mmf1din2012

BULETINUL INSTITUTULUI POLITEHNIC DIN IAI

Tomul LVIII (LXII)

Fasc. 1

MATEMATIC. MECANIC TEORETIC. FIZIC

2012 Editura POLITEHNIUM

BULETINUL INSTITUTULUI POLITEHNIC DIN IAI PUBLISHED BY

GHEORGHE ASACHI TECHNICAL UNIVERSITY OF IAI Editorial Office: Bd. D. Mangeron 63, 700050, Iai, ROMANIA

Tel. 40-232-278683; Fax: 40-232-237666; e-mail: [email protected]

Editorial Board

President: Prof. dr. eng. Ion Giurma, Member of the Academy of Agricultural Sciences and Forest, Rector of the Gheorghe Asachi Technical University of Iai

Editor-in-Chief: Prof. dr. eng. Carmen Teodosiu, Vice-Rector of the Gheorghe Asachi Technical University of Iai

Honorary Editors of the Bulletin: Prof. dr. eng. Alfred Braier, Prof. dr. eng. Hugo Rosman,

Prof. dr. eng. Mihail Voicu Corresponding Member of the Romanian Academy, President of the Gheorghe Asachi Technical University of Iai

Editors in Chief of the MATHEMATICS. THEORETICHAL MECHANICS. PHYSICS Section

Prof. dr. phys. Maricel Agop, Prof. dr. math. Narcisa Apreutesei-Dumitriu, Prof. dr. eng. Radu Ibnescu

Honorary Editors: Prof. dr. eng. Ioan Bogdan, Prof. dr. eng. Gheorghe Nag

Associated Editor: Associate Prof. dr. phys. Petru Edward Nica

Editorial Advisory Board

Prof.dr.math. Sergiu Aizicovici, University Ohio, U.S.A.

Assoc.prof.dr.phys. Liviu Leontie, Al. I. Cuza University, Iai

Assoc. prof. mat. Constantin Bcu, Unversity Delaware, Newark, Delaware, U.S.A.

Prof.dr.mat. Rodica Luca-Tudorache, Gheorghe Asachi Technical University of Iai

Prof.dr.phys. Masud Caichian, University of Helsinki, Finland

Acad.prof.dr.math. Radu Miron, Al. I. Cuza University of Iai

Prof.dr.eng. Daniel Condurache, Gheorghe Asachi Technical University of Iai

Prof.dr.phys. Viorel-Puiu Pun, University Politehnica of Bucureti

Assoc.prof.dr.phys. Dorin Condurache, Gheorghe Asachi Technical University of Iai

Assoc.prof.dr.mat. Lucia Pletea, Gheorghe Asachi Technical University of Iai

Prof.dr.math. Adrian Cordunenu, Gheorghe Asachi Technical University of Iai

Assoc.prof.dr.mat.Constantin Popovici,Gheorghe Asachi Technical University of Iai

Prof.em.dr.math. Constantin Corduneanu, University of Texas, Arlington, USA.

Prof.dr.phys.Themistocles Rassias, University of Athens, Greece

Prof.dr.math. Piergiulio Corsini, University of Udine, Italy

Prof.dr.mat. Behzad Djafari Rouhani, University of Texas at El Paso, USA

Prof.dr.math. Sever Dragomir, University Victoria, of Melbourne, Australia

Assoc.prof.dr. Phys. Cristina Stan, University Politehnica of Bucureti

Prof.dr.math. Constantin Fetecu, Gheorghe Asachi Technical University of Iai

Prof.dr.mat. Wenchang Tan, University Peking Beijing, China

Assoc.prof.dr.phys. Cristi Foca, University of Lille, France

Acad.prof.dr.eng. Petre P. Teodorescu, University of Bucureti

Acad.prof.dr.math. Tasawar Hayat, University Quaid-i-Azam of Islamabad, Pakistan

Prof.dr.mat. Anca Tureanu, University of Helsinki, Finland

Prof.dr.phys. Pavlos Ioannou, University of Athens, Greece

Prof.dr.phys. Dodu Ursu, Gheorghe Asachi Technical University of Iai

Prof.dr.eng. Nicolae Irimiciuc, Gheorghe Asachi Technical University of Iai

Dr.mat. Vitaly Volpert, CNRS, University Claude Bernard, Lyon, France

Assoc.prof.dr.math. Bogdan Kazmierczak, Inst. of Fundamental Research, Warshaw, Poland

Prof.dr.phys. Gheorghe Zet, Gheorghe Asachi Technical University of Iai

BULETINUL INSTITUTULUI POLITEHNIC DIN IAI BULLETIN OF THE POLYTECHNIC INSTITUTE OF IAI Tomul LVIII (LXII), Fasc. 1 2012

MATEMATIC. MECANIC TEORETIC. FIZIC

Pag.

MUGUR B. RU, Asupra unei teorii newtoniene alternative la teoria MOND(engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

CLIN GHEORGHE BUZEA, MARICEL AGOP i SIMONA RUXANDRA VOLOV, Un nou model al carcinogenezei i progresiei tumorale n cadrul dinamicii neliniare (I) (engl., rez. rom.). . . . . . . . . . . . . . .. . . . . .

9

DRAGO IANCU, CLIN GHEORGHE BUZEA i MARICEL AGOP, Un nou model al carcinogenezei i progresiei tumorale n cadrul dinamicii neliniare (II) (engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

MARICEL AGOP i CLIN GHEORGHE BUZEA, Un nou model al carcinogenezei i progresiei tumorale n cadrul dinamicii neliniare (III)(engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

CRISTINA-DELIA NECHIFOR i SEBASTIAN POPESCU, Studiul energeticii suprafeei (engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . . . .

79

IRINA RADINSCHI, CRISTIAN DAMOC i BOGDAN AIGNTOAIE, Test de fizic online n Adobe flash CS4 pentru mbuntirea performanelor studenilor (engl., rez. rom.) . . . . . . . . . . . . . . . . . . . . . .

89

MIHAI AVRAM, CONSTANTIN BUCAN, SILVIA MIU i MARIAN TNASE, Grup de generare a energiei hidraulice n concepie mecatronic (engl., rez. rom.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

S U M A R

BULETINUL INSTITUTULUI POLITEHNIC DIN IAI BULLETIN OF THE POLYTECHNIC INSTITUTE OF IAI Tomul LVIII (LXI), Fasc. 1 2012

MATHEMATICS. THEORETICAL MECHANICS. PHYSICS

Pp.

MUGUR B. RU, On a Newtonian Alternative Theory of MOND (English, Romanian summary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

CLIN GHEORGHE BUZEA, MARICEL AGOP and SIMONA RUXANDRA VOLOV, A New Carcinogenesis and Tumor Progression Model in the Framework of Non-linear Dynemics (I) (English, Romanian summary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

DRAGO IANCU, CLIN GHEORGHE BUZEA and MARICEL AGOP, A New Carcinogenesis and Tumor Progression Model in the Framework of Non-linear Dynemics (II) (English, Romanian summary). . . . . . . . . . .

33

MARICEL AGOP and CLIN GHEORGHE BUZEA, A New Carcinogenesisand Tumor Progression Model in the Framework of Non-linearDynemics (III) (English, Romanian summary). . . . . . . . . . . . . . . . . . . . .

