel assumes that the boy has an infinite supply of enritic cells to sen to the tumor site as ictate [2]. The secon term accounts for DC transfer from tumor to bloo as well as natural eath of DCs in the tumor. The transfer term here is the negative sie of the trafficking term we saw in Equation 1). The final term is another injection function, as DePillis et al. ran simulations with both intravenous an intratumoral DC injections. Simulations Simulations of the system inicate that all solutions ten to one of two equilibria. Figures 4 an 5, run for the same parameter values an ifferent initial conitions, epict both of these equilibria. 3 Equilibria Consier the region of the 9-imensional space where all the variables are non-negative. It is easy to see that this region is invariant. We are only intereste in solutions that live in this part of the space, since other solutions are not physically meaningful. Notice that the system is infinite-imensional because of the elay; initial conitions consist of the values 9

10 Figure 4: Initial conition: T = 10 5, all other variables 0 of the variables not just at t = 0 but at every value of t from τ to 0. We woul like to unerstan which initial conitions an values of the parameters will yiel solutions where T 0 as t. It may also be of interest to ientify those initial conitions where T remains small but we have not focuse on this matter in this paper. Although ultimately we woul like to unerstan what injection regimens prouce solutions where T 0, in this paper we have primarily restricte our analysis to the boy s native immune response where v bloo t) v tumor t) 0. Notice that DC on D spleen ) is iscontinuous when D spleen = 0. This means that there coul be more than one solution passing through a point in phase space where D spleen = 0. However, of particular interest to us are solutions where T > 0 for some t = t. Any such solution will have D spleen > 0 for t t, so DC on D spleen ) will equal one an remain one as t. For this reason, in our moel analysis an simulations, we replace DC on D spleen ) with 1. Since D spleen is the number of enritic cells, it oes not make sense for it to take on fractional values, so in future work it may be appropriate to replace this function with another function that is similar but continuous when D spleen = 0. In summary, we stuy 10

11 Figure 5: Initial conition: T = , all other variables 0 the following system: t D bloo = µ B D bloo + µ T B D tumor t Ea bloo = µ D spleen )Espleen a µ BBEbloo a t Em bloo = µ D spleen )Espleen m µ BBEbloo m ) t D spleen = MaxD1 e µ BS D bloo ) MaxD a D D spleen b DE Espleen a D spleen t Ea spleen = µ BSEEbloo a µ D spleen )Espleen a + b ad spleen Espleen m + a EaSE naive a EaS + r am )Espleen a D spleen t τ D )Espleen a + b t τ D) p θ D + D spleen t τ D ) t Em spleen = r amespleen a a E m Espleen m + b ad spleen Espleen m + µ D spleen )Espleen m + µ BSEEbloo m t Ea tumor = µ BT E T )Ebloo a a E at Etumor a cetumort a [ t T = r 1 T ) ] DE a k tumor, T ) T t D tumor = mt q + T µ T B + a D )D tumor 11) 12) 13) 14) 15) 16) 17) 18) 19) 11

12 We have primarily stuie this system when the values of the parameters are those that are given in the appenix. However, since we expect the parameters to vary from iniviual to iniviual an we certainly expect the values of the parameters to be ifferent for people than they are for mice, we are intereste in how the behavior of the solutions epens on the parameters. In particular, we have notice that the behavior of the solutions is sensitive to the relative sizes of an r. DePillis et al. have foun the equilibria of the system. We reprouce their work here for completeness. To etermine the equilibria we set the left han sie of equations 11)-19) equal to zero. From equation 18) we see that either T = 0 or r 1 T ) k = ) E a l tumor T s + E a tumor T ) l. 20) We first look for equilibria where T = 0. In this case we see from equations 17) an 19) respectively that D tumor = E a tumor = 0. From 11) we get that D bloo = 0 an from 14) that D spleen = 0 since we are only intereste in those equilibria that lie in that part of the space where all the variables are non-negative). From equation 12) we have Substituting this into equation 15) we get so E a spleen = Ebloo a = µnormal µ BB From equation 13) we see that Ebloo a = µnormal Espleen a µ. BB µ Normal µ Normal Substituting this into equation 16) we get Espleen m = r am a Em + µ Normal µ BSE µ BB µ Normal so Ebloo m = µnormal mu BB a Em + µ Normal a EaSE naive + a EaS + r am µ BSE µ BB µ Normal, a EaSE naive + a EaS + r am µ BSE µ BB µ Normal Ebloo m = µnormal Espleen m µ. BB r am ) µ BSE µ BB µ Normal µ Normal ) ) a EaSE naive + a EaS + r am µ BSE µ BB µ Normal µ Normal. ) a EaSE naive + a EaS + r am µ BSE µ BB µ Normal Thus, there exists a unique equilibrium with T = 0. We call this equilibrium T. When the parameters take on the values given Table 1 in the Appenix, it is the point:, ). 12

