CAR T cells for T-cell leukemias: Insights from mathematical models
Víctor M. Pérez-García, O. León-Triana, M. Rosa, A. Pérez-Martínez
CCAR T cells for T-cell leukemias:Insights from mathematical models
V´ıctor M. P´erez-Garc´ıa
Mathematical Oncology Laboratory (MOLAB), Departamento de Matem´aticas, E. T. S. I.Industriales and Instituto de Matem´atica Aplicada a la Ciencia y la Ingenier´ıa, Universidadde Castilla-La Mancha, 13071 Ciudad Real, Spain. [email protected]
Odelaisy Le´on-Triana
Mathematical Oncology Laboratory (MOLAB), Departamento de Matem´aticas, E. T. S. I.Industriales and Instituto de Matem´atica Aplicada a la Ciencia y la Ingenier´ıa, Universidadde Castilla-La Mancha, 13071 Ciudad Real, Spain. [email protected]
Mar´ıa Rosa
Department of Mathematics, Universidad de C´adiz, Puerto Real, C´adiz, Spain. [email protected]
Antonio P´erez-Mart´ınez
Translational Research Unit in Paediatric Haemato-Oncology, Hematopoietic Stem CellTransplantation and Cell Therapy, Hospital Universitario La Paz, Madrid, Spain andPaediatric Haemato-Oncology Department, Hospital Universitario La Paz, Madrid, Spain
Abstract
Immunotherapy has the potential to change the way all cancer types are treatedand cured. Cancer immunotherapies use elements of the patient immune systemto attack tumor cells. One of the most successful types of immunotherapy isCAR-T cells. This treatment works by extracting patients T-cells and addingto them an antigen receptor allowing tumor cells to be recognized and targeted.These new cells are called CAR-T cells and are re-infused back into the patientafter expansion in-vitro. This approach has been successfully used to treat B-cellmalignancies (B-cell leukemias and lymphomas). However, its application to thetreatment of T-cell leukemias faces several problems. One of these is fratricide,since the CAR-T cells target both tumor and other CAR-T cells. This leads tononlinear dynamical phenomena amenable to mathematical modeling.In this paper we construct a mathematical model describing the competitionof CAR-T, tumor and normal T-cells and studied some basic properties of themodel and its practical implications. Specifically, we found that the modelreproduced the observed difficulties for in-vitro expansion of the therapeuticcells found in the laboratory. The mathematical model predicted that CAR-Tcell expansion in the patient would be possible due to the initial presence of a
Preprint submitted to Elsevier April 30, 2020 a r X i v : . [ q - b i o . T O ] A p r arge number of targets. We also show that, in the context of our mathematicalapproach, CAR-T cells could control tumor growth but not eradicate the disease. Keywords:
Mathematical Oncology, T-cell leukemias, CAR-T cell therapies
1. Introduction
Cancer immunotherapies use elements of the patient immune system to at-tack tumor cells. These treatments encompass different therapeutic strategiestypically involving collecting a specific set of cells from patients, modifying themto produce some kind of attack on cancer cells, and reinjecting them. Some ex-amples are tumor-infiltrating lymphocytes, engineered T-cell receptor, chimericantigen receptor (CAR)-T cells, cytotoxic T-lymphocytes, natural killer cells,and mesenchymal stem cells [1].Of these, the most successful type of immunotherapy today is CAR-T cells.This treatment works by extracting patients T-cells and adding the CAR groupto them, allowing them to recognize and target the cells carrying an antigen ex-pressed in the tumor [2]. The case of B-cell leukemias expressing CD19 has beenparticularly successful since this antigen is only expressed by B-lymphocytes andB-lymphoid leukemia cells. The clinical use of CAR-T cells engineered to rec-ognize this antigen have led to the full recovery of a large fraction of AcuteLymphoblastic Leukemia patients [2, 3, 4, 5]. Good results have been reportedfor large B-cell lymphomas [6, 7] and multiple myelomas [8]. These successeshave led to the approval of CAR-T therapies directed against CD19 for treat-ment of acute lymphoblastic leukemias and diffuse large B-cell lymphomas [9].However, CAR-T cell therapies have not yet been as successful for solid tumors,for a variety of different reasons [10, 11].Mathematical modeling has the potential to help in finding optimal admin-istration protocols, provide a deeper understanding of the dynamics, help inthe design of clinical trials and more. The clinical relevance of CAR-T cellshas attracted the attention of applied mathematicians that have started toconstruct mathematical models and study different aspects of these therapies[12, 13, 14, 15, 16, 17].Given the success of CAR-T cells directed against CD19 in B-cell malignan-cies, new targets are being developed and tested. Specifically, there has been alot of interest in the possibility of using CAR-T cells for the treatment of T-cellmalignancies [18, 19, 21, 22]. However there are many challenges in translatingthis