Mathematics Meets Oncology

Transcription

1 .. Mathematics Meets Oncology Mathematical Oncology Philippe B. Laval Kennesaw State University November 12, 2011 Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

2 Outline.1 Introduction: motivation for the talk..2 Why mathematicians and cancer researchers should collaborate more..3 Overview of cancer..4 Some tumor growth models..5 Conclusion. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

3 Introduction: Motivation for the Talk This talk is motivated by two facts:.1 There is a need for mathematicians and cancer researchers to work together. This is echoed in many papers and books I have read. A recent search on the PubMed database showed that out of about 1.5 million papers listed on cancer research, only 5% were related to mathematical modeling. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

4 Introduction: Motivation for the Talk This talk is motivated by two facts:.1 There is a need for mathematicians and cancer researchers to work together. This is echoed in many papers and books I have read. A recent search on the PubMed database showed that out of about 1.5 million papers listed on cancer research, only 5% were related to mathematical modeling..2 "I love math and science, I want to make a difference but I don t want to teach. What can I do?" Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

5 Introduction: Motivation for the Talk This talk is motivated by two facts:.1 There is a need for mathematicians and cancer researchers to work together. This is echoed in many papers and books I have read. A recent search on the PubMed database showed that out of about 1.5 million papers listed on cancer research, only 5% were related to mathematical modeling..2 "I love math and science, I want to make a difference but I don t want to teach. What can I do?".3 It looks like we should arrange this marriage between mathematicians and cancer researchers! Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

6 Why Mathematicians and Oncologists Are Not Working Together Cancer is still not fully understood. It is hard to model a situation we do not understand. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

7 Why Mathematicians and Oncologists Are Not Working Together Cancer is still not fully understood. It is hard to model a situation we do not understand. The parts of cancer we do understand are extremely complex, thus difficult to model. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

8 Why Mathematicians and Oncologists Are Not Working Together Cancer is still not fully understood. It is hard to model a situation we do not understand. The parts of cancer we do understand are extremely complex, thus difficult to model. Mathematical models are too simplistic and cannot model realistically a disease as complex as cancer. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

9 Why Mathematicians and Oncologists Are Not Working Together Cancer is still not fully understood. It is hard to model a situation we do not understand. The parts of cancer we do understand are extremely complex, thus difficult to model. Mathematical models are too simplistic and cannot model realistically a disease as complex as cancer. The mathematics involved in these models is difficult and scares away non mathematicians. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

10 What Would Be Gained if They Worked Together Mathematicians are not going to find a cure for cancer. However, they can help in the following areas: There is an enormous amount of data generated by the various experiments and clinical trials being conducted around the world. This data needs to be analyzed. This is the work of statisticians. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

11 What Would Be Gained if They Worked Together Mathematicians are not going to find a cure for cancer. However, they can help in the following areas: There is an enormous amount of data generated by the various experiments and clinical trials being conducted around the world. This data needs to be analyzed. This is the work of statisticians. Once we understand a mechanism involved with cancer, we can model it. We can then run simulations using the models. These simulations allow us to: Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

12 What Would Be Gained if They Worked Together Mathematicians are not going to find a cure for cancer. However, they can help in the following areas: There is an enormous amount of data generated by the various experiments and clinical trials being conducted around the world. This data needs to be analyzed. This is the work of statisticians. Once we understand a mechanism involved with cancer, we can model it. We can then run simulations using the models. These simulations allow us to: Validate theories. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

13 What Would Be Gained if They Worked Together Mathematicians are not going to find a cure for cancer. However, they can help in the following areas: There is an enormous amount of data generated by the various experiments and clinical trials being conducted around the world. This data needs to be analyzed. This is the work of statisticians. Once we understand a mechanism involved with cancer, we can model it. We can then run simulations using the models. These simulations allow us to: Validate theories. Predict the outcome of a disease. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

14 What Would Be Gained if They Worked Together Mathematicians are not going to find a cure for cancer. However, they can help in the following areas: There is an enormous amount of data generated by the various experiments and clinical trials being conducted around the world. This data needs to be analyzed. This is the work of statisticians. Once we understand a mechanism involved with cancer, we can model it. We can then run simulations using the models. These simulations allow us to: Validate theories. Predict the outcome of a disease. Finding the optimal dosage for a certain medicine. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

