Yifan Wang 1, Will Cao 2
1 Indian Springs School, 190 Woodward Drive, Indian Springs, AL 35124, USA
2 Department of Biomedical Engineering, Duke University, Durham, NC 27705, USA
Chemotherapy has been an effective approach to treat cancer; however, chemotherapy’s effectiveness could be influenced by many unknown and often neglected factors. In this project, we build mathematical models in MATLAB to explain what impact will blood vessel with stenosis have on the regime design at certain tumor stages, and also make predictions through analysis of the transmission systems and tumor growth models. In particular, we assumed a cylindrical vessel model that considered tumor growth and white blood cells recovery rate. Based on our model results, the number and morphology of the stenosis indeed influence the chemotherapy efficiency. Therefore, chemotherapy treatment should be customized case to case. Using the model that we developed in this study, patients will receive drug therapy regime based on the analysis results. The therapy regime includes the duration of each chemotherapy treatment, as well as the time period between individual treatments.
Cancer of any kind has become the top killer in many countries during recent years, and it had been troubling hospital and patients for many decades by now . Cancer may be caused by carcinogens  or x-ray . Several methods of treatment like hormone therapy  and radiation therapy  have been discovered and one of the most prevailed and effective approaches towards curing cancer is chemotherapy [6, 7] which effectively slows down the reproduction rate of tumor cells. Even though better targeting solutions have been made over the past few years, there are still many side effects of taking chemotherapy. Chemical substances contained in chemotherapy will disrupt the organization of tumor cells with side effect like hair loss, blood disorder caused by the damage of bone marrow . Constraints based on white blood cell (WBC) [9, 10] count which has variations  are used to monitor the usage of drug in chemotherapy. Using mathematical model as a guidance to design dosing regimen has become a trending topic [12, 13].
Most of these modeling-guide projects have been focused on the correlation between the drug and the tumor growth [12-14] to find the most suitable drug dosage regime. However, our model would identify the differences between different people even if they have the same type of cancer at the same phase. To understand the dynamic of substances of the chemotherapy transferring process, we have built a model that considers tumor growth, blood flow rate and cells recovery rate. The parameters can be manipulated to predict results that may include numbers of cycles or the dosage of drugs used to make comparisons between different patients who have different blood vessel conditions.
The methodology used in this project can be summarized as a three-step process. The first step is to set up a tube model in MATLAB. We set the length of the tube to be 30cm, because it requires moderate computational power to monitor the whole drug delivering process. There are several assumptions made in the model of the blood vessel. The blood flowing through the vessel should be a perfect Newtonian fluid, thus the tube can be viewed as a blood vessel. The stenosis, a barricade build within the vessel that obstruct the flow of blood, is described as a cosine shape smooth stenosis changing the axial velocity of blood flowing through the vein. The axial velocity is the same at all points of the same distance away from the central axis of the artery. Because most of the people have diseased blood vessel, caused by either genetic defections or bad daily habits. All the set ups are used to compare the difference between one stenosis’s influence on the regime and double stenosis’ influence. All the specific conditions set up are for the convenience of calculations and accuracy of the model, as non-Newtonian fluid and rough stenosis requires more calculations and formulas.
We set up two blood vessel models with single cosine shaped stenosis and (Figure 1A) two cosine shaped stenosis (Figure 1B). We developed the algorithm in MATLAB. Given the functions , we used two dimensional partial differential equations with finite difference approximation. Numerical solution was solved through finite differentiation. The flow rate is a key factor to consider when designing the regime of killing tumors. Hence, one matrix is formed to contain the flow rate at each specific time and location.
Figure 1: A: single cosine shaped blood vessel model. This figure represents the upper half of a diseased blood vessel of Patient A with single stenosis. B: Two cosine shaped blood vessel model. This figure represents the upper half of a diseased blood vessel of Patient B with double stenosis.
We assumed one end of the vessel is connected with a tumor. Therefore, the vessel serves as a drug delivery system. The tumor growth and the drug delivery dynamics are recorded and quantified in the simulation. Assume that the dispersion of the drug happened simultaneously leading to the same concentration along the vessel. The time it takes for the drug molecules to flow through the 30 cm vessel is essential. We can attain the time of the transporting process by calculating the average of the velocities of tiny sections of distance that comprise the total length of the blood vessel. Using the Gompertz Growth Model [16, 17], we can configure the growth of the tumor with restraints due to nutrient deficiency over the time span of speculation (Figure 2A).