13 D bloo = 0 Espleen a = Ea tumor = 0 D spleen = 0 Ebloo m = T = 0 Ebloo a = Em spleen = D tumor = 0 To fin the other equilibria, notice that the right-han sie of equation 20) is always a number between 0 an. This means that the equation has a positive solution if an only if min{0, k1 /r)} < T k. For any such T, this equation is satisfie when E a tumor = ) sr 1 T 1/l k )) r 1 T T. k Substituting into equation 17) we see that it is satisfie if an only if E a bloo = a E at + ct )α + T ) µ BB T ) sr 1 T 1/l k )) r 1 T T. k Similarly, equation 19) is satisfie if an only if ) ) mt 1 D tumor = q + T µ T B + a D an then equation 11) is satisfie if an only if D bloo = mu ) ) T B mt 1. µ B q + T µ T B + a D Plugging these values back into equation 14), we obtain a quaratic equation from [2]: 0 = θ shut µ + µ)m + θ shut µ a D + µθ shut a D + θ shut b DE µ BB E a bloo µ M)D spleen + µ + b DE E a bloo µ BB)D spleen ) 2 where M = MaxD 1 e µ BS D bloo MaxD ). The quaratic equation has the general form ax 2 + bx + c = 0 where an a = µ + b DE E a bloo µ BB > 0 b = θ shut µ a D + µθ shut a D + θ shut b DE µ BB E a bloo µ M > 0 c = θ shut µ + µ) < 0. Discriminant = b 2 4ac > 0, so this quaratic equation always yiels two solutions, x 1 an x 2. However, x 1 x 2 = c a < 0, 13

14 which means one solution is negative while the other is positive. We are only intereste in the positive solution. From equation 16), we get a value for E m spleen, E m spleen = r am E a spleen a Em + b a D spleen + µ D spleen 1 µ BSE/µ BB ). Lastly, equation 13) gives Ebloo m in terms of Em spleen, E m bloo = µ BB µ D spleen ) Ea bloo. This completes the computation of all equilibrium values in term of T. Notice that equation 15) is never use through the calculation. By plugging in all values we will obtain another equation of T. Although it is not clear how many solutions of T it will yiel, simulation exclusively inicates there is only one equilibrium, meaning only one solution for T. Recall that we are intereste in those initial conitions for which T 0 as t. We shall show in section 6 that every such solution actually converges to the equilibrium where T = 0, T, so it suffices to etermine the basin of attraction of this equilibrium. Notice that at this equilibrium we have T = E a tumor = 0, so the secon term in equation 18) is not ifferentiable. This means that we cannot linearize about this equilibrium an investigate its stability in the usual way. Our general approach is outline in Section 4 below. However, there is one observation that we can make that oesn t require much machinery. This eals with the case when < r an is outline in the theorem below. Theorem 1. If < r, then T is unstable. Proof. Notice that T = [ r 1 T k [ r 1 T ) k ) Ea tumor T ) l s + Ea tumor ] T. T ) l ] T Since < r, 1 r > 0, an if 0 < T < k 1 ), then r T > [ r 1 so T is increasing. The result follows. ) ] k1 /r) T = 0 k The proof of Theorem 1 actually works in the full system where v tumot t) an v bloo t) may not be ientically 0. In other wors, when < r, enritic cell therapy cannot, by itself, kill off the tumor. 14