59

CRISTINA-DELIA NECHIFOR and SEBASTIAN POPESCU, Study on Surface Energy of Polypropilene UV-Photo-Grafted with Glucose(English, Romanian summary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

IRINA RADINSCHI, CRISTIAN DAMOC and BOGDAN AIGNTOAIE, Online Physics Test Developed in Adobe flash CS4 to Improve Students Knowledge (English, Romanian summary) . . . . . . . . . . . . . . . . . . . . . . .

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MIHAI AVRAM, CONSTANTIN BUCAN, SILVIA MIU and MARIAN TNASE, Mechatronic Design of a Hydraulic Power-Supply Unit (English, Romanian summary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

C O N T E N T S

BULETINUL INSTITUTULUI POLITEHNIC DIN IAI Publicat de

Universitatea Tehnic Gheorghe Asachi din Iai Tomul LVIII (LXII), Fasc. 1, 2012

Secia MATEMATIC. MECANIC TEORETIC. FIZIC

ON A NEWTONIAN ALTERNATIVE THEORY FOR MOND

BY

MUGUR B. RU

Al. I. Cuza University of Iai, Department of Physics

Received: November 28, 2011 Accepted for publication: December 5, 2011

Abstract. This paper is an attempt to find an alternative physical theory for MOND. The basis of this physical theory was found to be a gravitational potential generated by a repulsive energy. The influence of this energy appears as an additional term to Newtons gravity law. This term plays an important role under the condition that the asymptotic rotational speed of the galaxy is independent from its radius. The found results are surprising because there are also specific for MOND theory, but this time under ordinary Newtonian theory assumptions.

Key words: dark energy; MOND; gravitational potential; galaxy rotation curves; critical radius; asymptotic rotational speed.

1. Introduction

The Modified Newtonian Dynamics theory, MOND, was proposed by (Milgrom 1983) as an alternative manner to explain the form of the galaxy rotation curves. When the uniform velocity of galaxies was first observed it was unexpected because this fact came in conflict with Newtonian theory of gravity predictions. The farther out are the moving objects the lower velocities they had. Later in the 70s the theory of dark matter seemed to solve this mystery. According to this theory each galaxy contains an exterior halo, like a contracting shell, of a special type of matter, dark matter. Consequently the normal matter rotates more as a rigid body than as a fluid, as the Newtonian theory suggests.

e-mail: [email protected]

2 Mugur B. Ru

Milgrom (1983) has tried to solve this problem in a different way. He pointed out that Newtons law of gravity is valid when the gravitational acceleration is large, but it may have not held at low accelerations. Consequently he proposed a new form for Newtons law of gravity. A particle at a distance r from a large mass M is amenable to an acceleration given by

2

0

a GM a

a r

=

. (1)

The right term is the Newtonian gravitational acceleration, the left term is

an interpolating function which is 1 for large accelerations and 0/a a for low

accelerations ( 0a been a constant).

One immediate consequence of Eq. (1) is that the asymptotic rotational speed of a galaxy becomes independent of radius

1

40( )v GMa= . (2)

This is the general case and it has an axiomatic validity, especially for flat

galaxies. Certainly, the goal of this theory is very easy to accomplish if there is only one interpolating function. In reality there are many effective interpolating functions, been dependent on the type of galaxy rotation curve we study. For instance the spiral-type rotational galaxies curves can be explained with

2

( )1

x x

x=

+. (3)

This fact makes MOND theory a little peculiar, but there is nothing

peculiar in a phenomenological theory, except only its lack of physical basis. If we observe that Eq. (3) tends to x for law values of x, maybe we get used to its peculiarity. The most significant tests of MOND are provided by disc-galaxy rotation curves (Begeman et al., 1991; Sanders & Verheijen, 1998; Sanders, 1996 ; de Blok & McGaugh, 1998). However, the dark matter theory seems to be the favorite regarding the galaxy rotation problem. The scientific community prefers it instead MOND, although the correct in many aspects predictions of no matter each of it is the comparison basis for the other, (McGaugh, 2004; Famaey et al., 2007).

This paper is an attempt to find an alternative to MOND theory, in order to explain the form of the galaxies rotation curves.

Bul. Inst. Polit. Iai, t. LVIII (LXII), f. 1, 2012 3

2. Repulsive Forces and the Dynamics of the Galaxies

In the following we show that one can obtain a theory to explain the rotation curves of discoid galaxies, a theory based on an unknown form of energy. It is certainly a new situation in this regard; so far responsible for the above mentioned curves was the dark matter or in MOND theory acceptance the law (1). Therefore we have a gravitational potential of the form

11

GM A rr

+= ++

, (4)

which is the result of solving Poisson's equation

4 ( )ueG = + . (5)

The second term on the right side of Eq. (4) is an empirical term attached to the Newtonian gravitational potential, a term that describes the action of an unknown repulsive energy. We have therefore A> 0.

With this potential (4) we try to show that this undefined energy may cause the radial movement of the discoid galaxies observed in reality. From (4) we find the expression of the first derivative of this equation

2

GMg Arr

= + , (6)

which has a great importance in the development of reasoning in the following. Eq. (4) can not accept local maxima or minima only if the potential would

be as follows

1( )l l

l r l lA r B r = + , (7)

in which case it would be the result of Laplace's equation

0 = . (8)

Note that the potential (4) does not satisfy the condition (7), because 0, 1l , so there may be local configurations of potentials (4) where there

may be maxima or minima. Hence g = 0 in (6). From this condition we can determine the critical radius whence equality occurs

2

c

c

GMAr

r= . (9)

Therefore the expression of this critical radius will be

4 Mugur B. Ru

1

2c

GMr

A

+ =

. (10)

From (9) by expressing the constant A as a function of the critical radius we can determine in a more synthetic way the accelerations expression

2

21N

c

rg g

r

+

+

=

,

with 2N

GMg

r= and forces expression

2

21N

c

rF F

r

+

+

=

with 2N

GMmF

r= deriving from the potential (4). Note that for distances

smaller than the critical radius there will be attractive accelerations and forces, after the critical radius the accelerations and forces become repulsive.

On reaching the critical radius the equilibrium of the galaxy will be the result of the galaxy accelerations balance

2

2c c

v GMa

r r= =

whence we can determine the radial velocity of the galaxy as

( )1

1 12( 2)

2 2( 2)

v GM A ++= . (11)

From observations made on the motion of galaxies and taking into account the Tully-Fisher empirical expression it results

v L M ,

where L represent the luminosity and 1 1...3 4

= .

So, in our view we can determine the proper in two cases corresponding to the two expressions of the potential (4), supplemented with the condition (7). Therefore we have

31

Mv , (12)

corresponding to =1 and the case in which we have a particular interest


41

Mv , (13)

corresponding to =0. Note that both potentials obtained from (4) with expressions (12) and (13) provides the shape of the rotation curves of discoid galaxies close to observations. But the form (4) with (13) it leads us unexpectedly closer to the MOND theory. Indeed if we take 0aA = we obtain from (10) exactly

2

1

0

=

a

GMrc , (14)

the critical radius from which the small accelerations approximation occurs in the MOND theory. From (11) and (14) it results

04 GMav = , (15)

which is the well-known expression obtained in the MOND theory for the radial velocities independent from the galaxies radius. What is important here is that we got this result in the approximation

0agg N += ,

and not in the approximation

1/20( )Ng g a= ,

like in MOND theory. The difference lies in the fact that our theory has not worked out with an expression of a modified inertia like in MOND theory, Newton's Second Law remaining unchanged. So with (4) and (13) MOND theory results can be obtained without the need to change the law of inertia like in (1). We just need to change the definition of constant 0a as an artifact of

galaxies.

