*Mathematical Modelling, *Vol. 7, pp. 1513-1577,1986

**IN BIOLOGY AND MEDICINE**

Ion C. Baianu

Physical
Chemistry and NMR Laboratories

567
Bevier Hall,

.Urbana,
UIinois 61801

*(Received *12 *September *1985)

**1.
INTRODUCTION**

The applications of
computers to biological and biomedical problem solving goes back to the very
beginnings of computer science, automata theory [1], and mathematical biology
[2]. With the advent of more versatile and powerful computers, biological and
biomedical applications of computers have proliferated so rapidly that it would
be virtually impossible to compile a comprehensive review of all developments
in this field. Limitations of computer
simulations in biology have also come under close scrutiny, and claims have
been made that biological systems have limited information processing power
[3]. Such general conjectures do not, however, deter biologists and biomedical
researchers from developing new computer applications in biology and medicine. Microprocessors
are being widely employed in biological laboratories both for automatic data
acquisition/processing and modeling; one particular area, which is of great
biomedical interest, involves fast digital image processing and is already
established for routine clinical examinations in radiological and nuclear
medicine centers, Powerful techniques for biological research are routinely
employing dedicated, on-line microprocessors or array processors; among such
techniques are: Fourier-transform nuclear magnetic resonance (NMR), NMR imaging
(or tomography), x-ray tomography, x-ray diffraction, high performance liquid
chromatography, differential scanning calorimetry and
mass spectrometry. Networking of laboratory microprocessors linked to a
central, large memory computer is the next logical step in laboratory
automation. Previously unapproachable problems, such as molecular dynamics of
solutions, many-body interaction calculations and statistical mechanics of
biological processes are all likely to benefit from the increasing access to
the new generation of "supercomputers". In view of the large number,
diversity and complexity of computer applications in biology and medicine, we
could not review in any degree of detail all computer applications* *in these fields; instead, we shall be
selective and focus our discussion on suggestive computer models of biological
systems and those fundamental aspects of computer applications that are likely
to continue to make an impact on biological and biomedical research. Thus, we
shall consider unifying trends in mathematics, mathematical logics and computer
science that are relevant to computer modeling of biological and biomedical
systems. The latter are pitched at a more formal, abstract level than the
applications and, therefore, encompass a number of concepts drawn from the
abstract theory of sets and relations, network theory, automata theory, Boolean
and *n*-valued logics, abstract
algebra, topology and category theory. The purpose of these theoretical' sections is to provide the ans for
approaching a number of basic biological questions:

(1) What are the
essential characteristics of a biological organism as opposed to an automaton?

(2) Are
biological systems recursively computable?

(3)
What is the structure of the simplest (primordial) organism?

(4)
What are the basic structures of neural and genetic networks?

(5)
What are the common properties of classes of biological organisms? (6) Which
system representations are adequate for biodynamics?

(7)
What is the optimal strategy for modifying an organism through genetic
engineering? (8) What is the optimal simulation of a biological system with a
digital or analog computer?

(9)
What is life?

The
present analysis of relational theories in biology and computer simulation has
also inspired a number of new results which are presented here as
"Conjectures" since their proofs are too lengthy and too technical to
be included in this review. In order to maintain a self-contained
presentation-the definitions of the main concepts are given, with the exception
of a minimum of simple mathematical concepts.

**2. COMPUTER MODELS OF
BRANCHING PROCESSES AND TREELIKE MORPHOLOGY **

One
of the simplest but nontrivial applications of computers in biology and
medicine has been the generation of "trees" or patterns of branching.
Such patterns of branching are common to arteries, bronchi, trees and rivers,
and have attracted considerable attention[4-22].
Computer simulation of the geometry of trees, based on branching angles, length
ratio of branches and differential rates of growth, has been quite successful
introducing models which are closely resembling the morphology of biological
systems[7, -19]. In such models of trees, the branching ratio was found to be
variable and, therefore, of little descriptive value. A computer program that
generates dichotomously various branching trees was recently described[22]
and it was employed to investigate if the human bronchial tree could be
adequately modelled.