15 4 General Approach The chart below shows which variables each of the other variables epens on. Notice that the first six variables, D bloo through Espleen m, form the bloo an spleen compartments. They epen only on each other an one other variable, D tumor. Similarly, the last three variables, Etumor, a T, an D tumor form the tumor compartment. They epen only on each other an one other variable, Ebloo a. Dbloo Dspleen E a bloo E a spleen E m spleen E m slpeen E a tumor T t D bloo t D spleen t Ea bloo t Ea spleen t Em bloo t Em spleen t Ea tumor t T t D tumor Our approach is to stuy the behavior of the variables in the bloo an spleen compartments where we treat D tumor as an external function of time, an to stuy the three variables in the tumor compartment where we treat Ebloo a as an external function of time. Our hope is then to unerstan how these two subsystems fee into each other an thereby unerstan the solutions to the complete system of nine ifferential equations 11) through 19). We have ha partial success to ate. Dtumor 5 The Bloo an Spleen Compartments In this section, we take a closer look at the bloo an spleen compartments. We write the moel out explicitly below for future reference. 15

16 t D bloo = µ B D bloo + µ T B D tumor t) 21) t Ea bloo = µ D spleen )Espleen a µ BBEbloo a 22) t Em bloo = µ D spleen )Espleen m µ BBEbloo m 23) ) t D spleen = MaxD1 e µ BS D bloo ) MaxD a D D spleen b DE Espleen a D spleen 24) t Ea spleen = µ BSEEbloo a µ D spleen )Espleen a + b ad spleen Espleen m + a EaSE naive a EaS + r am )Espleen a D spleen t τ D )Espleen a + b t τ D) p 25) θ D + D spleen t τ D ) t Em spleen = r amespleen a a E m Espleen m + b ad spleen Espleen m + µ D spleen )Espleen m + µ BSE Ebloo m 26) Analysis of this system is ifficult because of the elay, but we have some partial results. Figures 6 an 7 show typical simulations where D tumor is constant. Notice that in both cases the solutions appear to converge to an equilibrium. In Figure 8 we zoom in on the solutions in 6. We can see that the solutions oscillate as we might expect because of the elay but they are at an amplitue that s very small compare to the global behavior. Our first result about this system is Theorem 2 which concerns the bouneness of the solutions. We suspect that for any boune function D tumor t), the conclusion of the theorem is still vali an simulations appear to support this conjecture. Inee, the moel woul not be physically reasonable if solutions became unboune, but a proof of this is important in orer to prove other results about the system. The proof in the case of a general boune function D tumor t) is elusive because we have to eal with the elay in Theorem 2 we are able to boun the elay term). It is sufficient to etermine bouneness of the solutions for functions D tumor t) that are boune because D tumor t) is always boune in the full 9-variable moel see Section 6). Let α = min{µ BB µ BSE, a D, a EaS, a Em }. Also let β = µ BSµ B MaxD αµ T B ln 1 Note here that for parameter values in the appenix, a D α 4b p MaxD ) a D α 4b pmaxd < 1. 16

17 Figure 6: D tumor = Theorem 2. If D tumor t) < β for all t then every solution to Equations 21) to 26) is boune for all time. Proof. Note that D bloo < µ BD bloo + µ T B c, it follows that D bloo t) D bloo 0)e µ Bt + µ T Bc µ B 1 e µ Bt ) µ T Bc µ B. Choose time T 1 such that for all t > T 1, Now notice that But for all t > T 1, D bloo t) < 2µ T Bc µ B = MaxD ln D spleen MaxD 1 e µ BS D bloo MaxD MaxD 1 e µ BS D bloo MaxD ) 1 ) a D α. 4b p MaxD ) a D D spleen. < MaxD 1 e µ BS 2µ T B c = MaxD = a Dα 4b p, MaxDµ B ) ad α ) 4b p MaxD 17

18 Figure 7: D tumor = so Thus, for all t > T 1, D spleen a Dα 4b p a D D spleen. D spleen t) D spleen T 1 )e a Dt T 1 ) + α 4b p 1 e a Dt T 1 ) ) α 4b p. Choose T 2 > T 1 such that for all t > T 2, D spleen t) < α 2b p. Let s = E a bloo + Em bloo + Ea spleen + Em spleen. Then for all t > T 2 + τ D, Choose M such that: s = µ BB + µ BSE )E a bloo + Em bloo ) a E ase a spleen a Em Espleen m b ad spleen Espleen m + a E ase naive D spleen t τ D ) + b p θ D + D spleen t) Ea spleen t τ D) αs + a EaSE naive + α 2 Ea spleen t τ D) 18