3. Dark Energy and the Dynamics of the Galaxies

We might ask now what is the nature of this unknown energy responsible for the observed shapes of rotation curves of galaxies. In (4), condition (12) leads to = 1. Although this form of the potential (4) may explain the dynamics of discoid galaxies, we still do not accidentally deflect the discussion on this path. It shows that in (4), with = 1, the additional part of the gravitational potential can be conceived as a Newtonian equivalent to the cosmological constant

6 Mugur B. Ru

2

3

cA = .

From this we can infer only that Ar would express a repulsive force due to expansion of the universe. It can be either dark energy or vacuum energy. Assuming that the expansion of the universe would somehow influence the dynamics of galaxies we reach the role of dark matter in this regard. So we can think of a galaxy as being surrounded by a halo composed of dark matter.

Expansion of the universe expressed by the force 2

3

cr tends to divide

galaxies but the dark matter they are "wrapped" opposes with a reactive force which is opposite to the force of expansion. In this way the galaxies keep their

form and their dynamics is the observed one. The force 2

3

cr leads to

observations of the form 3/1Mv but the form 4/1Mv can not be explained by the same reactive force. Things get more complicated if we consider 0aA = . The acceleration 0a is four orders of magnitude higher than

the force 2

3

cr (for example, our galaxy have 2010 mr = ). So dark energy,

that causes the expansion of the universe, is almost four orders of magnitude smaller than the energy that causes the galaxy dynamics. The acceleration 0a

can not be attributed to expansion of the universe but it can be related to an internal expansion force of the galaxy. Let us admit that this force would be due to the generation of new matter in the galaxy (exploding stars, matter that tends to be extended in the galaxy). In such conditions the expansion force of the galaxy will find its reaction in the blanket of dark matter that surrounds it. This will "push" the galaxy with a constant force, equal and opposite, of value 0a .

This might explain the anomaly of Pioneer 10 spacecraft. There is an attractive constant force in our galaxy, supplementary to the Newtonian one, which is indirectly due to expansion trend of the galaxy and directly due to the blanket of dark matter that surrounds it. Eq. (15) is valid because dark matter is opposing to the trend of the dispersion caused by the rotation of the galaxy.

Under these conditions Eqs. (14) and (15) are valid without the need to consider MOND theory. If we admit that 0a have not the same meaning as in

the MOND theory but totally due to other causes, having no connection with the expansion of the universe but only with internal dynamics of galaxies, is a constant specific to each galaxy in part, then all we have talked so far is valid. Otherwise the place of 0a may be taken by the general value A which can be

determined from experimental curves. Amazingly, if we do this we get to the


result 0aA = (which was also determined from experimental curves, as a mean

value). In this case 0a can not be conceived as in MOND theory but as a

galactic characteristic without any connection with the universe.

4. Conclusions

1. In this paper we propose an alternative Newtonian theory for MOND. 2. Under the hypothesis that for the shape of the radial velocity curves of

galaxies is responsible an unknown form of energy, this influence is found to be expressed by a supplementary potential which it must be added to the Newtonian gravitational potential.

3. Somewhat surprising, we found the results specific to MOND theory, but this time with an ordinary Newtonian dynamics.

REFERENCES

Begeman K. G. , Broeils A. H., Sanders R. H., Mon. Not. of the Ro. Astron. Soc., 249, 523 (1991).

de Blok W. J. G., McGaugh S. S., Astrophys. J., 508, 132 (1998). Famaey B. , Gentile G. , Bruneton J.-P., Zhao H. S., Phys. Rev. D, 75, 063002 (2007). McGaugh S. S., Astrophys. J., 609, 65 (2004). Milgrom M., Astrophys. J., 270, 371 (1983). Ru M. B., Time Invariance of the Fundamental Physical Constants. Bul. Inst. Polit.

Iai, LIV(LX), 2, s. Matematic. Mecanic teoretic. Fizic, 33-38 (2010). Sanders R. H., Astrophys. J., 473, 117 (1996). Sanders R. H., Verheijen M. T. W., Astrophys. J., 503, 97 (1998). Zwicky F., Helvetica Physica Acta, 6, 110-127 (1933). Zwicky F., Astrophys. J., 86, 217 (1937).

ASUPRA UNEI TEORII NEWTONIENE ALTERNATIVE LA TEORIA MOND

(Rezumat)

Aceast lucrare este o ncercare de a gsi o teorie fizic alternativ pentru teoria

MOND. La baza acestei teorii st un potenial gravitaional generat de o energie repulsiv. Influena acestei energii apare ca un termen adiional la legea gravitaiei newtoniene. Acest termen joac un rol important n condiiile n care viteza asimptotic de rotaie a unei galaxii este independent de raza sa. Rezultatele gsite sunt surprinztoare deoarece ele sunt de asemenea specifice teoriei MOND, ns de ast dat n cadrul unei teorii newtoniene obinuite.




A NEW CARCINOGENESIS AND TUMOR PROGRESSION MODEL IN THE FRAMEWORK

OF NON-LINEAR DYNAMICS (I)

BY

CLIN GHEORGHE BUZEA1, MARICEL AGOP2

and SIMONA RUXANDRA VOLOV3

1National Institute of Research and Development for Technical Physics, Iai 2 Gheorghe AsachiTechnical University of Iai

3University of Medicine and Pharmacy Gr. T. Popa, Iai

Received: December 28, 2011 Accepted for publication: January 15, 2011

Abstract.We propose a new physical concept of carcinogenesis and tumor

progression by the use of a natural environment where malignant tumors grow, space(-time) with non-integer fractal dimension, in quest for further applications of the newly discovered phenomenon of tumor self-seeding by circulating cancer cells (CTC). We assume that metastasic tumor cells move (through the systemic circulation) as a coherent wave, a chemically pumped travelling wave laser with oxygen. The extracellular matrix (ECM) and the tumor microenvironment (TME) are assumed as non-differential media endowed with holographic properties. As a result, the tumor self-seeding by CTC may prove mathematically, the fact that the CTC returning to the initial tumor site, assembling a new tumor or fueling the primary tumor growth is a particular case of complete holography (a characteristic of the living organisms). Moreover, we believe cancer hypoxia and tumor self-seeding by circulating cancer cells are in vivo proofs of this phenomenon. In this paper some general notions about cancer, mathematics of cancer (connections between cancer, non-differentiability and chaos) and information about holography are revealed.

Keywords: carcinogenesis, tumor, fractal, travelling wave, holography.

Corresponding author: e-mail: [email protected]

10 Clin Gheorghe Buzea et al.

1. Introduction

During the past quarter century, tremendous steps have been made in the diagnosis and treatment of cancer. Technology now allows the diagnosis and treatment of tumors of ever-diminishing size, as with breast cancers; ductal carcinoma in situ now comprises 2530% of all newly diagnosed breast cancers at most medical centers (Armstrong et al., 2000). With earlier detection, an understanding of growth patterns reflective of the natural biological characteristics of these tumors must also evolve. Surgeons have always led the fields of technological and basic scientific medical advances. Current concepts, be they either the physiological characteristics of shock, organ transplantation, antisepsis, wound healing, or gene therapy, have been forged by surgical investigators.