15 How We Can Fix It Involve more mathematicians in cancer research. It begins early on, at the undergraduate level. After briefly introducing what cancer is, we will present models which can be discussed in undergraduate classes. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

16 What is Cancer? The Cancer Process.1 The development and healthy life of a human being requires the cooperation of more than 50 trillion cells. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

17 What is Cancer? The Cancer Process.1 The development and healthy life of a human being requires the cooperation of more than 50 trillion cells..2 This cooperation is maintained by numerous checks and balances which determine when and how a cell will divide, differentiate or die. For every process which can take place, there are factors to promote it and factors to inhibit it. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

18 What is Cancer? The Cancer Process.1 The development and healthy life of a human being requires the cooperation of more than 50 trillion cells..2 This cooperation is maintained by numerous checks and balances which determine when and how a cell will divide, differentiate or die. For every process which can take place, there are factors to promote it and factors to inhibit it..3 The phenomenon of cancer can be defined on various levels. On the most basic level, cancer represents a collapse of this cooperation. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

19 What is Cancer? The Cancer Process.1 The development and healthy life of a human being requires the cooperation of more than 50 trillion cells..2 This cooperation is maintained by numerous checks and balances which determine when and how a cell will divide, differentiate or die. For every process which can take place, there are factors to promote it and factors to inhibit it..3 The phenomenon of cancer can be defined on various levels. On the most basic level, cancer represents a collapse of this cooperation..4 Though cancer is a whole collection of diseases, all cancer cells have common features: Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

20 What is Cancer? The Cancer Process.1 The development and healthy life of a human being requires the cooperation of more than 50 trillion cells..2 This cooperation is maintained by numerous checks and balances which determine when and how a cell will divide, differentiate or die. For every process which can take place, there are factors to promote it and factors to inhibit it..3 The phenomenon of cancer can be defined on various levels. On the most basic level, cancer represents a collapse of this cooperation..4 Though cancer is a whole collection of diseases, all cancer cells have common features: They grow without control. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

21 What is Cancer? The Cancer Process.1 The development and healthy life of a human being requires the cooperation of more than 50 trillion cells..2 This cooperation is maintained by numerous checks and balances which determine when and how a cell will divide, differentiate or die. For every process which can take place, there are factors to promote it and factors to inhibit it..3 The phenomenon of cancer can be defined on various levels. On the most basic level, cancer represents a collapse of this cooperation..4 Though cancer is a whole collection of diseases, all cancer cells have common features: They grow without control. They cease to perform their functions. Their main and only goal is to grow. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

22 What is Cancer? How a Cell Becomes Cancerous All cancers start from a single cell that undergoes many changes. Some of those changes are permanent alterations to the DNA called mutations. Luckily, our bodies have a host of defensive strategies for making sure damaged or mutated cells never get the chance to reproduce. Even if they can t always stop the process of cancer development, our natural defenses slow it down. This process takes years, even decades. Over our lifetimes, thousands and thousands of damaged cells get disposed of before they can cause any harm. But if a cell does manage to get past our defenses and start multiplying without control, it can form a mass of abnormal cells called a tumor. Not all tumors are dangerous. Those that arise and then go quiet are called benign. But malignant or cancerous tumors can spread into surrounding tissues, damaging nearby cells or organs. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

23 Stages of Tumor Growth There are three distinct stages (avascular, vascular, metastatic) to cancer development..1 Avascular: Tumor cells grow at a much faster rate than the host cells, requiring more and more nutrients and creating more and more waste to dispose of. After a while, the hosts cells cannot satisfy the demands of the tumor and the tumor starves. It cannot grow anymore. Unless something else happens, tumors will remain small (less than a few mm in diameter) and will not spread. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