19 Figure 8: Closer look at Espleen a graph from fig. 6 All the variables are less than M for τ D t T 2 + τ D, M > 2a E ase naive. α We shall show that s < M for all time. Suppose this is not the case. Let T 3 be the first time that s = M. By our choice of M, T 3 > T 2 + τ D an since this is the first time that s = M, E a spleen T 3 τ D ) < M. Thus s T 3 ) αm + a EaSE naive + α 2 M = α 2 M + a E ase naive < 0. Thus there exists a time T 4 < T 3 such that st 4 ) > M, which means that there is a time T 5 < T 4 < T 3 at which st 5 ) = M. This is a contraiction to the efinition of T 3. It follows that s < M for all time. Our next result concerns the case when D tumor t) 0. In this case we have a complete unerstaning of the system. This case is interesting in its own right since it correspons to the moel of Luewig et al. Theorem 3. The system efine by Equations 21) to 26) where D tumor 0 has a unique equilibrium that is globally asymptotically stable. Proof. Setting the left-han sie of equations 11) through 16) equal to 0, it is easy to see that the system has a unique equilibrium. It follows from 11) that D bloo 0 exponentially as t. Looking at equation 14), it is then clear that D spleen 0 as t. 19

20 Now, setting D bloo = D spleen = 0, we stuy the system of the other four variables. Let x = Ebloo a, Em bloo, Ea spleen, Em spleen ). Notice that where an x = a x + b µ BB 0 µ Normal 0 A = 0 µ BB 0 µ Normal µ BSE 0 µ Normal a EaS r am 0 0 µ BSE r am a Em µ Normal B = 0 0 a EaSE naive 0 The eigenvalues of A are: λ 1,2 = 1 [ µ BB + a Em + µ Normal ) 2 ] ± µ Normal + a Em µ BB ) 2 + 4µ Normal µ BSE λ 3,4 = 1 [ r am + a EaS + µ Normal + µ BB ) 2 ] ± r am + a EaS + µ Normal µ BB ) 2 + 4µ Normal µ BSE an when evaluate at the parameters in the appenix: λ 1 = λ 2 = λ 3 = λ 4 = These are all negative, therefore the equilibrium is globally asymptotically stable. Note: In the above proof, we set D bloo = D spleen = 0, whereas in the 6x6 system, both variables are approaching 0. We believe the result is the same with this ifference, but have not reache a satisfying proof as yet. When D tumor is constant but possibly not 0) it isn t har to see that the system escribe by Equations 11) to 19) still has a unique equilibrium an we suspect that this equilibrium is still globally asymptotically stable. Simulations support this conjecture but a proof remains elusive because of the elay term. The first step in proving that the equilibrium is globally asymptotically stable is to show that the solutions are boune. Looking back at equation 11), when D tumor is a constant, then 11) converges to a constant. 20

21 This means that D spleen is boune above. If we coul boun Espleen a then it woul follow that Ebloo a is boune. When Ea spleen is large, we can see from 14) that D spleen will become smaller which means that the coefficient of the growth term in equation 15) becomes very small so Espleen a must ecrease. However, to make this into a rigorous proof, we nee a better unerstaning of the elay. In orer to help us unerstan the affect of the elay term in the system efine by equations 11)-19) we have been looking at the general elay ifferential equation x t) = αxt) + βt)xt τ) + ft). We have some bouns on the growth rate of x when ft) 0 an β is a constant, but we have not been able to apply these results to the moel. 6 The Tumor Compartment The system below moels the behavior of the variables in the tumor compartment. For simplicity we enote Etumor a by x, T by y an D tumor by z. The variable Ebloo a is external to the system; we think of it simply as a function of time. Our goal is to unerstan the behavior of the system whatever function of time Ebloo a may be. The parameters all have the values given in the appenix except we explore ifferent values of an r. x t = µ y BB 1 + y Ea bloo t) a E at x cxy 27) ) 2 x 3 y y 1 t = ry y ) k z t = s + x y ) 2 y 28) 3 my q + y µ T B + a D )z 29) Notice that x an y only epen on each other an not on z. This permits us to stuy the 2-variable system consisting of x an y alone. Having unerstoo these solutions we then stuy the behavior of z. The system in x an y is particularly interesting because it contains the term x y ) 2 3 s+ x y ) 2 3 that is not ifferentiable at the origin. 6.1 The Two-Variable System in x an y Alone We start by consiering the case when Ebloo a t) = Ea bloo 11, show typical simulations in this case. is constant. Figures 9, 10, an 21