*Generation of trees by
the computer *

According to Horsfield and Thurlbeck[22], each branch is encoded in the computer by providing
the three-dimensional (3D) coordinates of the branch ends. Horsfield
and Cuming[5] order the branches by starting at the
peripheral ones, which are assigned "order “ and the order is increased by
1 unit at each junction [Fig. l(a)] after Horsfield
and lurlbeck[22]). The asymmetry of the branching is
represented by an asymmetry parameter (3) which is the difference in order
between the two daughter branches. An example asymmetry of branching which was
given by Horsfield and Thurlbeck
is reproduced Fig. l(b). A stem branch is generated by
inputing its coordinates and stating its Horsfield der; the stem
bifurcates in the x-y plane, the order of the major daughter branch being le
digit less than the parent branch by definition, while the order of the minor branch
defined by using a value of 3. By defining the angles of branching and the
lengths, the coordinates of the ends of the daughter branches can also be
calculated. The daughter branches bifurcate in turn until an order-1 branch is
generated recursively, and then bifurcation stops on that selected pathway. The
value of 8 for a given bifurcation is determined by a pseudorandom number
generated by a digital computer, and takes values between 0 and 9. The
probability for a given value of 8 to be realized in a given tree from e
pseudorandom string of numbers is defined on input; for example if 8 = 0, the
probability is zero.

__3. Computer Models of Neural
Networks.__

An
extensive review of neural networks with approximately 100 references up to
1986 is presented summarizing the results reported to be relevant to basic brain
control functions. Alternate approaches based on an enzymatic network in single
functional neurons by M. Conrad were also reviewed in detail, and were later considered
by other authors to lead to tthe possibility of
quantum processing and the emergence of consciousness.

__4.
COMPUTER MODELS OF CARCINOGENESIS AND CANCER CHEMOTHERAPY __

Computer
simulation studies of carcinogenesis are closely related to theoretical studies
of the cell cycle, the control of cell division and the growth of cell
populations [43-52].In a computer model of erythroleukemia,
Düchting[51] considered a control process ( cell
proliferation of the form shown in Fig. 8 (also see Fig. 9). The simulation of
this process was performed on an AEG- Telefunken
TR440 digital computer using an ASIM computer program. This program is written
in the block-oriented language for Analogous SIMulation.
The digital logic device in this model ascertains and registers the presence of
each cell in a specific compartment; the analog transfer elements were
integrators and switching components. The model is therefore a combination of
analog and digital devices, and the simulation process is in this case more
complex than in the more popular, digital-only models. This model mimicked
malignancy through an uncontrollable increase in compartment population, but as
many other computer models of carcinogenesis, is limited by the lack of a
detailed, experimental analysis of the parameters controlling carcinogenesis.
An attempt to introduce such parameters into a model of malignant
"stem" cell growth was recently made by Rittgen[53]. Rittgen's basic model where G1, S, G2, *M, Q1 *and *Q2 *are cell
cycle phases; Q1 and Q2 are the resting phases, while S is the synthesis phase.
Mitosis starts either after G2 or after *Q2,
*and the daughter cells begin in the resting phase *Q*1. The simulation was executed with a special stochastic system
[54]. With this model, it was possible to calculate the number of malignant
proliferating, maturing and mature cells, as a function of time. The simulated
malignant cell population growth was exponential, with growth velocities
depending on the cell cycle parameters.

__Computer Models and
Cancer Chemotherapy.__

A model
conceptually similar to the Rittgen simulation, but
simpler, was applied to the analysis of cancer chemotherapy by Chuang and Soong [55]). A FORTRAN
IV program was developed for a PDP 15!76 computer which was employed for
simulations of scheduled chemical treatments with cell-cycle specific, phase
specific and cycle non- specific drugs. It also allowed for Gompertzian
tumor growth and variation of kinetic parameters in relation to tumor size.
Typical simulated curves of synchronization and thymidine
blocking effects in cancer chemotherapy are discussed in Ref. [55]. Complications,
not considered in this model, can arise in cancer chemotherapy due to the fact
that tumor cells can begin to divide parasynchronously
following interruption of the treatment. The agreement between this model and
the two experimental animal tumor systems, L1210 leukemia and Lewis lung
carcinoma, cannot yet be considered as conclusive because of the paucity of
experimental data available. In an
interesting report by Swan and Vincent[56], the problem
of minimizing the total amount of cycle nonspecific cytotoxic
drug in the body of the patient was investigated. Their solution was in terms
of optimal control theory and their theoretical results were compared with
clinical data stored in a computer at the

