Proliferation and Nonlinear Dynamics of Childhood Acute Lymphoblastic Leukemia Revisited

Proliferation and Nonlinear Dynamics of Childhood Acute Lymphoblastic Leukemia Revisited

George I. Lambrou (University of Athens, Greece)
DOI: 10.4018/978-1-4666-8828-5.ch015
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Acute Lymphoblastic Leukaemia (ALL) is the most common neoplasm in children but the mechanisms underlying leukemogenesis along with the dynamics of leukemic cell proliferation are poorly understood. The importance in understanding the proliferation dynamics of leukaemia lies in the fact that our knowledge from the point of first appearance to the moment of clinical presentation, we know almost nothing. Further on, describing cell proliferation dynamics in a more mature, probably mathematical, way it could lead us to the understanding of disease ontogenesis and thus its aetion. This chapter reviews the current knowledge on proliferation dynamics and proliferation non-linear dynamics of the leukemic cell. Furthermore, we present some “in-house” experimental data that support the view that it is possible to model leukemic cell proliferation and explain how this has been performed in in vitro experiments.
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Acute Lymphoblastic Leukaemia-Disease Description and Preliminaries

Acute leukaemia mainly appears during childhood but it can also occur in adolescence manifesting a poor prognosis regardless of age. Progress in childhood leukaemia has been immense with an overall survival rate exceeding 75% the last decades (Carroll et al., 2003). Still there is an approximate 20% that relapses, which in many cases can prove fatal. In the majority of cases leukemia appears to have a greater incidence of chromosomal abnormalities compared to solid tumors (Saha, Young, & Freemont, 1998). Also, gene expression is aberrantly regulated and in certain cases fusion genes form, that are similarly aberrantly expressed. It has been reported that those genes, involved in leukemia progression, are very potent regulators of cell proliferation, differentiation, cell cycle progression and anti-apoptosis (Saha et al., 1998).

Yet, the question that might rise is why leukaemia and especially childhood leukaemia? Acute lymphoblastic leukaemia (ALL) is the most frequent occurring malignancy among childhood cancers (Severson & Ross, 1999). It originates from the undifferentiated lymphoblast, which abnormally ceases to develop into the mature lymphoid cell giving rise to a tumour. Hence, one of the most interesting characteristics of leukaemia is its trait of clonal expansion. That is, the almost uniform phenotype of cells giving rise to the tumour. But, what does this has to do with leukemogenesis? Necessarily, leukemia as a disease has a starting point and a diagnosis point. Between those two there is an immense lack of knowledge. This does not apply only to leukemia but to any neoplasm in general or even inflammation. For example, as mentioned, absolute lymphocyte count is a prognostic factor in childhood leukemia (De Angulo, Yuen, Palla, Anderson, & Zweidler-McKay, 2008). In that sense, cell counts thus proliferation, is tightly connected to disease prognosis. The next question that would arise is: what is the connection between the first steps of disease emergence and the presentation stage. That is the most difficult part to answer since we simply do not, and cannot, have the slightest clue about what happens between that time and the present. A necessary approach to this phenomenon would be the modeling approach. That is the understanding and prediction of the phenomenon on a physical and systems basis. In order for such an approach to succeed it must entail a range of “crafts” ranging from mechanics, systems theory and thermodynamics to mathematical analysis and chaos dynamics.

Key Terms in this Chapter

Gombertz Curve: named after Benjamin Gompertz, is a sigmoid function. It is a type of mathematical model for a time series, where growth is slowest at the start and end of a time period. The right-hand or future value asymptote of the function is approached much more gradually by the curve than the left-hand or lower valued asymptote, in contrast to the simple logistic function in which both asymptotes are approached by the curve symmetrically. It is a special case of the generalised logistic function.

Lactic Acid: is a chemical compound that plays a role in various biochemical processes. It was first isolated in 1780 by the Swedish chemist Carl Wilhelm Scheele . Lactic acid is a carboxylic acid with the chemical formula CH 3 CH(OH)CO 2 H. It has a hydroxyl group adjacent to the carboxyl group, making it an alpha hydroxy acid (AHA). It is produced during glycolysis, during glucose metabolism, especially in tumour cells.

Game Theory: Game theory is the study of strategic decision making. Specifically, it is “ the study of mathematical models of conflict and cooperation between intelligent rational decision-makers ”. An alternative term suggested “ as a more descriptive name for the discipline ” is interactive decision theory . Game theory is mainly used in economics, political science, and psychology, as well as logic, computer science, and biology. The subject first addressed zero-sum games, such that one person's gains exactly equal net losses of the other participant or participants. Today, however, game theory applies to a wide range of behavioural relations, and has developed into an umbrella term for the logical side of decision science, including both humans and non-humans (e.g. computers, animals).

Lyapunov Exponent: In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by: |dZ(t)|~e ?t |dZ 0 | , where ? is the Lyapunov exponent. The rate of separation can be different for different orientations of initial separation vector. Thus, there is a spectrum of Lyapunov exponents -equal in number to the dimensionality of the phase space. It is common to refer to the largest one as the Maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the system is chaotic (provided some other conditions are met, e.g., phase space compactness). Note that an arbitrary initial separation vector will typically contain some component in the direction associated with the MLE, and because of the exponential growth rate, the effect of the other exponents will be obliterated over time.

Henri Poincare: Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and a philosopher of science. He is often described as a polymath, and in mathematics as The Last Universalist by Eric Temple Bell, since he excelled in all fields of the discipline as it existed during his lifetime. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture , which was one of the most famous unsolved problems in mathematics until it was solved in 2002-2003. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology.

Acute Lymphoblastic Leukaemia: an acute form of leukaemia, or cancer of the white blood cells, characterized by the overproduction and accumulation of cancerous, immature white blood cells—known as lymphoblasts. In persons with ALL, lymphoblasts are overproduced in the bone marrow and continuously multiply, causing damage and death by inhibiting the production of normal cells—such as red and white blood cells and platelets—in the bone marrow and by spreading (infiltrating) to other organs. ALL is most common in childhood with a peak incidence at 2–5 years of age, and another peak in old age.

Warburg Effect: One of the first carcinogenetic theories proposed. In particular, a common characteristic of almost all types of tumours is the known Warburg effect described by Otto Warburg in 1924. This simple observation remains until today a fundamental trait of tumour biology i.e. that tumours perform a shift from oxidative phosphorylation to anaerobic glycolysis for their energy needs. At the time, Warburg considered this as the leading aetiology of cancer and of note that it was considered as such until the observation or finding that it is one of the side-effects of cancer. It is now known that the avoidance of the oxidative phosphorylation pathway, it is the one that leads to excessive lactate production causing this deteriorating effect cancerous cell have on the surrounding tissue.

Verhulst Function: A logistic function or logistic curve is a common “S” shape (sigmoid curve), with equation: INSERT SHAPE AU257: Pict Element 13 AU258: Anchored Object 147 AU259: Image 15 where e = the natural logarithm base (also known as Euler's number), x 0 = the x-value of the sigmoid's midpoint, L = the curve's maximum value, and k = the steepness of the curve. For values of x in the range of real numbers from -8 to +8, the S-curve shown on the right is obtained (with the graph of f approaching L as x approaches +8 and approaching zero as x approaches -8). The function was named in 1844–1845 by Pierre François Verhulst , who studied it in relation to population growth. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops.

Lorenz Attractor: The Lorenz system is a system of ordinary differential equations (the Lorenz equations , note it is not Lorentz ) first studied by Edward Lorenz . It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system which, when plotted, resemble a butterfly or figure eight.

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