22 Figure 9: > r, E a bloo = 1000 axis scale: 104 ) Figure 10: > r, E a bloo = 0.03 Notice that the origin 0, 0) is always an equilibrium of the system. The system oes not look locally linear near the origin because of the non-ifferentiable term. For some values of Ebloo a, an r, the origin is unstable, for others it is semi-stable, an for yet others it is unstable. We shall see that there is always another stable equilibrium where y is large near the carrying capacity of the tumor). Sometimes there is a thir unstable equilibrium. The thir equilibrium is helpful in etermining the basin of attraction of the origin. In what follows we o a null-cline analysis of the system in orer to quantify an provie proofs of these observations. A simple calculation shows that the x-nullcline is given by: x = µ BB Ebloo a y 1 + y)a EaT + cy). 22

23 Figure 11: < r, E a bloo = 1000 axis scale: 104 ) The x-axis y = 0) forms part of the y-nullcline. The other part is given by: ) 2 x 3 y s 1 y ) + k 1 y k ) [ s + ) 2 x 3 y y k r 1 y ) = k x y x y x y ) 2 ] 3 = r ) 2 3 = r s + x y x y ) 2 3 ) 2 x 3 y ) 2 3 ) 2 [ 3 s1 y = k ) r 1 y k ) x = [ s1 y k ) r 1 y k ) ] ] 3 2 y The graph in Figure 12 shows the nullclines when E a bloo = , an = Let an fy) = µ BBE a bloo 1 + y gy) = a EaT + cy) [ s1 y k ) r 1 y k ) We have alreay observe that the origin is always an equilibrium. A point x, y) is another equilibrium if an only if µ BB Ebloo a x = 1 + y)a EaT + cy) 23 ] 3 2

24 ) 3 Figure 12: Ebloo a > s 2 a EaT r 1, > r µ BB an fy) = gy). The graphs of f an g are shown in figures 13, 14, an 15 for three ifferent sets of values of Ebloo a, an r. Observation 1: Consier the equilibria of the system efine by Equations 27) through 28). 1) If r, the equilibria consist of the origin an exactly one other point. 2) If > r an ) 3 Ebloo a s 2 a EaT r 1, µ BB then the equilibria of the system consist of the origin an exactly one other point. 3) If > r an ) 3 Ebloo a > s 2 a EaT r 1, µ BB then the equilibria consist of the origin an exactly two other points. We see this as follows: Function f is efine, ecreasing for all y 0, an oes not epen on Ebloo a. It ecreases at an extremely high rate until some y that is relatively small compare to k. It 24

25 Figure 13: r Figure 14: > r, E a bloo ) 3 s 2 a EaT r 1 µ BB Figure 15: > r, E25 bloo a > s r 1 ) 3 2 a EaT µ BB

26 then flattens an goes to 0 as y goes to infinity. The function g is also ecreasing, though its rate of ecrease is closer to a constant than that of f see figures 13 through 15). Moreover, gk) = 0. 1) When < r, the function g is only efine for k1 r ) < y k an has an asymptote at y = k1 r ). In this case, there is only one intersection between f an g. 2) When > r, g is still ecreasing but now is efine for all 0 y k an g0) = a EaT ) 3 2 s r 1. By the inequality in the hypothesis we see that g0) f0). It follows that fy) intersects gy) only once. 3) In this case g is ecreasing an efine for all 0 y k an g0) = a EaT s r 1 as in the secon case above. This time, however, from the inequality in the hypothesis it follows that f0) > g0). As f escens quickly it crosses the graph of g once, then it intersects g again when it flattens out. Observation 2: Every solution to the system efine by equations 27) to 28) converges to some equilibrium. We ivie the iscussion into three cases. Case I: r Figure 16 shows a sketch of the nullclines an the irection fiels in all regions in this case. ) 3 2 All solutions go to equilibrium point I, except those that lie on the line T = 0, which go to the origin. ) Case II: > r an Ebloo a s r 1 a EaT µ BB Figure 17 shows a sketch of the nullclines an the irection fiels in all regions in this case. Everything in regions 1, 3 an 4 goes to equilibrium I. Points in the top left of region 2 enter region 1 an also go to equilibrium I. Points in the top right can go irectly to equilibrium I. Points lower own can enter region 4 an be sent to equilibrium I. Points further own can just keep moving southeast without ever hitting the y-nullcline an converge to the origin. 26

