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MBW:Optimal Chemotherapy Strategies

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Article review by Sigmund Stoerset, March 2010.

Article: Optimizing Drug Regimens in Cancer Chemotherapy by an Efficacy-Toxicity Mathematical Model, Illiadis A., Barbolosi D. Computers and Biomedical Research 2000; 33: 211-226.


Summary

This article looks at common clinically used protocols for chemotherapeutics for cancer. These are usually overruled by constraints on maximum drug concentration in plasma and total amount of drug introduced to a patient's body. By instead introducing constraints concerning total white blood cell counts in the patient's body, the authors are capable of optimizing a simulation resulting in largely reduced tumor sizes at the end of treatment, without putting the patient at risk. This is accomplished by introducing small infusion intervals, and optimizing drug infusion rate at each interval such that the tumor is minimized in size, without violating any of the constraints on white blood cell counts, which give a general indicator of the patients overall health or side effect response by the treatment. By the simulations in this study, the authors show that it is possible to monitor chemotherapeutic side effects by dynamically observing white blood cell counts, instead of statically observing plasma concentrations and total drug amount. This shows promising results as a new way of achieving optimized drug delivery protocols for minimizing tumor size.

Biological background

When a chemotherapeutic agent (a drug) is introduced into the blood stream, both cancer cells and normal, healthy cells are affected and killed. Because of this it is important to monitor the delivery of the drug to the patient, since overexposure could lead to dramatic side effects and death of the patient. Since the drug is highly present in the blood stream of the patient, monitoring the white blood cell concentration in plasma as a function of time, drug concentration and drug infusion rate could give a good idea of at which level of risk the patient is at each given time during the treatment. Also, by using this approach, a drug delivery regimen can be optimized such that cancer cell count is minimized at the end of treatment without putting the patient at risk during the cure.

Model characterization

This article studies the effect of various chemotherapy administration strategies on cancerous tumors, with an emphasis on patient safety. It proposes a new approach to chemotherapy treatment, monitoring white blood cell count in addition to total drug levels in the body. To support this strategy, the authors develop a staged cellular-scale model in which drug concentration is described by modified mass-action and delay differential equations.

History

Chemotherapy of cancer is a field which have been studied extensively over the past decades. Cancer is a very complex group of diseases which strikes numerous people around the world each year, and the wish to find a good and effective cure has driven research forward in this field. Chemotherapy of cancer is a way of killing cancer cells by introducing poison to the body. This poison mainly effect proliferating cells, which the bulk of most cancer tumors consists of. The poisons introduced can attack the cancer cells in many different ways. One method is to induce DNA-damage to activate the p53-pathway and induce apoptosis in the cells[1]. Other pathways can interact with the cell cycle and result in cell cycle arrest[2]. Common for all chemotherapeutic schemes is that they do not influence cancer cells alone. The chemotherapeutic poison is often introduced intravenously leading to a poison distribution throughout the body. This gives cell death of healthy cells around the body leading to severe side effects [3] often leading to great suffering of the patient. Because of this, research within chemotherapy have also been focused on how to distribute the chemotherapeutic in a best possible way such that side effects are minimized and tumor regression is maximized. This has been done using several constraints such as maximum drug exposure, maximum injection rate etc. Here, the constraints investigated are instead the white blood cell count per unit volume as a function of time, drug concentration and drug infusion rate.

Mathematical model

Outline

Figure 1- Compartment model of the system

The model is designed to give information of the chemotherapy's effect on both tumor cells and white blood cells. To do this, a compartment model is used to illustrate the stages the drug can be in in the body (figure 1). The model consists of four compartments. On the side of pharmacokinetics, the plasma drug concentration, c1(t), and the active drug concentration, c2(t) ares present, with volumes of distribution V1 and V2. The active drug concentration is the portion of the drug which can affect the tumor cells, while the drug in plasma will affect the white blood cells. The two other compartments in the model are present on the pharmacodynamic side of the model, and are the white blood cell count per unit volume, w(t), and the tumor cell count per unit volume, n(t). These compartments constitute the phamacodynamics of the model.

