September 25, 2017, Monday
University of Colorado at Boulder Search A to Z Campus Map University of Colorado at Boulder CU 
Search Links


MBW:Inducing Catastrophe in Malignant Growth

From MathBio

Jump to: navigation, search

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.

Overview

  • Mathematics used: Cosine-modulated power law, probability density function, and a growth law.
  • Biological system studied: A cancer mass is depicted as a system of power laws that can predict breast cancer growth during any time dependent therapeutic activity a(t).

Executive Summary

Biological 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

  • Catastrophe theory is a branch of mathematical topology and a division of bifurcation theory, which is the study of change in qualitative structure, and is applied in the study of chaos and nonlinear dynamics. Bifurcation theory “classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analyzing how the qualitative nature of equation solutions depend on the parameters” (2). This can create impulsive and abrupt changes. The tiniest of changes in a parameter can create a big difference. For example, equilibria could appear/disappear and even change from attracting to repelling; however, when the parameter space is large, the bifurcation point becomes well-defined.
  • The catastrophe theory was derived from the mathematical work of René Thom. Thom in the 1960’s. In the 70’s, Christopher Zeeman. Zeeman studied the Lyapunov function, in which the long-run stable equilibrium could be linked to the minimum of a continuous, potential function.
  • Nonlinear systems change suddenly in a limited amount of ways known as elementary catastrophes. The theory evaluates what happens with degenerate critical points of the potential function, which is where points of higher derivation equal zero, which are known as germs. The deteriorating critical points are analyzed through expansion of the potential function as a Taylor’s series. When the a degenerate point is structurally stable, any function or flow close to it follows the same dynamics and will not change under small perturbations, it acts as a centre for geometric structures of lower degeneracy. When the potential function is dependent on two or less variables, and four or less parameters, the Taylor expansion around the catastrophe germs go through diffeomorphism (smooth transformation/smooth inverse) into seven fundamental types of catastrophe.

The Seven Fundamental Types of Catastrophe

  • Fold catastrophe is a catastrophe that arises when there is one control factor and one behavior axis. When a is negative, there are two extremas, one stable, one unstable. When a increases, the stable point is followed. If a fold bifurcation is followed by a physical system, as it reaches zero, the stability of the one point is lost and a new behavior arises. This new bifurcation value of parameter a can be called the tipping point.
  • When there are two parameters, a and b, there becomes a cusp catastrophe. Instead of a parabolic shape, there is now a curve where stability is lost and the stable solution will jump to another outcome; however, the bifurcation curve will loop back onto itself which creates an additional branch where the other solution loses stability and jumps back to the original set. By varying the parameter b, hysteresis loops are observed, which is when a system follows one solution, jumps to the alternative solution, jumps back to the original and follows it back, and then jumps again back to the other. Hysteresis only occurs when a is negative, and as a increases, the loops tend to get smaller and disappear at a=0, creating one stable solution. If, on the other hand, you hold b constant and vary a, where b=0, a pitchfork bifurcation is observed. As a decreases, the stable solution splits into two stable and one unstable solutions.
  • The swallowtail catastrophe allows a three dimensional parameter space, and is made up of a bifurcation set of three surfaces of fold bifurcations. As the fold bifurcations are followed by the parameters, one minimum and one maximum of the potential function disappear. At the cusp, two minimums and one maximum are replaced by one minimum. The swallowtail point shows two minima and two maxima that meet at a sole value of x.
  • The butterfly catastrophe has a potential function that may have three or less different local minima’s separated by the loci of fold bifurcations. The three surfaces of fold bifurcation, two surfaces of cusp bifurcation, and the lines of swallowtail bifurcations meet and disappear. There is a single cusp lasting when a is positive.
  • The hyperbolic umbilic catastrophe can occur for three control factors and two behavior axes. The codimension of the equation is three and therefore has three unfolding parameters.