therapy for T-cell disease. The first one is fratricide, which refers to themutual killing of CAR T-cells. This phenomenon may prevent the generation,expansion and persistence of CAR-T cells. The second one is the prolongedand profound T-cell aplasia induced by the destruction of normal T-cells, thatexposes patients to severe opportunistic infections. The third one is the po-tential contamination of CAR T-cell products with malignant T-cells. Indeed,circulating tumor T-cells are often found in the peripheral blood of patients.Because tumor T-cells may harbor the same properties as normal T-cells, they2ay be harvested, transduced, expanded, and infused concomitantly with nor-mal T-cells as described recently in the context of B-cell leukemias [23]. Thus,developing CAR-T cells for T-cell malignancies requires avoiding contaminationof the CAR-T cell product with malignant transduced T-cells [18].To the best of our knowledge, no mathematical model has yet consideredCAR-T cell treatments for T-cell malignancies.In this paper we want to build the first minimal mathematical model de-scribing the dynamics of tumor cells in T-cell leukemias and normal T-cells plusa population of injected CAR-T cells. Our intention is to describe the effectof the fratricide mathematically and to obtain conclusions of practical interest.This interesting phenomenon, which involves a nonlinear self-interaction withinthe CAR-T cell compartment will be shown to place a limit on the productionof these cells in vitro. Our theoretical and simulation results support that CART-cells could be able to control tumor growth in vivo to a certain extent. Wewill show that it may not be possible to get rid of all tumor cells, but that thetreatment could be useful either as a bridge treatment or as a way of makingthe disease chronic.Our focus in this paper was to perform a preliminary exploration of the bio-logical problem and obtain conclusions of practical applicability, using numericalsimulations of the mathematical model as a test bed.The structure of the paper is as follows. First, in Sec. 2 we set out themathematical model, estimate its parameters and perform a basic study of someof its properties and study the model’s equilibria. Next, in Sec. 3 we considerdifferent scenarios including the generation of the CAR-T product in vitro andthe in-vivo dynamics. Finally, Sec. 4 discusses our findings and summarizes ourconclusions.
2. The Model
Our mathematical model accounts for the dynamics of several cell popula-tions: CAR-T cells C ( t ), leukemic T-cells L ( t ), and normal T-cells T ( t ). Theequations describing the dynamics of these populations are dCdt = ρ C ( T + L + C ) C − τ C C − αC + ρ I C, (1a) dLdt = ρ L L − αLC, (1b) dTdt = g ( T, L, C ) − αT C. (1c)CAR-T cells, described by Eq. (1a), have a finite lifespan τ C and prolifer-ate due to stimulation by target cells (either L ( t ) or T ( t ) or the CAR-T cellthemselves C ( t )). The parameter ρ C measures the stimulation of mitosis afterencounters with target cells. The parameter α in Eq. (1a) is a cell kill term3ccounting for the fratricide. It measures the probability that CAR-T cell en-counters lead to the death of one of the cells. Once the CAR-T cell identifies thetarget cell, killing and detachment are very fast processes [27]. We consider hereonly serial killing, excluding multiplexed killing, which would be a less relevantprocess and have a different kinetics.In line with models for CAR-T cell dynamics in B-cell leukemias [17], we didnot include a CAR-T cell death term due to encounters with target cells. Thereason is that CAR-T cells do not die after killing target cells [24, 25]. Also,T-cells do not divide in vivo spontaneously [26], their clonal expansion beingdependent on the stimulation with the target antigen, thus in vivo ρ I = 0. WhenCAR-T cells are expanded in-vitro cytokines are added externally forcing thecells to divide, thus in that context we will assume ρ I (cid:54) = 0.Leukemic cells [Eq. (1b)] proliferate with a rate ρ L and die to the encounterswith the CAR-T cells with the rate α .For the normal T-cell compartment we will only consider a simplified effec-tive description accounting for the different lineages expressing the same targetantigen in an aggregate form. These cells will be assumed to be killed at a rate α per cell assumed to be similar to that of the other subpopulations and will beproduced at a rate g ( T, L, C ). This function is expected to depend on the totalnumber of T-cells via cytokine signaling, on the effect of CAR-T cell on T-cellprogenitors, etc. In this paper we will assume g ( T, L, C ) to be very small andcontribute only to a minimal residual level of normal T-cells that would not berelevant for the nonlinear dynamics of the system. In what follows we will take g ( T, L, C ) = 0.Figure 1 summarizes the relationships between the different cell subpopula-tions and the assumptions behind our model.