Differential equations

The plasma concentration of the drug is modeled by


(1)\ {dc_{1}(t) \over dt}=-(k_{1}+k_{{12}})\cdot c_{1}(t)+{u(t) \over V_{1}}


where k1 and k12 are rate constants as defined in figure 1 (k12 is the link between plasma and active state of the drug and k1 is general elimination of the drug from the plasma through other pathways), u(t) is the drug infusion rate function, and V1 is the plasma volume of distribution (the total volume of plasma in the body). The active drug concentration is modeled by


(2)\ {dc_{2}(t) \over dt}=k_{{12}}\cdot {V_{1} \over v_{2}}\cdot c_{1}(t)-k_{2}\cdot c_{2}(t)


where k2 is the elimination rate constant of active drug and V2 is the volume of distribution of the cancer tumor. For both these equations it is important to remember that an interaction between the drug and a cancer or white blood cell does not lead to elimination of the drug. The drug affects the cell merely by its presence in the given volume of distribution. For both equations (2) and (3) the initial drug concentrations are set to zero:


(3)\ c_{1}(0)=0,c_{2}(0)=0


The tumor count is modeled by the following differential equation:


(4)\ {dn(t) \over dt}=\lambda \cdot n(t)\cdot ln[\theta /n(t)]-k\cdot [c_{2}(t)-c_{{MIN}}]\cdot n(t)\cdot H[c_{2}(t)-C_{{MIN}},n(0)=n_{0}


The first term is the tumor growth term modeled using a Gompertz curve growth type, which has previously been showed to fit cancer growth data quite accurate[4]. Here n(t) donates the number of cancer cells at time t and n0 is the number of cancer cells at time 0. θ is the carrying capacity of the tumor, that is the maximal size the tumor can achieve at any time with the available nutrients. When the tumor reaches this size, it will not grow anymore, and we see from equation (4) that when θ=n(t) the growth term of the equation equals zero. λ is a constant modeling the proliferation ability of the cancer cells. The larger the value of λ is, the faster the cells proliferate, and the result is increased tumor growth rate. The second term is the tumor cell-loss term, which is dependent of active drug concentration. H is the Heaviside step function, which leads to the cell-loss term equaling zero at all values of c2(t) < CMIN. CMIN is set as a threshold value of the active drug concentration. The drug will have no effect on the cancer cells under this active concentration. The cell-loss term is also proportional with the tumor cell count, n(t), and k is a proportionality constant which expresses the constant-cell-kill hypothesis expected to be valid when cells are sensitive to the drug. In short this states that the chemotherapeutic will not kill all tumor cells, but instead kill a certain fraction of the tumor cells when the chemotherapeutic is applied at a given concentration over a given period of time. The white blood cell count is modeled by


(5)\ {dw(t) \over dt}=r_{c}-v\cdot w(t)-\mu \cdot w(t)\cdot c_{1}(t-\tau )


where rc is constant production rate of white blood cells (with dimension volume-1\cdot time-1), and v is the fraction of white blood cells which dies per time. The steady state of this system with no drug present gives the initial white blood cell count w(0) = w0 = rc/v. The last term in equation (5) represents the loss of white blood cells because of drug present in the blood plasma. White blood cells are all produced from stem cells in the bone marrow and not by cell division during circulation. Because of this, it is reasonable to assume a constant production rate. This phenomenon will also introduce a delay from the white blood cells are produced in the bone marrow until they have matured and been transported into circulation in the blood system were they are susceptible for the drug. These considerations result in the last term of equation (5) where a delay, τ, is introduced into the plasma drug concentration. The interpretation of this would be that the white blood cells produced at time t, will not be susceptible for drug until a time t + τ in the future. Therefor is the delayed plasma drug concentration used. The total drug induced white blood cell-loss term is then proportional to the number of white blood cells and the delayed drug concentration. μ is a proportionality constant with dimensions concentration-1\cdot time-1.