27 Figure 16: The graph of the two nullclines when r It is not clear from the figure that these points exist, but simulations see for example Figure 10) suggest that there are always such points. Only points on the E a tumor axis go to the origin. Case III: > r an E a bloo > s r 1 ) a EaT µ BB. Figure 18 shows a sketch of the nullclines an the irection fiels in all regions in this case. The nullclines efine 5 ifferent regions in phase space. Everything in region 1 goes to equilibrium I. Everything in 4 an top half of 3 also goes to equilibrium I. In the top left of region 2, points go into region 1 an be sent to equilibrium I. In top right of region 2, points can go irectly to equilibrium I. Points lower own can enter region 4 an go to equilibrium I. Points even lower own in region 2 go to the origin. Everything in region 5 enters the bottom part of region 2. In the bottom part of region 3, points can hit the y-nullcline an enter region 5, an eventually go to the origin. Of particular interest to us is the basin of attraction of the origin. We saw in the observations above that when r the only solutions that converge to the origin are those where T = 0. When > r an ) Ebloo a s a EaT r 1 µ BB 27

28 Figure 17: The graph of the two nullclines with > r an E a bloo are labelle 1-4 s r 1 ) a EaT µ BB. Regions we see numerically that points that lie in a sliver where T is small compare to Etumor a are in the basin of attraction of the origin but we haven t been able to prove this rigorously or locate this sliver more precisely see the shae region in Figure??). When > r an ) Ebloo a > s r 1 a EaT µ BB there exists a thir equilibrium that can help us ientify the basin of attraction of the origin. Consier the shae region that consists of part of region 3, all of region 5 an part of region 2. All the points in this region lie in the basin of attraction of the origin. We see this as follows. The part in region 2 consists of all points that lie vertically below equilibrium II. Notice that this part of the shae region is invariant uner the flow, so therefore all solutions go to the origin. There are also some points outsie the shae part of region 2 that lie in the basin of attraction of the origin, but we are not able to ientify them precisely. Everything in region 5 enters the shae part of region 2 an goes to the origin. The points in the shae part of region 3 enter region 5 an go to the origin. Because we are especially intereste in the initial conition where y > 0 an x = 0, it is helpful to estimate how far the shae area in region 3 extens. Let 0, T 0 ) be the point in the phase plane where any solution passing through 0, T ) where T < T 0 converges to the origin an any solution passing through 0, T ) where T > T 0 converges to the other stable equilibirum. The charts below show the values of T 0 for ifferent values of Ebloo a an. 28

29 Figure 18: The graph of the two nullclines when > r an E a bloo > are labelle 1-5. s r 1 ) a EaT µ BB. Regions Parameter r has value = 1 E a bloo = 1 y 0 E bloo a

30 y Polynomial approximation reveals a strong linear relationship between y 0 an Ebloo a. It is well-approximate by the function: y 0 = E a bloo The coefficient of etermination is 1 an it gives great preication of y 0 in larger scale. Locally, the relationship between y 0 an can be best approximate by the following quaratic function with coefficient of etermination 1: y 0 = Unfortunately, it only offers a relatively accurate approximation for small values of an results in huge errors on a larger scale. Overall, we are able to numerically etermine the value of y 0 with 6.2 The Variable z T 0 = )9.4250E a bloo ) We have shown above that when Ebloo a is constant, along all solution curve for equation 27) an 28), y must converge some constant, y. Thus, where t D tumor = mt) µ T B + a D )D tumor mt) m = my q + y as y y. As mt) is converging to a constant, D tumor will also converges to a constant, namely m. This concies with our observation that D tumor always converge to a constant µ T B + a D in any simulations. 30