X(t) is the function of the tumor size at time t, K is the carrying capacity, the maximum size that the tumor would reach based on the environmental condition. The constant ? is the proliferation ability of the tumor.
A given time lapse in the model need to be long enough to observe the first cycle of a chemotherapy. If the actual duration of the first cycle of a chemotherapy exceeds the given time span, we wouldn’t know what happens after the time span, and an error would occur.
It is assumed that there is no drug tolerance and resistance, in other words, the patient won’t have a diminished response to the chemotherapy. Thus the death rate of tumor cells is constant once the type of the medicine is defined. There is a threshold number of drug molecules used in a chemotherapy to ensure the health of the patient as chemotherapy kills normal cells. Two constraints are set to calculate the threshold number of drug molecules: the maximum drug constraint of the medicine itself and the white blood cell count. The WBC count evaluates the number of WBC in a patient’s body, and is a major indicator of when chemotherapy could become harmful. By measuring the initial value of WBC and the minimum number of WBC, we can eventually record the process and terminate the code immediately once the WBC is lower than the safety line.
Figure 2: A: Tumor growth model. This figure shows the growth of the tumor number of cells plotted against time(s). B: White blood cell recovery model. This figure shows the recovery rate of white blood cell in a period of two weeks after the chemotherapy. C: Tumor recovery growth model. This figure shows the growth rate of a tumor after the chemotherapy over a period of 40 days. Drugs change the tumor growth function.
If the tumor cannot be killed by a one-time chemotherapy, the duration of chemotherapy needed is attained based on the time it needed for the white blood cells to grow back to the normal level (before the last chemotherapy). Assuming that there is no fluctuation to the number of white blood cell, we need to consider the recovering rate, due to the effects of chemotherapy; the reproduction of normal cells will be dragged down and gradually rise back to its normal speed (Figure 2B).
Here the WBC recovery rate is expressed as:
Y(t) is the function representing the recovery rate of the white blood cells, and M is the normal WBC recovery rate.
The reproduction mechanism of the tumor was disrupted by the drug molecules, which lead to a brand new function to describe its recovering velocity (Figure 2C).
Tumor recovery rate is expressed as:
Z(t) is the function representing the recovery rate of the tumor after the distortion of chemotherapy, and t is the number of days after the chemotherapy. N is the normal recovery rate of tumor cell if not disrupted by chemotherapy, and n is a constant to configure the function. The time units of functions Z(t) and Y(t) are the same, and through balancing the two functions, the optimal time to conduct the next cycle of a chemotherapy can be found.
The interval between consecutive chemotherapy is determined by the number of WBC. As soon as the number of WBC is back to normal, the next chemotherapy can be executed. One cycle after another, the program will end once the tumor is cured entirely. Lists of data include the number of cycles, the number of drug molecules used and the total time used for chemotherapy etc. These statistics will suggest whether the patient should keep taking the same regime or switch to another. The program can also determine whether the chemotherapy can cure the patient entirely or it is only a way to suspend life.
Results and Discussion
To test the program, we have two patients with different health conditions, but are prescribed with the same drug to treat tumors that have the same parameters (size and growth rate). Patient A has only one stenosis (Figure 1A) to obstruct the flowing of the drug to its designated place to treat cancer, while patient B has two stenosis (Figure 1B) in the blood vein. All three stenosis have the same horizontal and vertical length as shown in the figures. This result derived from the comparison between the flow rate and the numbers of cycle and molecules needed to treat the cancer of both patient is that stenosis has a fairly large impact chemotherapy. The effectiveness of the drug, was measured by its killing rate, which was 0.1 tumor cells/s and 1.5 WBC/s. Great fluctuation is shown in the tumor size graph caused by the drug during the first cycle chemotherapy (Figure 3A and B). After linearizing the fluctuation points, the tumor grows at a significantly slow rate. Comparing the size of the tumor under the influence of chemotherapy and the growth function of the natural tumor growth, the difference is more obvious (Figure 3C and D). Because of the high flow rate of the drug in a single stenosis blood vessel, it is efficient to stop the rapid growth of the tumor at an earlier stage as shown in the figure. The difference is tiny, but the ultimate effect on the number of cycles of chemotherapy and the dosage of drug needed is quite large. Patient A needs 5 million drug molecules to kill the tumor which takes about 51 cycles. While patient B needs around 6 million drug molecules to kill the tumor which takes about 59 cycles. One more cycle of chemotherapy may create a higher risk for the patients. This is why patients should be treated differently according to their own conditions instead of having the same prescription.
