Tumor Volume for a Mathematical Model of Anti-Angiogenesis ... - SIUE

Minimizing Tumor Volume for a Mathematical Model of Anti-Angiogenesis with Linear Pharmacokinetics Urszula Ledzewicz1 , Helmut Maurer2 , and Heinz Sch¨attler3 1

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Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Il, 62026-1653, USA, [email protected] Institut f¨ ur Numerische und Angewandte Mathematik, Rheinisch Westf¨ alische Wilhelms Universit¨ at M¨ unster, D-48149 M¨ unster, Germany, [email protected] Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo, 63130-4899, USA, [email protected]

1 Introduction Tumor anti-angiogenesis is a rather novel cancer treatment that limits a tumor’s growth by inhibiting it from developing the vascular network it needs for its further supply with nutrients and oxygen. Ideally, deprived of its sustenance, the tumor regresses. As with any novel approach, the underlying biological mechanisms are not fully understood and several important questions such as how to best schedule these therapies over time still need to be answered. Various anti-angiogenic agents have been and still are tested in clinical trials (e.g., [6, 11]). Naturally, the scope of these experiments is limited to simple structured protocols. Mathematical modelling and analysis can give valuable insights here into the structure of both optimal and suboptimal protocols and can thus become an important tool towards the overall aim of establishing robust and effective treatment protocols (e.g., [1, 9]). Mathematically, the various protocols can be considered as control functions defined over time and the tools and techniques from optimal control theory are uniquely suited to address these difficult scheduling problems. In previous research, for various formulations of the dynamics underlying antiangiogenic treatments that were based on a biologically validated model developed at Harvard Medical School [10] and one of its modifications formulated at the Cancer Research Institute at NIH [7], Ledzewicz et al. have considered the optimal control problem of how to schedule an a priori given amount of anti-angiogenic agents in order to minimize the tumor volume. Complete solutions in form of a regular synthesis of optimal controlled trajectories [2] (which

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Urszula Ledzewicz, Helmut Maurer, and Heinz Sch¨ attler

specifies the optimal controls and their corresponding trajectories for arbitrary initial conditions) have been given for the two main models in [13, 17]. Because of the great complexity of the underlying biological processes, in these papers the dosages and concentrations of the anti-angiogenic agents have been identified, a commonly made first modeling simplification. In reality these clearly are different relations studied as pharmacokinetics (PK) in the medical and pharmaceutical literature. The standard and most commonly used model for P K is a simple model of exponential growth and decay given by c˙ = −ρc + u, c(0) = 0, (1) where u denotes the dosage of the agent and c its concentration. The coefficient ρ is the clearance rate and is related to the half-life of the agents. The important question is to what extent optimal controls will be altered under the addition of pharmacokinetic equations, both qualitatively and quantitatively. In models for chemotherapy which we had considered earlier optimal controls were bang-bang and this structure was retained if a linear pharmacokinetic model of the form (1) was added [14, 16]. Thus in this case no qualitative changes and in fact also only minor quantitative changes arose. But the solutions to the mathematical models for tumor anti-angiogenesis are characterized by optimal singular arcs which are defined by highly nonlinear relations (see below, [13, 17]) and now significant qualitative changes in the concatenation structure of optimal controls occur. They lead to the presence of optimal chattering arcs once a pharmacokinetic model (1) is included. In this paper we describe these changes for the mathematical model proposed by Ergun et al. [7] and give numerical results that show that the minimal tumor values can very accurately be approximated by reasonably simple, piecewise constant controls.

