- Research
- Open Access
A patient-specific therapeutic approach for tumour cell population extinction and drug toxicity reduction using control systems-based dose-profile design
- Suhela Kapoor†^{1},
- VP Subramanyam Rallabandi†^{1},
- Chandrashekhar Sakode^{2},
- Radhakant Padhi^{2}Email author and
- Prasun K Roy^{1}Email author
https://doi.org/10.1186/1742-4682-10-68
© Kapoor et al.; licensee BioMed Central Ltd. 2013
- Received: 20 September 2013
- Accepted: 11 December 2013
- Published: 26 December 2013
Abstract
Background
When anti-tumour therapy is administered to a tumour-host environment, an asymptotic tapering extremity of the tumour cell distribution is noticed. This extremity harbors a small number of residual tumour cells that later lead to secondary malignances. Thus, a method is needed that would enable the malignant population to be completely eliminated within a desired time-frame, negating the possibility of recurrence and drug-induced toxicity.
Methods
In this study, we delineate a computational procedure using the inverse input-reconstruction approach to calculate the unknown drug stimulus input, when one desires a known output tissue-response (full tumour cell elimination, no excess toxicity). The asymptotic extremity is taken care of using a bias shift of tumour-cell distribution and guided control of drug administration, with toxicity limits enforced, during mutually-synchronized chemotherapy (as Temozolomide) and immunotherapy (Interleukin-2 and Cytotoxic T-lymphocyte).
Results
Quantitative modeling is done using representative characteristics of rapidly and slowly-growing tumours. Both were fully eliminated within 2 months with checks for recurrence and toxicity over a two-year time-line. The dose-time profile of the therapeutic agents has similar features across tumours: biphasic (lymphocytes), monophasic (chemotherapy) and stationary (interleukin), with terminal pulses of the three agents together ensuring elimination of all malignant cells. The model is then justified with clinical case studies and animal models of different neurooncological tumours like glioma, meningioma and glioblastoma.
Conclusion
The conflicting oncological objectives of tumour-cell extinction and host protection can be simultaneously accommodated using the techniques of drug input reconstruction by enforcing a bias shift and guided control over the drug dose-time profile. For translational applicability, the procedure can be adapted to accommodate varying patient parameters, and for corrective clinical monitoring, to implement full tumour extinction, while maintaining the health profile of the patient.
Keywords
- Tumour cell extinction
- Cancer therapy optimization
- Control system
- Chemotherapy
- Immunotherapy
- Glioma
- Astrocytoma
- Meningioma
- Oligodendroglioma
- Glioblastoma
Background
Though there has been considerable efforts in exploring newer modalities of cancer treatment in the last several decades, hopes have been belied by a fundamental reason, though not fully appreciated, namely the difficulty of eradicating a tumour mass due to the nature of reaction kinetics that govern the interaction of tumour cells with the therapeutic agents administered. These interactions are governed by first-order reaction processes (chemotherapy) and enzyme saturation kinetics (immunotherapy) [1]. Both these prescribed treatments cause an exponential decay of the tumour cell population leaving a finite definitive number of tumour cells at the asymptotic extremity of tumour versus drug distribution curve. By its very nature, an asymptotic tail implies the tumour cell population curve will contact the horizontal axis and become zero only when either the time duration or the drug dose is infinite. For instance, a typical tumour may have 10^{10} cells, and the surviving fraction, SF, of tumour cells after administration of a drug concentration D is SF = exp (−α D). Using the standard values of α = 0.02 and D = 150 mg for bleomycin (maximum dose tolerated) [2], the equation yields 1,765 cells surviving. This elucidates why many tumours recur, after appearing to initially shrink or regress under therapy.
An imminent question in neurooncology is the efficacy of the drug to be able to cross the “blood–brain barrier” (BBB) to enter the brain. A recent progress is the development of DNA alkylating drugs such as Temozolomide (TMZ) which while being highly targeted, can effectively cross the BBB [3]. However TMZ is known to be contraindicated in some cases such as in patients with severe myelosuppression. A few other common side effects are nausea and vomiting which are self-limiting or readily controlled with standard therapy. Temozolomide is a teratogenic compound and thus should not be used during pregnancy. It might very rarely leads to acute respiratory failure. However, no drug-related adverse CNS effects or alopecia are known to occur with temozolomide [3].
To complement the effects of chemotherapy, immunotherapy, is also being considered for its synergistic carcinolytic effects. Recent incisive findings of Wheeler et al. [4] indicate that a combinatorial therapy design utilizing immunotherapy with chemotherapy reduces tumour volume by 50% and appreciably extends the average 1 year survival duration of glioblastoma patients, which cannot be done by either chemotherapy or immunotherapy alone. An important aspect of immunotherapy is to utilize cytotoxic T-lymphocytes (CTL) which are CD8 + T cells, that are known to be carcinolytic. Activated CTL can be generated by administering immunomodulating factors,. This can be done by two means: (i) administering cytokines such as Interleukin-2 (IL-2) which can cross the blood–brain barrier, and (ii) injecting cellular agents such as tumour-infiltrating lymphocytes (TIL) prepared beforehand by sensitizing T-cells of the patient’s blood against the tumour (biopsy tissue). These TILs proceed satisfactorily to the tumour mass in the brain parenchyma. Also, IL-2 is well known to stimulate recruitment and proliferation of cytotoxic T-lymphocytes, such as CD8+ T cells, suggesting a novel neurooncological approach [5].
Here u_{ 1 }, u_{ 2 }, …. are the levels of the different antitumour agents, while B_{ 1 }, B_{ 2 }, …. are the weighting factors of the different agents. We use this principle to suitably orchestrate the temporal schedule of the drugs, so that that toxicity is minimized.
