
MBW:Inducing Catastrophe in Malignant GrowthFrom MathBio(Redirected from Inducing Catastrophe in Malignant Growth)
The following wiki, reviewed by Emily Schuck, explores the Article "Inducing Catastrophe in Malignant Growth" written by Robert Gatenby and B. Roy Frieden in 2008. ContentsOverview
Executive SummaryBiological systems depict many features of chaotic systems, which include fractal structures and strange attracters. When looking at tumors, cautious immunization may lead to profound changes in the stability and may allow the system’s ability to recognize those foreign cells. Cancer, a time dependent process, will show a catastrophe, a sudden change of state in the system, at some time t. Cancer abruptly changes the mass in cells, and according to therapies (i.e. chemotherapy, no therapy, etc.), the mass can be positive or negative. The catastrophe theory can be used to help identify the remission or replication of such cancer cells and can assist in the termination by predicting the pattern. A Brief History of the Catastrophe Theory
The Seven Fundamental Types of Catastrophe
Background of Gatenby and Frieden's studyLiving systems, most importantly, biological systems, are nonlinear. With infinite variation within parameters. Robert Gatenby is a major researcher on the link between catastrophe and cancer. According to him, the catastrophe theory can describe breast cancer growth during any timedependent therapeutic activity, a(t). This therapeutic activity must be actively imposed, for example, chemotherapy or something that can occur as an immune response. The catastrophe theory can predict the growth in cancer mass in time t as a cosinemodulated power law, using the Fibonacci constant of 1.618…. This modulation is able to predict the relapses and remissions of cancer growth. “These fairly well agree with clinical data on breast cancer recurrences following mastectomy. Two such studies of 3183 Italian women consistently show an immune system’s average activity level of about a = 2.8596 for the women. Fortunately, an optimum timevarying therapy program a(t) is found that affects a gradual approach to full remission over time, i.e. to a chronic disease. Both activity a(t) and cancer mass p(t) monotonically decrease with time, the activity a(t) as 1/(ln t) and mass remission as tˆ{−0.382}. These predicted growth effects have a biological basis in the known presence of multiple alleles during cancer growth” (6). Experimental MethodsThe problem that needs to be looked at then becomes one to answer the ability to track tumor growth without therapy. Gatenby took theoretical information, using Fisher Information, and devised an equation characterizing the developmental age of a tumor over observation time, (dp(t)/dt)2/p(t), where p(t) represents the probability law governing cancer mass over time. Through the evolvement of equations, Gatenby referred to two types of therapy, including therapeutic activity over time and a mastectomy with no future treatment. The therapeutic activity was looked at when a(t)=a, which was when there was constant activity, which allowed prediction of remission and relapse of growth. Then when a(t) decreases, an ‘asymptotic cure’ is related. Both of the above therapies were derived from the catastrophe theory. The Full Mathematical ModelAccording to Gatenby and Frieden's research, untreated breast cancer will grow as the following power law: where [Eq 1]. This power law depicts the probability density function for observation of cancer over time (t, t + dt), and is in addition, proportional to cancer mass. Here, , Fibonacci constant. Equation 1 was derived from the information obtained beforehand and the Fibonacci constant was used due to the fact that it was assumed that due to accumulating mutations, the tumor cell information will asymptotically approach a minimum value necessary for replication. Treated breast cancer has a better connection with the catastrophe theory due to the fact that therapy will abruptly alter the state of a cells total mass. Due to the assistance of the catastrophe theory, equation 1 will be modified to include therapeutic activity to become that of the form: [Eq 2]. A is the normalization constant, of units 1/time, to match the units of p(t), and the angle a(t)lnt/to (of cosine) allows the program a(t) of therapeutic activity to be included in growth law. Equation 2 now predicts cancer growth to have a free growth factor, , attained by the maximum tumor growth, as well as having a limitation to growth . The onset of therapeutic activity, a(t), acts as a catastrophe to the cancer, ultimately reducing the mass. The cosine dependence shows a periodic disease progression or recurrence where the mass, p(t), goes to zero, including a local maxima. When a(t) is a constant, the mass, p(t) is a periodic function with an instantaneous wavelength increasing with time.