The field of mathematics has undergone a similar evolution. Topology, fractals, chaos theory, and development of nonlinear descriptive methods have provided mathematicians new creative tools that permit the development of models of tumor growth and behavior at the microenvironmental level (Friedman & Reitich, 1999; Waliszewski et al.,1998). Specific formulas have been described for growth, angiogenesis (Orme & Chaplain, 1997), cell-to-cell adhesion (Perumpanani et al, 1997) and even pH regulation and drug delivery (Secomb et al, 2001). From a clinical viewpoint many of these formulas may seem oversimplified, but they collectively form an important foundation for descriptive insight.

What has been lacking is the linkage of these two naturally and mutually beneficial research endeavors. For oncologic surgeons, the ability to mathematically describe (or, even better, predict) patterns of tumor behavior provides an exciting, new, and precise method that may benefit both current and future therapies. For the mathematician, an understanding of the clinical factors essential for tumor development and metastasis provides realistic insight into these complex biological processes, in turn permitting the development of accurate, clinically relevant mathematical formulas.

In most medical centers, surgeons lead the team that provides comprehensive cancer care. Oncologic mathematics provides surgeons another opportunity to expand their leadership role and to better understand tumor behavior and optimize cancer treatment.

In this paper we aim to envisage a new concept of carcinogenesis and tumor progression. Consequently, we use the natural environment where malignant tumors grow, space(-time) with non-integer fractal dimension, in quest for further applications of the newly discovered and intriguing phenomenon of tumor self-seeding by circulating cancer cells (CTC). More precisely, we assume that the metastasic tumor cells move (through the systemic circulation, yet not necessarily only there) as a coherent wave, or even more precisely, a chemically pumped travelling wave laser with oxygen. The


extracellular matrix (ECM) and in particular, the tumor microenvironment (TME) are assumed as non-differential media endowed with holographic properties and may be good candidates for recording materials. As a result, the tumor self-seeding by CTC may prove mathematically, the fact that the CTC returning to the initial tumor site and fueling the primary tumor growth or even grow a new tumor is a particular case of complete holography (i.e. a hologram which does not represent only the virtual objects image, but it becomes the very object - which we believe, is a characteristic of the living organisms). We believe our findings may provide new opportunities to set up new targeted therapies that may slow down or even prevent tumor progression.

The work is structured as follows. After a short introduction, in Sec. 2 we provide some general notions about cancer, which will be addressed in the rest of the paper. In Sec. 3, the mathematics of cancer as it is today is reviewed shortly, and also a few words about the connections between cancer, non-differentiability and chaos are provided. Further, in Sec. 4, some information about holography: how it works and recording materials are revealed. This will be important for what will follow.

2. The Biology of Cancer

2.1 Cancer, What Should be Noticed

Cancer or malignant neoplasm is a class of diseases that rises from the anomalous behavior of normal tissue. Cancer cells are aberrant cells which have acquired malignant traits such as uncontrolled growth (cells continuously proliferate), tissue invasion (they intrude into normal tissue and they destroy it) and metastasis (they spread outside the location of the body where they were originally generated). Additionally, the term tumour or neoplasm is used to indicate an abnormal swelling of tissue caused by an excessive cell proliferation.

A tumour can be of benign or malignant nature, while benign tumours are self-limiting, do not express patterns of invasion, and they do not metastasize, malignant tumours do possess all these characteristics. The term malignant tumour is also used as synonym for cancer, although some cancers, such as leukemia, do not form tumours.

Cancer cells develop these malignant features because of genetic mutations, accumulated during the organism lifetime. Cancer is in fact a multi-step chance process that transforms a normal cell into a tumour cell, after having collected a set of 58 crucial genetic alterations (Fodde et al., 2001; Hahn et. al., 1999; Kinzler et al., 1996) as schematically shown in Fig. 1.

A newborn malignant cell, expressing aberrant traits, can lead to the formation of cancer and, in most of the cases, of a tumour. Without treatment, the destructive behavior of such colony of cells is usually lethal for the patient. The probabilistic nature of this disease and the increase in life expectancy had


made cancer the second cause of death in the industrialised countries (see any cancer statistics). Nevertheless, cancer it is not a modern disease and it was known since the antiquity: Egyptians of the New Kingdom (Olson et al., 2002), Greeks (Porter, 1997) and Romans (Hajdu, 2004) accurately described medical treatments for tumour removal. It is only within the last two centuries however, that due to the higher standards of living, cancer has become one of the main life-threatening diseases.

Fig. 1 Acquisition of the tumourigenic phenotype by a population of normal cells

through multiple genetic mutations.

2.2. Distinguishing Traits of Cancer

The tumourigenic properties, generically discussed in the previous

section, have shown to be common to almost all cancers. They have been studied since the dawn of cancer research and they can be enumerated and defined with a relatively high accuracy. These hallmarks are a set of characteristic traits typical of cancer cells that are essential for the formation of a macroscopic malignant neoplasm (Hanahan & Weinberg, 2000):

Self-Sufficiency in Growth Signals. All cells communicate through signals. A biological signal is, in most of the cases, a protein able to deliver a particular piece of information by binding uniquely to specific receptors on the cell surface. Normal cells need mitogenic growth signals to proliferate (signals that allow and stimulate cell proliferation). Those signals are regulated by the homeostasis of the tissue and they guarantee a correct balance between cell proliferation and death, according to the needs of the organism. In order to lead to cancer, tumour cells may develop the ability of self-generating such signals in one way or another. One possible way is a genetic aberration in one of the fundamental genes responsible for the building of the signaling pathway, for instance the RAS oncogene (Kinzler et al., 1996; Medema et al., 1993). As consequence, the associated component of the signaling system would become constitutively active and hence, independent by the signal molecule. A second option is the self-production of growth factors that would stimulate growth by paracrine signaling, where a cells stimulates the neighbours and vice-versa or even autocrine signaling, when the cell stimulates its own receptors as shown in Fig. 2.


Fig. 2 Example of self-signaling (autocrine): the cell produces its own growth factors which stimulate the growth receptors on the surface (Dr. W.H. Moolenaar, Netherlands

Cancer Institute).

Insensitivity to Antigrowth Signals. As counterparts of growth factors, homeostasis employs growth inhibiting signals as well. These signals act similarly to their antagonists but they promote cell cycle arrest or cell quiescence, rather than proliferation. An example of a crucial gene involved in anti-growth pathways is the retinoblastoma protein (pRb). The retinoblastoma protein is capable of altering the function of the E2F transcription factors and control the expression of the bank of genes essential for the transition from GAP-1 phase to DNA Synthesis phase of the cell cycle (Weinberg, 1995). The disruption of such pathway results in the insensitivity of the cell to anti-growth signals.

Evading Apoptosis. Apoptosis is a mechanism of controlled cell death. Through special signals, a cell has the capacity of terminating itself in a highly regulated way. A normal cell dying by apoptosis undergoes a sequence of events such as condensation, fragmentation and phagocytosis. This avoids the cell to free potentially dangerous enzymes and proteins stored inside its cytoplasm and its nucleus. During apoptosis the cell membrane is kept intact while, in 30120 minutes the cell is fragmented in small parts or apoptotic bodies, still protected by pieces of membrane. Those cell leftovers are successively phagocytated by macrophages within the next 24 hours (Wyllie et al., 1980). Apoptosis is a common mechanism of cell death and takes part in the homeostasis of a healthy organism as well as in its embryogenesis and in its morphogenesis. When any cell violates such homeostasis, an apoptotic signal is delivered to it. Therefore, in order for cancer cells to develop into a malignant lesion, it is necessary to deactivate apoptotic signal pathways. A mutation in the p53 tumour suppressor gene (TSG) is one of the most common ways to acquire resistance to apoptosis because p53 regulates the whole signaling process of programmed cell death. Indeed, more than 50% of human cancers carry a mutation in the p53 tumour suppressor gene (Harris, 1996).