Figure 3: A: Tumor size model. This figure represents the change of the tumor size during Patient A’s first cycle of chemotherapy (Single stenosis). B: Tumor size model. This figure represents the change of the tumor size during Patient B’s first cycle of chemotherapy (Double stenosis). C: Tumor size comparison model. This figure represents the comparison between natural growth of tumor and the influence of tumor size under chemotherapy of Patient A. Magnifying the impact of chemotherapy and stenosis have on tumor growth model (Single stenosis). D: Tumor size comparison model. This figure represents the comparison between natural growth of tumor and the influence of tumor size under chemotherapy of Patient B. Magnifying the impact of chemotherapy and stenosis have on tumor growth model. (Double stenosis)
In order to prove the effect is universal, in other words, not affected by the type of drug used in the chemotherapy. Patient A and B are put under the same circumstances, but the drug that they take is more powerful, increasing the chances of one shot cure, meaning small tumors can be entirely killed with only one cycle of chemotherapy. The condition may be idealized, but the aim is to find out how large the impact of the stenosis on the amount of drug that the patient need to take. The parameter of the drug is killing 10 tumor cells/s and 0.1 WBC/s. The result of Patient A to treat the tumor is 518 seconds and the total drug molecules are 51,140. While the total time of Patient B is 563 seconds and the total drug molecules needed are 58,446. From the graph of the tumor size change during the chemotherapy (Figure 4A and B), the conclusion is that it is faster to terminate the tumor in a comparatively healthier blood vessel (Figure 4C and D).
Figure 4: A: Tumor size model. This figure represents the change of the tumor size during Patient A’s first cycle of chemotherapy using the second type of drug (Single stenosis). B: Tumor size model. This figure represents the change of the tumor size during Patient B’s first cycle of chemotherapy using the second type of drug (Double stenosis). C: Tumor size comparison model. This figure represents the comparison between natural growth of tumor and the influence of tumor size under chemotherapy of Patient A. Magnifying the impact of chemotherapy and stenosis have on tumor growth model (Single stenosis). D: Tumor size comparison model. This figure represents the comparison between natural growth of tumor and the influence of tumor size under chemotherapy of Patient B. Magnifying the impact of chemotherapy and stenosis have on tumor growth model (Double stenosis).
The model also idealistically predicts whether the chemotherapy would have a chance of killing the tumor. Now, Patient C is introduced in the model. Patient C has the same blood vessel stenosis with Patient A, however, is treated with a drug that mildly kill tumor and WBC. We create a new model and the only thing changes is that the drug kills 0.001 tumor cells/s and 0.1 WBC/s, while the patient’s tumor has a high proliferation index of 0.005. Here, proliferation index is used to evaluate the rate of tumor cell growth. The higher the index is; the faster the tumor grows. The result is that the tumor growth is way higher than the maximum number of tumor cells killed by the drug after calculations (Figure 5A). The size of the tumor is bigger and the effect of the drug is smaller when comparing patient C to patient A, whose condition is the same as C except the difference in the parameter of chemotherapy (Figure 5B). In this case, Patient C should be discouraged to use the less effective chemotherapy because it would not be beneficial to him. Using a model like this may find out the optimum regime for a patient rather than wasting time on inefficient chemotherapies.
Figure 5: A: Tumor growth model. The figure shows the change of the size of the tumor during first cycle chemotherapy of Patient A (Single stenosis). B: Tumor size comparison model, this figure shows the comparison between natural growth and the growth under the influence of chemotherapy of Patient A. Magnifying the impact that stenosis and drug efficiency have on tumor size change (Single stenosis).
The influence of diseased blood vessel on the efficiency of chemotherapy is significant from the scientific calculations and the comparison between patients with single stenosis and double stenosis. Thus using this model, it would be easier for the patient to get the approximately correct and helpful time duration of chemotherapy and the length of the interval between two chemotherapies, which can save the patients a lot of time and money. In addition, the model indirectly suggests a good life style can be beneficial to a good life quality. As stenosis of the blood vessel can be caused by unhealthy eating habits in daily life.
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