Member # 1064
posted 29. June 2004 02:36
DYNAMICS OF ENDOCRINE FEEDBACK
Chinmoy K Bose MD,PhD; Bidyut K Sarkar MS; Sanhita Sarkar MBBS; Banani Bose MBBS,DRP.
"It's an experience like no other experience I can describe, the best thing that can happen to a scientist, realizing that something that's happened in his or her mind exactly corresponds to something that happens in nature. It’s startling every time it occurs. One is surprised that a construct of one's own mind can actually be realized in the honest-to-goodness world out there. A great shock and great, great joy." - Leo Kadanoff.
“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightening travel in a straight line.” –Benoit Mandelbrot
Idea of feedback mechanism in endocrine medicine has remained ill conceived specially in the light of modern control theory where energy transfer is understood in a system wise fashion. The idea is showing reductionism 1 i.e. trying to be discrete molecular biologically whilst visualization is not getting desired accuracy. Text books has even gone with this wind of discreteness, some even drastically cut the subject to bare minimum probably agreeing to the sanctity of its holiness of omnipotence. Roles of end product inhibition, regulatory enzyme, allosteric configuration etc. are described to explain feedback, but confusion in the brain goes round. It is not astonishing that modern control theory with all its mathematical facets has already gained a perceptible entry in our literature 2. What Mr Brincat 3 (1994) thought as philosophical in chaos and found suitable for understanding to gynaecologists and obstetricians was only the beginning. This subject is fast gaining ground and even mathematics, like molecular biology in last decade can not be ignored anymore and be kept in abeyance. The time is coming that a little mathematics, whatever needed for this, is to be learnt and the subject to be introduced now. This is because another paradigm shift has started and is under way like that of molecular biology which has already immensely influenced medicine including gynaecology and obstetrics. In our endocrine prone discipline diagnostic assessment of hypothalamo-pituitary ovarian axis in varied clinical condition sometimes remains insufficient, inefficient and even obscure. Their laboratory results are influenced by their function as well as by the physiological context of our body. Though they have undoubted value in screening axis disorders, debates over different hormonal manipulations go on, like breast cancer and sex hormone, ovarian cancer and hormone, colonic cancer and contraceptive pill use, understanding of different hypo and hypergonadotrophic state 4, to name a few. Hence, introduction of control theory dynamics of endocrine system is an utter necessity. A brief discussion towards this jump or shift could be worthwhile and rewarding as well.
Feedback mechanism is essentially a physical term, not biochemical, as is evident, and Uvarov and Chapman 5 (1974) understands the theme in 1974 which describes this as " In general, (It is) the coupling of the output of a process to the input. In negative feedback a rise in the output energy is arranged to cause a decrease in the input energy, as for example, a governor. In positive feedback a rise in the output energy is caused to reinforce the input energy. In particular these terms are applied to electronic amplifiers, in which a portion of the output energy is used to reduce or increase the amplification, by reacting on an earlier stage according the relative phase of the return ". Since this term feedback being used in biology of endocrine system it makes us understand the biochemical phenomenon in two points of two endocrine glands or between gland and target tissue in a homeostasis of our internal milieu which is, so to say, more or less a thermodynamically stable system. But if we consider it as information processing in biological system, phase will mean " a separate part of a heterogeneous body or system as for example a mixture of ice and water is a two phase system while a solution of salt in water is a system of one phase 5”. Feedback can be closed loop like in electric circuit or open loop like environmental biodiversity. Thus in biology it is an open loop system with not only an input and output but also an array of internal variable influencing the outcome.
In case of endocrine glands until now 2,6 at least in thyroid gland milieu TSH patterns were mainly classified comparatively on simple measures like amplitude and rate. This rate of change is data producing and can be appreciated in a statistical system in our hoemeostatis by understanding differential equation i.e. calculus. Only problem here is that the relationship cannot be as simple as one to one, one to two or three or even four relations, we can hardly imagine range of relationships causing a network leading to a successful chaotic system having its influence in any organ system like hypothalamo-pituitary-ovarian axis in gynaecology and obstetrics. Chaos theory is however discussed in general here before 3 . Visual, tactile, auditory and complex infinite numbers of metabolic phenomena influence this relation. Infinite time integration of those differential equations of feedback control has become possible by computer only. A small jeopardy in some unnoticeable corner in this homeostatis in one such differential equation will have a butterfly and is capable to give rise to wild oscillation explaining unexplained fallout of disease and treatment including unexplained death. Computer assisted integration of even a few equations assuming other data 2 as much as possible will generate different fractal dimensions and graphic and attracter. We will use those in our known statistical system to evaluate different situations and act likewise. Instead of being single input single output (SISO) it becomes multi output multi input (MIMO) system. Hence, differential linear relation of steady state must break. Frequency domain estimation of simple linear calculation fails and nonlinear calculation in time domain becomes necessary. Nonlinearity and chaos emerges with this idea of time domain. Time domain includes time discrete and time continuous (time series} calculation. Time series may occur in state space with countable variables like in engines and in phase space where it is impossible to find out all variables. Our body is a phase space.