24 Stages of Tumor Growth There are three distinct stages (avascular, vascular, metastatic) to cancer development..1 Avascular: Tumor cells grow at a much faster rate than the host cells, requiring more and more nutrients and creating more and more waste to dispose of. After a while, the hosts cells cannot satisfy the demands of the tumor and the tumor starves. It cannot grow anymore. Unless something else happens, tumors will remain small (less than a few mm in diameter) and will not spread..2 Vascular: Many tumor cells have the ability to develop their own blood supply (angiogenesis). Since they are not starved, tumors which reach this stage can continue growing and invade surrounding tissues. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

25 Stages of Tumor Growth There are three distinct stages (avascular, vascular, metastatic) to cancer development..1 Avascular: Tumor cells grow at a much faster rate than the host cells, requiring more and more nutrients and creating more and more waste to dispose of. After a while, the hosts cells cannot satisfy the demands of the tumor and the tumor starves. It cannot grow anymore. Unless something else happens, tumors will remain small (less than a few mm in diameter) and will not spread..2 Vascular: Many tumor cells have the ability to develop their own blood supply (angiogenesis). Since they are not starved, tumors which reach this stage can continue growing and invade surrounding tissues..3 Metastatic: Once a tumor has acquired its own blood supply, some of its cell can escape the primary tumor via the circulatory system (metastasis) and set up secondary tumors elsewhere in the body. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

26 Angiogenesis? Illustration of Angiogenesis Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

27 Tumor Growth Models We will only present models, not discuss them in detail or solve them. The goal is to give a sample of the methods used and show that simple cancer models can be presented early on in undergraduate classes. We will present models using ODE s as well as PDE s. The ODE models view cancer as a population of cells and study how the population evolves. Thus, traditional population models can be used. The PDE models are more spatially structured. They allow to make a dynamic description of spatial variations in the system. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

28 Avascular Tumor Growth - ODE Models Homogeneous Solid Tumors - Logistic Growth As noted above, avascular tumors cannot grow beyond a certain size. This is because as the tumor increases in size, competition for nutrients and space cannot be neglected anymore. Therefore, we cannot use the exponential growth law. Instead, we use the logistic growth law. ) dn (1 dt = kn NN N (0) = N 0 > 0 The solution is N (t) = N N 0 N 0 + (N N 0 ) e kt N as t where: N (t) is the number of cells within a tumor. k > 0 is the net rate at which the cells proliferate. N > 0 is the carrying capacity of the population. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

29 Avascular Tumor Growth - ODE Models Homogeneous Solid Tumors - Modified Logistic Growth The symmetry of the previous solution about its inflection point makes it difficult to fit experimental data. So a modified version is used instead. The solution is N (t) = N ( dn dt = k ( ( N α N 1 N N (0) = N 0 > 0 N α 0 ) α ) N α 0 + (Nα N α 0 ) e kt Note that when α = 1, we have the logistic model. The Gompertzian law is recovered in the limit as α 0 +. ) 1 α Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

30 Avascular Tumor Growth - ODE Models Gompertzian Law Named after Benjamin Gompertz ( ) and adapted in the 1960 s to tumor growth. dn dt = kn ln N N N (0) = N 0 > 0 The solution is N (t) = N e ln N 0 N e kt N as t Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

31 Avascular Tumor Growth - ODE Models Comparing the Solutions ( N We plotted g (N 0, α, N, k, t) = ) 1 0 N α for N0 α + (Nα N0 α) e kt N 0 = 0.1, N = 1, k = 0.1 and various values of α shown in the table: α Color red black green blue Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

32 Avascular Tumor Growth - ODE Models Treatment of Homogeneous Solid Tumors Consider a tumor which would follow logistic growth without therapy. The patient undergoes chemotherapy. The patient is injected a drug which kills cancer cells upon contact. Let A (t) be the drug concentration within the tumor, µ is the rate at which the drug kills the tumor, λ is the decay rate of the drug, γ is the rate at which the drug becomes ineffective after it has killed tumor cells and a (t) is the rate at which the drug is being delivered. Then, the following model can be used: ) dn (1 dt = kn NN µan da = a (t) λa γan dt N (0) = N 0 and A (0) = A 0 Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

33 Avascular Tumor Growth - ODE Models Treatment of Homogeneous Solid Tumors To keep the system fairly simple, we can consider two methods of infusion..1 Continuous infusion: a (t) = a t 0 Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