2 A Mathematical Model for Tumor Anti-Angiogenesis We consider a mathematical model for tumor anti-angiogenesis that was formulated by Ergun, Camphausen and Wein in [7] and is a modification of the model by Hahnfeldt et al. from [10]. In both models the spatial aspects of the underlying consumption-diffusion processes that stimulate and inhibit angiogenesis are incorporated into a non-spatial 2-compartment model with the primary tumor volume p and its carrying capacity q as variables. The tumor dynamics is modeled by a Gompertzian function, p (2) p˙ = −ξp ln q with ξ denoting a tumor growth parameter. The carrying capacity q is variable and in [7] its dynamics is modelled as 2

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q˙ = bq 3 − dq 3 − µq − γuq,

(3)

Suboptimal Controls for a Model of Tumor-Antiangiogenesis with PK

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where b (birth) and d (death), respectively, are endogeneous stimulation and inhibition parameters for the endothelial support, the term µq represents natural death terms and γuq stands for additional exogenous inhibition. The variable u represents the control in the system and corresponds to the angiogenic dose rate while γ is a constant that represents the anti-angiogenic killing parameter. The particular inhibition and stimulation terms chosen in 4 2 this model, I(q) = dq 3 and S(q) = bq 3 , are a modification of the original terms in [10] in the sense that the tumor’s stimulation of the carrying capacity now becomes proportional to the tumor radius, no longer its surface area. Also, compared with [10] the dynamics of the vascular support has been slowed down leading to an overall improved balance in the substitution of stimulation and inhibition (see [7]). Anti-angiogenic agents are “biological drugs” that need to be grown in a lab and are very expensive and limited. From a practical point of view, it is therefore of importance how given amounts of these agents, Z T u(t)dt ≤ ymax , (4) 0

can be administered to have “optimal” effect. Taking as objective to maximize the possible tumor reduction and adding an extra variable y that keeps track of the total amounts of agent that have been given, this problem takes the following form: [C] for a free terminal time T , minimize the objective J(u) = p(T ) subject to the dynamics p˙ = −ξp ln pq , p(0) = p0 , (5) 2

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q˙ = bq 3 − dq 3 − µq − γuq,

y˙ = u,

q(0) = q0 ,

(6)

y(0) = 0

(7)

over all Lebesgue measurable functions u : [0, T ] → [0, umax ] for which the corresponding trajectory satisfies the end-point constraints y(T ) ≤ ymax . It is easily seen that for any admissible control u and arbitrary positive initial conditions p0 and q0 the solution (p, q, y) to the corresponding differential equation exists for all times t > 0 and both p and q remain positive [15]. Hence no state space constraints need to be imposed. Necessary conditions for optimality are given by the Pontryagin Maximum Principle (e.g., see [3, 4]) and these conditions identify the constant controls u = 0 (no dose) and u = umax (maximum dose), the so-called bang controls, as well as a time-varying feedback control, a so-called singular control, as candidates for optimality. Using Lie-algebraic calculations, analytic formulas for the singular control and the corresponding trajectory can be given. Proposition 1. [15, 13] The singular control is given in feedback form as

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1 usin (q) = ψ(q) = G

!

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(8)

−µ .

This control is locally optimal if the state of the system lies on the corresponding singular arc S defined in (p, q)-space by ! 1 2 b − dq 3 − µq 3 psin = psin (q) = q exp 3 . (9) 2 b + dq 3 This curve is an admissible trajectory (i.e., the singular control takes values between 0 and umax ) for q`∗ ≤ q q ≤ qu∗ where q`∗ and qu∗ are the unique solutions b 3 to the equation ψ(q) = a in (0, d ).

Fig. 1 gives the graph of the singular control (8) on the left and the corresponding singular arc defined by (9) is shown on the right. 25

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Fig. 1. Singular control (left) and singular arc, the corresponding trajectory (right)

Overall, optimal controls then need to be synthesized from bang and singular controls. This requires to analyze concatenations of these structures. Based on the formulas above a full synthesis of optimal controlled trajectories was given in [13]. Such a synthesis provides a complete “road map” of how optimal protocols look like depending on the initial condition in the problem, both qualitatively and quantitatively. Examples of projections of optimal trajectories into the (p, q)-space are given in Fig. 2. The admissible singular arc is shown as a solid blue curve which becomes dotted after the saturation points. Trajectories corresponding to u ≡ a are marked as solid green curves whereas trajectories corresponding to u ≡ 0 are also in green, but marked as dashed curves. The dotted black line on the graph is the diagonal, p = q. We highlighted with bold one specific, characteristic example of the synthesis. Initially the optimal control is given by a full dose u = a segment until the corresponding trajectory hits the singular arc. At that time the optimal control becomes singular following the singular arc until all inhibitors become


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exhausted. Since the singular arc lies in the region p > q, the tumor volume still shrinks along u = 0 until the trajectory reaches the diagonal p = q at the final time T when the minimum tumor volume is realized. This structure as0 is the most typical form of optimal controls. (For a more precise description of the synthesis, see [13])

tumor volume, p

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Fig. 2. Synthesis of optimal trajectories (vertical axis p, horizontal axis q

3 Addition of a Pharmacokinetic Model We now add the standard linear pharmacokinetic model (1) to the mathematical model and replace the control u in (6) by the concentration c, but otherwise preserve the same formulation. Thus the optimal control problem now becomes to minimize p(T ) subject to p˙ = −ξp ln pq , p(0) = p0 , (10) 2

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q˙ = bq 3 − dq 3 − µq − γcq,

q(0) = q0 ,

(11)

c˙ = −ρc + u, y˙ = u,

c(0) = 0, y(0) = 0.

(12) (13)

Once more the conditions of the Maximum Principle identify bang and singular controls as candidates. In [18] it is shown for a more general controllinear nonlinear system of a form that includes problem [C] that the optimality status of a singular arc is preserved under the addition of a linear pharmacokinetic model. In fact, the very same equations that define the singular control

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and arc in the model without PK remain valid verbatim, albeit with a different interpretation. The singular curve (9) is preserved as a vertical surface in (p, q, c)-space and the singular arc is now defined as the intersection of this cylindrical surface with the graph of the function c = ψ(q) defined by (8), see Fig. 3.

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Fig. 3. Vertical singular surface in (p, q, c)-space (left) and intersection with the concentration c = ψ(q) (right)

However, under the addition of pharmacokinetic equations of the form (1) the so-called order of the singular arc increases from 1 to 2. It is well-known that a smooth singular control with values in the interior of the control set for which the Kelley condition (a high order necessary condition for optimality of singular controls of order 2, [21]) is satisfied, cannot be concatenated optimally with either of the constant bang controls u = 0 or u = umax [3, 20, 21]. These concatenations are now accomplished by means of chattering controls. This fact is also known as the Fuller phenomenon in the optimal control literature [8]. The structure of optimal controlled trajectories therefore clearly changes. The construction of an optimal synthesis that contains chattering arcs is quite a challenging task [21] and has not yet been completed for this model. However, practically the relevant question is what effect these changes actually have on the value of the objective. Chattering controls are not practical and thus the question about realizable suboptimal structures arises.

4 Suboptimal Approximations In this section we give numerical results which show that it is possible to give simple suboptimal controls that achieve a tumor volume which gives an excellent approximation of the optimal value. In our calculations we use


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the following parameter values taken from [10] that are based on biologically validated data for the anti-angiogenic agent angiostatin: ξ = 0.192 per day, b = 5.85 per day, d = 0.00873 per mm2 per day, γ = 0.15 kg per mg of dose per day with concentration in mg of dose per kg. For illustrative purposes we also chose a small positive value for µ, µ = 0.02 per day. The upper limit of the dosage was taken as umax = 15 mg per kg and ymax = 60 mg per kg. The half-life k of the agent is k = 0.38 per day [10]. The variables p and q are volumes measured in mm3 and the initial conditions for our numerical calculations are p0 = 8, 000 mm3 and q0 = 10, 000 mm3 . The optimal control package NUDOCCCS due to B¨ uskens [5] is implemented to compute a solution of the discretized control problem using nonlinear programming methods. We chose a time grid with N = 400 points and a high order Runge-Kutta integration method. Fig. 4 shows the graph of a numerically computed ‘optimal’ chattering control on the left and gives the corresponding concentration c on the right. The highly irregular structure of the control on the left is caused by the fact that the theoretically optimal control chatters and has a singular middle segment. Due to numerical inaccuracies, as the intervals shrink to 0, the actual control values no longer are at their upper and lower values ±1. But the value of the objective is within the desired error tolerance. The final time is T = 11.6406 and the minimum tumor volume is given by p(T ) = 78.5326. 16