However, the value of resultant level of hormone in blood in no way can be underestimated because we have to see it as an outcome of this very complex system. There are scientists who are calculating this feedback in a time discrete multi steady state system and they are not less interesting 7. They are more important in cellular and subcellular mechanism, particularly genomic automanipulations. The Discrete Time (DT) domain is a timed extension of the Synchronous Dataflow (SDF) domain. Although not completely backward compatible with SDF, DT keeps most of the desirable properties of SDF-like static scheduling, regular/periodic execution, bounded memory usage, and a guarantee that deadlock will never occur. In addition, DT has some desirable temporal properties such as uniformly-timed token flow and causality 8.
In this time domain, continuous state space model helps in protein chemistry 9,10 . But to understand why our dynamical, complex world, our lives, our whether and our experience never repeat but follow a pattern we have to move to nonlinear multivariate model in time series where differential interrelations are repeated (iterated) producing specific patterns in a phase space. It is also like a shape or a system which does never repeat but follow a pattern with different repeating shades and textures like trees, mountains and clouds 11. This ‘self similar’ feature of a dynamic system having non-integer dimension are described as fractal. Whilst the topological dimension of a line is always 1 and that of a surface always 2, the fractal dimension may be any real number between 1 and 2. The fractal dimension D is defined by
log ( L2 / L1 )
D = ---------------
log ( S1 / S2 )
where L1, L2 are the measured lengths of the curves (in units), and S1, S2 are the sizes of the units (i.e. the scales) used in the measurements. Like distance here, change of any vector or its topology like dynamic chemical reaction may give rise to dimension unique to that system. Several other dimensions like capacity dimension, correlation dimensions are described according to the need of the subject or problem. It gives insight to understanding of many biological structure and function. They will have specific dimensions peculiar to a particular system. This direction of understanding which needs computer to be developed and which we may call "more sophisticated" approaches is using methods from nonlinear systems to measure the complexities of the signal patterns. It has been applied to many endocrine control systems, e.g. to the release of PTH in the calcium phosphate homeostasis 12, Growth hormone, prolactin 13, to detect early ovarian cancer based on very low dimensionality adaptive texture feature vectors from cell nuclei from monolayers and histological sections 14. Gough has used fractal dimension in foetal heart rate variability15 . Cerebellar study contains fractal based analysis 16.
We can now take up with mathematical modern control.
The Birth of Mathematical Control Theory
The design of feedback control systems up through the Industrial Revolution was by trial-and-error mixed with a great deal of engineering intuition. Thus, it was more of an art than a science. The stability of feedback control systems was analyzed by using mathematics in the mid 1800’s. Since automatic control theory has formal language in mathematics only; period before this time is the prehistory of control theory. In 1840, the British Astronomer Royal at Greenwich, G.B. Airy, developed a feedback device for pointing a telescope. His device was a speed control system, which turned the telescope automatically to compensate for the earth's rotation, affording the ability to study a given star for an extended time. Unfortunately, Airy discovered that by improper design of the feedback Control loop, wild oscillations were introduced into the system. He was the first to discuss the instability of closed-loop systems, and the first to use differential equations in their analysis 17. The theory of differential equations was by then well developed, due to the discovery of the infinitesimal calculus by I. Newton (1642-1727) and G.W. Leibniz (1646-1716), and the work of the brothers Bernoulli (late 1600's and early 1700's), J.F. Riccati (1676-1754), and others. The use of differential equations in analyzing the motion of dynamical systems was established by J.L. Lagrange (1736-1813) and W.R. Hamilton (1805-1865).
Modern controls design is fundamentally a time-domain technique. An exact state-space model of the system to be controlled is required. This is a first-order vector differential equation of the form
dx/dt = Ax + Bu
y = Cx
Where x(t) is a vector of internal variables or system states where infinitesimal data is possible in a biosystem, u(t) is a vector of control inputs, and y(t) is a vector of measured outputs (cybernetic systems).