The zeroes show cancer remissions and maxima the relapses of the cancer Constant Therapeutic ActivityFor the simplest of cases, constant activity, a(t), is analyzed. This consistency relates the use of constant concentrations of therapy and constant natural killer cells. This predicted growth will follow an increasing oscillatory curve. When no therapy is given, a=0, there will be a constantly increasing curve which acts as an upper bound due to the fact that optimization is found when a(t)=0. This relationship is shown in the figure below: . As the curves diverge, the freegrowth, untreated cancer increases asymptotically, whereas the therapeutic oscillation shows that even though the tumors stop growing, the growth starts again, recurring in wavelike fashion. The peak of the curve will increase each recurrence until the patient dies; however, the increased spacing between recurrences shows relapses and remissions become rarer. The value of 5 is given to a because it was shown to be a realistic result for maximal density therapy. If the cancer cells were to synchronize, rather than mutate after every recurrence, a cure could be discovered; however, the cells undergo dephasing and resistant phenotypes are created TimeVarying Therapeutic Schedule
, where [Eq 4]; furthermore, Xo denotes the therapy time duration, and min represents the smaller of terms in brackets. Min is put into place so an excessive value of a is avoided and the patient remains unharmed. Now, equation 4 is substituted into equation 2: [Eq 5]. Equation 5 depends on whether or not is smaller. From this equation, the mass goes to zero, full remission, as time goes to infinity. Through Taylor Series' expansion, an estimated rate can be found: . Remission is approached constantly and slowly, as fast as . This remission is “possible because the fittest population is never brought to complete extinction” (6). A complete cure is not obtained by this treatment; however, it might complete a transformation of disease from a metastasizing process. This overall remission may also show the background of other cancers that do not metastasize, such as a form of prostate cancer. The optimal growth curves in the figure below are shown for three levels of activity. All three curves show a full remission by 2.5to years. The lower the activity level is at t/to=0, the slower the branching is made to the lowest curve, so it is more beneficial to begin with a larger dose; however, the lower the dose, the higher the peak value of cancer mass is. So if the dose is too weak, there may be a peak value of mass fatal to the patient. . Although this looks very promising, full remission can not occur in observed growth curves because there is an uncertainty in the onset time of the cancer and there will be a blurred curve Mastectomy with No Further TreatmentIn the case where there includes a mastectomy with no future treatment, the catastrophe theory comes up again because there is an abrupt change in the system, mass, and will follow the following growth law: [Eq 7]. The catastrophe in equation 7 is at time t=0 with time uncertainty, for Gatenby’s studies cited. This shows that the mastectomy has a more profound catastrophe than the drug effect, equation 2, due to the power of k being in the denominator. There is however, a major difference between equations 2 and 7. equation 2 represents cancer mass at time t, whereas equation 7 shows the probability of relapse of cancer, and the unknown medical constants k, b, c in equation 7 make up for the difference. Gatenby’s longterm study showed an occurrence rate of cancer relapse in women who had a mastectomy with no future treatment. Below, the graphs show a maximum at 1.5 yrs, and a minimum at 4 yrs with another max at 5 yrs. The curves should be the same only differing in sampling. The parameters they used were k=1.618, m=.8, b=.9189, and c=2.2663. Clinical Relapse Probability Clinical Relapse Failure Theoretical relapse probability according to catastrophe theory
Brief Discussion of ResultsAfter looking through the results of the studies and the models for therapy, it was found that an effective activity level a must be present to achieve full remission. After finding the maximum value of t=1.5, it was possible to solve for a value a: . If this number had been known before, scientists could have predicted the features of clinical curves representing the figures showing clinical relapse probability and failure. This number defines the effective activity level of the immune system of Italian women having undergone a mastectomy. Conclusion
External Links and References
Mathematical Biology and Medicine 25 (2008): 26783. <http://imammb.oxfordjournals.org/cgi/reprint/25/3/267>.
Breast Cancer Awareness and Support 