__A
microprocessor model of perturbed cell renewal __

Duchting[62]
re-approached the problem of computer simulation of carcinogenesis a the more
basic level of perturbed cell renewal by considering the interactions between adjacent
cells on a two-dimensional grid. Such questions were also considered previously
by Gardner[63], Lindenmayer[64], Reshodko
and Bures[65], Ransom[66] and Arbib[67] The approach
is close to what Arbib describes as a "tessellation" model, and
involve~ basic concepts from automata theory (see also Section 5). Duchting's simulation of disturbed cell renewal [62] was
carried out by means of an Intel 8080 microprocessor and we expect that his
model could also be programmed on the now popular IBM PC/ ATT microprocessors. The organization of the programs run by the
Intel 8080 for this simulation was produced by Duchting[62]. This
simulation yielded some interesting results, such a' the onset of metastasis
after "surgery" even if only one "malignant" cell is left
amongst the "normal" cells of the grid (Fig. 6 in Ref. [62]); in the
case of no surgery , the mode predicts that normal cells would eliminate the
few malignant cells present. Related to this tessellation approach to population
growth, Lieberman considered in an earlier report [68] a stochastic model in
which the population distribution is confined within a limited space. The
simulation was carried out with an IBM Model 360 and showed that the size and
abundance of organisms are linked by a logarithmic relationship if the
organisms are limited by a single resource. It would be interesting to adapt
this model to the study of tumor growth, under conditions of limited nutrient
supply since the tumor cell proliferation is strictly dependent upon the local
availability of nutrients supplied by tumor vessels[69].
The tumor vascularization itself is, however, induced
by the elaboration of a tumor antigenic factor (T AV by the tumor cells[70]. In a detailed model of tumor growth, Liotta *et al.[71] *considered both vascularization
and necrosis of tumors by taking into account both diffusion an( proliferation
of tumor cells. Coupled diffusion equations with a nonlinear source and sine
terms described the proliferation, migration and necrosis of tumor cells.
According to Liotta *et al.[71], *their diffusion model is superior to lumped parameter
models of tumor growth such as that .of Saidel *et al.[72] *because "the
lumped-parameter simulation doe: not yield any information about the spatial
distribution of the tumor cells and vessels in the tumor. The results of the diffusion model are
qualitatively similar to those determined by the experiment (Figs. I and 3,
respectively, in Ref. [71]). One major limitation of this diffusion model of
tumor growth is that the tumor was assumed to be spherically sym metric. Other
limitations of the model are discussed in Ref. [71].

__5. AUTOMATA THEORY AND COMPUTABLE
MODELS OF BIOLOGICAL SYSTEMS__

The
formal theory of automata or sequential machines is considered in the context
of network models of biological systems. The collection of discrete automata semigroups is organized as an abstract category whose
algebraic, universal properties have been determined and that presents realizability problems ressembling
those of the simplest biomathematical network models

__6. GENERAL COMPUTABILITY
QUESTION FOR BIODYNAMICS, NEUROSCIENCES AND COGNITIVE-RELATED FIELDS__

__Conjecture:__

Generalized,
algebraic-symbolic computations of biodynamics may
become possible with a topological semigroup ‘computers’(Baianu, 1971a, b), such as a quantum computer.

On
the other hand, existing digital computers are known to be limited in their
ability to compute complex biodynamics such as t cell network dynamics.