Our body environ is a place where we cannot think of a closed loop. Everything fluid and solid creates biochemical phases and energy transfer is in open loop system in sub pituitary endocrine system. Closed loop being basic is explained here, but of course anyone interested will start from basic physical here to reach biosystem where number of equations to be understood increase enormously. This fails multivariate estimations18 and chaos with resultant fractal based statistics intervenes. It helps us with providing different dimensions or constant to realize difference of two situations numerically. The power of modern control has its roots in the fact that the state-space model can as well represent a MIMO system as a SISO system. That is, u(t) and y(t) are generally vectors whose entries are the individual scalar inputs and outputs. Thus, A, B, C are matrices whose elements describe the system’s dynamical interconnections.
Modern controls techniques were first firmly established for linear systems.
Extensions to nonlinear systems can be made using the Lyapunov 19 (1907) approach, which extends easily to MIMO systems, dynamic programming, and other techniques. Open-loop optimal controls designs can be determined for nonlinear systems by solving nonlinear two-point boundary-value problems.
To achieve suitable closed-loop properties, a feedback control of the form
u = -Kx may be used. The feedback gain K is a matrix whose elements are the individual control gains in the system. Since all the states are used for feedback, this is called state-variable feedback. Multiple feedback gains and large systems are easily handled in this framework. Thus, if there are n state components (where n can be very large in an aerospace or power distribution system) and m scalar controls, so that u(t) is an m-vector, then K is an mxn matrix with mn entries, corresponding to mn control loops. Further deduction (function of function of function etc.) takes us to the end of modern physical control and by repeating (iterating) function infinitesimally we get fractal dimensions and graphics. Obviously this attempt of ours is only to create interest in fractal based calculation for interested persons.
We can demodulate the above theme very easily in nonlinear area from a little absolute perspective. It’s also a link between input and output. By the quality of working link at that moment, i.e. whether serial, parallel, negative feedback, positive feedback, behaviour of the system is determined at random. Here, the control is, of course, nonlinear (Poincaré). Comparing with linear system, here, we can’t get a cross-section of the system at our will, having some values to work with; rather we have to have a picturesque overview of the whole system (in phase space – of dimension depending on number of variables controlling) as a categorical object for qualitative study to understand the dynamism, as like the study of other chaotic system 20. It is comparable with the study of a system from the dynamic shape of its shadow (Homomorphism). This is practically feasible in case of a thermodynamically stable system like the biological system within us. Like every physical system it is undergoing entropy, perpetuating disorderly state. It is obvious also because in our probabilistic existence the chances of getting disorderly states completely outweigh the chance of encountering orderly state in our future. What in our biological system is responsible to give feedback, go opposite to the direction of thermodynamical time i.e. direction of entropy. Information theory says us that only our complex biological pattern (relics of investment of energy) is the basis of that order which appears to negates the entropy (called negentropy) and eventuates, what we call, feedback.
Calculation of endocrine function in system of chaos and fractal will generate new dimension in nonlinear system like fractal dimension, in single or multiple time series giving us scope to differentiate and evaluate self similar features or Poincaré c section indicating fractal attractor. But it has to sustain test of time to come out as an efficacious method. Other processes like time discrete, state space model, Markov chain, Monte Carlo path (Korn et al 1988), are already being tried alongside. Markov chain Monte Carlo statistics are more used in assessing cost effectiveness and relative efficacy of different hospitals 21(CabasÃ©s Hita 2000) where as random walk technique is used in researches on gait 22, cerebellar function, ophthalmology 23 etc. But, undoubtedly, fractal based statistical calculations are more modern and very much useful in indicated cases.
Previous behaviourally isomorphic way using different classes of equations (linear, logarithmic, exponential or polynomial) though deliver probable ways in which the system might be realized, this approach also exposes the models to charges of being arbitrary 24,25. Neuroendocrinal hierarchy of pituitary thyroid axis has already disproved theory of pulsatility of input of TRH at the pituitary yielding corresponding TSH pulse 26 where as mechanism of causing the fast oscillations is also unknown.
Pituitary ovarian axis as yet unexplained by this new system, attracts concentration. People will speculate how accurate it will be to have the glimpses of this complex calculation and some will doubt its efficacy. However trial run with many different systems with this new mathematics serving as statistics at hand will go on unhindered revealing more accurate picture with more approximate differential reach. It will eventually endow human hands with more power and edge leading to lesser sufferings of the humanity; we hope.
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