Abstract

In this paper, a new stochastic framework to determine optimal combination therapies in colon cancer-induced immune response is presented. The dynamics of colon cancer are described through an It¨o stochastic process, whose probability density function evolution is governed by the Fokker-Planck equation. An open-loop control optimization problem is pro- posed to determine the optimal combination therapies. Numerical results with combination therapies comprising of the chemotherapy drug Doxorubicin and immunotherapy drug IL-2 validate the proposed framework.

Keywords: Fokker-Planck optimization, non-linear conjugate gradient, immunotherapy, chemotherapy

3.1 Introduction

Colon cancer is a leading cause of global cancer related deaths [11]. The lack of early symptoms forces the detection of colon cancer to take place at the metastatic phase of the cancer [29]. Thus, it becomes important to devise fast and accurate treatment strategies for a cure. In this context, combination therapies have been clinically shown to be an effective strategy for combating cancer in comparison to monotherapy (see [18]). Conventional monotherapeutic techniques are indiscriminate in choosing actively growing cells that lead to the death of not only cancerous cells but also healthy cells. E.g., chemotherapy drugs can be toxic and lead to multiple side effects and risks by weakening the patient’s immune system, targeting the bone-marrow cells [22]. This leads to increased susceptibility to secondary infections. However, with combination therapies, even though it might still be toxic if there is a presence of a chemotherapy drug, the toxicity effect is significantly diminished due to different targets being affected. Moreover, since the effect of combination therapies work in synergy, lower dosages of the individual drugs in the combination therapy are required for treatment, which further reduces the toxic effects [19].

To develop optimal combination therapies in colon cancer, it is important to understand relevant biomarkers that govern the progression of the cancer [8]. One such biomarker is the immune response to the cancerous cells. It has been observed that onsite immune reactions at the tumor locations and prognosis are highly correlated and independent of the size of the tumor [12]. Furthermore, a high expression of various immune pathways like Th1 and Th17 are associated with a poor prognosis or prolonged disease-free survival for patients with colon cancer [32]. Thus, it becomes important to develop optimal personalized treatments for colon cancer patients, taking into account the various types of immune cells, their numbers, and interaction with the cancer cells and each other. Since drugs for treatment of colon cancer like chemotherapeutic drugs and immunotherapies are quite costly, it is expensive to perform in-vitro and vivo experimental studies for testing effects of such drugs. An alternative cost effective option is to develop computational frameworks for testing optimal combination drug dosages [7]. This can be done using pharmacokinetic models that are given by a set of ordinary or partial differential equations (ODE or PDE).

There are several dynamic models that have been developed to describe the immune response and interactions in colon cancer and associated therapies. In [4], a combined compartmental model with that of irinotecan is used to describe the physiology of colon cancer. A set of ODEs were used to represent a 3D structure of colon cancer in [14]. In [15], the authors use describe the phenomenon of tumorigenesis in colon cancer using mathematical modeling. The authors in [17], describe the initiation of colon cancer and its link with colitis using a mathematical framework. In [23], the authors use a pharmacokinetic cellular automata model to incorporate the cytotoxic effects of chemotherapy drugs. In [13], dynamical systems are used to describe multiple pathways in colon cancer. For a detailed review of various mathematical models for colon cancer-induced immune response, we refer the reader to [3].

We contribute to the field of pharmacokinetic cancer research by presenting an effective approach to develop personalized therapies for colon cancer-induced immune response. The starting point of this estimation process is the dynamic model for colon cancer induced immune response, given in [10]. The model describes the evolution of four variables: the tumor cell count, concentration of the Natural Killer cells, concentration of the CD8⁺ cells, and concentration of the remaining lymphocytes. We extend the dynamic model in [10] to an Itô stochastic ODE system that takes into account the randomness of the colon cancer-induced immune response dynamics. We also model a combination therapy comprising of a chemotherapy drug and an immunotherapy drug as a control in the stochastic ODE system. The goal is to obtain an optimal combination therapy strategy that drives the cancer-induced immune response state. Since solving an optimal control problem using stochastic state equations is difficult, we use a more convenient framework for obtaining optimal combination therapies through the Fokker-Planck (FP) equations that govern the evolution of the joint probability density function (PDF) associated with the random variables in the stochastic process. Such FP control frameworks have been used for problems arising in control of collective and crowd motion [24, 25], investigating pedestrian motion from a game theoretic perspective [26], reconstructing cell-membrane potentials and mean field control problems [5], and controlling production of Subtilin [31]. Very recently, in [27, 28], FP frameworks were used for parameter estimation in colon cancer induced dynamic systems. Until now, this is the first work that considers the FP framework to devise optimal combination treatment strategies for colon cancer induced immune response.

In the next section, we describe an FP control framework for devising optimal treatments in colon cancer induced immune response systems. Section 3.3 is concerned with the theoretical properties of the FP optimization problems. In Section 3.4, we describe the numerical discretization of the optimality system. In Section 3.5, we obtain the optimal combination therapies comprising of Doxorubicin and IL-2 for two types of simulated colon cancer patients and show the correspondence of the results with experimental findings. We end with a section of conclusions.

3.2 The Control Framework Based on Fokker-Planck Equations

The interaction between immune cells and colon cancer growth in a patient can be modeled using a coupled system of ODEs. The starting point is the set of equations given in the paper by dePillis [10]. The following are the variables associated with different types of cell populations that appear as model variables.

1. T(τ)- tumor cell population number (cells).
2. N(τ)- natural killer (NK) cells concentration per liter of blood (cells/L).
3. L(τ)- cytotoxic T lymphocytes (CD8⁺) concentration per liter of blood (cells/L).
4. C(τ)- other lymphocytes concentration per liter of blood (cells/L).

The governing system of ODEs representing the dynamics of the above defined cell populations is given as follows.

(3.1)

where represent the initial conditions for T, N, L, and C, respectively, and u₁, u₂ represent dosages of Doxorubicin and IL-2, respectively.

The parameters of the system (3.1) are defined in [10]. To stabilize the solutions of the numerical methods, we non-dimensionalize the ODE system (3.1) using the following non-dimensionalized state variables and parameters:

(3.2)

Then, the transformed non-dimensional ODE system is given as follows:

(3.3)

where

The compact form of the aforementioned system of ODEs, given in (3.3), is as follows:

(3.4)

where

We extend the ODE system (3.3) to include stochasticity present in the dynamics. For this purpose, we consider the Itô stochastic differential equation corresponding to (3.3).

(3.5)

where dW_i, i = 1, 2, 3, 4 are one-dimensional Wiener processes and σ_i, i = 1, 2, 3, 4 are positive constants. Equation (3.5) can be written using a compact notation as follows:

(3.6)

where

is a four-dimensional Wiener process vector with stochastically independent components and

is the dispersion matrix.

We now describe the PDF of the stochastic process (3.6), confined in a Lipschitz domain Ω by virtue of a reflecting barrier on ∂Ω. This is motivated by the maximum cell carrying capacity.

Thus, Define f (x, t) as the PDF for the stochastic process described by (3.6), i.e., f (x, t) is the probability of X(t) assuming the value x at time t. Then, the PDF of X(t) evolves through the following Fokker-Planck (FP) equations.

(3.7)

where f₀(x) is non-negative with mass equals one and in the admissible set

Here, the FP domain is Q = Ω × (0, T_f ), where T_f is the final time and f₀(x) represents the distribution of the initial state X₀ of the process. The FP equation (3.7) is associated with the no-flux boundary conditions. To describe this, we write (3.7) in flux form as

(3.8)

where the components of the flux H is given as follows:

(3.9)

Then, the no-flux boundary conditions are

(3.10)

with as the unit outward normal on ∂Ω.

To obtain the optimal combination therapy function vector U, we solve the following optimization problem:

(3.11)

subject to the FP system (3.7),(3.10), where the desired PDF is f ^⋆(x, t).

3.3 Theoretical Results

In this section we describe some theoretical results related to the minimization problem (3.11). One can also find similar results in [1, 24, 25]. For this purpose, we denote the FP system (3.7),(3.10) as E(f₀, U) = 0. The existence and uniqueness of solutions of (3.7) is given in the following proposition.

Proposition 3.1. Assume f₀ ∈ H¹(Ω) with f₀ non-negative and U ∈ U_ad. Then, there exists a unique non-negative solution of E(f₀, U ) = 0, given by f ∈ L²([0, T_f]; H¹(Ω)) ∩ C([0, T_f]; L²(Ω)).

Under the assumptions of higher regularity of ∂Ω and the boundary of Ω, one can also obtain H²(Ω) regularity of the solution of (3.7) (see [30]). Next, we state the conservativeness property of (3.7), which can be proved using straightforward applications of weak formulation, integration by parts, and divergence theorem for the flux H.

Proposition 3.2. The FP system given in (3.7),(3.10) is conservative. The next proposition states and proves the L² stability property of (3.7).

Proposition 3.3. The FP system (3.7),(3.10) solution, given by f, satisfies the following L² stability property:

(3.12)

where N = sup_Ω×U |F (x, θ)|.

Proof. Multiplying (3.7) with the test function ψ = f (·, t) and an integration by parts gives:

(3.13)

The last term in (3.13) can be estimated using the Young’s inequality, 2bd ≤ kb² + d²/k, with k = || σ⁻¹ ||₂, which is the L² matrix norm of σ⁻¹. We then obtain the following:

Applying the Gronwall’s inequality gives the desired result.

The above results imply that the map Λ: U_ad → C([0, T_f]; H¹(Ω)), given by f = Λ(U), is continuous and Fréchet differentiable. The main theoretical result in this work, which shows that there is an optimal open-loop control U^⋆, is given below:

Theorem 3.1. Let f₀ ∈ H¹(Ω) and let J be given as in (3.11). Then, there exists (f ^*, U ^∗) ∈ C([0, T_f]; H¹(Ω)) × U_ad with f ^∗ being a solution to E(f₀, U ^∗) = 0 and U ^∗ minimizing J in U_ad.

Proof. Since J is bounded, a minimizing sequence (U^m) exists in U_ad. Moreover, J is being coercive and sequentially weakens lower semi-continuous in U_ad, implying the boundedness of this sequence. Due to the fact that U_ad is a closed and convex subset of a Hilbert space, the sequence (U^m) contains a convergent subsequence in U_ad, such that . Correspondingly, the sequences where are bounded in L²([0, T_f]; H¹(Ω)), L²([0, T_f]; H⁻¹(Ω)), respectively. This implies the weak convergence of the sequences to f ^* and ∂_t f ^*, respectively.

We next use the compactness result of Aubin-Lions [16] to obtain strongly convergent subsequence in L²([0, T_f], L²(Ω)). Thus, the sequence in L²([0, T_f], L²(Ω)) is weakly convergent. This implies that f ^* = Λ(U ^*) and (f ^*, U ^*) are minimizers of J.

For the minimization problem (3.11), the optimality system can now be written as

The optimality system comprises of three sets of equations: the forward or state equation that governs the FP dynamics (FOR), the adjoint equation (ADJ), and the optimality condition (OPT). In the next section, we describe numerical schemes to implement the optimality system.

3.4 Numerical Schemes

We consider the mesh {Ω_h}_h>0 given by

where and N_xi is the number of discretization points along the i^th coordinate direction. Define δt = T_f /N_t to be the temporal discretization step, where N_t denotes the maximum number of temporal steps. This gives us the discretized domain for Ω as follows

The value of f (x, t) on Q_h,δt is denoted as . To solve the forward Fokker-Planck equation (3.7), we use the scheme described in [27], that is comprised of the spatial discretization using the Chang-Cooper (CC) method [6]. The temporal derivative discretization is done using the fourstep, alternate direction implicit Douglas-Gunn (DG4) method. The combined scheme is referred to as the DG4-CC scheme. It has been shown in [27] that this scheme is positive, conservative, stable, and second order convergent in the L¹ norm. Furthermore, we use the temporal D-G scheme for the time discretization in the first term, one sided finite difference discretization for the second term, and central difference for the third term on the left hand side of the adjoint equation (ADJ).

To solve the optimization problem (3.11), we use a projected non-linear conjugate gradient scheme (PNCG), as described in [28, 27]. Such a scheme has also been used for solving optimization problems related to controlling stochastic crowd motion [24, 25], studying avoidance behavior of pedestrians using game theory [26], and cure rate models [20, 21]. The PNCG scheme is described below.

Algorithm 3.1 (PNCG Scheme).

1. Input: Initial guess u₀. Compute . Set k = 0 and maximum number of iterations k_max, tolerance TOL.
2. If (k < k_max), do
3. Compute u_k+1 = P_U [u_k + α_k d_k] with α_k obtained by the Armijo line-search method.
4. Evaluate .
5. Evaluate
6. Compute
7. Check for If yes, terminate.
8. Update k = k + 1.
9. End if.

3.5 Results

This section describes the numerical results for obtaining optimal treatment strategies with the aforementioned FP framework. We choose our domain Ω = (0, 6)⁴ and discretize it using N_xi = 51 points for i = 1, 2, 3, 4. The final time T_f is chosen to be 10 and the maximum number of time steps N_t is chosen to be 200. The values of the constants used in converting the ODE system (3.1) to its non-dimensional form given in (3.3) are given as k₁ = 10⁻¹⁰, k₂ = 10⁻⁵, k₃ = 10⁻⁷, k₄ = 10⁻⁸, and k₅ = 1. To obtain the target PDF f ^*(x, t), we simulate the ODE system (3.3) with the set to 0 and with the value of the non-dimensional parameters (2.1,1.1,1.25) that represents a patient with strong immune response [10]. The values of the other parameters are taken from [10]. After we obtain the trajectories of T^¯, N^¯, L^¯, and C, we then choose 20 time points t_i, and at each point (T^¯(t_i), N^¯(t_i), L^¯(t_i), C^¯(t_i)) assign a Gaussian PDF given by a normal distribution with variance 0.05. We finally perform a 5D interpolation to obtain the desired PDF f ^*(x, t). Based on a statistical analysis of the dataset given in [10], we choose

The regularization parameters are chosen to be α = 1 and ν₁ = ν₂ = 0.01. The maximum tolerable dosage for Doxorubicin is taken to be 7 mg/day and for IL-2 is taken to be 7.2×10⁵ IU IL-2/l/day, which implies D₁ = 7 and D₂ = 0.072.

In Test Case 1, we simulate a tumor patient with a value of (T (0), N (0), L(0), C(0)) = (2 × 10⁹, 10⁵, 10⁷, 10⁸). This implies (T¯(0), N¯ (0), L¯(0), C¯(0)) = (0.2, 1, 1, 1)). We also choose the values of the non-dimensional parameters that represent patients with weak immune systems. The goal is to determine optimal drug dosages such that the tumor profile is given by the target PDF f ^*.

Figure 3.1 presents the simulation results obtained with our framework. We observe that without the combination therapy, the mean tumor cell count will keep on increasing until it reaches the cell carrying capacity, ultimately leading to patient death. With the combination therapy, the mean tumor cell count is brought back to diminishing levels. We also note the optimal dosage patterns for both Doxorubicin and IL-2 over time. Traditionally, Doxorubicin is administered every 21 days with a dosage of 142.5 mg [9], that translates to a total dosage of 70 mg for 10 days. However, from Figure 3.1, we note that the total dosage of Doxorubicin over 10 days is far less than 70 mg. Moreover, the observed daily dosage of IL-2 from Figure 3.1 is less than the standard dosage of IL-2 over a day, which is 2.1 × 10⁶ IU IL-2/l. This suggests that optimal combination therapies administered daily leads to lower total dosages and, thus, lower toxicity effects. In Test Case 2, we choose the same initial values of and C, along with the values of the non-dimensional parameters that represent patients with moderately strong immune systems.

Figure 3.2 shows the plots of the tumor profiles without and with treatment and the optimal daily dosages of Doxorubicin and IL-2. We observe a similar behavior as Test Case 1. In addition, we note that the dosages are smaller compared to the patient in Test Case 1 since the immune system is moderately strong. This shows the effectiveness of our framework in obtaining optimal dosages in colon cancer.

3.6 Conclusion

In this paper, we presented a new stochastic frameworks to determine optimal combination therapies in colon cancer-induced immune response. We considered the tumor and immune system response dynamics as proposed in [10] and extend it to a stochastic process to account for incorporating randomness in the dynamics. We characterized the state of the stochastic process using the PDF, whose evolution is governed by the FP equation. We then solved an optimal control problem with open loop controls to obtain the optimal combination dosages involving chemotherapy and immunotherapy. Numerical results demonstrate the feasibility of our proposed framework to obtain small dosages of the combination drugs, leading to lower toxicity whilst preserving the effectiveness to eliminate the tumor.

Acknowledgments

The authors were supported by the National Institutes of Health (Grant Number: R21CA242933).

S. Roy was also partially supported by the Interdisciplinary Research Program, University of Texas at Arlington, Grant number: 2021-772. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

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