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Fig. 4. A numerically computed ‘optimal’ chattering control (left) with corresponding concentration c (right)

For the same parameter values, Fig. 5 gives an example of a suboptimal control that is computed taking a control of the following simple bang-bang structure:    umax for 0 ≤ t < t1  0 for t1 ≤ t < t2 u(t) = . u for t3 ≤ t < t3  max   0 for t4 ≤ t ≤ T

Thus both the chattering and singular arcs are completely eliminated at the expense of two adjacent bang-bang arcs that become larger. The switching times t1 , t2 , t3 and the final time T are the free optimization variables. Using the arc-parametrization method developed in [19] and the code NUDOCCCS

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[5], we obtain the optimal switching times t1 = 1.78835, t2 = 4.60461, t3 = 6.86696 and the final time T = 11.3101. This suboptimal control approximation gives the minimal tumor volume p(T ) = 78.8853. It is surprising that this rather crude approximation of the chattering control gives a final tumor volume that is very close to the minimal tumor volume p(T ) = 78.5326 for the ”chattering control” in Fig. 4. On the right of Fig. 5 the corresponding concentration is shown. 16

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Fig. 5. A simple suboptimal bang-bang control with four arcs (left) and corresponding concentration c (right)

Tumor volumes that are close to identical with those realized by the optimal control can be achieved with a slightly more refined control structure given by  umax for 0 ≤ t < t1      0 for t1 ≤ t < t2 v for t2 ≤ t < t3 . u(t) =    umax for t3 ≤ t < t4   0 for t4 ≤ t ≤ T

Both chattering arcs are approximated by a simple bang-bang control that switches once from umax to 0 and the singular segment is approximated by a constant control v over the full singular interval. This particular choice is probably the simplest reasonable approximation to the control structure that the theory predicts as optimal: a chattering control followed by a singular control and one more chattering control. Here the switching times ti , i = 1, . . . , t4 , the final time T , and the value v of the control are free optimization variables. Again, we use the arc-parametrization method [19] and the code NUDOCCCS [5] to compute the optimal switching times t1 = 1.46665, t2 = 3.08056, t3 = 5.98530, t4 = 7.35795, the final time T = 11.6388 and the constant control v is given by v = 6.24784. This gives the minimal tumor volume p(T ) = 78.5239 for the suboptimal approximation which, for practical purposes, is identical with the minimal tumor volume p(T ) = 78.5326 for the “chattering control” in Fig. 4. The computations also show that second order sufficient conditions for the underlying optimization problem are satisfied and hence we have found a strict (local) minimum. On the right of Fig. 6 the


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corresponding concentration is shown. Overall the behavior is very similar as in case of the chattering control, but the system has a much smoother and thus for many aspects preferable response. Like in the case of the problem when P K is not modelled [12], it appears that the differences in the minimum tumor volumes that can be achieved on the level of suboptimal controls are negligible. 16

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Fig. 6. A suboptimal piecewise constant control (left) and corresponding concentration c (right)

5 Conclusion For a model of tumor anti-angiogenesis we have shown that, based on the structure of optimal controls, excellent simple suboptimal protocols can be constructed that realize tumor volumes close to the optimal ones. This holds both for the model without and with a linear pharmacokinetic model (1). Obviously, our numerical results are only for one special case, but we expect similar good approximations to be valid for a broad range of parameters. The significance of knowing the optimal solutions is bifold: it sets the benchmark to which suboptimal ones will be compared and in suggests the simpler structures for the approximating controls.

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