31 iscussion We have etermine the x,y-nullclines for equation 27) an 28) previously. When Ebloo a increases, the y-nullcline remains unchange an the x-nullcline is stretche in the x-irection. This causes the equilibrium to move up-right along the y-nullcline an away from the origin. This will inee increase the area of basin of attraction we ientifie earlier, which is boune by y-nullcline an the horizontal ray starting from equilibrium to the right. As for the part of basin attraction that lays above the y-nullcline, we coul not elaborate the effect of increasing Ebloo a ue to the lack of an explicit expression of the bounary. Nonetheless, we know that T 0 will move upwars along y-axis because of its linear epenency on Ebloo a. Moreover, for every appropriate Ea bloo, we will always have a unique solution that goes into its unstable equilibrium, appearing as the bounary of basin of attraction. If Ebloo a increases, all vectors in the fiel will have a smaller slope, ue to a larger x-component. It follows that the solution going into the new unstable equilibrium will lay above the ol one, resulting in an expansion of basin of attraction. This inteprets the phenomena that a solution growing towars the carrying-capacity equilibrium will ocasionally be eflecte half-way an pulle back into the origin equilibrium. In this case, the expansion of basin of attraction ue to inreasing Ebloo a is rapi enough to catch up with the growth of tumor. 7 Conclusion/Future Work Thus far, we have been able to prove the stability of the bloo an spleen compartment with respect to Luewig s moel in the 6 6 system analysis. However, in the bloo an spleen compartment, D tumor t) is not constant in general, an we are moving towars proving the bouneness of the six variables in that system, which will bring us closer to the general stability of that system. As we mentione before, our reasoning tells us that the bloo an spleen compartment is globally asymptotically stable given a constant number of enritic cells in the tumor. In the tumor compartment, we have been able to conuct nullcline analysis in orer to etermine the basin of attraction of the equilibrium at T, so as to put the patient into a situation where they can kill off tumor cells on their own, without any enritic cell vaccinations. In our future work, we woul like further our unerstaning of the stability of the T equilibrium. Ultimately, we woul like to etermine the DC injections that will put the patient into the basin of attraction at the T equilibrium. We woul also like to etermine which parameters have the greatest effect on the basin of attraction of T, since we o not expect all patients to have the same parameter values. Acknowlegments: The authors woul like to thank Anrea Bertozzi for organizing the 2012 summer Research Experience for Unergrauates at UCLA. The authors were supporte by the California Research Training Program in Computational an Applie Mathematics through the NSF grant DMS

32 A Parameter Values The parameters use in the moel are escribe in the following table. These parameter values are taken from DePillis et al. Table 1: Parameter Values Parameter name Description Value Units α Component of µ BT E 1 cell µ BS Transfer rate of DCs from the /ay bloo to spleen µ B Rate of DC emigration from /ay bloo µ BB Total elimination rate of CTL 5.8 1/ay from bloo µ BB Scale an shifte elimination Calculate 1/ay rate of CTL from bloo µ Normal Normal DC transfer rate from /ay spleen to bloo µ DC reuce transfer rate from /ay spleen to bloo µ BSE Transfer rate of activate /ay CTLs from bloo to spleen µ BSE = µ BSE Q spleen /Q bloo Scale transfer rate of activate Calculate 1/ay CTLs from bloo to spleen µ LB Transfer rate of DCs from the /ay liver to the bloo µ BL Transfer rate of DCs from the 0.1 1/ay bloo to the liver µ BT E = µ BB T ) T-epenent rate at which effector Calculate 1/ay cells enter the tumor compartment from the bloo µ T B Rate of transfer of DC from /ay tumor to bloo Q bloo Murine bloo volume 3 ml Q spleen Murine spleen volume 0.1 ml Q liver Murine liver volume 0.5 ml a D Natural eath rate of DCs in the spleen /ay 32

33 Parameter name Description Value Units bde Elimination rate of DCs by activate /cell ay CTLs per concentra- tion) b DE = b DE /Q spleen Per cell elimination rate of Calculate 1/ay DCs by activate CTLs a EaS Natural eath rate of activate /ay CTLs in spleen E naive Number of naive CTL cells 370 cells contributing to primary clonal expansion b p Maximal expansion factor of 85 1/ay activate CTL τ D Duration of pre-programme 0.5 ays CTL ivisions θ D Threshol in DC ensity in cell/ml the spleen for half maximal proliferation rate of CTL θ D = θ D Q spleen Scale threshol in DC ensity Calculate cell in the spleen for half max- imal proliferation rate of CTL r am Reversion rate of activate /ay CTL to memory CTL ba Activation rate of memory ml/cell ay) CTL concentration by DCs b a = b a /Q spleen Per cell activation rate of Calculate 1/cell ay) memory CTLs by DCs a Em Natural eath rate of memory /ay CTLs in the spleen θ shut Threshol in DC ensity in 13 cells/ml the spleen for half maximal transfer rate from spleen to bloo θ shut = θ shut Q spleen Scale threshol in DC ensity in the spleen for half maximal transfer rate from spleen to bloo Calculate cells 33

34 Parameter name Description Value Units a EaT Natural eath rate of activate log2)/1.5 1/ay CTLs in tumor com- partment r Tumor growth rate /ay k Carrying capacity of tumor cells c Rate at which activate CTLs /cell ay) are inactivate by tumor cells Steepness coefficient of the /ay fractional tumor kill by CTLs s Value of D necessary for half 1.4 unitless maximal activate CTL toxicity l Immune strength scaling exponent 2/3 unitless m Steepness coefficient of native cells/ml immune response q Value of T necessary for half maximal native immune response 100 cells B Equilibrium Values Here are the approximate values of the equilibrium points for varying values for E a bloo an in the 2 2 system analysis. Besies the equilibrium at T from which we recall is at T = 0), we have two other equilibria, namely the intermeiate equilibrium near the origin), an the equilibrium near k carrying capacity). Note: When r, then the equilibrium at T is unstable. Table 2: Equilibrium points where = Ebloo a Intermeiate Equilibrium Equilibrium Near k 0 none , ) 10 none , ) 5 10 none , ) none , ) none , ) none , ) none , ) 34

35 Table 3: Equilibrium points where = r = Ebloo a Intermeiate Equilibrium Equilibrium Near k 0 none , ) , ) , ) , ) , ) , ) , ) , ) , ) none , ) , ) , ) Table 4: Equilibrium points where = Ebloo a Intermeiate Equilibrium Equilibrium Near k 0 none , ) , ) , ) , ) , ) , ) , ) , ) , ) , ) , ) , ) , ) References [1] van en Driessche P. Zou X. Cooke, K. Interaction of maturation elay an nonlinear birth in population an epiemic moels. Journal of Mathematical Biology, 394): , [2] Lisette epillis, Angela Gallegos, Amy Raunskaya, Chris DeBoever, Helen Wu, an Megan Hunter. A moel of enritic cell therapy for melanoma [3] A. Gallegos an A. Raunskaya. Do Longer Delays Matter? The Effect of Prolonging Delay in CTL Activation. ArXiv e-prints, July [4] Sang Bae Han, Se Hyung Park, Young Jin Jeon, Young Kook Kim, Hwan Mook Kim, an Kyu Hwan Yang. Proigiosin blocks t cell activation by inhibiting interleukin-2r expression an elays progression of autoimmune iabetes an collagen-inuce arthritis. Journal of Pharmacology an Experimental Therapeutics, 2992): , [5] Tae-Hyung Lee, Young-Hun Cho, an Min-Geol Lee. Larger numbers of immature enritic cells augment an anti-tumor effect against establishe murine melanoma cells. Biotechnology Letters, 29: , /s y. [6] Burkhar Luewig, Philippe Krebs, Tobias Junt, Helen Metters, NevilleJ. For, RoyM. Anerson, an Gennay Bocharov. Determining control parameters for enritic cellcytotoxic t lymphocyte interaction. European Journal of Immunology, 349): ,

36 Table 5: Equilibrium points where = Ebloo a Intermeiate Equilibrium Equilibrium Near k 0 none , ) , ) , ) , ) , ) , ) , ) , ) , ) , ) , ) , ) , ) Table 6: Equilibrium points where = Ebloo a Intermeiate Equilibrium Equilibrium Near k 0 none , ) , ) , ) , ) , ) , ) , ) , ) , ) , ) , ) , ) , ) [7] Olivier Preynat-Seauve, Emmanuel Contassot, Prisca Schuler, Lars E French, an Bertran Huar. Melanoma-infiltrating enritic cells inuce protective antitumor responses meiate by t cells. Melanoma Research, 173): ,