Background of Gatenby and Frieden's study

Living systems, most importantly, biological systems, are non-linear. 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 time-dependent 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 cosine-modulated 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 time-varying 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 Methods

The 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 Model

According to Gatenby and Frieden's research, untreated breast cancer will grow as the following power law: p(t)_{{0<t<T}}=Ct^{\phi } where C=(\phi +1)T^{{-\phi +1}} [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, \phi =1.618, 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:

p(t)_{{t_{0}<t<T}}=A(t/t_{0})^{\phi }cos^{2}(a(t)ln(t/t_{0}) [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, A(t/t_{0})^{\phi }, attained by the maximum tumor growth, as well as having a limitation to growth

cos2(a(t)ln(t/t_{0})).

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.

Zeroes=t_{0}e^{{(n+(1/2))(\pi /2)}}

Maxima=t_{0}e^{{(n\pi )/a)}},n=0,1,2

The zeroes show cancer remissions and maxima the relapses of the cancer

Constant Therapeutic Activity

For 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: Picture1.jpg. As the curves diverge, the free-growth, untreated cancer increases asymptotically, whereas the therapeutic oscillation shows that even though the tumors stop growing, the growth starts again, recurring in wave-like 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

Time-Varying Therapeutic Schedule

  • After the simple case, Gatenby looked at the case when there was a time-varying therapy schedule:

a(x)_{{1<x<X_{0}}}=(\pi /2)min[a_{0},(ln(x+\Delta x))^{{-1}},

where x=t/t_{0},\Delta x=constant [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:

p(x)_{{1<x<X_{0}}}=Ax^{{\phi }}cos^{2}[(\pi /2)(ln(x)/ln(x+\Delta x))]or=Ax^{{\phi }}cos^{2}[(\pi /2)a_{0}ln(x)] [Eq 5].

Equation 5 depends on whether or not [ln(x+\Delta x)]^{{-1}}ora_{0} 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:

p(x)=lim_{{xinfinity}}((A\pi ^{2})/4)x^{{\phi }}(\Delta x/xlnx)^{2}=x^{{\phi -2}}=x^{{-0.382}}.

Remission is approached constantly and slowly, as fast as x^{{-0.382}}. 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. Picture2.jpg. 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 Treatment

In 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:

p(t)=|cos(b-cln(mt))|/(mt+\Delta t)^{k},k>0 [Eq 7].

The catastrophe in equation 7 is at time t=0 with time uncertainty, \Delta t=0.5years 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 long-term 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.

Picture3.jpg Clinical Relapse Probability Picture4.jpg Clinical Relapse Failure Picture5.jpg Theoretical relapse probability according to catastrophe theory

  • As you can see, the theoretical relapse plot agrees well with the study.

Brief Discussion of Results

After 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: e^{{\pi /2}}=m_{1}/t_{0}=1.5/0.5=3.0,a=\pi /ln3=2.8596. 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

  • Through the conducted study, it is shown that the effect therapy has on tumor growth can be depicted with the catastrophe theory. This theory defines a system before or after an abrupt change in the system. It is signaled by a sudden increase or decrease followed by oscillation. The oscillation represents the recurrence and remission of cancer cells.
  • The models in equations 2 and 7, and the figure depicting no treatment, show that conventional cancer therapy will characteristically fail “due to a combination of destruction of the dominant population and the resulting selection for resisting phenotypes” (6).
  • The therapy works at first, but when the cells have a chance to metastasize, the mutated genes will multiply; however, when the treatment is given at different levels, the population of sensitive genes will have a minimum presence and continually limit growth of therapy resistant strains.
  • In the end, it was suggested that the catastrophe theory can also be related to other types of cancer, such as prostate cancer, or any other type that does not metastasize. It is obvious that mathematics is now becoming an ever-growing aid in the growth and development of biology.

External Links and References

Mathematical Biology and Medicine 25 (2008): 267-83. <http://imammb.oxfordjournals.org/cgi/reprint/25/3/267>.

Breast Cancer Awareness and Support