34 Avascular Tumor Growth - ODE Models Treatment of Homogeneous Solid Tumors To keep the system fairly simple, we can consider two methods of infusion..1 Continuous infusion:.2 Periodic infusion: a (t) = a (t) = a t 0 { a if n < t < n + τ 0 if n + τ < t < n + 1 Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

35 Tumor Growth - PDE Models Gatenby, 1996 This next model explains how tumor cells can grow by killing surrounding healthy tissue. They do so by modifying the microenvironmental ph. Cancer cells resist better to an environment with excess acid than healthy cells. N 1 (x, t): density of normal tissue with growth rate r 1 and carrying capacity K 1. N 2 (x, t): density of neoplastic tissue with growth rate r 2 and carrying capacity K 2. L (x, t): excess concentration of H + ions. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

36 Tumor Growth - PDE Models Gatenby, 1996 The behavior of the normal tissue is determined by:.1 Logistic growth of N 1..2 Population competition with neoplastic tissue with Lotka-Volterra competition strength parameter α Interaction of N 1 with excess H + ions leading to a death rate proportional to L..4 Cellular diffusion with an N 2 -dependant coefficient D N1 [N 2 ] The behavior of the neoplastic tissue is similar, except for dying from excess H + ions. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

37 Tumor Growth - PDE Models Gatenby, 1996 We assume that the excess H + ions satisfy:.1 They are produced at a rate proportional to N 2 with rate r 3..2 They diffuse chemically with diffusion coefficient D 3..3 An uptake term is included to account for the body trying to eliminate the excess acid with reabsorption rate d 3. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

38 Tumor Growth - PDE Models Gatenby, The Model Putting the above together, we obtain ( N 1 = r 1 N 1 1 N ) 1 N 2 α 12 d 1 LN 1 + (D N1 [N 2 ] N 1 ) t ( K 1 K 2 N 2 = r 2 N 2 1 N ) 2 N 1 α 21 + (D N2 [N 1 ] N 2 ) t K 2 K 1 L t = r 3N 2 d 3 L + D 3 L Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

39 Tumor Growth - Angiogenesis How can we take into account angiogenesis?.1 Incorporate angiogenesis into existing models: All the above models used a constant carrying capacity. With angiogenesis, the carrying capacity will increase. We make it a function of time. Which function to use is determined by available data. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

40 Tumor Growth - Angiogenesis How can we take into account angiogenesis?.1 Incorporate angiogenesis into existing models: All the above models used a constant carrying capacity. With angiogenesis, the carrying capacity will increase. We make it a function of time. Which function to use is determined by available data..2 Create new models: Incorporate the process of angiogenesis in the model. This requires adding variables for all the chemical components involved and the equations which describe this process. The resulting models are much more complicated. They are still work in progress as angiogenesis is extremely complex and not fully understood. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

41 Conclusion.1 Cancer is one of the main causes of mortality in the world, thus cancer research is one of our main priorities. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

42 Conclusion.1 Cancer is one of the main causes of mortality in the world, thus cancer research is one of our main priorities..2 Mathematicians are not going to find a cure for cancer. However, they are needed in cancer research to help analyze the enormous amount of data generated by the various experiments and clinical trials being conducted around the world. They are also needed to help develop and validate new mathematical models. These models can in turn be used to run simulations, test theories, help determine the optimum dosage of new medicines being developed. Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23

43 Conclusion.1 Cancer is one of the main causes of mortality in the world, thus cancer research is one of our main priorities..2 Mathematicians are not going to find a cure for cancer. However, they are needed in cancer research to help analyze the enormous amount of data generated by the various experiments and clinical trials being conducted around the world. They are also needed to help develop and validate new mathematical models. These models can in turn be used to run simulations, test theories, help determine the optimum dosage of new medicines being developed..3 If you love mathematics and science and you want to make a difference but you do not want to teach, a career in cancer research may be for you. You are needed. We can adapt John F. Kennedy s quote to say: Don t ask what cancer research can do for you, but what you can do for cancer research! Philippe B. Laval (Kennesaw State University)Mathematics Meets Oncology November 12, / 23