The in-vitro expansion during the CAR-T cell production can be describedby setting L = T = 0 in Eqs. (1), and taking ρ I (cid:54) = 0, thus dCdt = ˆ ρC − ˆ αC . (2)The parameter ˆ ρ = ρ I − /τ C > α = α − ρ C ≥
0, since the kill rate is expected to be larger than the stimulationrate due to the different speeds of the killing and replication processes.Eq. (2) is a logistic equation, that for positive initial values satisfies that C −→ t →∞ C ∗ = ˆ ρ/ ˆ α. (3)This result is in line with the observation that CAR-T cells targeting T-cellantigens cannot be expanded beyond a certain value [19]. Here we show thatthis value will depend on the cytokine stimulation provided and the fratricidalcell killing rate. 4 igure 1: Cellular populations and biological processes included in the mathemat-ical model (1) . Normal T ( t ) and leukemic L ( t ) T-lymphocytes are killed by CAR-T cells C ( t ) at a rate α and stimulate CAR-T cell proliferation at a rate ρ C , both per cell. Tumorlymphocytes proliferate at a rate ρ L . As a result of fratricide and self-stimulation, CAR-Tcells are eliminated at a rate α − ρ C per CAR-T cell. CAR-T cell finite lifetime τ C also resultsin cell loss. Some parameters in model Eq. (1) can be estimated from biological data.Firstly, the typical lifetime of activated CAR-T cells τ c is in the range 14-28days [20]. T-cell leukemias are typically rather aggressive tumors with smalldoubling times that can be estimated to be around ρ L = 1/40 day − in vivo[28], although chronic forms of the disease could have much smaller numbers[29]. Finally, α and ρ C can be expected to be in the range of B-cell leukemias,where they have been found to be around 10 − day − cell − [17]. One wouldexpect ρ C to be of the order of or smaller than α , since it corresponds to thenumber of new cells generated by each encounter of CAR-T cells with targetcells.As to the initial data, the total number of T-lymphocytes in the humanbody is around 10 and typical tumor loads in acute T-cell leukemias can bein a similar range [30]. Most CAR-T administration regimes are preceded by alympho-depleting treatment that creates a favorable cytokine profile, favoringthe growth of injected cells [31, 32]. Thus, the previous numbers are substan-tially reduced once the treatment is started. We will take our initial data to bearound ∼ for tumor and normal T-cells.Finally, the number of CAR-T cells injected would depend on the maximalexpansion obtained in vitro, which could range from as low as 10 when fratricideis present to larger numbers around 10 depending on the strategies used to5vercome it. Theorem 1.
For any non-negative initial data ( C , L , T ) and all the param-eters of the model being positive, the solutions to Eqs. (1) exist for t > , arenon-negative and unique.Proof. The ODE system (1) has bounded coefficients and the right-hand sideof the system is a continuous function of (
C, L, T ), thus the local existence ofsolutions follows from classical ODE theory. Since the partial derivatives of thevelocity field are continuous and bounded, uniqueness follows from the Picard-Lindelof theorem.Let us rewrite Eqs. (1) when g = 0 as˙ C = [ ρ C ( T + L + C ) − /τ C − αC ] C, (4a)˙ L = ( ρ L − αC ) L, (4b)˙ T = − ( αC ) T, (4c)then we may write C ( t ) = C exp (cid:18)(cid:90) tt (cid:20) ρ C T ( t (cid:48) ) + ρ C L ( t (cid:48) ) + ( ρ C − α ) C ( t (cid:48) ) − τ C (cid:21) dt (cid:48) (cid:19) , (5a) L ( t ) = L ( t ) exp (cid:18)(cid:90) tt ( ρ L − αC ( t (cid:48) )) dt (cid:48) (cid:19) , (5b) T ( t ) = T ( t ) exp (cid:18) − (cid:90) tt αC ( t (cid:48) ) dt (cid:48) (cid:19) , (5c)which leads to the positivity of solutions. Definition.
The sum of all cell populations studied will be denoted by S ( t ) , i.e. S ( t ) = C ( t ) + T ( t ) + L ( t ) . Theorem 2.
Let C ( t ) , L ( t ) , T ( t ) be solutions of Eqs. (1) with initial data C ( t ) = C > , L ( t ) = L > , T ( t ) = T > , S ( t ) = S > . If H1 ρ C > α H2 ( ρ C − α ) τ C S > ,then S ( t ) increases monotonically with time and lim t →∞ S ( t ) = ∞ .Proof. Let us first sum the three equations Eq. (1) to obtain dSdt = ( ρ C − α ) SC − τ C + ρ L L. L ( t ) implies that, S defined as the solution of dSdt = ( ρ C − α ) SC − τ C, (6)satisfying S ( t ) = S with C ( t ) = C , is a subsolution of S ( t ), i.e. satisfying S ( t ) < S ( t ) , ∀ t > t . Clearly, under our hypothesis dSdt (cid:12)(cid:12)(cid:12)(cid:12) t = t = (cid:20) ( ρ C − α ) S − τ (cid:21) C > , but then, using Eqs. (6), this leads to dS/dt > t > t . Moreover, fromEq. (1a) and using the fact that S ( t ) > C ( t ), for all t > t we get dCdt = ρ C SC − τ C C − αC > (cid:20) ( ρ C − α ) S − τ C (cid:21) C, (7)where we have used ρ C SC − τ C C − αC > ρ C SC − τ C C − αC ( C + L + T ).This means that C ( t ) > C for any non-zero initial data, then dSdt > ( ρ C − α ) S − τ C ≡ Q > , (8)then S ( t ) > S ( t ) > Q t + S , which proves the unboundedness of the totalpopulation S ( t ), i.e. the fact that lim t →∞ S ( t ) = ∞ .Taking reasonable initial numbers (see Sec. 2.3) we will always initially bein the regime ( ρ C − α ) τ C S > The equilibria of Eqs. (1) in the case of interest g = 0, are given by theequations 0 = ρ C ( T + L + C ) C − τ C C − αC , (9a)0 = ρ L L − αLC, (9b)0 = − αT C. (9c)Eq. (9c) leads to either T = 0 or C = 0. The latter leads to L = 0 using Eq.(9b) and the former to either L = 0 or C = ρ L /α . Then using Eq. (9a) allowsus to obtain the expressions for the three equilibrium points of Eqs. (1) E = (0 , , T ∗ ) , (10a) E = (cid:18) ρ L α , ρ C τ C + ρ L ρ C − ρ L α , (cid:19) . (10b) E = (cid:18) τ c ( ρ C − α ) , , (cid:19) . (10c)7or any T ∗ . The Jacobian of the differential equations (1) is J = ρ C − α ) C − /τ C + ρ C ( T + L ) ρ C C ρ C C − αL ρ L − αC − αT − αC . (11)Let us now use Eq. (11) to study the local stability of the different equilibriagiven by Eqs. (10). Firstly, for E we get J ( E ) = ρ C T ∗ − /τ C ρ L − αT ∗ . (12)The eigenvalues of J ( E ) are λ = 0 , λ = ρ L , λ = ρ C T ∗ − /τ C , (13)thus the equilibrium point E is unstable. For the second