Limitless Replicative Potential. Even with all the anti-growth and anti-apoptosis pathways triggered off, a cell could not generate a vast population


able to form a tumour. That is because of the intrinsic proliferation limit of all mammalian cells. All chromosomes have an ending cap called telomere, a T-loop non-coding DNA sequence (2...50 Kb) that prevents the end of the chromosomes from attaching to other genetic material. At every mithosis the cell loses a small part of its telomeres because of the impossibility for DNA duplication enzymes, for instance DNA-polymerase, to continue working until the very end of the genome (Fig. 3). This limitation is due to the fact that enzymes like DNA-polymerase always move in the 53 direction of the DNA sequence, so when the side of the replication is opposite, a small part of the genome is lost. The shortening of the telomeres induces cell senescence, a state of cellular elderly where division no longer occurs. This avoids genetically unstable cells to replicate. Senescence starts after the so called Hayflick limit (Hayflick, 1997) of about 50 cell divisions. In cancer cells instead, the disabling of the pRb and the p53 pathways allows unlimited replication, until the point when the telomeres are completely absent. Once having entirely consumed the telomeres, the cell population is believed to undergo a phase of massive genomic instability, causing extended cell death. The high selective pressure induced by this crisis may permit specific resistant clones to emerge (Fig. 4). Those survivor cells would be immortalised (unlimited proliferative potential) by finding ways to maintain their telomeres long enough. A possible way is the overexpression of the telomerase gene (Shay & Bacchetti, 1997) which appears to take place in 8590 % of cancers. Telomerase is a telomere-rebuilding enzyme normally expressed in germline cells and stem cells, in which immortalisation is an essential feature. Once immortalised, malignant cells have made a further step towards the formation of cancer.

Fig. 3 Illustration of the end replication problem: at both sides of the copying, the

leading DNA strand has lost part of the telomeric sequence, which stops at the 5 end of the parental strand, whereas the lagging strand results completed until the very end

(Dr. R. Beijersbergen, Netherlands Cancer Institute).

Sustained Angiogenesis. In order for a cell to survive normally, it must rely within 100 m from a capillary blood vessel (Hanahan & Weinberg, 2000).


For this reason, the initial exponential growth of a newborn malignant neoplasm causes a shortage of nutrients among cancer cells. Local pre-existent vascularisation is never enough to sustain growth for more than 108 cells. The colony must therefore develop angiogenesis-triggering capabilities (Bouck et al., 1996; Hanahan & Folkman, 1996). Angiogenesis is the process of formation of new blood vessels in response to a stimulus secreted by poor vascularised tissues. Angiogenesis is important for the organism morphogenesis and even later maintains the correct supply of nutrients for all tissues. Fast growing cells, such as cancer cells, start soon to starve, and have the need of additional blood supply in order to keep expanding. A possible solution is the production by cancer cells of vascular endothelial growth factors (VEGF) and fibroblast growth factors (FGF1/2) which bind to the transmembrane receptors of endothelial cells (cells covering the interior surface of blood vessels) stimulating their growth towards the signal concentration gradient (Veikkola & Alitalo, 1999). Angiogenesis is the principal mechanism that transforms a microscopic malignancy into a macroscopic tumour and, also in the later stages, it is necessary for a lesion to grow and sustain itself. This implies that angiogenesis is an important target for anti-cancer drugs like thrombospondin-1 (Bull et al., 1994) and bevacizumab (Shih & Lindley, 2006), also known as avastin.

Fig. 4 The progressive shortening of the telomeres leads to a massive cell death due to the induced genomic instability (death by genomic catastrophe while duplicating). From

such process of intense genetic mutation and selection an immortalised clone could emerge (Dr. R. Beijersbergen, Netherlands Cancer Institute).

Tissue Invasion and Metastasis. The most dangerous and destructive

features of cancer are tissue invasion and the consequent metastasis. Its ability of forming distant colonies or metastases all over the body represents the cause of 90% of all cancer related deaths (Sporn, 1996). Normal cells are usually


unable to travel outside their own tissue due to their necessity to be anchored and reside among similar cells. An eventual detachment from the extracellular matrix or ECM (a complex structure of proteins and specific cells forming the tissue scaffold and microenvironment see Sec. 6.1) would occur in a form of apoptosis called anoikis (Frisch & Screaton, 2001). Contrary to their normal counterparts, cancer cells are able to survive the loss of anchorage, to travel through the vascular system and form distant tumours elsewhere (Fig. 5). The traits expressed by invasive and metastatic cancer cells are principally loss of cell-to-cell adhesion, anchorage-independence, chemotaxis (migration towards a diffusible substance gradient), haptotaxis (migration towards a non-diffusible substance gradient) and production of matrix degrading enzymes (e.g. Matrix metalloproteinase) which cleave the extracellular matrix (Fidler, 2003; Matrisian, 1992; Mignatti & Rifkin, 1993) making space for invasion and freeing growth and angiogenic factors trapped inside.

Fig. 5 Tissue invasion is a multi-step process that requires the cancer cell to have

developed many malignant traits, necessary for the formation of new distant colonies called metastases (Fidler, 2003).

3. Mathematics of Cancer

In comparison to molecular biology, cell biology, and drug delivery research, mathematics has so far contributed relatively little to the area. A search in the PubMed bibliographic database (http://www.ncbi.nlm.


nih.gov/PubMed/) shows that out of 1.5 million papers in the area of cancer research, approximately 5% are related to mathematical modeling. However, it is clear that mathematics could make a huge contribution to many areas of experimental cancer investigation since there is now a wealth of experimental data which requires systematic analysis.

Nevertheless, over the last decade, the activity in mathematical modeling and computational simulation of cancer has increased dramatically (e.g., reviews such as (Araujo et al., 2004; Byrne et al., 2006; Adam, 1996; Bellomo et al., 2003; Quaranta et al., 2005; Sanga et al., 2006). A variety of modeling strategies have been developed, each focusing on one or more aspects of cancer. Cellular automata and agent-based modeling, where individual cells are simulated and updated based upon a set of biophysical rules, have been developed to study genetic instability, natural selection, carcinogenesis, and interactions of individual cells with each other and the microenvironment. Because these methods are based on a series of rules for each cell, it is simple to translate biological processes (e.g., mutation pathways) into rules for the model. However, these models can be difficult to study analytically, and computational costs can increase rapidly with the number of cells. Because a 1-mm tumor spheroid may have several hundred thousand cells, these methods could become unwieldy when studying tumors of any significant size. See (Alarcn et al., 2005; Anderson, 2003; Mallett et al., 2006) for examples of cellular automata modeling and (Abbott et al., 2006; Mansury et al., 2002) for examples of agent-based modeling. In larger-scale systems where the cancer cell population is on the order of 106 or more, continuum methods may provide a more suitable modeling technique. Early work, including (Byrne et al., 1996; Byrne et al., 1996; Greenspan et al., 1976), used ordinary differential equations to model cancer as a homogeneous population, as well as partial differential equation models restricted to spherical geometries. Linear and weakly nonlinear analyses have been performed to assess the stability of spherical tumors to asymmetric perturbations (Araujo et al., 2004; Byrne et al., 2006), (Byrne et al., 2002; Chaplain et al., 2001; Cristini et al., 2003; Li et al., 2007) in order to characterize the degree of aggression. Various interactions of a tumor with the microenvironment, such as stress-induced limitations to growth, have also been studied (Ambrosi et al., 2002; Ambrosi et al., 2002; Ambrosi et al., 2002; Araujo et al., 2004; Araujo et al., 2005; Jones et al., 2000; Roose et al., 2003). Most of the modeling has considered single-phase (e.g., single cell species) tumors, although multiphase mixture models have also been developed to provide a more detailed account of tumor heterogeneity (Ambrosi et al., 2002; Byrne et al., 2003; Chaplain et al., 2006).