We may mention that various attempts at modeling the immune system interaction with neoplastic tumours have been previously made [8–10]. These models have efficiently characterized the computational dynamics of drug versus tumour interaction via the immune system. Using the background of the existing models, in our model we have tried to delineate the kinetics and dynamics of immune modulation responsible for the paradoxical clinical phenomenon of tumour dormancy, prolonged arrest and oscillations of tumour-size [11]. A unitary approach to the dual behaviour of tumour progression and tumour regression has recently been explained [12], where the neoplastic process has been elucidated as systems biology-based abnormality. The tumour regression approach that we report in the present work is to our knowledge, the first endeavor to elucidate a quantitative methodology to delineate the dose-time profile of administration of the antitumour agents (chemotherapy, interleukin, lymphocytes) with neuroncological cases as examples, so as to enforce the tumour cell population to zero, thus enabling full tumour elimination. For this, we develop an interdisciplinary approach, utilizing input reconstruction analysis and bias shift.
Methods
Inverse construction of drug input for obtaining desired tumour response
So, in the equation (2), one puts T = 0 at time t_{ 1 }, which is used to calculate the temporal profile of dosages of the combination drugs (D_{N}^{†}). Thus, if one administers the temporal dosages profile (D_{N}^{†}), then the tumour cell population definitively becomes zero in time t_{ 1 }, indicating tumour elimination. The perspective of inverse analysis approach [eq. (2)] has been well used in other fields in the form of stimulus reconstruction approach. This enables one to accurately quantify an antecedent unknown stimulus, using the knowledge of the observed response pattern. In our case, the methodology enables one to reconstruct the input drug stimulus, which will accurately produce complete tumour elimination as the output in finite time.
Problem of asymptotic drug reaction kinetics
As in the case of a biochemical reactor, conditions may arise where the synthesis rate R following first-order reaction, needs to be controlled and stopped at a definitive time [13]. To explore the possibility of inducing the reactor's processing to stop (i.e., R = 0), one may omit the input feed of reactants which causes the reaction to slow, but not become zero even after any length of time leaving a small asymptotic level of reaction rate R persisting at a time t, since first-order kinetics indicates R′ + κ R = 0.
The asymptoticity in both the above cases is due to the fact that both the dynamic equations are similar and first-order, viz. y′ + k y = 0. Here the y-axis denotes the system parameter, as reaction rate or altitude, while the x-axis denotes time. The equation implies that at infinite time, the curve meets the y-axis, if the coordinates at this meeting point is t^{ † } and y^{ † }, then y^{ † } = 0, as t^{ † } = ∞. Nevertheless, in the exponential systems above, we elucidate that the systems dynamics can be halted in a finite time (i.e., y = 0, at specific time t_{ p }), if one implements the concept of an adjustable or tunable bias shift, denoted by y*. This enables the performance curve to approach a value y → y* (a proposed pre-determined negative value), as t → ∞ [Figure 2B]. This ensures that the curve trajectory intersects the time axis, and has an exact value y = 0 robustly, at a definite time t_{ p } [point P, in Figure 2B]. We have earlier methodologically simulated and validated realistically the procedure of bias shift, while respecting the requisite bounds or constraints imposed on the system [14]; thereby the exponential curve was induced to become zero within specific time interval, the error in the validation process being within 2.5%.
Using bias shift as baseline regulating principle
Eq. (6) is the condition for complete tumour regression, and hence should be followed by the tumour cell compartment in Figure 1. Thereafter, proceeding upwards through the successive compartments, such as administration of chemotherapy then the immunotherapy, we can obtain the dose-time profiles of temozolomide, interleukin-2 and tumour-infiltrating lymphocyte injections, for regression of the tumour within the time duration t_{ d }. Thereafter, the tumour shall not recur as there are no surviving tumour cells. As a precaution, we continue the therapies for an extra sufficiently long period before fully stopping.
We have earlier used the inverse solution approach, but without bias shifting, for two treatment scenarios: (i) controlling chemotherapy infusion (imatinib) in myeloid leukemia [16], and (ii) regulating therapeutic infusion for treatment of ionic metabolic or hypocalceamic imbalance (control target error < 5%) [17]. In these cases the tumour cell population regressed and the blood ionic level tended to approach the desired value with high stability and asymptotically, but complete tumour elimination could not be obtained at a definitive time. In the present paper, we have remedied this problem by using adjustable bias shifting.
Computational model of multimodal therapy
Values of the biological and pathophysiological parameters for the tumour system
Symbol | Numerical value | Characteristic significance | Reference | Units |
---|---|---|---|---|
Therapeutic parameters | ||||
p _{I} | 1.25 × 10^{-1} | Interleukin 2-induced CD8+T-cell recruitment rate (maximum value) | Kirschner et al.[10] | per day (rate value) |
g _{I} | 2.00 × 10^{7} | Interleukin 2-induced CD8+T-cell recruitment (steepness value of the curve) | Kirschner et al.[10] | cell^{2} (cell-cell interaction) |
μ_{I} | 1.0 × 10^{1} | Decay rate of Interleukin-2 drug | Kirschner et al.[10] | per day |
Birth/death parameters | ||||
α | 7.5 × 10^{8} | Circulating lymphocyte birth rate (constant source) | de Pillis et al.[18] | cells per day |
β | 1.20 × 10^{-2} | Lymphocyte death and differentiation rate | de Pillis et al.[18] | per day |
γ | 9.0 × 10^{-1} | Chemotherapy drug decay rate | Calabresi et al.[20] | per day |
k _{ T } | 9.0 × 10^{-1} | Chemotherapy-induced tumour cell lysis | Perry [1] | per day |
k _{ N } , k _{ L } , k _{ C } | 6.0 × 10^{-1} | Chemotherapy-induced lysis of NK cells, CD8+T-cells, and Circulating lymphocytes, respectively | Perry [1] | per day |
r _{ 1 } | 1.1 × 10^{-7} | CD8+T cell generation rate, induced by tumour cell lysis by NK cell | per cell per day | |
r _{ 2 } | 6.5 × 10^{-11} | CD8+T cell generation rate, induced by tumour cell - circulating lymphocyte lysis | de Pillis et al.[18] | per cell per day |
Pathophysiological parameters | ||||
a | 4.31 × 10^{-1} (high grade tumour); 3.01x10^{-1} (low grade tumour) | Rate of tumour growth | per day | |
b | 2.17 × 10^{-8} (high grade tumour); 1.02 × 10^{-8} (low grade tumour) | Deceleration effect of logistic growth of tumour | per cell | |
c | 6.41 × 10^{-11} | NK cell-induced lysis of (non)-ligand-transduced tumour cell | Diefenbach et al.[24] | per cell per day |
d | 2.34 | CD8+T cell-induced fractional tumour cell lysis (saturation value); priming by ligand-transduced cell | Dudley et al.[25] | per day |
e | 2.08 × 10^{-3} | Lymphocyte fraction converting to NK cells | Kuznetsov et al.[9] | per day |
f | 4.12 × 10^{-2} | NK cell death rate | Kuznetsov [9] | per day |
g | 4.98 × 10^{-1} | Ligand-transduced tumour cell-induced NK cell recruitment rate (maximum value) | Dudley et al.[25] | per day |
h | 2.02 × 10^{7} | NK cell recruitment by tumour cell (steepness index of recruitment curve) | Kuznetsov et al.[9] | cell^{2} (cell-cell interaction) |
l | 2.09 | CD8+T cell-induced tumour cell lysis (exponent value) | Dudley et al.[25] | Dimensionless (exponent) |
s | 8.39 × 10^{-2} | CD8+T cell-induced tumour cell lysis (Value of Steepness index of term D denoting lysis) | Dudley et al.[25] | Dimensionless (exponent) |
p | 3.42 × 10^{-7} | Tumour-cell induced inactivation of NK cell | Diefenbach et al.[24] | per cell per day |
m | 2.04 × 10^{-1} | CD8+T cell death rate | Yates et al.[21] | per day |
j | 2.49 × 10^{-2} | Recruitment rate of CD8+T cells (max. value), cells primed with ligand-transduced tumour cells | Diefenbach et al.[24] | per day |
k | 3.66 × 10^{7} | CD8+Tcell recruitment curve (steepness index), cells primed with ligand-transduced tumour cells | Diefenbach et al.[24] | cell^{2} (cell-cell interaction) |
q | 1.42 × 10^{-6} | Tumour cell-induced CD8+T cell inactivation rate | Kuznetsov et al.[9] | per cell per day |
u | 3.00 × 10^{-10} | CD8+T-cell regulation by NK-cell | de Pillis et al.[18] | per cell^{2} per day (cell-cell interaction rate) |
Numerical values of the variables from the patient’s clinical data
- (i)
The pharmacological/cell birth/death/interaction parameters: a, b, c, d, e, f, p, m, q, u, r _{ 1 } , r _{ 2 }
- (ii)
Drug input/output parameters: α, β, γ, μ
- (iii)
Michaelis-Menten or saturation parameters: g, h, j, k, p _{ I } , g _{ I } , D, d, l, s;
- (iv)
Temporal lysis parameters of different cells: k _{ T }, k _{ N }, k _{ L }, k _{ C }.
- (a)
v _{M} (t), v _{I} (t), v _{L} (t) are the daily injected dosages (input rate per day) of chemotherapy (temozolomide), immunotherapy (interleukin-2), and cytotherapy (TIL cells) respectively.
- (b)
d, the saturation level of fractional tumour cell kill by the cytotoxic T cells,
- (c)
l, the power-law exponent of fractional tumour cell kill by CD8+ T cells,
- (d)
s, the steepness coefficient of the cytotoxic T cell – tumour cell interaction curve
Stationary condition of the system in tumour elimination
This equation implies that there is a definitive tumour-free stationary state. At this equilibrium point there is no presence of any malignant cell, i.e., the cytotoxic T-cell population T_{E} = 0, L_{E} = 0, while there are specific levels of the immune cells, NK cells and circulating lymphocytes, that correlate with the healthy state (N_{E} = eα/βf, C_{E} = α/β). Substituting the values of the parameters α, β, e and f from Table 1 and taking the average blood volume of 6 liters for an adult, we arrive at the natural killer cell equilibrium population = 430,000 cells for the whole subject and circulating lymphocytes population = 72 billion for the same. It is this equilibrium state that the system tends to settle to, in the occasion of successful therapy.
System design formulation of multimodal therapy
- (i)
System state variables are the parameters indicating the internal biological condition of host-tumour interaction in the patient. The system state variables in this case are (1) Tumour cells (T), (2) Natural killer cells (N), (3) Circulating lymphocytes (C), (4) Cytotoxic T-Lymphocytes (L), as well as (5) blood concentrations of Interleukin (I) and of (6) temozolomide chemotherapy (M) [Figure 1]. The state variables are denoted as functions X _{ n }, where n = 1 to 6.
- (ii)
Control variables of system include the parameter values the external administrations of the three therapeutic agents, viz. the dose input rates of the injected chemotherapy (temozolomide) and immunomodulants (interleukin-2 and tumour-infiltrating lymphocytes), which are represented by v _{M} (t), v _{I} (t), and v _{L} (t) respectively. These parameters vary with time and are measured in body-weight normalized rate units, e.g. in mg. of a drug per kg of body weight (or, per square metre of body surface) of the patient, given per day. These control variables are denoted as functions U _{ m }, where m = 1 to 3.
- (iii)
Constraint conditions of system indicate the quantitative requirements or thresholds that cannot be crossed application of the therapy. All the state variables need to be maintained within minimum and maximum values given in Table 2. For instance, the minimum and maximum values of the different cellular parameters should be within the physiological range that is necessary for homeostatic maintenance of the internal environment, or milieu interne of the patient. Too high a value of an entity can be toxic to the system, while, in some cases, there is a minimum value required so that the host can have proper immune surveillance to ward off foreign cells or microorganisms.
Bound limits of biological and therapeutic parameters applicable to human cases
Parameter | Lower limit | Upper limit | Reference |
---|---|---|---|
Circulating Lymphocyte, C (total population in the individual) | 2.32 × 10^{9} cells (for not more than 6 months duration) | 4.5 × 10^{11} cells | Macintyre et al.[26] (upper limit); Jarosz et al.[27] (lower limit) |
Natural Killer Cells, N (total population in the individual) | Negligible value (for not more than 3½ months duration) | 5.85 × 10^{10} cells | Berrington et al.[28] (upper limit); Jawahar et al.[29] (lower limit) |
Tumour-specific cytotoxic (CD8+) T-cells (CTL), L (total population of these cells in the individual) | Negligible value * | 6.05 × 10^{10} cells | Dudley et al.[25] (upper limit); Roitt et al.[30] (lower limit) |
Temozolomide infusion dosage rate, v_{M} | 0 | 200 mg/m^{2}/day (4.45mg/kg/day) | Perry [1] |
Interleukin-2 infusion dosage rate, v_{ I } | 0 | 7.2 × 10^{4} I.U/kg/day | Perry [1] |
Tumour Infiltrating lymphocyte (TIL) cumulative dosage (over full therapy duration, i.e., ∑ v_{ L }) | 0 | 13.7 × 10^{10} cells | Dudley et al.[25] |
Using control systems practice, to construct the rate equation of any one particular state variable X_{ p } (a biological parameter), we express its time derivative or rate parameter X_{ p }′, as a function of (i) the values of the various biological parameters that determine the system, namely X_{ n }, and (ii) the effect of the proximal causal control variables (drug factors, U_{ m }) on the biological variables (X_{ n }).
We shall now explain the construction of the control design, in terms of objective, operation and toxicity-minimization for each compartment.
Level 1: tumour cell compartment
Objective
The aim is to find the desired values, M* and L* of this compartment’s input parameters (Temozolomide blood level and Cytotoxic T-cell population), which will drive the compartment’s output (tumour cell population curve T) towards the line T*, until the T value becomes zero at point P in time t_{ P } (Figure 2B).
Operation of the compartment
- (i)
the function f _{ T } (X _{ n }) which denotes the values of the various biological parameters that determine the tumour cell population system, namely the state parameters X _{ n. } This explains the tumour cell alteration function due to two intrinsic biological (non-pharmacological) processes: (a) tumour cell growth term, due to logistic or saturable growth, (b) tumour cell elimination term due to natural killer cells.
- (ii)
the functions g _{T 1} and g _{T 2} that imply the biological effect of the compartment’s input therapy variables that show a dose–response saturation behavior U _{ r } , which is produced by the two antitumour entities, cytotoxic T-cell population L, and temozolomide concentration in blood, M.
Description of therapy system terms
Terms | Description | Reference formulation |
---|---|---|
U _{ M } | Therapeutic efficiency factor of Chemotherapy | U_{ M } = {1 − exp(−M)} [Eq. (10)] |
U _{ L } | Therapeutic efficiency factor of Cytotoxic T-lymphocytes | U_{ L } = d L^{ l }/(s T^{ l } + L^{ l }) [Eq. (10)] |
U _{ I } | Therapeutic efficiency factor of Interleukin-2 | U_{ I } = p_{ I }LI/(g_{ I } + I) [Eq. (27)] |
b _{ T } | Total tumour cell lysis effect from the two cytotoxic agents, Chemotherapy (temozolomide) and Cytotoxic T-cells | b_{ T } = U_{ M }g_{T 1} + U_{ L }g_{T 2} [Eq. (12)] |
b _{ L } | Total cytotoxic T-cell activation effect from the two Immuno-modulating agents, Interleukin and Tumour-infiltrating lymphocyte | b_{ L } = U_{ I } + v_{ L } [Eq. (29)] |
Eq. (12) furnishes the relationship of the blood concentration of the antitumour agents (U_{ M } and U_{ L }), which, if implemented, will ensure that the tumour cells undergo complete extinction.
Eq. (12) furnishes the relationship of the blood concentration of the antitumour agents (U_{ M } and U_{ L }), which, if implemented, will ensure the tumour cells undergo complete extinction.
Minimization of toxicity of antitumour therapy
where the right sided expression within the second brackets {….} in eq. (16) incorporates the left side of the constraint equation [eq. 14(A)], as required by the Lagrange’s method. To pursue the minimization of the augmented cost function, we differentiate J with respect to the two variables U_{M} and U_{L}. Thereby, we find out the minimization conditions, i.e., ∂J/∂U_{ M } = 0, and ∂J/∂U_{ L } = 0, whence we get the expressions for the two variables:
Desired input values of the therapy levels
In the last two equations, for the terms U_{M} and U_{L}, we substitute their values from eq. (17) and (18) respectively. Thence, we arrive at the desired values of the temozolomide blood level and cytotoxic T-cell population, which, if implemented, will regress the tumour fully:
Level 2: cytotoxic T-cell compartment
Objective
Here the goal is to find the desired values, I* and v_{ L }*, of this compartment’s input parameters (interleukin-2 blood level and tumour-infiltrating lymphocyte injection dose-rate), which would drive the compartments output, namely the cytotoxic T-cell population, L to its desired value L* mentioned in the last paragraph. This enforced driving needs to be faster than the earlier compartment (tumour cell compartment) and requires to be done till time t_{r}, point P, when all the tumour cells have become eliminated (Figure 2B).
Operation of this compartment
Note here that f_{ L }(X) is the performance function of the cytotoxic T-cell compartment, a part of the the right-side of the eq. (26) except its last two terms, which are the therapy-input terms, dependent on the interleukin and tumour-infiltrating lymphocyte injected, the two terms being denoted as b_{ L }, In eq. (26), the last term v_{L} is the Tumour-infiltrating lymphocyte injection dose-rate, while the second-last term [p_{ I }LI/(g_{ I }+I)], now denoted as U_{I}, is the therapeutic efficiency relationship of interleukin-2. Evidently from eq. (26), the term b_{ L } signifies the total cytotoxic T-cell activation effect by the two immunotherapeutic inputs: the interleukin efficiency term U_{ I } and the tumour-infiltrating lymphocyte administration term v_{L} (Table 3). Indeed, the said equation indicates that the relaxation decay term b_{ L } has positive value if the tumour regresses (or zero, if the tumour is arrested and thereby stable), i.e. b_{ T } ≥ 0. Actually, eq. (29) furnishes the relationship of the characteristics of the antitumour agents, interleukin and tumour infiltrating lymphocytes (U_{I} and v_{L}), which if implemented, will ensure the full elimination of tumour.
Minimization of toxicity of antitumour therapy
where r_{L 1} and r_{L 2} are the sensitivity weights due to the two aforesaid therapeutic moieties in this compartment. For minimization, the constraint requirement [U_{ I } + v_{ L } = b_{ L }] from [eq. (29)] should be obeyed. Solving using the Lagrange multiplier method, we arrive at:.
Desired input values of the therapy level
Substituting U_{I} from eq. (32) to eq. (35), we get the desired value of tumour-infiltrating lymphocyte injection dose-rate which will eliminate the tumour:
Level 3: temozolomide injection compartment
Objective
Here the goal is to find the desired value, v_{ M }*, of this compartment’s input parameter (temozolomide injection dose-rate), which would drive the compartment’s output, namely the patient’s blood level of temozolomide M to its desired value M* as given in eq. (22). This driving needs to be done faster than the preceding compartment (T-cell compartment), and is to be done by time t_{ p }, point P (Figure 1).
Operation of this compartment
Level 4: interleukin-2 injection compartment
Level 5: tumour-infiltrating lymphocyte injection compartment
Determining tumour regression rate constant, bias shift and therapeutic weights
Delineating the rate parameters κ_{T}, κ_{M}, κ_{L}, κ_{I} and bias T*
These are rate constants of the tumour cell compartment, temozolomide compartment, cytotoxic T-cell compartment and interleukin compartment, respectively, as regression occurs under the action of multi-modal therapy. These are calculated from the desired rate of tumour regression, expressed as settling time t_{s} of the regression process (the time duration in which 90% of tumour has regressed), and is taken to be around 1–2 months. The tumour regression rate constant κ_{ T } = 4/t_{s}; so if t_{s} is 60 days, κ_{T} = 0.067 per day. On the other hand, the dynamics of the successively preceding compartments (e.g. chemo-therapy and cytotoxic T-cell compartments) need to be faster, as they causally influence the tumour cell compartment, and thus need to change more rapidly if they are to have a controlling influence on the tumour cell compartment (Figure 1). Thus, the time constants of the modular stages will be lesser, and hence the process rate constant will be higher. So, we can choose the rate constants κ_{ L } and κ_{ M } of cytotoxic T-cell compartment and temozolomide injection compartment respectively, such that they exceed κ_{ T }. Similarly, the rate constant κ_{ I } of interleukin compartment (that causally acts on the T-cell compartment), is chosen to be higher that κ_{ L }.
In the above example, since, κ_{ T } = 0.067/day, we get κ_{ L } = 0.201/day, κ_{ M } = 0.201/day, and κ_{ I } = 0.603/day. It may be noted that the tumour elimination time t_{ p } (Figure 2) when 100% tumour has regressed, is longer than t_{s}, and can be selected as 10 days more, i.e. t_{ p } = 70 days. If the initial pre-therapy tumour cell population is estimated as T_{ 0 } , then substituting the values of T_{ 0 }, κ_{ T } and t_{ p } in the right-sided expression in eq. (4), provides the value of bias shift T*.
Selection of therapeutic weights r_{T1} , r_{T2} and r_{L1} , r_{L2}
These adaptable parameters, r_{ T1 }, r_{ T2 }, r_{ L1 }, r_{ L2 } are needed for minimizing the toxicity cost of the therapy. It is these parameters that give a control to the investigator for maneuvering the tumour elimination process, under adjustable dosing of the drugs. Initially, the values of the tuning parameters r_{ T1 }, r_{ T2 }, r_{ L1 }, r_{ L2 } which appear in derivation, are chosen by specific quantitative conditions (see Additional file 2: Supporting Analysis). To recapitulate, r_{ T1 } and r_{ T2 } are respectively the toxicity cost weighting factors of temozolomide and cytotoxic-T-cell, acting on the tumour cell population compartment, producing the cost J_{T} which is to be optimized.
For such optimization problems, it is known that the important characteristic to be considered is the ratio r_{ T1 }: r_{ T2 }[15]. Thus, we can take the parameter r_{ T1 } to have a normalized value of unity (i.e. r_{ T1 } = 1), thereby the task is to suitably choose or optimize the value of the other tuning parameter r_{ T2 }. Correspondingly, r_{ L1 } and r_{ L2 } are respectively the toxicity cost weighting factors of interleukin-2 and tumour-infiltrating lymphocytes, which act on the cytotoxic T-cell compartment and produces the toxicity cost J_{L}. Similarly, this cost can be minimized by normalizing r_{ L1 } = 1, and then optimizing r_{ L1 }. All the tuning parameter values must be greater than or equal to zero. Using the upper and lower bounds that needs to be followed by the cellular and pharmacological variables (Table 2), we can calculate the numerical values of the therapeutic weight parameters (Additional file 2: item B.3 and Table 1 there).
Efficiency of different combinations of therapeutic agents
- (i)
The dosage parameters of all three drugs remain positive throughout the time duration: If so, all are given.
- (ii)
One drug becomes negative or imaginary for a particular time interval: this drug is stopped during that interval, and the other two drugs are continued as per the modeling procedure with the first omitted
- (iii)
Two drugs become negative or imaginary: here the remaining drug with a positive drug dosage rate is administered.
- (iv)
All the three drugs violate the positivity condition: then no drug is given, until one or more drug rates become positive at a subsequent time, upon which the administration of the drug/s is resumed.
Results and discussion
Numerical simulations for tumour elimination
Low-grade tumour
We also plot the blood concentrations of the cytotoxic T-cells, chemotherapeutic agent temozolomide and interleukin-2, required to eliminate the tumour (Figure 7B-7D). Also displayed are plots of the population of circulating lymphocytes and NK cells to check that there is no significant toxicity as side-effects of the therapy (Figure 7E-7F). We also observe that these values are also well within the corresponding bounds of the human host system (Table 2). Finally Figure 7G-7I show the injected dose rates, as required, day-wise, for each of the three therapeutic agents, that enables full tumour elimination. The circulating lymphocytes and NK cells takes around 500 days to reach the steady state values of 76 billion and 510,000 cells respectively. Note that these values closely corroborate with the stationary points solved theoretically earlier, which are 72 billion and 430,000 cells respectively [see eq. (6A) and subheading “Stationary condition of the system” there]. We find that the extreme values of these variables in the graphs are well within the bounds of the human host system (Table 2). For instance, the maximal dose-rate of temozolomide chemotherapy and of tumour-infiltrating lymphocytes are less than 10% and less than 1% respectively, of the upper bound of these agents in Table 2. Further, the cumulative dose of the interleukin (from the dose-rate graph calculated as area under the curve in Figure 7D), is below 1% of the interleukin upper bound in Table 2.
High-grade tumour
- (i)
the dosages of the therapeutic agents drops to zero from the tumour extinction time onwards
- (ii)
none of the cell populations or dosing of therapeutic agents crosses the bounds (Table 2), and
- (iii)
the dosages of the three agents are approximately the same fraction of the upper bound (10%, 1% and 1%).
We also discern that the therapy period is considerably longer in high-grade tumour, basically since the tumour growth rate is about 50% more than the slow-growing tumour (value of parameter α, in Table 1). Further, since these therapeutic dosages are a much smaller fraction of the upper bounds, the dosage profile can be well tolerated by patients. Actually, the overall patterns of the graphs (Figures 7 and 8) are comparable, across both the high and low grade tumours.
Robustness of tumour elimination procedure
Robustness study with biological parameter variation
Expt. No. | Coefficient of Variation | Percentage of success out of 500 cases |
---|---|---|
1 | 0% | 100% |
2 | 1% | 99.98% |
3 | 2.5% | 99.87% |
4 | 5% | 99.5% |
5 | 7.5% | 98.88% |
6 | 10% | 98.01% |
Unitary pattern in tumour regression process
If one compares the corresponding graphs in Figures 7 and 8, there are evident similarities in the temporal profiles of the therapeutic agents needed in both high-grade and low-grade tumours to enable elimination of malignancy. The common patterns valid across both tumours are elucidated below.
Profile A: terminal therapy pulse and cytotoxic lymphocyte persistence
One notes that (i) no injections of any of the three therapeutic agents are required any time after the time point of extinction of tumour, which does not recur later (Figure 7G-7I and Figure 8G-8I); (ii) the blood levels of the three therapeutic agents and the populations of the natural killer cells and circulating lymphocyte stay within the requisite limits, assuring that there would be no significant toxicity to the patient (Figure 7B-7F and Figure 8B-8F). One can make two pertinent observations from the simulation results. Firstly, for one to make the exponential decay curve of tumour cell population hit the T = 0 baseline (i.e. the x-axis of Figure 2B) at a definitive time point t_{ p } , one needs to inject a terminal pulse of each of three therapeutic dosages before the end of treatment protocol in both the low-grade and high-grade tumours, as shown by arrows in the six graphs in Figure 7G-7I and Figure 8G-8I. The conjoint effect of the pulses of all three agents ensure that in the last stage, all the tumour cells in the exponential extremity of cell population decay curve, do become eliminated. Secondly, the persistence of the cytotoxic T lymphocytes in blood for a over a week after tumour has been eliminated (Figures 7B and 8B), can have a beneficial effect, such that it can act as a vigilant anti-tumour measure against recurrence, by acting for an appreciable time after tumour extinction. After that duration elapses, this lymphocyte population tapers off.
Profile B: common temporal paradigm of therapeutic agents needed
An insight into the mechanism of tumour regression may be obtained by investigating the commonalities in the pattern of the temporal variation in the blood levels of the therapeutic agents and population of the protective cells, which seem to be common across both high-grade and low-grade tumours. From Figure 7B-7D and Figure 8B-8D, we discern the following temporal patterns of the aforesaid entities, the pattern being similar for both rapidly and slowly growing tumours:
(i) Temozolomide concentration (chemotherapy): monophasic activation pattern
(ii) Cytotoxic T-lymphocyte (CTL) concentration (cytotherapy): biphasic activation pattern
This graph has bimodal peaks for both the high-grade and low-grade tumour (Figures 7B and 8B, arrows; Figure 9B). We observe that for tumour elimination, there is a need to have a peak of CTL, both, around the initial and the final phases of the therapy. The initial peak concentration in required to forcefully target and guide the trajectory of the cancer cell population curve (T) towards the zero baseline (Figures 7A and 8A). The final peak concentration of CTL, occurs before the time of tumour elimination, and is necessary to sufficiently eliminate the tumour cells and depress their population trajectory so that the same hits the zero baseline at the definitive selected time point P (Figure 1B). Depending on the tumour system dynamics, the height or activation-level of the second peak may be lower (Figure 7B), or higher (Figure 8B), than the height or activation-level of first peak. In case of Cytotoxic T-cell concentration, the first peak is due to therapeutic agents of tumour-infiltrating lymphocytes and interleukin, whose dosing starts as an impulse stimulus from the initial time. The impulse of these agents also enhance the generation of CTL population. The second peak in cytotoxic T-cells (point B in Figure 8B) occurs due to the later dip or decay of chemotherapy concentration along the hump of the M curve (point K in Figure 8C). The chemotherapeutic agent is toxic to and diminishes all the cellular constituents, including CTLs, and, hence, a decrease of chemotherapy induces a rise of CTL then. This same pattern of primary and secondary induction of T-cell population also occurs in the other tumour (Figure 7B-7C).
(iii) Interleukin-2 concentration (immunotherapy): stationary activation pattern
This agent is needed at a substantially high level (at the upper bound, just below toxicity level), so as to induce a higher level of immune activation that would enable complete elimination of glioma cells (Figures 7D, 8D and 9C). Indeed it is well known from clinical experience in neurooncological immunotherapy that interleukin-2 administered at significantly augmented dose, induces long-lasting immunomodulation to act against those malignant cells that bypass usual therapeutic intervention [36]. Actually, the graph for the high- and low-grade tumours has rapidly rising high amplitude, as the interleukin-2 level is truncated and kept stationary within toxicity limit.
(iv) Circulating lymphocyte population: saturating activation pattern
This population initially increases and then plateaus in both tumours (Figures 7F, 8F and 9D), to approach the saturation level of the long-term steady state as mentioned earlier.
(v) Natural killer cell population: saturating activation pattern
Similar to the Circulating lymphocytes, the NK cells show saturation behaviour for both high and low grade tumours (Figures 7E, 8E and 9D).
Experimental, biological and clinical corroboration
- (a)
Reducing the tumour cell proliferation, as by chemotherapy or chemical alkylation (DNA damage);
- (b)
Increasing the tumour cell lysis by anti-tumour lymphocytes, which are activated by cytokine modulation (for instance tumour infiltrating lymphocytes, interleukin-2 etc.).
Further, the model developed shows that to induce tumour regression, the three antitumour entities should have three distinct temporal profiles: (1) biphasic intensity for lymphocyte activation, (2) monophasic intensity for activation of chemomodulative DNA damage (chemical alkylation), (3) stationary intensity of cytokine activation (interleukin-2). The experimental studies in animals [37–39] and clinical situation [40, 41] corroborate these findings. Thus our results show that complete elimination of tumour can be attained by (i) the five-pattern profile: activation of antitumour lymphocyte (bimodality), chemomodulative DNA damage (unimodality), interleukin (stationary), natural killer cell (saturation), and circulating lymphocyte (saturation), (ii) the two kinetic conditions: bias shift and exponential decrementing dynamics [the tumour trajectory formula of eq. (4)].
Translational applicability
There are two aspects where the approach could be improved. Firstly, one can increase the system robustness, which diminishes as the individual patient-specific fluctuations increases (see earlier subsection on Robustness). For real-time implementation, we can use the neuro-adaptive controller, which can reliably follow the desired mathematical trajectory of the tumour cell population curve (Figure 2B), and can be well adapted to different values and fluctuations of biological parameters of different patients. We have used such a controller in devising imatinib chemotherapy dosing in chronic myeloid leukaemia [16], and 100% robustness was obtained even when the maximal tumour density varied from 150,000 to 400,000 cells/mm^{3}, indicating about 250% variation on the baseline level. Secondly, for proper monitoring of a patient, the declining tumour load T can be weekly or semiweekly estimated non-invasively, by MRI amide-proton transfer imaging, which maps the cell proliferation intensity by amide mobility [42]. This emerging technology holds high potential, as this method is an efficient one to distinguish between various oncological conditions such as tumour recurrence, tumour haemorrhage or tumour necrosis.
For monitoring while the therapy is in progress, one need to have at suitable time intervals, the overall measurements of the serum temozolomide and interleukin-2 levels, as well as the populations of cytotoxic T-cells, natural killer cells and circulating lymphocyte in the blood (Figure 10). Only approximate values of these parameters or of the tumour cell population are required, as even large variations of these parameters can be accommodated by the neuro-adaptive controller mentioned above. During practical implementation, it may not be possible to measure all the biological parameters, neither a parameter can be estimated at all the desired time points as there may be discontinuous or missed sessions. Here, one can use suitable quantitative procedures as the Kalman’s technique or constraint filter method which has been used in immunology to estimate the likely values of biological parameters at the discrete or missed sessions, given their measured values in other sessions [43, 44]. Thus, it would suffice to monitor the tumour parameters weekly or twice-weekly, to enable the therapy control system tracking and adapting function, as demarcated in Figure 10.
One may observe that during therapy (Figures 7E-7F and 8E-8F), the circulating lymphocytes and natural killer cells increase but there is no hazard, as the populations are within the upper physiological limits of Table 2. Moreover, for making the methodology more suited to personalized medicine so that we can pre-select the most sensitive drugs beforehand, we can utilize the emerging method of tumour-graft technology whereby one can grow a therapeutically faithful model of the individual patient’s tumour biopsy tissue on experimental mice [45]. Tumour graft platforms can test different drug combinations, and pre-select the most sensitive drug before starting the treatment, the accuracy of selecting effective drug molecule being 86%. Lymphocytic (CTL) immunomodulation can be of tactical utility, as it exhibits a range of unique behaviors [46], that chemotherapeutic drugs cannot, such as (i) the cells can migrate to the antigen-bearing primary or secondary growths of tumour, even in hidden tissue depths, (ii) CTLs can continue to multiply automatically in response to immunogenic proteins of malignant cells, until all those cells become extinct.
Conclusion
We have formulated the distinctive dose-time relationship of chemotherapy and of immunotherapy (interleukin-2 and cytotoxic lymphocyte), such that their orchestrated functioning, that incorporates a biphasic temporal profile for T-cell, ensures that all the tumour cells are eliminated within a desired time. Excess toxicity to the host is avoided, as the circulating lymphocyte and natural killer cells in blood are protected. The approach is patient-specific as the formulation depends on the tumour load, the levels of cytotoxic T lymphocytes, cytokine interleukin, natural killer cells and circulating lymphocytes. All these parameters vary with the individual patients, and hence the different therapeutic dose-time profile obtained for each specific patient will be optimally suited from her. The formulation put forward use of an innovative approach of bias shift, control systems analysis, performance cost minimization and inverse construction of drug input. The limitation of the proposed model is that we have not considered the effect of tumour cells resistant to the chemotherapy drug and also not taken care of angiogenesis process where the drug permeability to the tumour cells will hinder. A broad-based gamut of findings from animal experimentation and clinical investigations are shown to corroborate with the tumour extinction approach developed. An unanticipated but noteworthy finding is the importance of giving a terminal pulse of the therapeutic agents before the end of the therapy, so that all the tumour cells become extinct and there is no extra drug-induced toxicity, the natural killer cells and circulating lymphocytes being within the physiological limits. This is in contrast to the generally prevailing view in clinical medicine, which advocates tapering off of the therapy in the later stages. This tapering may cause tumour recurrence in clinical praxis, as there is no intensive spike of the therapeutic agents to eradicate all the malignant cells.
To summarize, information from temporal dynamics of both the endogenous and exogenous tumour regression has been used to explore the mechanism and elucidation of integrative functioning of various therapeutic modalities, whose combined effect eliminates the malignant tumour as corroborated with experimental findings. The method proposed can be of wide-ranging application, and can be adapted for application to conditions where there is involvement of chemotherapy and/or immunotherapy. The procedure delineated is also applicable to other tumour systems, as it offers a principled approach to tumour containment and thus an incisive prospect for probing towards further biological and clinical situations.
Endnotes
^{ a } Clarifications on upper/lower limits of biological and therapeutic parameters (Table 2 ).
Circulating lymphocyte population: The normal blood volume under active circulation is 3.5-4.5 litres. The higher limit of lymphocytes in individuals can be up to 100 × 10^{9} cells/litre [26]. Taking the upper bound of the blood volume, this translates to the higher bound of lymphocyte in a person to be 4.5 × 10^{11} cells. In contrast, the lowest lymphocyte count for patients for duration of 6–12 months across therapy can be 663–1160 cells/μl [27]. Taking the lesser value of the cell count and the lower level of blood volume, we get minimal bound of lymphocytes as 2.32 × 10^{9} cells, which the patient can tolerate up to 6 months.
Natural Killer Cells population: The upper limit of NK cell is 13% of lymphocyte population, with CD56/CD16 surface protein being the marker for these lymphocytes [28]. As the higher bound of lymphocyte population in the earlier paragraph is 4.5 × 10^{11} cells, we have the maximum value of NK cell population in the individual to be 5.85 × 10^{10} cells. On the other hand, the lower limit of NK cells (CD56/CD16 lymphocytes) is 0, occurring in people having natural killer cell deficiency condition [29], and a period of 3½ months have been noted for elapse of this condition, before considerable infection can set in [29]. Hence we also mention this time duration for the NK cells lower bound.
Tumour-specific Cytotoxic (CD8+) T-cell population: The upper limit of this cell population for a patient is 20150 cells/mm^{3}[25]. Using a blood volume of 4.5 litres, the total T-cell (CD8) population will be 6.05 × 10^{10} cells. Furthermore, tumour-specific cytotoxic T-cells, that are specifically active against a particular malignant lesion, has been known to come into play if the tumour is present, and to decay away if the tumour undergoes elimination [30]. Hence, one mentions the lower bound of these T cells to be 0.
Temozolomide chemotherapy infusion dosage rate: Maximum daily dosage [1] of temozolomide is 200 mg/m^{2}/day, i.e., 4.45 mg/kg body. wt. per day. One may choose not to give it, so the lower limit is 0.
Interleukin-2 immunotherapy infusion dosage rate: Maximum infusion given is 7.2 × 10^{4} International Units (I.U.)/kg/day [1]. The lower bound can be set to nil, as above.
Tumour-infiltrating lymphocytes (TIL) immunotherapy cumulative dosage: The maximum cumulative dose for a patient over the whole duration of therapy is 13.7 × 10^{10} cells [25]. Likewise, the lower limit is zero.
^{ b } Determination of initial number of the cells for numerical experimentation.
Tumour cells: The number of cells in a tumour which becomes clinically detectable is 10^{8} occupying a volume of 1 cm^{3}, out of these 10^{7} cells are malignant, while the rest are stromal cells [47]. For our quantitative experimentation, we take double the amount of tumour cells to have a safety factor of 2, thus giving 2 × 10^{8} malignant cells as our initial condition.
Circulating lymphocytes: We also note the range of normal values: leucocyte count = 4000 to 11000/mm^{3}, the fraction of lymphocytes are 15-40%, and actively pulsating blood volume under circulation is 3.5-4.5 litres. To be cautious for ensuring tumour regression, we will consider the lower values in the range. By multiplying the requisite aforesaid values, we have the circulating lymphocyte population in the person as 2.1 × 10^{9} cells. Again to be on safer side, we take half of the population for our study (i.e. 1.05 × 10^{9} cells, say 10^{9} cells).
Natural killer cells: The fraction of NK cells is 1-13% of circulating lymphocytes [28], we take the lower value for calculation, and use the circulating lymphocyte value of 1 × 10^{9} cells given above. Further, there is available data on the antitumour effector factor of NK cells during study of endogenous tumour regression, namely 1:50 as target tumour cell: effector NK cell ratio, i.e. a value of 2% [48]. Multiplying these values, we get the effective population of tumour-targeted NK cells as 2 × 10^{5} cells. As earlier, we will consider half this value to be on the safe side, i.e. NK cell population = 10^{5} cells.
Cytotoxic T-cells: The fraction of cytotoxic CD8+ T-cells is 1-3% of circulating lymphocytes [49]. The cytotoxicity factor of activated tumour-infiltrating cytotoxic T cells (CTL) is 9.5% as regression process is underway [48]. One also knows that if the tumour is invasive and spreads, the fatigue factor come into force, which can cause the CTL efficiency (as estimated by cytokine production) to decrease to 12.2% of the level as compared to when there is no invasion of tumour [50]. Using these factors, we arrive at the effective population of cytotoxic T cells as 1.15 × 10^{5} cells, say 10^{5} cells. As per the lower end, we have taken half, i.e. 5 × 10^{4} cells as the CTL population.
Notes
Declarations
Acknowledgements
The research was supported by core funds from National Brain Research Centre, Ministry of Science & Technology, Govt. of India. SK, VPSR and CS are respectively supported by funding received from educational grant of Dept. of Biotechnology, Dept. of Information Technology, and Ministry of Human Resource Development, Government of India.
Authors’ Affiliations
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