equilibrium point weget J ( E ) = ρ L (cid:0) ρ C α − (cid:1) ρ C ρ L /α ρ C ρ L /α − α ( ρ L + 1 /τ C ) /ρ C + ρ L − ρ L . (14)Thus λ = − ρ L < λ , satisfy the equation λ + λ (cid:16) − ρ C α (cid:17) ρ L + ρ L (cid:16) − ρ C α (cid:17) + ρ L τ = 0 , (15)which leads to the eigenvalues λ ± = 12 ρ L (cid:16) ρ C α − (cid:17) ± D / , (16)with the discriminant D being given by Dρ L = (cid:16) − ρ C α (cid:17) − (cid:16) − ρ C α (cid:17) − ρ L τ C . (17)Let us consider the case ρ C /α < < − ρ C α < . Since (1 − ρ C /α ) < − ρ C /α , we get Dρ L < − (cid:16) − ρ C α (cid:17) − ρ L τ C < . Thus, the equilibrium is a stable node-focus.Finally, for E we get J ( E ) = 1 τ C ρ C ρ C − α ρ C ρ C − α ρ L τ c + αα − ρ C
00 0 αα − ρ C . (18)8he eigenvalues of J ( E ) are λ = 1 /τ C > , (19a) λ = ρ L τ c + α/ ( α − ρ C ) > , (19b) λ = α/ ( α − ρ C ) > , (19c)thus E is an unstable node.In conclusion, there is only one stable equilibrium point E given by Eqs.(10) of node-focus type, which can be an attractor for the dynamics of thesystem (1).
3. Applications.
To obtain further insight into the global dynamics of solutions of Eqs. (1)we simulated different initial data in the biologically feasible parameter andinitial data regions. In all cases studied, we found an oscillatory behavior of thesolutions towards the stable node-focus point E after a fast reduction of theinitial normal T-cell number.Figure 2 provides a typical example of the dynamics. There we see howtumor grows for a short time, typically 10-15 days, while CAR-T cells expand.The CAR-T cell expansion persists over more than four orders of magnitude incell number (Figure 2(c)), with a peak at about 15 days after injection (Figure2(b)). This leads to a substantial decrease of the tumor load and T-cell aplasia(Figure 2(a,b)). For this parameter set, tumor was not controlled for longperiods of times and relapse was noticeable a few months after the injectiondate of CAR-T cells. After relapse oscillations of leukemic and CAR-T cells areobserved in their course towards the equilibrium, in this case corresponding to3 × CAR-T cells and 2 . × tumor cells. Interestingly, the number oftumor cells in this case is one order of magnitude smaller than the initial tumorload (2 × cells), which supports the possibility of CAR-T cells effectivelycontrolling tumor to clinically acceptable levels.The numerical results used to construct Figure 2(c), show that in less thantwo months after injection, treatment was able to reduce tumor load from theinitial level of 2 × cells down to a minimum level of 2 . × cells, i.e. adecrease of about two orders of magnitude. The asymptotic equilibrium values of leukemic cells and CAR-T cells aregiven by E , i.e. L = 1 ρ C (cid:18) ρ L + 1 τ C (cid:19) − ρ L ρ C α , (20a) C = ρ L α . (20b)9 igure 2: Typical dynamics of cell populations governed by Eq. (1) . Results of asimulation are shown for parameter values τ C = 14 days, ρ L = 1 /
60 day − , α = 5 . × − day − cell − , ρ C = α/
2, and initial data T = 10 , L = 2 × , C = 10 cells. (a-c) Dynamics of the populations of CAR-T (green), tumor (red) and normal T cells (blue).Dynamics are depicted on the time intervals t ∈ [0 , t ∈ [0 ,
50] (b) , and inlinear (a,b) and logarithmic (c) scales. (d) Trajectory of the solution in the phase space. (e)Projection of the selected part of the trajectory on the ( T ( t ) , C ( t )) plane. igure 3: Higher mitotic stimulation rates provide better tumor control.
Dynamicsof the leukemic population governed by Eq. (1) for initial data T = 10 , L = 2 × , C =10 cells, and parameter values τ C = 14 days, ρ L = 1 /
60 day − , α = 5 . × − day − cell − . The different curves correspond to stimulation rates ρ C = 0 . α (blue solid line), ρ C = 0 . α (red dashed line), ρ C = 0 . α (green, dash-dotted line), ρ C = 0 . α (black dottedline). Interestingly, the equilibrium level of CAR-T cells does not depend on the mi-totic stimulation rate ρ C , but only on the growth and death rates of leukemiccells. However, the most important thing, due to the clinical implications, arethe leukemia equilibrium levels L , and the maximum leukemic load max t L ( t ).Maxima would typically be attained in time during the CAR-T cell expansionstage.Let us note that dL ( ρ C ) dρ C = − ρ C (cid:18) ρ L + 1 τ C (cid:19) − ρ L α < , (21)this means that L ( ρ C ) is a monotonically decreasing function. Since ρ C α > L ( ρ C ) over the range ρ C ∈ [0 , α ] would be obtained when ρ C = α . Figure 3 confirms that the asymptotic values of L decrease with themitotic stimulation rate ρ C and thus larger values of the mitotic stimulationrate lead to better tumor control. However, going beyond ρ C = α destabilizesthe system, as discussed in Sec. 2.4. Thus, it may be necessary to control indetail the CAR-T manufacturing process to get both high mitotic stimulationrates while at the same time not getting too close to the instability regime.11 .3. Initial number of CAR-T cells injected does not affect the outcome of ther-apy We next studied the effect of the number of CAR-T cells initially injectedon the system’s dynamics. To do so, we performed an extensive number of sim-ulations over the biologically feasible range and found a very weak dependenceof the dynamics on the number of injected CAR-T cells. An example is shownin Fig. 4 for a broad range of cells initially injected ranging from 10 to 10 .Although there was a difference of two orders of magnitude in C , it led to asmall variation in the time to peak expansion of a few days, a negligible increaseof the maximum CAR-T and leukemic cell number, and a minor differences inthe times to tumor relapse. Finally, we studied the dynamics under modifications of the tumor prolifera-tion rate in the whole feasible range for fast-growing leukemias ρ C ∈ [1 / , / E , which are both proportional to ρ L according to Eq. (16).As expected from the expression for E , and the values of the parameters,there was a weak dependence of the number of leukemic cells in the equilibriumon ρ L in the range of relevance (Figure 5(c)), given analytically by L = 1 . × +4 . × ρ L , with ρ L ∈ [1 / , /
20] day − . Thus the major contributionto the asymptotic tumor cell count was L ∼ / ( ρ C τ C ).Although the therapy had a substantial effect, logarithmic scale plots (Figure5d) show the persistence of measurable disease for all times.Let us define the maximum tumor cell load reduction achieved by the treat-ment as R = max t ( T ( t )) / min t ( T ( t )) . (22)In the simulations shown in Figure 5, this quantity was found to be R ( ρ L =1 /
20) = 65, R ( ρ L = 1 /
30) = 72, R ( ρ L = 1 /
40) = 78, R ( ρ L = 1 /
50) = 82, R ( ρ L = 1 /
60) = 88, thus always smaller than 100 (two orders of magnitude).It is easy to see that CAR-T cells decreased in number over time, as did thetumor load, but they were always above the numbers of cells initially injected.In fact, for most tumor proliferation rates the number of CAR-T cells was morethan one order of magnitude above the level injected.12 igure 4:
Initial number of CAR-T cells injected does not affect the outcome oftherapy.
Dynamics of the number of CAR-T cells (a,b) and leukemic cells (c,d) governed byEq. (1) over the time range [0,300] days (a,c). We also show the details of the initial responseof the treatment over the time interval [0,30] (b,d). Initial data used in the simulationswere T = 10 , L = 2 × cells, and parameter values τ C = 14 days, ρ L = 1 /
60 day − , α = 5 . × − day − cell − . The curves correspond to different values of C = 10 (bluesolid line), C = 10 (red dashed line), C = 10 (green dash-dotted line). igure 5: Tumor proliferation rate did not affect either the initial response orthe asymptotic values, but did influence relapse time.
Dynamics of the leukemicpopulation governed by Eq. (1) for initial data T = 10 , L = 2 × , C = 10 cells, andparameter values τ C = 14 days, α = 5 . × − day − cell − , ρ C = 0 . α , for differentvalues of ρ L . The different curves correspond to different values of ρ L = 1 /
60 (blue line), ρ L = 1 /
50 (red line), ρ L = 1 /
40 (green line), ρ L = 1 /
30 (magenta line), ρ L = 1 /
20 (cyanline) (a) Dynamics over the time range [0,800] days. (b) Details of the dynamics for the timeinterval [0,25]. (c) Dependence of the asymptotic tumor values obtained from Eq. (20a). (d)Tumor cell number evolution in logarithmic scale. The rate between maximum and minimumtumor load is indicated with an arrow for the case ρ L = 1 /
20. (e) Evolution of the number ofCAR-T cells. igure 6: CAR-T cell reinjection does not improve the therapy outcome.
Variationsof the maximum tumor load at the first relapse when reinjecting C = 10 CAR-T cells fordifferent reinjection times using initial data T = 10 , L = 2 × , C = 10 cells, andparameter values τ C = 14 days, α = 5 . × − day − cell − , ρ C = 0 . α . The differentcurves correspond to different values of the tumor proliferation rate ρ L = 1 /
60 (blue line), ρ L = 1 /
50 (red line), ρ L = 1 /
40 (green line), ρ L = 1 /
30 (magenta line), ρ L = 1 /
20 (cyanline).
An interesting question is if one could control relapses by acting on the tumorby reinjecting CAR T cells. To study that we performed extensive numericalsimulations over the parameter range of interest. Figure 6 summarizes someresults where we simulated the reinjection of C = 10 CAR-T cells at differenttimes post-injection for different tumor growth rates and quantified the varia-tions in maximum tumor load with respect to the case without reinjection. Inthe best scenario, corresponding to slowly growing tumors, the improvementin the peak tumor cell number at relapse was around 2%. Thus, the CAR-Treinjection did not substantially improve the outcome for any delay nor tumorproliferation rate.
4. Discussion and Conclusion
CAR-T cell therapies for B-cell malignancies is one the most resoundingsuccesses of the current immunotherapies, driving a strong interest on the topic[9]. Several mathematical models have been constructed describing the observeddynamics [12, 13, 14, 15, 16, 17]. This has lead to very considerable interestin extending these therapies to other hematological tumors, such as malignantT-cell leukemias.As stated in the introduction, one of the challenges faced by these treatmentsis fratricide, i.e. the fact that CAR-T cells, belonging to the T-cell lineage andexpressing common antigens with the leukemic cells, would themselves becometargets of the therapy. This poses the very interesting question of what theoutcome of such a therapy would be, given that it poses challenges even for15AR-T cell production in vitro. Interestingly, our simple mathematical modelcaptured the difficulties for CAR-T cell expansion in vitro, with a limit in cellproduction given by Eq. (3). Thus, the maximum number of CAR-T cells thatcan be produced in vitro depends on the stimulation provided by the cytokinesand the excess CAR-T killing efficiency over the mitotic stimulation.One might naively think that in-vivo expansion would also be limited, nothaving a substantial effect on the disease. However, this is not true. Oursimulations showed that when they are injected, even in the small numbers thatcan be obtained in vitro, the CAR-T cells find many targets initially on boththe healthy and tumor T-cells. During this initial stage, the CAR-T populationis amplified even in the presence of fratricide. We also found in silico that theoutcome of the therapy did not depend on the number of CAR-T cells injected.A relapse was always observed in the framework of our model simulationsand the number of tumor cells was initially reduced by a factor smaller than 100,with the persistence of measurable disease for all times, so the treatment didnot eradicate the disease in our numerical simulations. However, CAR-T cellswere able to control tumor growth after two weeks and then, even in spite ofthe oscillations, the high initial tumor loads were never found to appear again..Relapse time after the CAR-T treatment was found to depend strongly onthe