Recently, nonlinear modeling has been performed to study the effects of morphology instabilities on both avascular and vascular solid tumor growth. (Cristini et al., 2003) used boundary integral methods to perform the first fully nonlinear simulations of a continuum model of tumor growth in the avascular


and vascular growth stages with arbitrary boundaries. These investigations of the nonlinear regime of shape instabilities predicted encapsulation of external, non-cancerous tissue by morphologically unstable tumors. (Li et al., 2007) extended the model from (Cristini et al., 2003) in 3-D via an adaptive boundary integral method. (Zheng et al., 2005) built upon the model in (Cristini et al., 2003) to include angiogenesis and an extratumoral microenvironment by developing and coupling a level set implementation with a hybrid continuum-discrete angiogenesis model originally developed by (Anderson & Chaplain, 1998). As in (Cristini et al., 2003), (Zheng et al., 2005) found that low-nutrient (e.g., hypoxic) conditions could lead to morphological instability. Their work served as a building block for recent studies of the effect of chemotherapy on tumor growth (Sinek et al., 2004) and for studies of morphological instability and invasion (Cristini et al., 2005; Frieboes et al., 2006; Hogea et al., 2006) have also begun exploring tumor growth and angiogenesis using a level set method coupled with a continuum model of angiogenesis. Macklin & Lowengrub used a ghost cell/level set method for evolving interfaces to study tumor growth in heterogeneous tissue and further studied tumor growth as a function of the microenvironment (Macklin & Lowengrub, 2007). Finally, (Wise et al.; Frieboes et al., 2007) have developed a diffuse interface implementation of solid tumor growth to study the evolution of multiple tumor cell species, which was employed in (Frieboes et al., 2007) to model the 3-D vascularized growth of malignant gliomas (brain tumors).

In the case of biological systems, the fractal structure of space in which cells interact and differentiate is essential for their self-organization and emergence of the hierarchical network of multiple cross-interacting cells, sensitive to external and internal conditions. Hence, the biological phenomena take place in the space whose dimensions are not represented only by integer numbers (1, 2, 3, etc.) of Euclidean space. In particular, malignant tumors (Jones et al., 2000; Roose et al., 2003; Byrne et al., 2003; Chaplain et al., 2006) grow in a space with non-integer fractal dimension. More precisely, it was proved that the analytical formulae describing the time-dependence of the temporal fractal dimension and scaling factor very well reproduce the growth of the FlexnerJobling rats tumor in particular and growth of other rats tumors in general. The results of some test calculations indicated that the formula derived for the time-dependent temporal fractal dimension and the scaling factor satisfactory describe the experimental data obtained by Schrek for the Brown-Pearce rabbits tumor growth in the fractal space-time (Waliszewski et al., 1998; Waliszewski et al., 1999; Waliszewski et al., 2000; Waliszewski et al., 2001).

In our assertion fractal space(-time) consists in developing the consequences of the withdrawal of space(-time) differentiabilitys hypothesis and acquiring a fractal geometry, namely, space(-time) becomes explicitly dependent on the observation scale (Nottale, 1993).


On the other hand, of great use in our further reasonings, will be the fact that in many biological systems it is possible to empirically demonstrate the presence of attractors that operate starting from different initial conditions (Ivancevic). Some of these attractors are points, some are closed curves, while the others have noninteger, fractal dimension and are termed strange attractors (Guarini et al., 1993). It has been proposed that a prerequisite for proper simulating tumor growth by computer is to establish whether typical tumor growth patterns are fractal. The fractal dimension of tumor outlines was empirically determined using the box-counting method (Sedivy, 1996). In particular, fractal analysis of a breast carcinoma was performed using a morphometric method, which is the box-counting method applied to the mammogram as well as to the histologic section of a breast carcinoma (Sedivy & Windischberger, 1998).

If tumor growth is chaotic, this could explain the unreliability of treatment and prediction of tumor evolution. More importantly, if chaos is established, this could be used to adjust strategies for fighting cancer. Treatment could include some form of chaos control and/or anti-control.

4. A Few Words about Holography

A hologram is usually recorded on a photographic plate or a flat piece of film, but produces a three-dimensional image. In addition, making a hologram does not involve recording an image in the conventional sense. To resolve this apparent paradox and understand how holography works, we have to start from first principles.

In conventional imaging techniques, such as photography, what is recorded is merely the intensity distribution in the original scene. As a result, all information about the optical paths to different parts of the scene is lost.

The unique characteristic of holography is the idea of recording both the phase and the amplitude of the light waves from an object. Since all recording materials respond only to the intensity in the image, it is necessary to convert the phase information into variations of intensity. Holography does this by using coherent illumination and introducing, as shown in Fig. 6, a reference beam derived from the same source. The photographic film records the interference pattern produced by this reference beam and the light waves scattered by the object.

Since the intensity at any point in this interference pattern also depends on the phase of the object wave, the resulting recording (the hologram) contains information on the phase as well as the amplitude of the object wave. If the hologram is illuminated once again with the original reference wave, as shown in Fig. 7, it reconstructs the original object wave.

An observer looking through the hologram sees a perfect three-dimensional image. This image exhibits all the effects of perspective, and depth of focus when photographed, that characterized the original object.


Fig. 6 Hologram recording: the interference pattern produced by the reference wave

and the object wave is recorded.

Fig. 7 Image reconstruction: light diffracted by the hologram reconstructs

the object wave.

4.1. Early Development

In Gabors historical demonstration of holographic imaging (Gabor, 1948), a transparency consisting of opaque lines on a clear background was illuminated with a collimated beam of monochromatic light, and the interference pattern produced by the directly transmitted beam (the reference wave) and the light scattered by the lines on the transparency was recorded on a photographic plate. When the hologram (a positive transparency made from this photographic negative) was illuminated with the original collimated beam, it produced two diffracted waves, one reconstructing an image of the object in its original location, and the other, with the same amplitude but the opposite phase, forming a second, conjugate image.

A major drawback of this technique was the poor quality of the reconstructed image, because it was degraded by the conjugate image, which


was superimposed on it, as well as by scattered light from the directly transmitted beam.

The twin-image problem was finally solved when Leith and Upatnieks (Leith & Upatnieks, 1962; Leith & Upatnieks, 1963; Leith & Upatnieks, 1964) developed the off-axis reference beam technique shown schematically in Figs. 6 and 7. They used a separate reference wave incident on the photographic plate at an appreciable angle to the object wave. As a result, when the hologram was illuminated with the original reference beam, the two images were separated by large enough angles from the directly transmitted beam, and from each other, to ensure that they did not overlap.

The development of the off-axis technique, followed by the invention of the laser, which provided a powerful source of coherent light, resulted in a surge of activity in holography that led to several important applications.

4.2. The In-line Hologram

We consider the optical system shown in Fig. 8 in which the object (a transparency containing small opaque details on a clear background) is illuminated by a collimated beam of monochromatic light along an axis normal to the photographic plate.

Fig. 8 Optical system used to record an in-line hologram.

The light incident on the photographic plate then contains two components. The first is the directly transmitted wave, which is a plane wave whose amplitude and phase do not vary across the photographic plate. Its complex amplitude can, therefore, be written as a real constant r. The second is a weak scattered wave whose complex amplitude at any point (x, y) on the photographic plate can be written as o(x, y), where |o(x, y)|


2 22( , ) ( , ) ( , ) ( , ) ( , ),I x y r o x y r o x y ro x y ro x y= + = + + +

(1)

where o*(x, y) is the complex conjugate of o(x, y).

A positive transparency (the hologram) is then made by contact printing from this recording. If we assume that this transparency is processed so that its amplitude transmittance (the ratio of the transmitted amplitude to that incident on it) can be written as

0t t ,TI= + (2) where t0 is a constant background transmittance, T is the exposure time and is a parameter determined by the photographic material used and the processing conditions, the amplitude transmittance of the hologram is

22

0t( , ) t [ ( , ) ( , ) ( , )].x y T r o x y ro x y ro x y= + + + + (3)

Finally, the hologram is illuminated, as shown in Fig. 9, with the same

collimated beam of monochromatic light used to make the original recording. Since the complex amplitude at any point in this beam is, apart from a constant factor, the same as that in the original reference beam, the complex amplitude transmitted by the hologram can be written as

22

0

2 2

( , ) t( , ) (t ) ( , )

( , ) ( , ).

u x y r x y r Tr Tr o x y

Tr o x y Tr o x y

= = + + +

+ +

(4)

The right-hand side of (4) contains four terms. The first of these,

r(t0+Tr2), which represents a uniformly attenuated plane wave, corresponds to the directly transmitted beam.

The second term, Tr |o(x, y)|2, is extremely small, compared to the other terms, and can be neglected.

The third term, Tr2o(x, y), is, except for a constant factor, identical with the complex amplitude of the scattered wave from the object and reconstructs an image of the object in its original position. Since this image is formed behind the hologram, and the reconstructed wave appears to diverge from it, it is a virtual image.

The fourth term, Tr2o*(x, y), represents a wave similar to the object wave, but with the opposite curvature. This wave converges to form a real image (the conjugate image) at the same distance in front of the hologram.

With an in-line hologram, an observer viewing one image sees it superimposed on the out-of-focus twin image as well as a strong coherent background. Another drawback is that the object must have a high average transmittance for the second term on the right-hand side of (4) to be negligible.


As a result, it is possible to form images of fine opaque lines on a transparent background, but not vice versa. Finally, the hologram must be a positive transparency. If the initial recording is used directly, in (2) is negative, and the reconstructed image resembles a photographic negative of the object.

Fig. 9 Optical system used to reconstruct the image with an in-line hologram,

showing the formation of the twin images.

4.3. Off-axis Holograms

To understand the formation of an image by an off-axis hologram, we consider the recording arrangement shown in Fig. 10, in which (for simplicity) the reference beam is a collimated beam of uniform intensity, derived from the same source as that used to illuminate the object.

Fig. 10 The off-axis hologram: recording.

The complex amplitude at any point (x, y) on the photographic plate due to the reference beam can then be written as

( , ) exp(i2 ),r x y r x= (5) where =(sin)/, since only the phase of the reference beam varies across the photographic plate, while that due to the object beam, for which both the amplitude and phase vary, can be written as


( , ) ( , ) exp[ i ( , )].o x y o x y x y=

(6) The resultant intensity is, therefore,

2 2 2

22

( , ) ( , ) ( , ) ( , ) ( , )

( , ) exp[ ( , )]exp( i2 ) ( , ) exp[i ( , )]exp(i2 )

( , ) 2 ( , ) cos[2 ( , )].

I x y r x y o x y r x y o x y

r o x y i x y x r o x y x y x

r o x y r o x y x x y

= + = + +

+ + =

= + + +

(7)

The amplitude and phase of the object wave are encoded as amplitude and phase modulation, respectively, of a set of interference fringes equivalent to a carrier with a spatial frequency of .

If, as in (2), we assume that the amplitude transmittance of the processed photographic plate is a linear function of the intensity, the resultant amplitude transmittance of the hologram is

' 20t( , ) t ( , ) ( , ) exp[ i ( , )]exp( i2 )

( , ) exp[i ( , )]exp(i2 ),

x y T o x y Tr o x y x y x

Tr o x y x y x

= + + +

+

(8)

where t`0 = t0 +Tr2 is a constant background transmittance. When the hologram is illuminated once again with the original reference

beam, as shown in Fig. 11, the complex amplitude of the transmitted wave can be written as

2'

0

2 2

( , ) ( , )t( , ) t exp(i2 ) ( , ) exp(i2 )

( , ) ( , )exp(i4 ).

u x y r x y x y r x Tr o x y x

Tr o x y Tr o x y x

= = + +

+ + (9)

The first term on the right-hand side of (9) corresponds to the directly

transmitted beam, while the second term yields a halo surrounding it, with approximately twice the angular spread of the object. The third term is identical to the original object wave, except for a constant factor Tr2, and produces a virtual image of the object in its original position. The fourth term corresponds to the conjugate image which, in this case, is a real image. If the offset angle of the reference beam is made large enough, the virtual image can be separated from the directly transmitted beam and the conjugate image.

In this arrangement, corresponding points on the real and virtual images are located at equal distances from the hologram, but on opposite sides of it. Since the depth of the real image is inverted, it is called a pseudoscopic image, as opposed to the normal, or orthoscopic, virtual image. It should also be noted that the sign of only affects the phase of the reconstructed image, so that a positive image is always obtained, even if the hologram recording is a photographic negative.


Fig. 11 The off-axis hologram: image reconstruction.

4.4. Recording Materials

Several recording materials have been used for holography (Smith, 1977). Table 1 lists the principal characteristics of those that have been found most useful.

Table 1

Recording materials for holography

Material

Exposure J/m2

Resolution mm-1

Processing

Type

max (diffraction efficiency)

Photographic 1.5 5000 Normal Amplitude 0.06 Bleach Phase 0.60

DCG (dichromated gelatin)

102 10000 Wet Phase 0.90

Photoresists 102 3000 Wet Phase 0.30 Photopolymers 10-104 5000 Dry Phase 0.90

PTP (photothermoplastics)

10-1 500-1200 Dry Phase 0.30

BSO ( Bi12SiO20

photorefractive crystals)

10 10000 None Phase 0.20

High-resolution photographic plates and films were the first materials used to record holograms. They are still used widely because of their relatively high sensitivity when compared to other hologram recording materials (Bjelkhagen, 1993). In addition, they can be dye sensitized so that their spectral sensitivity matches the most commonly used laser wavelengths.


The silver-halide sensitized gelatin technique makes it possible to combine the high sensitivity of photographic materials with the high diffraction efficiency, low scattering and high light-stability of DCG (dichromated gelatin) (Pennington et al., 1971).

In positive photoresists, such as Shipley AZ-1350, the areas exposed to light become soluble and are washed away during development to produce a relief image (Bartolini, 1977).

Several organic materials can be activated by a photosensitizer to produce refractive index changes, due to photopolymerization, when exposed to light (Booth, 1977). A commercial photopolymer is also available coated on a polyester film base (DuPont OmniDex) that can be used to produce volume phase holograms with high diffraction efficiency (Smothers et al., 1990).