Constraints

Several constraints are put on the model to insure that the patient can handle the toxic side effects of the drug. Firstly, the maximum plasma drug concentration must never exceed a given value:


(6)\ c_{1}(t_{i})<=C_{{MAX}}


where ti represents the time interval of maximum plasma drug concentration in one given administration protocol. Also, a maximum drug exposure is introduced, meaning that the total amount of drug introduced to the patient over the whole treatment period must not exceed a given limit. This can be expressed by


(7)\ {da(t) \over dt}=c_{1}(t),a(0)=0


This is commonly known as area under the curve, AUC, because a(t) equals the integral from 0 to infinity over the concentration distribution of c1(t). Equation (7) leads to


(8)\ a(T)<=A_{{MAX}}


where T is at the time point where all drug is completely eliminated from the body (the integral is zero from point T until infinity, and therefore the integration can be stopped at this point). Also constraints on the white blood cell counts are introduced. First, a minimum value of w(t) is defined, under which the white blood cell count never is allowed to fall:


(9)\ w(t_{j})>=W_{D}


where tj represents the time interval of lowest white blood cell count. The second constraint puts a limit on how long the white blood cell count can stay under a given upper value, WU. The time tU(T) over which the white blood cell count is under WU must be less or equal to TU:


(10)\ t_{u}(T)<=T_{u}

Drug delivery

The optimization problem considers how to best infuse the drug over a period of time, tN, divided into N time treatment intervals, such that the tumor size is minimized without violating any of the given constraints. The infusion rates can vary between each of the time intervals such that


(11a)\ u(t)={d_{i} \over t_{i}}\ for\ 0<=t<t_{1},


(11b)\ u(t)={d_{i} \over t_{i}}\ for\ t_{{i-1}}<=t<t_{i},i=2,...,N,


is fulfilled. di donates the total amount of drug infused in the time interval from ti-1 to ti. By examining different drug regimens, the optimal drug delivery scheme can be found using this given model.


The model and results

Parameter values

There are quite many parameters involved with this model, and to get accurate results, it is important to chose values as close to reality as possible. The values used in this simulation are mainly taken from literature, but assumptions on some of the parameter values are done as well. Table 1 includes all parameter values with comments on each value.


Parameter Value Dimension Comment
V1 25 L Volume of plasma in adult human
V2 15 L No data available. Assume effective volume of distribution equal to 15 L
CL 50 L\cdot day-1 Total clearance from the body per day
k1+k12 2 day-1 Values of V1 and CL gives total elimination, k1+k12, equal to 2 per day.
k1 1.6 day-1 No data available. Assumed that k12 is one fifth of k1
k12 0.4 day-1 No data available. Assumed that k12 is one fifth of k1
k2 0.8 day-1 No data available. Assumed to be one half of k1
\theta 1012 cells Largest tumor size, modeling value, corresponding to 1 kg of tumor mass
n0 30*109 cells Starting tumor size, modeling value, corresponding to 30 g of tumor mass
\lambda 3*10-3 day-1 Assumed time for doubling of tumor mass without drug to be 75 days, which gives this value.
CMIN 0.1 μg\cdot mL-1 Assumed value
k 30 L\cdot g-1\cdot day-1 Assumed value
W0 8*109 L-1 White blood cell count in normal adult human
v 0.15 day-1 Normal turnover value for adult human
rc 1.2*109 L-1\cdot day-1 Given from values for v and W0
\mu 80 L\cdot g-1\cdot day-1 Assumed value
\tau 5 days Assumed value, associated with μ
CMAX 10 μg\cdot mL-1 Modeling value
WD 2*109 L-1 Modeling value
TU 7 days Modeling value
\delta t 0.52 days Sampling interval for infusion protocol
AMAX 30 μg\cdot day\cdot mL-1 Modeling value
tN 21 days Treatment period, taken from the clinical protocols
T 31 days Treatment period plus to times the white blood cell delayed response

Clinical protocols

Using the model described it was possible to simulate both tumor cell count an white blood cell count using conventional clinical protocols for chemotherapeutic drug infusion. The protocols simulated were


(1) 500 mg\cdot m-2 infused over 1 day.

(2) 2 – h infusions of 100 mg\cdot m-2\cdot day-1 for the first 5 days of cycle.

(3) 1 – h infusions of 150 mg\cdot m-2\cdot day-1 for days 1, 3, and 5.

(4) Continuous infusion for 3 day at the rate of 125 mg\cdot m-2\cdot day-1.

(5) Continuous infusion over 21 days at the rate of 25 mg\cdot m-2\cdot day-1.


Figure 2- Simulation of the clinical protocals. a) w(t), b) n(t)

The simulation results for cancer cell count, n(t), and white blood cell count, w(t), using these 5 protocols in the described model are presented in figure 2a and 2b respectively. Considering white blood cell counts, the results show that w(t) never drops below the lower limit WD for any of the protocols. All protocols also hold for constraint (8), a(T) <= AMAX. Cancer cell counts decrease for all five protocols, and decreases from an initial cancer mass of 30 g to values from 25.3 g to 28.1 g by the end of treatment. The simulation shows quite different results for protocol 1 through 4 and protocol 5. The four first protocols all involve high infusion rates early in the treatment cycle and a long recovery step, while protocol 5 involves low continuous infusion over the whole treatment period. The result show that for protocol 1 through 4, both white blood cell and cancer cell count fall quickly during the beginning of treatment and then rise slowly. By the end of treatment, the white blood cell count had rose to almost initial values. For protocol 5, both counts fall slowly until end of treatment, and then slowly rise again. In this protocol, the white blood cell count did not reach initial values by the end of treatment.

Optimized protocols

The model was next used to optimize infusion protocol to minimize tumor size without violating the constraints of the model. Two different modeling problems were designed:

Figure 3 - Simulation of the optimized system. a) Optimal drug rates, u*(t) (magnified by 10), c1(t) and w(t-τ). b) Optimal drug rates, u*(t) (magnified by 80), c2(t) and n(t).


(P1) Given the drug infusion function, (11), determine the amount of di at each interval which minimizes the minimal tumor size over the time interval 0 < t > T while satisfying the constraints (6), (9) and (10).


(P2) Given the drug infusion function, (11), determine the amount of di at each interval which minimizes the minimal tumor size over the time interval 0 < t > T while satisfying the constraints (6) and (8).


By this, problem 1 concerns the constraints on white blood cell counts, while problem 2 concerns the constraint on total drug exposure. Both problems involve constraint (6) which limits the maximum plasma drug concentration allowed. The two problems both seek to minimize the minimum tumor size, but have different constraints overruling the simulations. The optimized result for problem 1 is given in figure 3a and 3b. Figure 3a shows the optimized drug infusion rates, u*(t), the white blood cell count (shifted τ backwards), w(t-τ), and the plasma drug concentration, c1(t). Figure 3b shows the optimized drug infusion rates, u*(t), the cancer cell count, n(t), and the active drug concentration, c2(t).

Analysis

Inspection of problem 1

Inspection of the optimized infusion rates show an initial high infusion rate leading to a fast increase of c1(t) until it reaches the maximum allowed value, CMAX. The white blood cell count drops in the same region until it reaches its minimum value, WD. To avoid violation of the constraints, u*(t) is lowered in the following time intervals, and finally put equal to zero, to rise the white blood cell count above WU to avoid violating constraint (10). The infusion until this break is the first phase of the treatment. The break continuous for two days, before the second phase of treatment starts, where infusion rates in general are lower than in phase one. In phase two, both white blood cell count and plasma drug concentration are kept at levels above and below their limit values. Moving our attention to figure 3b we see that the cancer cell count first increases slightly until the drug reaches the active compartment, and after that decreases until after the end of treatment. The minimum cancer cell count is found at t* = 23.6 days and has the value of n(t*) = 5.1 g. The minimum tumor size is actually achieved after the end of treatment. The total amount of drug infused over the treatment period was 4.39 g.

Inspection of problem 2

Optimized infusion rates for problem 2 were found be high rates for the first 6 days and then no infusion until the end of the treatment period. For this optimization, the minimum tumor size was found to be n(t*) = 16.6 g and t* = 9.0 days. The total amount of drug infused over the treatment period was 1.5 g.

Discussion

We see immediately that the minimum tumor size is achieved with the procedure of problem 1, and this result is dramatically better then the result achieved in problem 2 (4.39 g versus 16.6 g minimal tumor mass). The total amount of drug infused is also allowed to be elevated in problem 1 compared to problem 2. This is because of the different means of measuring maximal allowed toxicity of the drug in the different procedures. The area under the curve approach of problem 2 is a static approach for measuring maximal toxicity. The infusion has to stop when a(t) reaches AMAX. Problem 1 on the other hand solves this issue by dynamically measuring the response in the white blood cell count and putting constraints on which white blood cell levels that are allowed. By this no constraint is put on the maximum amount of infused drug, and the result is a larger amount of drug infused, and a smaller minimum tumor size. The simulation does indeed show promising results of designing drug regimens using the dynamics of white blood cell count as limiting constraints instead of the static area under the curve, AUC, approach. To arrive at the optimized protocol, a combination of the clinical protocols were used. It starts with high infusion doses and lower the doses to low constant doses later in the treatment. By this, both the initial fast drop in tumor size, n(t), observed using clinical protocols 1 through 4 and the slow decay observed using protocol 5 is achieved, without violating the constraints on white blood cell counts. The results in an exponential-like decay of tumor size. These findings will hopefully contribute to the consideration of drug toxicity surveillance using dynamic models such as the one presented here, instead of solely relying on static data.

Recent Extension

In a 2003 paper, Mechanistic models for myelosuppression,[5] Lena E. Friberg and Mats O. Karlsson cite the Illiadis paper in a review of chemotherapeutic treatment optimization models. Their paper analyzes five different models and analyzes their strengths and shortcomings. They argue that mechanistic models, based on physiological realities rather than simply empirical data, are more helpful in determining optimal treatments because they more reliably allow for extrapolation. However, they note that empirical data, such as that obtained through monitoring blood levels, is still necessary for the development of such models.

Friberg's citation of the Illiadis paper references it as a "first approach to integrate both pharmacokinetic-pharmacodynamic relations of effect and toxicity in the same model to optimize therapy.' However, they criticize the great extent to which the Illiadis paper bases its model on empirical data, and propose that the model would be more robust if it had a stronger foundation in the underlying mechanisms of drug-tumor interactions. The Friberg paper supports the use of delay equations in describing drug effects, and indicates that drug administration schedules are a valuable topic of research in combatting cancer. They recognize the Illiadis paper is an early contribution to this area.

External links

  1. Lowe SW et al, p53 is required for radiation-induced apoptosis in mouse thymocytes. Nature 1993; 362: 847-849
  2. Wikipedia.com: Chemotherapy, types.
  3. Ramirez LY et at, Potential Chemotherapy Side Effects: What Do Oncologists Tell Parents? Pediatr Blood Cancer 2009;52:497–502
  4. Bezwoda WR. High dose chemotherapy with hematopoietic rescue in breast cancer: From theory to practice. Cancer Chemother. Pharmacol. 1997; 40; 79-87
  5. Friberg LE and MO Karlsson, Mechanistic models for myelosuppression Investigational New Drugs 2003;21:183–194