proliferation rate. This makes us wonder if a combination with a post-CARchemotherapy could prove useful in delaying tumor regrowth. The oppositestrategy, i.e. first giving chemotherapy and then CAR-T before relapse, wouldnot be recommended, however, since chemotherapy would be expected to reducethe number of target cells and then CAR-T would lack the substrate to expand.Another question related to these treatments is whether re-challenging withCAR-T cells at any given time could be beneficial. For instance one may wonderif that process could be used to delay or even eliminate tumor relapse. On thebasis of our computational results, additional injections were found to have nosubstantial effect on the dynamics. The reason is that CAR-T cells decreased innumber transiently in time after reaching peak expansion, but their levels werealways above the numbers of cells initially injected, typically by more than anorder of magnitude. This means that injecting small numbers of CAR-T cellsin relation to those already present would not have a substantial effect on thedynamics.The fact that the equilibrium L ∼ / ( ρ C τ C ) implies that there are two waysto improve the long-term efficacy of CAR-T cell therapy for T-cell leukemias.The first would be to improve the persistence of the CAR-T cells, somethingthat has been done for B-cell leukemias by using CD19 CAR (CAT) with loweraffinity than FMC63, the high-affinity binder used in many clinical studies [20].The second would be to improve the mitotic stimulation rate, but keeping inmind the restriction α > ρ C .In conclusion, in this paper we have developed a mathematical model of thedynamics of leukemic cells, healthy T-cells and CAR-T cells, after the ther-apeutic injection of the latter population. The mathematical model showedthe potential of the treatment to control, but not eradicate, the disease. Thiswould result in a chronification of the disease that could last for a long time, or it16ould buy some time to try alternative therapeutic strategies. Our work is a firstsimple mathematical attempt to cast light on the potential outcomes of thesetreatments. There are different types of T-cell malignancies and specifically T-cell leukemias, and the particular features of each type could be incorporatedinto more detailed models including additional biological details. We hope ourwork will stimulate further work in this exciting sub-field of immunotherapy. Acknowledgements
This work has been partially supported by the Junta de Comunidades deCastilla-La Mancha (grant number SBPLY/17/180501/000154), the James S.Mc. Donnell Foundation (USA) 21st Century Science Initiative in Mathemat-ical and Complex Systems Approaches for Brain Cancer (Collaborative award220020450), Junta de Andaluca group FQM-201, Fundacin Espaola para la Cien-cia y la Tecnologa (FECYT, project PR214 from the University of Cdiz) and theAsociacin Pablo Ugarte (APU). OLT is supported by a PhD Fellowship fromthe University of Castilla-La Mancha research plan.We would like to acknowledge Gabriel F. Calvo, Carmen Ortega-Sabater,Juan Belmonte Beitia (MOLAB, University of Castilla-La Mancha, Spain),Manuel Ram´ırez-Orellana (Hospital Universitario Ni˜no Jes´us, Madrid, Spain)and Soukaina Sabir (University Mohamed V, Morocco) for discussions.17 eferences [1] Rafei H, Mehta RS, Rezvani K (2019) Editorial: Cellular Therapies in Can-cer. Front Immunol. 10:2788.[2] Feins S, Kong W, Williams EF, Milone MC, Fraietta JF (2019) An introduc-tion to chimeric antigen receptor (CAR) T-cell immunotherapy for humancancer. Am J Hematol. 94(S1):S3-S9.[3] Maude SL, Laetsch TW, Buechner J, Rives S, Boyer M, Bittencourt H, BaderP, Verneris MR, Stefanski HE, Myers GD, Qayed M, De Moerloose B, Hira-matsu H, Schlis K, Davis KL, Martin PL, Nemecek ER, Yanik GA, PetersC, Baruchel R, Boissel N, Mechinaud F, Balduzzi A, Krueger J, June CH,Levine BL, Wood P, Taran T, Leung M, Mueller KT, Zhang Y, Kapildeb S,Lebwohl D, Pulsipher MA, Grupp SA(2018) Tisagenlecleucel in children andyoung adults with B-cell lymphoblastic leukemia. N Engl J Med. 378:439-448.[4] Pan J, Yang JF, Deng BP, Zhao XJ, Zhang X, Lin YH, Wu YN, Deng ZL,Zhang YL, Liu SH, Wu T, Lu PH, Lu DP, Chang AH, Tong CR (2017) Highefficacy and safety of low-dose CD19 −−