Photothermoplastics (PTP) - a hologram can be recorded in a multilayer structure consisting of a glass or Mylar substrate coated with a thin, transparent, conducting layer of indium oxide, a photoconductor, and a thermoplastic (Lin & Beauchamp, 1970; Urbach, 1977).

When a photorefractive crystal is exposed to a spatially varying light pattern, electrons are liberated in the illuminated areas. These electrons migrate to adjacent dark regions and are trapped there. The spatially varying electric field produced by this space-charge pattern modulates the refractive index through the electro-optic effect, producing the equivalent of a phase grating. The spacecharge pattern can be erased by uniformly illuminating the crystal, after which another recording can be made (Huignard & Micheron, 1976; Huignard, 1981).

In order to maximize the visibility of the interference fringes formed by the object and reference beams, while recording a hologram, it is essential to use coherent illumination. In addition to being spatially coherent, the coherence length of the light must be much greater than the maximum value of the optical path difference between the object and the reference beams in the recording system. Lasers are therefore employed almost universally as light sources for recording holograms.

Consequently, to get a hologram, one needs a laser, which provides a powerful source of coherent light, and a recording material which records the interference pattern produced by a reference beam and the light waves scattered by the object.

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UN NOU MODEL AL CARCINOGENEZEI I PROGRESIEI TUMORALE N CADRUL DINAMICII NELINIARE (I)

(Rezumat)

Se propune un nou concept fizic al carcinogenezei i progresiei tumorale prin

utilizarea unui mediu natural n care tumorile maligne se pot dezvolta, adic spaiul-timp fractal, cu scopul de a gsi aplicaii ulterioare ale noului fenomen descoperit, autonsmnarea tumoral prin celule canceroase circulante. n prezenta lucrare sunt preezentate noiuni generale despre cancer, matematica cancerului (conexiunea ntre cancer, nedifereniabilitate i haos) i holografie.




A NEW CARCINOGENESIS AND TUMOR PROGRESSION MODEL IN THE FRAMEWORK

OF NON-LINEAR DYNAMICS (II)

BY

DRAGO IANCU1, CLIN GHEORGHE BUZEA2 and MARICEL AGOP3

University of Medicine and Pharmacy Gr. T. Popa, Iai National Institute of Research and Development for Technical Physics, Iai

Gheorghe Asachi Technical University of Iai

Received: December 28, 2011 Accepted for publication: January 15, 2011

Abstract. Starting from a basic model for solid tumors growth, a chaotic multi-scale cancer-invasion model is manufactured, which embeds a Lorenz attractor in its solutions. Furthermore we show how a laser can be expressed in terms of a Lorenz system and correspondences between the laser and the above mentioned chaotic multi-scale cancer-invasion model are proposed. Also, we show that the basic model for solid tumors growth admits a travelling wave solution and we suggest that metastatic tumor cells which move through the systemic circulation, and not necessarily there, are similar to a coherent wave, i.e. a travelling wave, chemically pumped oxygen type laser.

Keywords: carcinogenesis, tumor, fractal, travelling wave, holography.

1. Basic Model

1.1. The PDE Cancer-invasion Model

We consider and present in what follows in extenso, the basic mathematical model of growth of a generic solid tumour, which is assumed just been vascularised, i.e. a blood supply has been established. Let us focus on four key variables involved in tumour cell invasion, in order to produce a minimal

Corresponding author: e-mail: [email protected]

34 Drago Iancu et al.

model, namely tumour cell density (denoted by n), matrix-degradative enzymes (MDE) concentration (denoted by m), the complex mixture of macromolecules from the extracellular materials (MM) concentration (denoted by f ) and the oxygen concentration (denoted by c). Each of the four variables (n, m, f, c) is a function of the spatial variable x and time t. Initially, we define a system of coupled non-linear partial differential equations to model tumour invasion of surrounding tissue.

We will assume that the ECM consists of a mixture of MM (e.g. collagen, fibronectin, laminin and vitronectin) only and not any other cells. Most of the MM of the ECM which are important for cell adhesion, spreading and motility are fixed or bound to the surrounding tissue. MDEs are important at many stages of tumour growth, invasion and metastasis, and the manner in which they interact with inhibitors, growth factors and tumour cells is very complex. However, it is well known that the tumour cells produce MDEs which degrade the ECM locally. As well as making space into which tumour cells may move by simple diffusion (random motility), we assume that this also results in a gradient of these bound cell-adhesion molecules, such as fibronectin. Therefore, while the ECM may constitute a barrier to normal cell movement, it also provides a substrate to which cells may adhere and upon which they may move. Most mammalian cell types require at least some elements of the ECM to be present for growth and survival and will indeed migrate up a gradient of bound (i.e. non-diffusible) cell-adhesion molecules in culture in vitro (Carter, 1965; Quigley et al., 1983; Lacovara et al., 1984; McCarthy et al., 1984; Klominek et al., 1993; Lawrence et al., 1996).

By definition, haptotaxis is the directed migratory response of cells to gradients of fixed or bound chemicals (i.e. non-diffusible chemicals). While it has not yet been explicitly demonstrated that haptotaxis occurs in an in vivo situation, given the structure of human tissue, it is not unreasonable to assume that haptotaxis is a major component of directed movement in tumour cell invasion. Indeed, there has been much recent effort to characterise such directed movement (Klominek et al., 1993; Lawrence et al., 1996; Debruyne et al., 2002). We therefore refer to this directed movement of tumour cells in this model as haptotaxis, i.e. a response to gradients of bound MM such as fibronectin. To incorporate this response in the mathematical model, we take the haptotactic flux to be Jhapto = n f , where > 0 is the (constant) haptotactic coefficient.

As mentioned above, the only other contribution to tumour cell motility in the model is assumed to be random motion. To describe the random motility of the tumour cells, we assume a flux of the form Jrand = Dnn, where Dn is the constant random motility coefficient.

We only model the tumour cell migration at this level, as all other tumour cell processes, such as proliferation, adhesion and death will be considered at


the single cell level within the hybrid discrete-continuum model. The conservation equation for the tumour cell density n is therefore given by

( )rand hapto 0,nt

+ + =

J J

and hence the partial differential equation governing tumour cell motion (in the absence of cell proliferation) is

( )2nn

D n n ft

=

. (1)

The ECM is known to contain many MM, including fibronectin, laminin

and collagen, which can be degraded by MDEs (Stetler-Stevenson et al., 1996; Chambers et al., 1997). We assume that the MDEs degrade ECM upon contact and hence the degradation process is modelled by the following simple equation

f

mft

=

, (2)

where is a positive constant.

Active MDEs are produced (or activated) by the tumour cells, diffuse throughout the tissue and undergo some form of decay (either passive or active). The equation governing the evolution of MDE concentration is therefore given by

2 ( , ) ( , , ),m

mD m g n m h n m f

t

= +

(3)

where Dm is a positive constant, the MDE diffusion coefficient, g is a function modelling the production of active MDEs by the tumour cells and h is a function modelling the MDE decay. For simplicity we assume that there is a linear relationship between the density of tumour cells and the level of active MDEs in the surrounding tissues (regardless of the amount of enzyme precursors secreted and the presence of endogenous inhibitors) and so these functions are taken to be g = n (MDE production by the tumour cells) and h = =m (natural decay), respectively.

It is well known that solid tumours need oxygen to grow and invade.

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