**7.
**__Łukasiewicz____ Algebraic Logic Networks of Genomes__

A
detailed review of both Boolean and Łukasiewicz
logic networks of the genome, or genetic network models are presented with a view
to future applications such as the dynamic applications to the human genome
analysis. Related spin offs may occur in n-state models of non-random,
nonlinear neural networks by modeling cognitive systems in categories of Łukasiewicz Logic Algebras.

__8. (M,R)-Systems
Models and the simplest Metabolic-Repair-Replication Models__

__ __

Generalizations of Robert Rosen’s (M,R)-system
models are discussed in terms of general categories whose objects are not
restricted to sets, by endowing such objects with algebraic and topological
structures as in the theory of organismic supercategories (Baianu, 1970; 1971; 1973, 1974; 1980;
1983; 1985). Further extensions of (M,R)-systems to self-replication
and reverse-transcription are also constructed and their categorical-algebraic
properties are derived.

__CONCLUSION____S__

Several
answers provided to the questions posed in the introduction are summarized and
conclusions are drawn concerning

the future directions of computer modeling and automata theory
in biology and medicine, such as the nature of the diagnostic, cognitive processes
currently employed in medicine that could benefit from the Luksiewicz
Logic Algebraic models developed in the context of non-random, nonlinear Genetic
Networks. The computability Conjecture for Biodynamic Models and networks is
again stated in the broader cognitive context of the medical sciences that will
increasingly depend on automatic

processes and computations for data analysis and
diagnostics.

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__Applications
of the Theory of Categories, Functors and Natural
Transformations, N-categories, (Abelian or otherwise)
to:__

*Cognitive Systems Automata
Theory/ Sequential Machines, Bioinformatics, Complex Biological Systems
/Complex Systems Biology, Computer Simulations and Modeling, Dynamical Systems
, Quantum Dynamics, Quantum Field Theory, Quantum Groups,Topological
Quantum Field Theory (TQFT), Quantum Automata, Graph Transformations, Logic,
Mathematical Modeling, etc.
*

2. Rosen, Robert. 1964. Abstract Biological Systems as Sequential Machines, Bull. Math. Biophys., 26: 103-111; 239-246; 27:11-14;28:141-148.

3. Arbib, M. 1966. Categories of (M,R)-Systems. Bull. Math. Biophys., 28: 511-517.

4. Cazanescu, D. 1967. On the Category of Abstract Sequential Machines.

5. Rosen, Robert. 1968. On Analogous Systems. Bull. Math. Biophys.,

6. Baianu, I.C. and Marinescu, M. 1968. Organismic Supercategories:I. Proposals for a General Unitary Theory of Systems. Bull. Math. Biophys., 30: 625-635.

7. Comorozan,S. and Baianu, I.C. 1969. Abstract Representations of Biological Systems in Supercategories. Bull. Math. Biophys., 31: 59-71.

8. Baianu,

9. Baianu,

10. Baianu,

11. Baianu,

12. Rosen, Robert. 1973. On the Dynamical realization of (M,R)-Systems. Bull. Math. Biology., 35:1-10.

13. Baianu,

14. Baianu,

15. Baianu, I.C. 1977. A Logical Model of Genetic Activities in Lukasiewicz Algebras: The Non-Linear Theory., Bull. Math. Biol.,39:249-258.

16. Baianu, I.C. 1980. Natural Transformations of Organismic Structures. Bull.Math. Biology, 42:431-446.

17. Warner, M. 1982. Representations of (M,R)-Systems by Categories of Automata., Bull. Math. Biol., 44:661-668.

18. Baianu, I.C.1983. Natural Transformations Models in Molecular Biology. SIAM Natl. Meeting, Denver, CO,

19. Baianu, I.C. 1984. A Molecular-Set-Variable Model of Structural and Regulatory Activities in Metabolic and Genetic Systems., Fed. Proc. Amer. Soc. Experim. Biol. 43:917.

19. Baianu, I.C. 1987. Computer Models and Automata Theory in Biology and Medicine. In: "Mathematical models in Medicine.",vol.7., M. Witten, Ed., Pergamon Press: