**Implications of the "Subquantum Level" in Carcinogenesis and Tumor Progression via Scale Relativity Theory**

Daniel Timofte, Lucian Eva, Decebal Vasincu, Călin Gh. Buzea, Maricel Agop and Radu Florin Popa

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59233

#### **1. Introduction**

24 ime knjige

(2001).

217.

[38] K. Huang, *Statistical Mechanics*, Wiley, New York, Second edition (1963).

[41] E.H. Lieb, *Communications in Mathematical Physics* 62 (1978) 35.

[42] F. Lado, *Physics Letters* A 312 (2003) 101.

398 Selected Topics in Applications of Quantum Mechanics

[47] N. Ullah, *Physical Review* E 49 (1994) 1743.

6347; 30 (1997) 6327; 28 (1995) 6345.

(New York: Academic, 1965) p 229.

[50] M.M. Nieto, *Physical Review* D22 (1980) 391.

[44] A. Feldman, A.H. Kahn, Phys. Rev. B 1 (1970) 4584.

[45] F. Pennini, A. Plastino, *Physics Letter* A 349 (2006) 15.

[46] F. Pennini and S. Curilef Commun. Theor. Phys. 53 (2010) 535.

[48] F.J. Arranz, F. Borondo, R. M. Benito, *Physical Review* E 54 (1996) 2458.

[39] L.E. Reichl, *A Modern course in statistical physics*, Wiley, New York, Second edition, (1998).

[40] M. Janssen, *Fluctuation and localization in messoscopic electronic systems*, World Scientific,

[43] T. Chakraborty, P. Pietiläinen, *The Quantum Hall Effects*, Springer-Verlag, Berlin, (1995).

[49] J.T. Titantah, M. N. Hounkonnou, *Journal of Physics A: Math. Gen.* 32 (1999) 897; 30 (1997)

[51] J.J. Sakurai, Modern Quantum Mechanicsed S.F. Tuan (Reading, MA: Addison-Wesley, 1994) p

[52] J. Schwinger, In Quantum Theory of Angular Momentumed, L C Biedenharn and H van Dam

[53] C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics vol.1 John Wiley & Sons, (1977).

[54] J.A. Morales, E. Deumens and Y Öhrn, *Journal of Mathematical Physics* 40 (1999) 776.

The last 25 years witnessed tremendous achievements in cancer diagnose and treatment. Technology currently permits small size tumors (like breast cancers) diagnosis and treatment, ductal cancer in situ currently including 25...30% of all freshly diagnosed breast cancers at the majority of medical centers [1]. Thus, early detection now allows the understanding of growth patterns. Surgeons are in the front line of technological and basic scientific medical advances. Current ideas, such as the physiological characteristics of shock, organ transplantation, antisepsis, wound healing, or sequence medical care, are cast by surgical investigators.

The field of mathematics suffered an identical evolution. Revolutionary mathematical branches such as topology, fractals, chaos theory, and development of nonlinear descriptive strategies have provided mathematicians new inventive tools to create growth models and to behaviors at the small environmental level [2, 3]. Growth, angiogenesis [4], cell-to-cell adhesion [5], hydrogen ion concentration regulation and drug delivery [6] can now systematically be described using specific formulas. From a clinical viewpoint several of these formulas could seem simple, however they put together a very important foundation for descriptive insight.

What is currently lacking is a connection between these two naturally and mutual analysis endeavors. For oncology surgeons, the ability to mathematically analyze and predict patterns of growth provides precise techniques that are beneficent for both current and future therapies. For mathematicians, defining the clinical factors essential for growth development and metastasis can provide realistic insight into these biological processes, successively allowing the event of correct, clinically relevant mathematical formulas. In almost every dedicated

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

medical institutions, the teams that have comprehensive cancer are being led by surgeons. Mathematics that can be applied in the oncology field provide a chance to expand the leader‐ ship role of the surgeons and to raise awareness about the significance of understanding growth behavior and, also improve cancer treatments.

The present study aims at defining 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, questing 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 metastatic tumor cells move (through the systemic circulation, yet not necessarily only there) as a coherent wave, or even more precisely, a chemically pumped travelling laser wave 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 be proved 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 object's 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.

According to the Pribram-Bohm's holographic theory (http://en.wikipedia.org/wiki/Holo‐ nomic\_brain\_theory) of the brain, the intercellular and intracellular communication implies the existence of a fractal medium equivalent to the vacuum between the elementary particles. Since vacuum dynamics is studied using quantum mechanics, it is only natural that the status of the fractal medium implies the use of a quantum type mathematical formalism (i.e. Scale Relativity theory either in its Schrödinger type representation or the fractal hydrodynamic one) applicable to different resolution scales (mesoscopic for intercellular communications or nanoscopic for intracellular ones). In Nottale's interpretation, each resolution scale is charac‐ terized by a Planck type constant *h* ¯

$$\overline{h} = \hbar \left(\frac{dt}{\pi}\right)^{(2fD\_{\overline{r}})-1}$$

where *ħ* is the standard Planck's constant, *dt* is time's resolution scale, *τ* the time's reference scale and *D*F is the fractal dimension of the motion curve.

#### **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 destroy it) and metastasis (they spread outside the location of the body where they were originally generated). Additionally, the term tumor or neoplasm is used to indicate an abnormal swelling of tissue caused by an excessive cell proliferation.

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

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 tumor cell, after having collected a set of 5...8 crucial genetic alterations [7-9] as schematically shown in Fig. 1. **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 destroy it) and metastasis (they spread outside the location of the body where they were originally generated). Additionally, the term tumor or neoplasm is used to indicate an abnormal swelling of tissue caused by an excessive cell

A newborn malignant cell, expressing aberrant traits, can lead to the formation of cancer and, in most of the cases, of a tumor. 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 industrialized 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 [10], Greeks [11] and Romans [12] accurately described medical treatments for tumor 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. proliferation. A tumor can be of benign or malignant nature, while benign tumors are self-limiting, do not express patterns of invasion, and they do not metastasize, malignant tumors do possess all these characteristics. The term malignant tumor is also used as synonym for cancer, although some cancers, such as leukemia, do not form tumors. 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 tumor cell, after having collected a set of 5…8 crucial genetic alterations [7-9] 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 tumor. 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 industrialized 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 [10], Greeks [11] and Romans [12] accurately described medical treatments for tumor 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** í Acquisition of the tumorigenic phenotype by a population of normal cells through multiple genetic mutations. **Figure 1.** Acquisition of the tumorigenic phenotype by a population of normal cells through multiple genetic mutations.

#### **2.2. Distinguishing Traits of Cancer**  The tumorigenic properties, generically discussed in the previous section, have shown to be common to almost all cancers. **2.2. Distinguishing traits of cancer**

stimulates its own receptors as shown in Fig. 2.

medical institutions, the teams that have comprehensive cancer are being led by surgeons. Mathematics that can be applied in the oncology field provide a chance to expand the leader‐ ship role of the surgeons and to raise awareness about the significance of understanding

The present study aims at defining 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, questing 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 metastatic tumor cells move (through the systemic circulation, yet not necessarily only there) as a coherent wave, or even more precisely, a chemically pumped travelling laser wave 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 be proved 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 object's 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.

¯

scale and *D*F is the fractal dimension of the motion curve.

According to the Pribram-Bohm's holographic theory (http://en.wikipedia.org/wiki/Holo‐ nomic\_brain\_theory) of the brain, the intercellular and intracellular communication implies the existence of a fractal medium equivalent to the vacuum between the elementary particles. Since vacuum dynamics is studied using quantum mechanics, it is only natural that the status of the fractal medium implies the use of a quantum type mathematical formalism (i.e. Scale Relativity theory either in its Schrödinger type representation or the fractal hydrodynamic one) applicable to different resolution scales (mesoscopic for intercellular communications or nanoscopic for intracellular ones). In Nottale's interpretation, each resolution scale is charac‐

(2 1 ) *DF*

*dt h*

æ ö = ç ÷ è ø t h

where *ħ* is the standard Planck's constant, *dt* is time's resolution scale, *τ* the time's reference

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

growth behavior and, also improve cancer treatments.

400 Selected Topics in Applications of Quantum Mechanics

terized by a Planck type constant *h*

**2. The biology of cancer**

**2.1. Cancer, what should be noticed**

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 [13]: *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 The tumorigenic 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.  In order to lead to cancer, tumor 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 [9,14]. 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 neighbors and vice-versa or even autocrine signaling, when the cell

2

**Fig. 2** í 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).

stimulates its own receptors as shown in Fig. 2.

cancer, although some cancers, such as leukemia, do not form tumors.

crucial genetic alterations [7-9] as schematically shown in Fig. 1.

the motion curve.

proliferation.

**2. The Biology of Cancer**

**2.1 Cancer, What Should be Noticed**

*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, tumor 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 [9,14]. As consequence, the associated component of the signaling system would become constitutively active and hence, independ‐ ent 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 neighbors and vice-versa or even autocrine signaling, when the cell stimulates its own receptors as shown in Fig. 2. **Fig. 1** í Acquisition of the tumorigenic phenotype by a population of normal cells through multiple genetic mutations. **2.2. Distinguishing Traits of Cancer**  The tumorigenic 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 [13]: *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, tumor 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 [9,14]. 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

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 destroy it) and metastasis (they spread outside the location of the body where they were originally generated). Additionally, the term tumor or neoplasm is used to indicate an abnormal swelling of tissue caused by an excessive cell

A tumor can be of benign or malignant nature, while benign tumors are self-limiting, do not express patterns of invasion, and they do not metastasize, malignant tumors do possess all these characteristics. The term malignant tumor is also used as synonym for

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 tumor cell, after having collected a set of 5…8

A newborn malignant cell, expressing aberrant traits, can lead to the formation of cancer and, in most of the cases, of a tumor. 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 industrialized 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 [10], Greeks [11] and Romans [12] accurately described medical treatments for tumor 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.

growth by paracrine signaling, where a cells stimulates the neighbors and vice-versa or even autocrine signaling, when the cell

**Fig. 2** í 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). **Figure 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).

2 *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 retinoblas‐ toma 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 [15]. 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 phagocy‐ tosis. 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 30...120 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 [16]. 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 tumor 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 tumor suppressor gene [17]. *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 [15]. The disruption of such pathway results in the insensitivity of the cell to anti-growth signals.

*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, tumor 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 [9,14]. As consequence, the associated component of the signaling system would become constitutively active and hence, independ‐ ent 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 neighbors and vice-versa or even autocrine signaling, when the cell stimulates its own receptors as shown in Fig. 2.

**Fig. 1** í Acquisition of the tumorigenic phenotype by a population of normal cells through multiple genetic mutations.

The tumorigenic 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

*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, tumor 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 [9,14]. 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 neighbors and vice-versa or even autocrine signaling, when the cell

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 destroy it) and metastasis (they spread outside the location of the body where they were originally generated). Additionally, the term tumor or neoplasm is used to indicate an abnormal swelling of tissue caused by an excessive cell

A tumor can be of benign or malignant nature, while benign tumors are self-limiting, do not express patterns of invasion, and they do not metastasize, malignant tumors do possess all these characteristics. The term malignant tumor is also used as synonym for

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 tumor cell, after having collected a set of 5…8

A newborn malignant cell, expressing aberrant traits, can lead to the formation of cancer and, in most of the cases, of a tumor. 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 industrialized 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 [10], Greeks [11] and Romans [12] accurately described medical treatments for tumor 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.

the motion curve.

proliferation.

**2. The Biology of Cancer**

**2.1 Cancer, What Should be Noticed**

**2.2. Distinguishing Traits of Cancer**

stimulates its own receptors as shown in Fig. 2.

cell to anti-growth signals.

neoplasm [13]:

cancer, although some cancers, such as leukemia, do not form tumors.

crucial genetic alterations [7-9] as schematically shown in Fig. 1.

402 Selected Topics in Applications of Quantum Mechanics

2

*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 phagocy‐ tosis. 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 30...120 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 [16]. Apoptosis is a common mechanism of cell death and takes part in the homeostasis

*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 retinoblas‐ toma 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 [15]. The disruption of such pathway results in the insensitivity of the

**Fig. 2** í 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). **Figure 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).

*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 tumor. 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 mitosis 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 5'...3' 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 [18] 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 telo‐ meres, 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 immortalized (unlimited proliferative potential) by finding ways to maintain their telomeres long enough. A possible way is the over expression of the telomerase gene [19] which appears to take place in 85...90 % of cancers. Telomerase is a telomere-rebuilding enzyme normally expressed in germ line cells and stem cells, in which immortalization is an essential feature. Once immor‐ talized, malignant cells have made a further step towards the formation of cancer. *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 30…120 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 [16]. 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 tumor 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 tumor suppressor gene [17]. *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 tumor. 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 mitosis 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 5'…3' 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 [18] 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 immortalized (unlimited proliferative potential) by finding ways to maintain their telomeres long enough. A possible way is the over expression of the telomerase gene [19] which appears to take place in 85…90 % of cancers. Telomerase is a telomererebuilding enzyme normally expressed in germ line cells and stem cells, in which immortalization is an essential feature. Once immortalized, malignant cells have made a further step towards the formation of cancer.

**Fig. 3** í 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). **Figure 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 complet‐ ed until the very end (Dr. R. Beijersbergen, Netherlands Cancer Institute).

angiogenesis-triggering capabilities [20, 21]. 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 [22]. Angiogenesis is the principal mechanism that transforms a microscopic malignancy into a macroscopic tumor 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

Local pre-existent vascularisation is never enough to sustain growth for more than 108

thrombospondin-1 [23] and bevacizumab [24], also known as avastin.

*Sustained Angiogenesis.* In order for a cell to survive normally, it must rely within 100 Pm from a capillary blood vessel [13]. For this reason, the initial exponential growth of a newborn malignant neoplasm causes a shortage of nutrients among cancer cells.

cells. The colony must therefore develop

3

*Sustained Angiogenesis.* In order for a cell to survive normally, it must rely within 100 μm from a capillary blood vessel [13]. 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 [20, 21]. 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 [22]. Angiogenesis is the principal mechanism that transforms a microscopic malignancy into a macroscopic tumor 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 [23] and bevacizumab [24], also known as avastin.

**Fig. 4** í 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 immortalized 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 **Figure 4.** The progressive shortening of the telomeres leads to a massive cell death due to the induced genomic insta‐ bility (death by genomic catastrophe while duplicating). From such process of intense genetic mutation and selection an immortalized clone could emerge (Dr. R. Beijersbergen, Netherlands Cancer Institute).

metastasis. Its ability of forming distant colonies or metastases all over the body represents the cause of 90% of all cancer related deaths [25]. 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 [26]. Contrary to their normal counterparts, cancer cells are able to survive the loss of anchorage, to travel through the vascular system and form distant tumors 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 [27-29] making space for invasion and freeing growth and angiogenic factors trapped inside. *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 [25]. 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 microenviron‐ ment – see Sec. 6.1) would occur in a form of apoptosis called anoikis [26]. Contrary to their normal counterparts, cancer cells are able to survive the loss of anchorage, to travel through the vascular system and form distant tumors elsewhere (Fig. 5). The traits expressed by invasive and metastatic cancer cells are principally loss of cell-to-cell adhesion, anchorage-

4

**Fig. 5** í 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 [27].

In comparison to biology, cell biology, and drug delivery analysis, mathematics has, to date, made comparatively very few contributions to this field of research. A statistical analysis of the PubMed platform list information (http://www.ncbi.nlm. nih.gov/PubMed/) showed that out of 1.5 million works that deal with cancer analysis, only 5% are associated with mathematical modeling. However, it is clear that mathematics could contribute significantly to areas of experimental cancer analysis since there is

Even in these conditions, in the last decade, mathematical modeling and machine simulation of cancer has multiplied dramatically (e.g., reviews like [30-35]. A broad range of strategies were developed, specializing in one or additional aspects of cancer. For example, genetic instability, natural selection or interactions of individual cell with each other or the environment have been modeled using methods of cellular automata and agent-based modeling. These discrete methods have the disadvantage of being difficult to use when we deal with tumors of significant size. (see [36-38] for samples of cellular automata modeling and [39,40] for

**3. Mathematics of Cancer**

currently a wealth of experimental information which needs a systematic analysis.

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 [27-29] making space for invasion and freeing growth and angiogenic factors trapped inside. 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 [26]. Contrary to their normal counterparts, cancer cells are able to survive the loss of anchorage, to travel through the vascular system and form distant tumors 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 [27-29] making space for invasion and freeing growth and angiogenic factors trapped inside.

**Fig. 4** í 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 immortalized clone could

deaths [25]. Normal cells are usually unable to travel outside their own tissue due to their necessity to be anchored and reside among

**Fig. 5** í 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 [27]. **Figure 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 [27].

#### In comparison to biology, cell biology, and drug delivery analysis, mathematics has, to date, made comparatively very few contributions to this field of research. A statistical analysis of the PubMed platform list information (http://www.ncbi.nlm. **3. Mathematics of cancer**

**3. Mathematics of Cancer**

*Sustained Angiogenesis.* In order for a cell to survive normally, it must rely within 100 μm from a capillary blood vessel [13]. 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 [20, 21]. 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 [22]. Angiogenesis is the principal mechanism that transforms a microscopic malignancy into a macroscopic tumor 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 [23] and bevacizumab [24], also known as avastin.

404 Selected Topics in Applications of Quantum Mechanics

**Fig. 4** í 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 immortalized 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 [25]. 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 [26]. Contrary to their normal counterparts, cancer cells are able to survive the loss of anchorage, to travel through the vascular system and form distant tumors 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

**Figure 4.** The progressive shortening of the telomeres leads to a massive cell death due to the induced genomic insta‐ bility (death by genomic catastrophe while duplicating). From such process of intense genetic mutation and selection

*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 [25]. 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 microenviron‐ ment – see Sec. 6.1) would occur in a form of apoptosis called anoikis [26]. Contrary to their normal counterparts, cancer cells are able to survive the loss of anchorage, to travel through the vascular system and form distant tumors elsewhere (Fig. 5). The traits expressed by invasive and metastatic cancer cells are principally loss of cell-to-cell adhesion, anchorage-

extracellular matrix [27-29] making space for invasion and freeing growth and angiogenic factors trapped inside.

an immortalized clone could emerge (Dr. R. Beijersbergen, Netherlands Cancer Institute).

4

**Fig. 5** í 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 [27].

In comparison to biology, cell biology, and drug delivery analysis, mathematics has, to date, made comparatively very few contributions to this field of research. A statistical analysis of the PubMed platform list information (http://www.ncbi.nlm. nih.gov/PubMed/) showed that out of 1.5 million works that deal with cancer analysis, only 5% are associated with mathematical modeling. However, it is clear that mathematics could contribute significantly to areas of experimental cancer analysis since there is

Even in these conditions, in the last decade, mathematical modeling and machine simulation of cancer has multiplied dramatically (e.g., reviews like [30-35]. A broad range of strategies were developed, specializing in one or additional aspects of cancer. For example, genetic instability, natural selection or interactions of individual cell with each other or the environment have been modeled using methods of cellular automata and agent-based modeling. These discrete methods have the disadvantage of being difficult to use when we deal with tumors of significant size. (see [36-38] for samples of cellular automata modeling and [39,40] for

**3. Mathematics of Cancer**

currently a wealth of experimental information which needs a systematic analysis.

modeling. However, it is clear that mathematics could contribute significantly to areas of experimental cancer analysis since there is currently a wealth of experimental information which needs a systematic analysis. Even in these conditions, in the last decade, mathematical modeling and machine simulation of cancer has multiplied dramatically (e.g., reviews like [30-35]. A broad range of strategies were developed, specializing in one or additional aspects of cancer. For example, genetic instability, natural selection or interactions of individual cell with each other or the environment have been modeled using methods of cellular automata and agent-based modeling. These discrete methods have the disadvantage of being difficult to use when we deal with tumors of significant size. (see [36-38] for samples of cellular automata modeling and [39,40] for In comparison to biology, cell biology, and drug delivery analysis, mathematics has, to date, made comparatively very few contributions to this field of research. A statistical analysis of the PubMed platform list information (http://www.ncbi.nlm. nih.gov/PubMed/) showed that out of 1.5 million works that deal with cancer analysis, only 5% are associated with mathe‐ matical modeling. However, it is clear that mathematics could contribute significantly to areas of experimental cancer analysis since there is currently a wealth of experimental information which needs a systematic analysis.

nih.gov/PubMed/) showed that out of 1.5 million works that deal with cancer analysis, only 5% are associated with mathematical

4 Even in these conditions, in the last decade, mathematical modeling and machine simulation of cancer has multiplied dramatically (e.g., reviews like [30-35]. A broad range of strategies were developed, specializing in one or additional aspects of cancer. For example, genetic instability, natural selection or interactions of individual cell with each other or the environ‐ ment have been modeled using methods of cellular automata and agent-based modeling. These discrete methods have the disadvantage of being difficult to use when we deal with tumors of significant size. (see [36-38] for samples of cellular automata modeling and [39,40] for samples of agent-based modeling). In systems at larger scales, the neoplasic cell population is of the order of 106 or more, making these discrete methods unfitted. For these situations, the continuum methods provide the best approach. Early work, as well as [41-43], used ODE to model cancer as a uniform population and partial differential equation models restricted to spherical geometries. To assess the stability of spherical tumors to asymmetric perturbations and to characterize the degree of aggression [31, 44-48] use linear and weakly nonlinear analysis. The interactions of a growth with the microenvironment, like stress-induced limita‐ tions to growth, are studied in [30,49-54]. For the sake of simplicity, most of the modeling has considered single-phase (e.g., single cell species) tumors. To provide an elaborate account of growth non-uniformity, [50, 55, 56] have been developed a mixture of models.

The results of morphology instabilities on each avascular and vascular solid neoplasm growth have been recently studied using non-linear modeling. With the help of boundary integral methods, Cristini et al [47] performed the first absolutely nonlinear simulations of a time model of neoplasm growth within the avascular and vascular growth stages with arbitrary bounda‐ ries. The model from [47] has been extended in 3D by Li et al. [48] via adaptive boundary integral technique. The inclusion of angiogenesis and extratumoral environment has been performed by Zheng et al., [57]. By developing and coupling a level set implementation with a hybrid continuum-discrete growing model originally developed by Anderson & Chaplain [58]they 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 impact of therapy on neoplasm growth [59] and for studies of morphological instability and invasion [60-62]. Macklin & Lowengrub used a ghost cell/level set technique for evolving interfaces to check neoplasm growth in heterogeneous tissue and additional studied neoplasm growth as a function of the microenvironment [63]. Wise et al. [64] and Frieboes et al. [65] have developed a diffuse interface implementation of solid neoplasm growth for the study of the evolution of multiple neoplasm cell species, that was used in [65] to model the 3-D vascularised growth of malignant gliomas (brain tumors).

In biological systems, the fractal structure of area in which cells act and differentiate is important for their organization and emergence of the hierarchical network of multiple crossinteracting cells, sensitive to external and internal conditions. The biological phenomena occur within the area whose dimensions aren't represented solely by integers (1, 2, 3, etc.) of Euclidean space. Particularly, malignant tumors [53-56] grow in a space with non-integer dimension, i.e. fractal dimension. The analytical formulae describing the time-dependence of the temporal fractal dimension and scaling reproduce the expansion of the Flexner–Jobling rat's neoplasm in particular and growth of different rat's tumors generally. The results of some calculations indicated that the formula derived for the time-dependent temporal fractal dimension and the scaling factor describe the experimental data obtained by Schrek for the Brown-Pearce rabbit's neoplasm growth within the fractal time-space [3, 66-68].

In our assertion, fractal space(-time) consists in developing the consequences of the withdrawal of space(-time) differentiability's hypothesis and acquiring a fractal geometry, namely space( time) becomes explicitly dependent on the observation scale [69].

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 non–integer, fractal dimension and are termed "strange attractors" [70]. 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 [71]. 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 histological section of a breast carcinoma [72].

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 anticontrol.

#### **4. A few words about holography**

These discrete methods have the disadvantage of being difficult to use when we deal with tumors of significant size. (see [36-38] for samples of cellular automata modeling and [39,40] for samples of agent-based modeling). In systems at larger scales, the neoplasic cell population

continuum methods provide the best approach. Early work, as well as [41-43], used ODE to model cancer as a uniform population and partial differential equation models restricted to spherical geometries. To assess the stability of spherical tumors to asymmetric perturbations and to characterize the degree of aggression [31, 44-48] use linear and weakly nonlinear analysis. The interactions of a growth with the microenvironment, like stress-induced limita‐ tions to growth, are studied in [30,49-54]. For the sake of simplicity, most of the modeling has considered single-phase (e.g., single cell species) tumors. To provide an elaborate account of

The results of morphology instabilities on each avascular and vascular solid neoplasm growth have been recently studied using non-linear modeling. With the help of boundary integral methods, Cristini et al [47] performed the first absolutely nonlinear simulations of a time model of neoplasm growth within the avascular and vascular growth stages with arbitrary bounda‐ ries. The model from [47] has been extended in 3D by Li et al. [48] via adaptive boundary integral technique. The inclusion of angiogenesis and extratumoral environment has been performed by Zheng et al., [57]. By developing and coupling a level set implementation with a hybrid continuum-discrete growing model originally developed by Anderson & Chaplain [58]they 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 impact of therapy on neoplasm growth [59] and for studies of morphological instability and invasion [60-62]. Macklin & Lowengrub used a ghost cell/level set technique for evolving interfaces to check neoplasm growth in heterogeneous tissue and additional studied neoplasm growth as a function of the microenvironment [63]. Wise et al. [64] and Frieboes et al. [65] have developed a diffuse interface implementation of solid neoplasm growth for the study of the evolution of multiple neoplasm cell species, that was used in [65] to model the 3-D vascularised growth of

In biological systems, the fractal structure of area in which cells act and differentiate is important for their organization and emergence of the hierarchical network of multiple crossinteracting cells, sensitive to external and internal conditions. The biological phenomena occur within the area whose dimensions aren't represented solely by integers (1, 2, 3, etc.) of Euclidean space. Particularly, malignant tumors [53-56] grow in a space with non-integer dimension, i.e. fractal dimension. The analytical formulae describing the time-dependence of the temporal fractal dimension and scaling reproduce the expansion of the Flexner–Jobling rat's neoplasm in particular and growth of different rat's tumors generally. The results of some calculations indicated that the formula derived for the time-dependent temporal fractal dimension and the scaling factor describe the experimental data obtained by Schrek for the

In our assertion, fractal space(-time) consists in developing the consequences of the withdrawal of space(-time) differentiability's hypothesis and acquiring a fractal geometry, namely space(-

Brown-Pearce rabbit's neoplasm growth within the fractal time-space [3, 66-68].

time) becomes explicitly dependent on the observation scale [69].

growth non-uniformity, [50, 55, 56] have been developed a mixture of models.

or more, making these discrete methods unfitted. For these situations, the

is of the order of 106

406 Selected Topics in Applications of Quantum Mechanics

malignant gliomas (brain tumors).

"Although it generates a three-dimensional image, a hologram is most often recorded on a photographic plate or a flat piece of film. Moreover, producing a hologram does not imply, in the conventional sense, the recording of an image. To better understand this apparent paradox and, as a result, the way holography works, we have to begin with the main principles.

In conventional imaging techniques, e.g. photography, what is being recorded is merely the intensity distribution in the original scene. Thus, all information about the optical paths to different parts of the scene is lost.

The unique property of holography is the method 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 mandatory to convert the phase information into intensity varia‐ tions. Holography accomplishes 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 beam and the light waves scattered by the object in cause.

Since the intensity at any given point in this pattern of interference 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 character‐ ized the original object.

**Fig. 6** í Hologram recording: the interference pattern produced by the reference wave and the object wave is recorded. **Figure 6.** Hologram recording: the interference pattern produced by the reference wave and the object wave is recorded.

An observer looking through the hologram sees a perfect three-dimensional image. This image exhibits all the effects of

#### **4.1. Early development** perspective, and depth of focus when photographed, that characterized the original object.

**4.1. Early Development**

**4.2. The In-line Hologram**

plate.

beam.

Gabor's historical experiment of holographic imaging [73] consisted in a transparency formed of opaque lines on a clear background which was illuminated with a collimated beam of monochromatic light, the interference pattern produced by the directly transmitted beam (the reference wave) and the light scattered by the lines on the transparency being recorded on a photographic plate. When the hologram (i.e. a positive transparency made from this photo‐ graphic negative) was illuminated with the original collimated beam, it produced two diffracted waves, one which reconstructed an image of the object in its original location, and the other, with identical amplitude but an opposite phase, which formed a second, *conjugate* image. **Fig. 6** í Hologram recording: the interference pattern produced by the reference wave and the object wave is recorded. 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. **4.1. Early Development**  Gabor's historical experiment of holographic imaging [73] consisted in a transparency formed of opaque lines on a clear background which was illuminated with a collimated beam of monochromatic light, the interference pattern produced by the directly transmitted beam (the reference wave) and the light scattered by the lines on the transparency being recorded on a photographic plate. When the hologram (i.e. a positive transparency made from this photographic negative) was illuminated with the original collimated Gabor's historical experiment of holographic imaging [73] consisted in a transparency formed of opaque lines on a clear background which was illuminated with a collimated beam of monochromatic light, the interference pattern produced by the directly transmitted beam (the reference wave) and the light scattered by the lines on the transparency being recorded on a photographic plate. When the hologram (i.e. a positive transparency made from this photographic negative) was illuminated with the original collimated beam, it produced two diffracted waves, one which reconstructed an image of the object in its original location, and the other, with identical amplitude but an opposite phase, which formed a second, *conjugate* image.

beam, it produced two diffracted waves, one which reconstructed an image of the object in its original location, and the other, with

**Fig. 7** í Image reconstruction: light diffracted by the hologram reconstructs the object wave. **Figure 7.** Image reconstruction: light diffracted by the hologram reconstructs the object wave. not overlap. The improvement of the off-axis technique, and, in equal measure, 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 crucial applications.

identical amplitude but an opposite phase, which formed a second, *conjugate* image.

was degraded by the conjugate image that was superimposing on it as well as by the scattered light from the directly transmitted beam. Leith and Upatnieks [74-76] found a solution to the above-mentioned problem developing a off-axis reference beam technique presented 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 An important flaw of this method of image reconstruction was the poor quality of the resulting image, due to the fact that it was degraded by the conjugate image that was superimposing on it as well as by the scattered light from the directly transmitted beam. **4.2. The In-line Hologram**  Let us now look upon the optical system presented 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.

images were separated by large enough angles from the directly transmitted beam, and from each other, thus ensuring that the images

An important flaw of this method of image reconstruction was the poor quality of the resulting image, due to the fact that it

images were separated by large enough angles from the directly transmitted beam, and from each other, thus ensuring that the images not overlap. The improvement of the off-axis technique, and, in equal measure, 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 crucial applications. Leith and Upatnieks [74-76] found a solution to the above-mentioned problem developing a off-axis reference beam technique presented schematically in Figs. 6 and 7. They used a

6

Let us now look upon the optical system presented 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

6

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, thus ensuring that the images not overlap.

The improvement of the off-axis technique, and, in equal measure, the invention of the laser, which provided a powerful source of coherent light, resulted in a surge of activity in holog‐ raphy that led to several crucial applications.

#### **4.2. The in-line hologram**

**4.1. Early development**

408 Selected Topics in Applications of Quantum Mechanics

**4.1. Early Development**

image.

beam.

plate.

beam.

not overlap.

plate.

not overlap.

**4.2. The In-line Hologram**

**4.2. The In-line Hologram**

**4.1. Early Development**

Gabor's historical experiment of holographic imaging [73] consisted in a transparency formed of opaque lines on a clear background which was illuminated with a collimated beam of monochromatic light, the interference pattern produced by the directly transmitted beam (the reference wave) and the light scattered by the lines on the transparency being recorded on a photographic plate. When the hologram (i.e. a positive transparency made from this photo‐ graphic negative) was illuminated with the original collimated beam, it produced two diffracted waves, one which reconstructed an image of the object in its original location, and the other, with identical amplitude but an opposite phase, which formed a second, *conjugate*

**Fig. 6** í Hologram recording: the interference pattern produced by the reference wave and the object wave is recorded.

**Figure 6.** Hologram recording: the interference pattern produced by the reference wave and the object wave is recorded.

An observer looking through the hologram sees a perfect three-dimensional image. This image exhibits all the effects of

Gabor's historical experiment of holographic imaging [73] consisted in a transparency formed of opaque lines on a clear background which was illuminated with a collimated beam of monochromatic light, the interference pattern produced by the directly transmitted beam (the reference wave) and the light scattered by the lines on the transparency being recorded on a photographic plate. When the hologram (i.e. a positive transparency made from this photographic negative) was illuminated with the original collimated beam, it produced two diffracted waves, one which reconstructed an image of the object in its original location, and the other, with

**Fig. 6** í Hologram recording: the interference pattern produced by the reference wave and the object wave is recorded.

perspective, and depth of focus when photographed, that characterized the original object.

identical amplitude but an opposite phase, which formed a second, *conjugate* image.

perspective, and depth of focus when photographed, that characterized the original object.

identical amplitude but an opposite phase, which formed a second, *conjugate* image.

An observer looking through the hologram sees a perfect three-dimensional image. This image exhibits all the effects of

Gabor's historical experiment of holographic imaging [73] consisted in a transparency formed of opaque lines on a clear background which was illuminated with a collimated beam of monochromatic light, the interference pattern produced by the directly transmitted beam (the reference wave) and the light scattered by the lines on the transparency being recorded on a photographic plate. When the hologram (i.e. a positive transparency made from this photographic negative) was illuminated with the original collimated beam, it produced two diffracted waves, one which reconstructed an image of the object in its original location, and the other, with

6

**Fig. 7** í Image reconstruction: light diffracted by the hologram reconstructs the object wave.

**Fig. 7** í Image reconstruction: light diffracted by the hologram reconstructs the object wave.

An important flaw of this method of image reconstruction was the poor quality of the resulting image, due to the fact that it was degraded by the conjugate image that was superimposing on it as well as by the scattered light from the directly transmitted

Leith and Upatnieks [74-76] found a solution to the above-mentioned problem developing a off-axis reference beam technique presented 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, thus ensuring that the images

An important flaw of this method of image reconstruction was the poor quality of the resulting image, due to the fact that it was degraded by the conjugate image that was superimposing

Leith and Upatnieks [74-76] found a solution to the above-mentioned problem developing a off-axis reference beam technique presented schematically in Figs. 6 and 7. They used a

An important flaw of this method of image reconstruction was the poor quality of the resulting image, due to the fact that it was degraded by the conjugate image that was superimposing on it as well as by the scattered light from the directly transmitted

The improvement of the off-axis technique, and, in equal measure, the invention of the laser, which provided a powerful

Leith and Upatnieks [74-76] found a solution to the above-mentioned problem developing a off-axis reference beam technique presented 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, thus ensuring that the images

Let us now look upon the optical system presented 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

The improvement of the off-axis technique, and, in equal measure, the invention of the laser, which provided a powerful

Let us now look upon the optical system presented 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

6

source of coherent light, resulted in a surge of activity in holography that led to several crucial applications.

**Figure 7.** Image reconstruction: light diffracted by the hologram reconstructs the object wave.

source of coherent light, resulted in a surge of activity in holography that led to several crucial applications.

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

Let us now look upon the optical system presented 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.

We can observe two components of the incident light. The first is the directly transmitted wave, which is a plane wave whose amplitude and phase do not vary across the photographic plate. Thus, its complex amplitude can be noted as a real constant *r*. The **Figure 8.** Optical system used to record an in-line hologram.

where |*o*(*x, y*)| << *r*. From these it can be shown that the resulting complex amplitude is the sum of these two complex amplitudes, and because of that the intensity at this point is ( , ) ( , ) ( , ) ( , ) ( , ) 2 2 2 *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. Therefore it can be assumed 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 We can observe two components of the incident light. The first is the directly transmitted wave, which is a plane wave whose amplitude and phase do not vary across the photographic plate. Thus, its complex amplitude can be noted as a real constant *r*. The second one is a weak scattered wave whose complex amplitude at any point (*x, y*) on the photographic plate can be noted as *o*(*x, y*), where |*o*(*x, y*)| << *r*.

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

second one is a weak scattered wave whose complex amplitude at any point (*x, y*) on the photographic plate can be noted as *o*(*x, y*),

�*TI* 0 (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 ( , ) [ ( , ) ( , ) ( , )] 2 2 0 *t x y t T r o x y ro x y ro x y* \* � � � � � � (3) From these it can be shown that the resulting complex amplitude is the sum of these two complex amplitudes, and because of that the intensity at this point is

Then, the hologram is illuminated, as shown in Fig. 9, with the same collimated beam of monochromatic light employed to

�*Tr*2

the object and has the property of reconstructing an image of the object in its original position. Due to the fact that this image forms

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 problem is that the object must have a high average transmittance in order for the second term on the right-hand side of (4) to be negligible. Thus, 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,

*t* � *t* �

7

$$I(\mathbf{x}, \mathbf{y}) = \left| r + o(\mathbf{x}, \mathbf{y}) \right|^2 = r^2 + \left| o(\mathbf{x}, \mathbf{y}) \right|^2 + r o(\mathbf{x}, \mathbf{y}) + r o^\*(\mathbf{x}, \mathbf{y}) \tag{1}$$

*o*(*x, y*), is, except for a constant factor, identical with the complex amplitude of the scattered wave from

*o*\*(*x, y*), represents a wave similar to the object wave, but having an opposite curvature. This wave

), which represents a uniformly attenuated plane wave,

(4)

�in (2) is

 corresponds to the directly transmitted beam. The second, �*Tr* |*o*(*x, y*)|2 , can be neglected, because is extremely small, compared to the other terms. where *o*\*(*x, y*) is the complex conjugate of *o*(*x, y*).

The third term,

The fourth term,

�*Tr*2

> �*Tr*2

The right-hand side of (4) contains four terms. The first, *r*(t0+

behind the hologram, and the reconstructed wave appears to diverge from it, it is a virtual image.

negative, and the reconstructed image can be considered a photographic negative of the object.

converges to form a real image (the conjugate image) at the same distance in front of the hologram.

A 'positive' transparency (the hologram) is then made by contact printing from this recording. Therefore it can be assumed that this transparency is processed so that its amplitude trans‐ mittance (the ratio of the transmitted amplitude to that incident on it) can be written as

$$t = t\_0 + \mathbb{B}TI \tag{2}$$

where t0 is a constant background transmittance, *T* is the exposure time and *β* is a parameter determined by the photographic material used and the processing conditions, the amplitude transmittance of the hologram is

$$t(\mathbf{x}, \mathbf{y}) = t\_0 + \beta T \left[ r^2 + \left| o(\mathbf{x}, \mathbf{y}) \right|^2 + r o(\mathbf{x}, \mathbf{y}) + r o^\*(\mathbf{x}, \mathbf{y}) \right] \tag{3}$$

Then, the hologram is illuminated, as shown in Fig. 9, with the same collimated beam of monochromatic light employed to produce the original recording. Since the complex ampli‐ tude at any point in this beam is, aside from a constant factor, the same as that in the original reference beam, the complex amplitude transmitted by the hologram can be written as

$$\ln(\mathbf{x}, \mathbf{y}) = r t(\mathbf{x}, \mathbf{y}) = r(t\_0 + \beta \, T r^2) + \beta \, T r \left| o(\mathbf{x}, \mathbf{y}) \right|^2 + \beta \, T r^2 o(\mathbf{x}, \mathbf{y}) + \beta \, T r^2 o^\*(\mathbf{x}, \mathbf{y}) \tag{4}$$

The right-hand side of (4) contains four terms. The first, *r*(t0+*βTr*2 ), which represents a uniformly attenuated plane wave, corresponds to the directly transmitted beam.

The second, *βTr* |*o*(*x, y*)|2 , can be neglected, because is extremely small, compared to the other terms.

The third term, *βTr*2 *o*(*x, y*), is, except for a constant factor, identical with the complex amplitude of the scattered wave from the object and has the property of reconstructing an image of the object in its original position. Due to the fact that this image forms behind the hologram, and the reconstructed wave appears to diverge from it, it is a virtual image.

The fourth term, *βTr*2 *o*\*(*x, y*), represents a wave similar to the object wave, but having an 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-offocus twin image as well as a strong coherent background. Another problem is that the object must have a high average transmittance in order for the second term on the right-hand side of (4) to be negligible. Thus, 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, *β* in (2) is negative, and the reconstructed image can be considered a photographic negative of the object.

Implications of the "Subquantum Level" in Carcinogenesis and Tumor Progression via Scale Relativity Theory http://dx.doi.org/10.5772/59233 411

**Fig. 9** í Optical system used to reconstruct the image with an in-line hologram, showing the formation of the twin images. **Figure 9.** Optical system used to reconstruct the image with an in-line hologram, showing the formation of the twin images.

#### In order to understand the formation of an image by an off-axis hologram, we must 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 **4.3. Off-axis holograms**

**4.3. Off-axis Holograms**

the one used to illuminate the object.

where x=(sin�)/�

where x=(sin�)/�

where t`0 = t0 +

where t`0 = t0 +

�*Tr*2

except for a constant factor

except for a constant factor

�*Tr*2

amplitude of the transmitted wave can be written as

amplitude of the transmitted wave can be written as

interference fringes equivalent to a carrier with a spatial frequency of

intensity, the resultant amplitude transmittance of the hologram is

�*Tr*2

�*Tr*2

A 'positive' transparency (the hologram) is then made by contact printing from this recording. Therefore it can be assumed that this transparency is processed so that its amplitude trans‐ mittance (the ratio of the transmitted amplitude to that incident on it) can be written as

where t0 is a constant background transmittance, *T* is the exposure time and *β* is a parameter determined by the photographic material used and the processing conditions, the amplitude

Then, the hologram is illuminated, as shown in Fig. 9, with the same collimated beam of monochromatic light employed to produce the original recording. Since the complex ampli‐ tude at any point in this beam is, aside from a constant factor, the same as that in the original reference beam, the complex amplitude transmitted by the hologram can be written as

of the scattered wave from the object and has the property of reconstructing an image of the object in its original position. Due to the fact that this image forms behind the hologram, and

opposite curvature. This wave converges to form a real image (the conjugate image) at the

With an in-line hologram, an observer viewing one image sees it superimposed on the out-offocus twin image as well as a strong coherent background. Another problem is that the object must have a high average transmittance in order for the second term on the right-hand side of (4) to be negligible. Thus, 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, *β* in (2) is negative, and the reconstructed image can be

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

2 2 2 2 0 *u x y rt x y r t Tr Tr o x y Tr o x y Tr o x y* (,) (,) ( ) (,) (,) (,) \* = = +b +b +b +b (4)

, can be neglected, because is extremely small, compared to the other

*o*(*x, y*), is, except for a constant factor, identical with the complex amplitude

*o*\*(*x, y*), represents a wave similar to the object wave, but having an

2 2

The right-hand side of (4) contains four terms. The first, *r*(t0+*βTr*2

attenuated plane wave, corresponds to the directly transmitted beam.

the reconstructed wave appears to diverge from it, it is a virtual image.

transmittance of the hologram is

410 Selected Topics in Applications of Quantum Mechanics

The second, *βTr* |*o*(*x, y*)|2

The third term, *βTr*2

The fourth term, *βTr*2

same distance in front of the hologram.

considered a photographic negative of the object.

terms.

0 *t t TI* = +b (2)

), which represents a uniformly

the one used to illuminate the object. In order to understand the formation of an image by an off-axis hologram, we must 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 the one used to illuminate the object. **Fig. 9** í Optical system used to reconstruct the image with an in-line hologram, showing the formation of the twin images. **4.3. Off-axis Holograms** In order to understand the formation of an image by an off-axis hologram, we must 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

which both the amplitude and phase vary, we can write the following: *o*(*x*, *y*) = *o*(*x*, *y*) exp� � � *i***Fig. 10** í The off-axis hologram: `recording. **Figure 10.** The off-axis hologram: `recording.

2 2

�

�

�

�

�

is a constant background transmittance.

�

is a constant background transmittance.

2 2

(*x*, *y*) (6) The resultant intensity is, therefore, 2 2 2 The complex amplitude at any point (*x, y*) on the photographic plate due to the reference beam can then be written as *r*(*x*, *y*) = *r* exp(*i*2px*x*) (5) The complex amplitude at any point (*x, y*) on the photographic plate due to the reference beam can then be written as

. Since only the phase of the reference beam varies across the photographic plat and because of the object beam, for

. Since only the phase of the reference beam varies across the photographic plat and because of the object beam, for

�

$$r(\mathbf{x}, y) = r \exp(i2\pi \mathbf{\tilde{x}} \mathbf{\tilde{x}}) \tag{5}$$

�

�

� px

px

px

px

2

2

(8)

(7)

(9)

(8)

(9)

 (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 x*.* 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 The resultant intensity is, therefore, � � � � ( , ) exp ( , ) exp( 2 ) ( , ) exp ( , ) exp( 2 ) ( , ) ( , ) ( , ) ( , ) ( , ) 2 2 2 *r o x y i x y i x r o x y i x y i x I x y r x y o x y r x y o x y* �px�px� � � � = = � = � � where *ξ*=(sin*θ*)/*λ*. Since only the phase of the reference beam varies across the photographic plat and because of the object beam, for which both the amplitude and phase vary, we can write the following:

> �

The amplitude and phase of the object wave are encoded as amplitude and phase modulation, respectively, of a set of

x*.* If, as in (2), we assume that the amplitude transmittance of the processed photographic plate is a linear function of the

When the hologram is illuminated for a second time with the original reference beam, as shown in Fig. 11, the complex

0

The first term on the right-hand side of (9) corresponds to the directly transmitted beam, while the second term generates a halo surrounding it, with approximately twice the angular spread of the object. The third term is identical to the original object wave,

( , ) ( , ) ( , ) exp[ ( , )]exp[ 2 ] 2

px

( , ) ( , ) ( , ) exp( 2 ) ( , ) exp( 2 )

= � � � � � �

px

, and produces a virtual image of the object in its original position. The fourth term corresponds to

�

= = � � �

px

*u x y r x y t x y t r i x Tr o x y i x*

px

( , ) ( , ) ( , ) exp( 2 ) ( , ) exp( 2 )

px

, and produces a virtual image of the object in its original position. The fourth term corresponds to

= = � � �

px

*u x y r x y t x y t r i x Tr o x y i x*

( , ) ( , ) ( , ) exp[ ( , )]exp[ 2 ] 2

*t x y t T o x y Tr o x y i x y i x*

= � � � � � �

*t x y t T o x y Tr o x y i x y i x*

px �

( , ) exp[ ( , )]exp[ 2 ]

px �

*Tr o x y i x y i x*

�

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

= � = � �

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

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

�

= � � �

0

= � � �

0

�

*r o x y r o x y x x y*

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

2 2

� � �

2 2

� � �

�

�

> �

�

*Tr o x y i x y i x*

*r o x y r o x y x x y*

8

The first term on the right-hand side of (9) corresponds to the directly transmitted beam, while the second term generates a halo surrounding it, with approximately twice the angular spread of the object. The third term is identical to the original object wave,

0

( , ) ( , ) exp( 4 )

*Tr o x y Tr o x y i x*

8

( , ) ( , ) exp( 4 )

When the hologram is illuminated for a second time with the original reference beam, as shown in Fig. 11, the complex

�

*Tr o x y Tr o x y i x*

( , ) exp[ ( , )]exp[ 2 ]

$$\rho(\mathbf{x}, y) = \left| \rho(\mathbf{x}, y) \exp\left[ -i\phi(\mathbf{x}, y) \right] \right| \tag{6}$$

The resultant intensity is, therefore,

$$\begin{aligned} \left| I(\mathbf{x}, \mathbf{y}) = \left| r(\mathbf{x}, \mathbf{y}) + o(\mathbf{x}, \mathbf{y}) \right|^2 = \left| r(\mathbf{x}, \mathbf{y}) \right|^2 + \left| o(\mathbf{x}, \mathbf{y}) \right|^2 + \\ \left| r + r \right| o(\mathbf{x}, \mathbf{y}) \right| \exp\left[ -i\phi(\mathbf{x}, \mathbf{y}) \right] \exp(-i2\pi\xi\mathbf{x}) + r \left| o(\mathbf{x}, \mathbf{y}) \right| \exp\left[ i\phi(\mathbf{x}, \mathbf{y}) \right] \exp(i2\pi\xi\mathbf{x}) = \\ \mathbf{x} = r^2 + \left| o(\mathbf{x}, \mathbf{y}) \right|^2 + 2r \left| o(\mathbf{x}, \mathbf{y}) \right| \cos[2\pi\xi\mathbf{x} + \phi(\mathbf{x}, \mathbf{y})] \end{aligned} \tag{7}$$

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 *ξ.*

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

$$\begin{aligned} t(\mathbf{x}, y) &= t'\_0 + \\$T \Big| o(\mathbf{x}, y) \Big|^2 + \\$T r \Big| o(\mathbf{x}, y) \Big| \exp[-i\phi(\mathbf{x}, y)] \exp[-i2\pi \xi \mathbf{x}] + \\ t + \\$T r \Big| o(\mathbf{x}, y) \Big| \exp[i\phi(\mathbf{x}, y)] \exp[i2\pi \xi \mathbf{x}] \end{aligned} \tag{8}$$

where *t*0 ' = t0 +*βTr*2 is a constant background transmittance.

When the hologram is illuminated for a second time with the original reference beam, as shown in Fig. 11, the complex amplitude of the transmitted wave can be written as

$$\begin{aligned} u(\mathbf{x}, \mathbf{y}) &= r(\mathbf{x}, \mathbf{y})t(\mathbf{x}, \mathbf{y}) = t\_0' r \exp(i2\pi \mathbb{E}\mathbf{x}) + \beta \operatorname{Tr} \left| o(\mathbf{x}, \mathbf{y}) \right|^2 \exp(i2\pi \mathbb{E}\mathbf{x}) + \\ &+ \beta \operatorname{Tr}^2 o(\mathbf{x}, \mathbf{y}) + \beta \operatorname{Tr}^2 o^\*(\mathbf{x}, \mathbf{y}) \exp(i4\pi \mathbb{E}\mathbf{x}) \end{aligned} \tag{9}$$

The first term on the right-hand side of (9) corresponds to the directly transmitted beam, while the second term generates 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 *βTr*2 , 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 taken large enough, the virtual image can be separated from the directly transmitted beam and the conjugate image.

In this setup, corresponding points on the real and virtual images are located at equal distances from the hologram, but on opposite sides of it. Because 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 mentioned that the phase of the reconstructed image is influenced only by the sign of *β*, always resulting in a "positive" image, even in the case of the hologram recording being a photographic negative.

image can be separated from the directly transmitted beam and the conjugate image. In this setup, corresponding points on the real and virtual images are located at equal distances from the hologram, but on Implications of the "Subquantum Level" in Carcinogenesis and Tumor Progression via Scale Relativity Theory http://dx.doi.org/10.5772/59233 413

> E,

the conjugate image which, in this case, is a real image. If the offset angle of the reference beam is taken large enough, the virtual

opposite sides of it. Because 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 mentioned that the phase of the reconstructed image is influenced only by the sign of

always resulting in a "positive" image, even in the case of the hologram recording being a photographic negative.

**Fig. 11** í The off-axis hologram: image reconstruction. **Figure 11.** The off-axis hologram: image reconstruction.

Exposure J/m2

#### Several recording materials have been used for holography [77]. Table 1 lists the principal characteristics of those that have been found most useful. **4.4. Recording materials**

Material

**4.4. Recording Materials**

*oxy oxy i xy* ( , ) ( , )exp ( , ) = -f é ù

222

( , ) exp ( , ) exp( 2 ) ( , ) exp ( , ) exp( 2 )

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

= +b +b ¢ - f - px +

When the hologram is illuminated for a second time with the original reference beam, as shown

= = px + b ¢ px +

The first term on the right-hand side of (9) corresponds to the directly transmitted beam, while the second term generates 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

In this setup, corresponding points on the real and virtual images are located at equal distances from the hologram, but on opposite sides of it. Because 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 mentioned that the phase of the reconstructed image is influenced only by the sign of *β*, always resulting in a "positive" image, even in the case of the hologram recording

, 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 taken large enough, the virtual image can be separated from the directly

+b f px (8)

+b + b px (9)

2

0 ( , ) ( , ) ( , ) exp[ ( , )]exp[ 2 ]

( , ) ( , ) ( , ) exp( 2 ) ( , ) exp( 2 )

*u x y r x y t x y t r i x Tr o x y i x*

*t x y t T o x y Tr o x y i x y i x*

is a constant background transmittance.

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

0

( , ) ( , )exp( 4 )

*Tr o x y Tr o x y i x* \*

*roxy i xy i x roxy i xy i x*

+ é ù - f - px + é ù f px = ë û ë û

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

*r oxy roxy x xy*

=+ + px + f

*Ixy rxy oxy rxy oxy*

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

2

( , ) exp[ ( , )]exp[ 2 ]

*Tr o x y i x y i x*

2 2

transmitted beam and the conjugate image.

being a photographic negative.

=+ = + +

The resultant intensity is, therefore,

412 Selected Topics in Applications of Quantum Mechanics

2 2

*ξ.*

where *t*0 '

*βTr*2

= t0 +*βTr*2

ë û (6)

(7)

**Table 1** *Recording materials for holography*  Several recording materials have been used for holography [77]. Table 1 lists the principal characteristics of those that have been found most useful.

mm-1 Processing Type

Kmax (diffraction

Resolution

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 [80]. **Table 1.** Recording materials for holography

Several organic materials can be activated by a photosensitizer to produce refractive index changes, because they suffer photopolymerization, when exposed to light [81]. A commercial photopolymer, coated on a polyester film base (DuPont OmniDex) that can be used to produce volume phase holograms with high diffraction efficiency is being currently produced [82]. 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 [83,84]. High-resolution photographic plates and films were the first materials used to record holo‐ grams. These are used widely even now, due to the fact that they exhibit relatively high sensitivity when compared to other hologram recording materials [78]. Moreover, they can be

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, being trapped there. The spatially varying electric field produced by this space-

stability of DCG (dichromated gelatin) [79] was made possible by the silver-halide sensitized gelatin technique.

9

dye sensitized so that their spectral sensitivity matches the most commonly used laser wavelengths.

Combining the high sensitivity of photographic materials with the high diffraction efficiency, low scattering and high light-stability of DCG (dichromated gelatin) [79] was made possible by the silver-halide sensitized gelatin technique.

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 [80].

Several organic materials can be activated by a photosensitizer to produce refractive index changes, because they suffer photopolymerization, when exposed to light [81]. A commercial photopolymer, coated on a polyester film base (DuPont OmniDex) that can be used to produce volume phase holograms with high diffraction efficiency is being currently produced [82].

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 [83,84].

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, being 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 space charge pattern can be removed by uniformly illuminating the crystal, after which another recording can take place [85,86].

It is essential to use coherent illumination for maximizing the visibility of the interference fringes formed by the object and reference beams, in the process of recording a hologram. 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, as a result, employed almost universally as light sources for recording holograms." (The text in quotation marks was reproduced from Hari‐ haran P. [165]).

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.

### **5. Tumor-associated ECM or tumor microenvironment, nonlinear medium with holographic properties**

#### **5.1. Extracellular matrix and tumor microenvironment**

Within tissue, cells are surrounded by a meshwork of proteins and proteoglycans collectively called the extracellular matrix (ECM), which compartmentalizes tissues. The ECM is divided into two distinct layers:

The most obvious function of the ECM is to provide structural support, shape, and stability for tissues. It does this by functioning as a base for cell anchorage. This base consists of three main structural components collagen, fibronectin, and elastic fibers, which bind to one another building a protein lattice upon which cells adhere.

Cell adherence to the ECM lattice provides cells support needed for cell migration. This is particularly important during embryonic development when cells are required to migrate into surrounding regions and differentiate into specific tissues [87]. A less obvious yet possibly more important function of the ECM in regards to tissue homeostasis and disease is its ability to mediate intracellular signaling. The ECM affects signaling through three main mechanisms:

**i.** cell – ECM interaction;

dye sensitized so that their spectral sensitivity matches the most commonly used laser

Combining the high sensitivity of photographic materials with the high diffraction efficiency, low scattering and high light-stability of DCG (dichromated gelatin) [79] was made possible

In positive photoresists, such as Shipley AZ-1350, the areas exposed to light become soluble

Several organic materials can be activated by a photosensitizer to produce refractive index changes, because they suffer photopolymerization, when exposed to light [81]. A commercial photopolymer, coated on a polyester film base (DuPont OmniDex) that can be used to produce volume phase holograms with high diffraction efficiency is being currently produced [82].

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,

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, being 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 space charge pattern can be removed by uniformly illuminating the crystal,

It is essential to use coherent illumination for maximizing the visibility of the interference fringes formed by the object and reference beams, in the process of recording a hologram. 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, as a result, employed almost universally as light sources for recording holograms." (The text in quotation marks was reproduced from Hari‐

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

**5. Tumor-associated ECM or tumor microenvironment, nonlinear medium**

Within tissue, cells are surrounded by a meshwork of proteins and proteoglycans collectively called the extracellular matrix (ECM), which compartmentalizes tissues. The ECM is divided

and are washed away during development to produce a relief image [80].

wavelengths.

haran P. [165]).

**with holographic properties**

into two distinct layers:

by the silver-halide sensitized gelatin technique.

414 Selected Topics in Applications of Quantum Mechanics

a photoconductor, and a thermoplastic [83,84].

after which another recording can take place [85,86].

a reference beam and the light waves scattered by the object.

**5.1. Extracellular matrix and tumor microenvironment**

The ECM also affects cellular activity by serving as a reservoir for proteins required for proper tissue function and repair. This includes a plethora of growth factors and proteases. These pleiotropic molecules have been shown to robustly affect proliferation, survival and migration in numerous cell types. Once growth factors are secreted from cells, they often become embedded within the ECM and require ECM degradation by proteases such as elastase to release the active protein allowing it to interact with surrounding and transduce downstream signaling. The ability of the ECM to control the bioavailability of growth factors provides another means of regulating cellular activities and further explains how alterations in the makeup of the ECM as observed in diseases such as cancer affect cell response.

Matricellular proteins also reside in the ECM. They are a unique family of proteins that do not function as structural proteins but rather orchestrate the deposition of the ECM and mediate cell-cell and cell-ECM interactions. To do this, matricellular proteins interact directly with cell surface receptors, structural proteins, growth factors and proteases found within the ECM [90]. Their expression is found in every tissue begins early in development, persists throughout adulthood and is increased during tissue remodeling events. Matricellular proteins are critical regulators of many aspects of cell function including differentiation, survival, proliferation and migration making them necessary for proper tissue function. Not surprisingly, given their affect on cell-ECM mediated signaling pathways, matricellular proteins have been shown to strongly influence tumor growth.

For tumor cells to metastasize, the local ECM must be remodeled to create an environment conducive to tumor survival and progression. This includes altering the architecture and composition of the tumor-associated ECM or tumor microenvironment (TME) to facilitate tumor cell dissemination [91]. Changes in ECM architecture are primarily carried out by enzymes such as MMPs which assist in remodeling of the TME by degrading structural proteins such as collagen and fibronectin allowing tumor cells to freely navigate through the surrounding ECM. MMPs and other proteases assist in destruction of the first barrier tumor cells face to successful metastasis, the basement membrane. They degrade the underlying basement membrane allowing tumor cells to escape the primary tumor and invade into surrounding non-neoplasic tissues. MMPs continue to breakdown barriers in the surrounding ECM clearing a path to blood vessels where tumor cells will intravasate into the circulatory system and seed secondary tumors [92]. Destruction of the ECM by proteases also promotes tumor progression by facilitating the release of angiogenic and mitogenic factors bound within the ECM [93]. In a surprising unexpected twist, studies revealed that the breakdown of ECM proteins by MMPs was more complex than anticipated. In fact it is a highly organized process which results in the generation of both protumor and antitumor cleavage products [94].

Presence of the ECM is required for cellular survival therefore increased degradation of the ECM within the TME must be balanced by an increase in ECM synthesis. The development of a tumor, much like a wound, provokes a robust inflammatory response causing an influx of mast cells, macrophages and neutrophiles into the TME [95].

We may summarize that the extracellular matrix generates signaling cues that regulate cell behavior and orchestrate functions of cells in tissue formation and homeostasis. Microenvir‐ onmental signaling, a process that determines cell shape, motility, growth, survival and differentiation is highly influenced by the ECM properties: composition, three-dimensional organization and proteolytic remodeling. Recent studies have shown that misregulation of cell–ECM interactions can contribute to many diseases, including developmental, immune, haemostasis, degenerative and malignant disorders.

Consequently, the structure and the behavior of the tumor-associated ECM allows us to think of it as a non-differential medium, and as will be shown below, a medium which holds the properties of a hologram (capacity to memorize, interference abilities) and may become a source of forces. In other words, ECM and TME are very suitable candidates for a 'recording material'.

#### **5.2. Tumor-associated ECM as a non-differential fractal medium**

We can simplify the dynamics of a biological system supposing that the motions on ECM take place on continuous but non-differentiable curves, i.e. fractal curves (for example, the Peano curve, the Koch curve or the Weierstrass curve [69,96,97].

Once this hypothesis has been accepted, some consequences of non-differentiability by SRT are evident [69,96]: i) the physical quantities that are used in describing the biological system dynamics are fractal functions, i.e. functions dependent both on spatial coordinates and time as well as on the scale resolution, δ*t*/*τ* (identified here with d*t*/*τ* by means of the substitution principle [69,96]. We mention that in the standard biophysics, the physical quantities describ‐ ing the dynamics of a biological system are continuous, but differentiable functions depending only on spatial coordinates and time; ii) the dynamics of the biological systems are given by the fractal operator *d* ^ /*dt* [98]:

$$\frac{d\hat{d}}{dt} = \frac{\partial}{\partial t} + \hat{V} \cdot \nabla - \mathrm{i}\frac{\lambda}{\pi} \left(\frac{dt}{\pi}\right)^{\left(\frac{2}{D\_r}\right) - 1} \Delta \tag{10}$$

where

surface receptors, structural proteins, growth factors and proteases found within the ECM [90]. Their expression is found in every tissue begins early in development, persists throughout adulthood and is increased during tissue remodeling events. Matricellular proteins are critical regulators of many aspects of cell function including differentiation, survival, proliferation and migration making them necessary for proper tissue function. Not surprisingly, given their affect on cell-ECM mediated signaling pathways, matricellular proteins have been shown to

For tumor cells to metastasize, the local ECM must be remodeled to create an environment conducive to tumor survival and progression. This includes altering the architecture and composition of the tumor-associated ECM or tumor microenvironment (TME) to facilitate tumor cell dissemination [91]. Changes in ECM architecture are primarily carried out by enzymes such as MMPs which assist in remodeling of the TME by degrading structural proteins such as collagen and fibronectin allowing tumor cells to freely navigate through the surrounding ECM. MMPs and other proteases assist in destruction of the first barrier tumor cells face to successful metastasis, the basement membrane. They degrade the underlying basement membrane allowing tumor cells to escape the primary tumor and invade into surrounding non-neoplasic tissues. MMPs continue to breakdown barriers in the surrounding ECM clearing a path to blood vessels where tumor cells will intravasate into the circulatory system and seed secondary tumors [92]. Destruction of the ECM by proteases also promotes tumor progression by facilitating the release of angiogenic and mitogenic factors bound within the ECM [93]. In a surprising unexpected twist, studies revealed that the breakdown of ECM proteins by MMPs was more complex than anticipated. In fact it is a highly organized process which results in the generation of both protumor and antitumor cleavage products [94].

Presence of the ECM is required for cellular survival therefore increased degradation of the ECM within the TME must be balanced by an increase in ECM synthesis. The development of a tumor, much like a wound, provokes a robust inflammatory response causing an influx of

We may summarize that the extracellular matrix generates signaling cues that regulate cell behavior and orchestrate functions of cells in tissue formation and homeostasis. Microenvir‐ onmental signaling, a process that determines cell shape, motility, growth, survival and differentiation is highly influenced by the ECM properties: composition, three-dimensional organization and proteolytic remodeling. Recent studies have shown that misregulation of cell–ECM interactions can contribute to many diseases, including developmental, immune,

Consequently, the structure and the behavior of the tumor-associated ECM allows us to think of it as a non-differential medium, and as will be shown below, a medium which holds the properties of a hologram (capacity to memorize, interference abilities) and may become a source of forces. In other words, ECM and TME are very suitable candidates for

mast cells, macrophages and neutrophiles into the TME [95].

haemostasis, degenerative and malignant disorders.

a 'recording material'.

strongly influence tumor growth.

416 Selected Topics in Applications of Quantum Mechanics

$$
\hat{\mathbf{V}} = \mathbf{V}\_D - i\mathbf{V}\_F \tag{11}
$$

is the complex velocity, *VD* is the differentiable and resolution scale independent velocity, *VF* is the non-differentiable and resolution scale dependent velocity, *V* **^** ∇ is the convective term,

$$\frac{\lambda^2}{\pi} \left(\frac{dt}{\pi}\right)^{\left(\frac{2}{D\_r}\right)-1} \Delta = \frac{\lambda^2}{\pi} \left(\frac{dt}{\pi}\right)^{\left(\frac{2}{D\_r}\right)-1} \left(\frac{\hat{\mathcal{O}}^2}{\hat{\mathcal{O}}\mathbf{x}^2} + \frac{\hat{\mathcal{O}}^2}{\hat{\mathcal{O}}\mathbf{y}^2} + \frac{\hat{\mathcal{O}}^2}{\hat{\mathcal{O}}\mathbf{z}^2}\right) \tag{12}$$

is the dissipative term, *DF* is the fractal dimension of the movement curve, *λ* is the space scale, *τ* is the time scale and *λ*2 /*τ* is a coefficient specific to the fractal – non - fractal transition. For *DF* any definition can be used (the Hausdorff – Besikovici fractal dimension, the Kolmogorov fractal dimension, etc. [97], but once such definition is accepted for *DF*, it has to remain constant over the entire analysis of the complex fluid dynamics. In a particular case, for motions on Peano curves, *DF* = 2 [97] of the complex fluid entities, the fractal operator (1) is reduced to Nottale's operator (*d* ^ */dt*)*Ν*

$$\frac{\hat{d}}{dt} = \frac{\partial}{\partial t} + \hat{\mathbf{V}} \cdot \nabla - iD\_N \Delta$$

where *DN* = *λ2 /τ* is the Nottale's coefficient associated to the fractal-non-fractal transition.

Applying the fractal operator (10) to the complex velocity (11) and accepting the principle of scale covariance [69,96] in the form:

$$\frac{d\hat{\mathbf{V}}}{dt} = -\nabla U \tag{13}$$

we obtain the motion equation:

$$\frac{d\hat{\mathbf{V}}}{dt} = \frac{\partial \hat{\mathbf{V}}}{\partial t} + \left(\hat{\mathbf{V}} \cdot \nabla\right) \hat{\mathbf{V}} - i \frac{\lambda^2}{\pi} \left(\frac{dt}{\pi}\right)^{\left(\frac{2}{D\_r}\right) - 1} \Delta \hat{\mathbf{V}} = -\nabla U \tag{14}$$

where *U* is an external scalar potential. Equation (14) is a Navier – Stokes type equation. It means that at any point of a fractal path, the local acceleration term, ∂*t V* **^** *,* the non-linearly (convective) term, (*V* **^** ∇) *V* **^** , the dissipative term, (*λ* 2 / *τ*)(*dt* / *τ*) ( 2 *DF* )−1 *ΔV* **^** , and the external free term ∇*U* make their balance. Therefore, the biological fluid is assimilated to a "rheological" fractal fluid, whose dynamics are described by the complex velocities field, *V* **^** , and by the imaginary viscosity type coefficient, *i*(*λ* 2 / *τ*)(*dt* / *τ*) ( 2 *DF* )−1 . The "rheology" of the fractal fluid can provide hysteretic properties to the biological fluid (the fractal fluid has a hysteresis cycle, memory, etc. [98-100].

For irrotational motions of the biological system entities

$$\begin{aligned} \nabla \times \hat{\mathbf{V}} &= 0, \qquad & \mathbf{a} \\ \nabla \times \hat{\mathbf{V}}\_D &= 0, \qquad & \mathbf{b} \\ \nabla \times \hat{\mathbf{V}}\_F &= 0 \qquad & \mathbf{c} \end{aligned} \tag{15}$$

we can choose *V* **^** of the form

$$\hat{\mathbf{V}} = -i\frac{\lambda^2}{\pi} \left(\frac{dt}{\pi}\right)^{\left(\frac{2}{D\_\mathbf{r}}\right)-1} \nabla \ln \psi \tag{16}$$

where *ϕ* ≡ ln*ψ* is the velocity scalar potential. By substituting (16) in (14) and using the method described in [98-100], it results

$$\frac{d\hat{\mathbf{U}}}{dt} = -i\frac{\lambda^2}{\pi} \left(\frac{dt}{\pi}\right)^{\left(\frac{2}{D\_r}\right)-1} \nabla \left[\frac{\partial \ln \Psi}{\partial t} - i\frac{\lambda^2}{\pi} \left(\frac{dt}{\pi}\right)^{\left(\frac{2}{D\_r}\right)-1} \frac{\nabla \Psi}{\Psi} + \mathcal{U}\right] = 0\tag{17}$$

This equation can be integrated in a universal way and yields

Implications of the "Subquantum Level" in Carcinogenesis and Tumor Progression via Scale Relativity Theory http://dx.doi.org/10.5772/59233 419

$$\frac{\lambda^4}{\pi^2} \left( \frac{dt}{\pi} \right)^{\left(\frac{4}{D\_r}\right) - 2} \Delta \Psi + i \frac{\lambda^2}{\pi} \left( \frac{dt}{\pi} \right)^{\left(\frac{2}{D\_r}\right) - 1} \frac{\partial \Psi}{\partial t} - \frac{\mathcal{U}}{2} \Psi = 0 \tag{18}$$

up to an arbitrary phase factor which may be set to zero by an appropriate selection of the phase of *ψ.* Relation (18) is a Schrödinger type equation. For motions on Peano curves, *DF* = 2 [97] at Compton scale, which implies *λ2* /*τ* = *ħ* /2*m*0 [69,96], with *ħ* the reduced Plank constant and *m0* the rest mass of the biological entities, the relation (18) becomes the standard Schrö‐ dinger equation:

$$\frac{\hbar^2}{2m\_0}\Delta\Psi + i\hbar\frac{\partial\Psi}{\partial t} - \frac{U}{2m\_0}\Psi = 0$$

If *ψ* = √*ρ eiS,* with √*ρ* the amplitude and *S* the phase of *ψ,* the complex velocity field (16) takes the explicit form:

$$\begin{aligned} \hat{\mathbf{V}} &= \frac{\lambda^2}{\tau} \Big( \frac{dt}{\tau} \bigg)^{\left(\frac{2}{D\_{\mathrm{r}}}\right)^{-1}} \nabla S - i \frac{\lambda^2}{2\tau} \Big( \frac{dt}{\tau} \bigg)^{\left(\frac{2}{D\_{\mathrm{r}}}\right)^{-1}} \nabla \ln \rho & \mathbf{a} \\ \mathbf{V}\_{D} &= \frac{\lambda^2}{\tau} \bigg( \frac{dt}{\tau} \bigg)^{\left(\frac{2}{D\_{\mathrm{r}}}\right)^{-1}} \nabla S & \mathbf{b} & \mathbf{a} \\ \mathbf{V}\_{F} &= \frac{\lambda^2}{2\tau} \bigg( \frac{dt}{\tau} \bigg)^{\left(\frac{2}{D\_{\mathrm{r}}}\right)^{-1}} \nabla \ln \rho & \mathbf{c} \end{aligned} \tag{19}$$

By substituting (19a-c) in (14) and separating the real and the imaginary parts, up to an arbitrary phase factor which may be set to zero by appropriate selection of the phase of *ψ,* we obtain:

$$\frac{\partial \mathbf{V}\_D}{\partial t} + \left(\mathbf{V}\_D \cdot \nabla\right) \mathbf{V}\_D = -\nabla \left(Q + \mathcal{U}\right) \tag{20}$$

$$\frac{\partial \mathbf{\hat{p}}}{\partial t} + \nabla \cdot \left(\rho \mathbf{V}\_D\right) = 0 \tag{21}$$

with *Q* the specific fractal potential

ˆ ˆ *d U dt*

( )

*dt t*

**^** ∇) *V*

imaginary viscosity type coefficient, *i*(*λ* 2 / *τ*)(*dt* / *τ*)

**^** of the form

described in [98-100], it results

**V**

For irrotational motions of the biological system entities

*dt t*

This equation can be integrated in a universal way and yields

2 1 ˆ 2 ˆ ˆ ˆ ˆ ˆ *d dt DF*

¶ t t è ø

means that at any point of a fractal path, the local acceleration term, ∂*t*

ç ÷- ç ÷ ¶ l æ öè ø = + ×Ñ - ç ÷ D = -Ñ

fractal fluid, whose dynamics are described by the complex velocities field, *V*

*D F*

Ñ´ = Ñ´ = Ñ´ =

*i*

**V V V** *i U*

**V V V V V** (14)

( 2 *DF* )−1 *ΔV*

**V** (16)

æ ö

where *U* is an external scalar potential. Equation (14) is a Navier – Stokes type equation. It

term ∇*U* make their balance. Therefore, the biological fluid is assimilated to a "rheological"

provide hysteretic properties to the biological fluid (the fractal fluid has a hysteresis cycle,

ˆ 0, a ˆ 0, b ˆ 0 c

2 1 2 ˆ ln *dt DF*

2 2 1 1 ˆ 2 2 ˆ ln 0 *d dt D D F F dt i iU*

æ ö æ ö ç ÷- - ç ÷ ç ÷ ç ÷ è ø è ø

l æ ö ê ú ¶y l æ ö Ñy =- Ñ - ç ÷ ç ÷ + = tt ¶ tt y è ø è ø ë û

æ ö ç ÷- ç ÷ l æ öè ø =- Ñ y ç ÷ t t è ø

where *ϕ* ≡ ln*ψ* is the velocity scalar potential. By substituting (16) in (14) and using the method

é ù

( 2 *DF* )−1

**^** , the dissipative term, (*λ* 2 / *τ*)(*dt* / *τ*)

we obtain the motion equation:

418 Selected Topics in Applications of Quantum Mechanics

(convective) term, (*V*

memory, etc. [98-100].

we can choose *V*

= -Ñ **V** (13)

*V* **^** *,*

. The "rheology" of the fractal fluid can

the non-linearly

**^** , and by the

(15)

(17)

**^** , and the external free

$$Q = -2\frac{\lambda^4}{\tau^2} \left(\frac{dt}{\tau}\right)^{\left(\frac{4}{D\_r}\right)^{-2}} \frac{\Delta\sqrt{\rho}}{\sqrt{\rho}} = -\frac{\mathbf{V}\_F^2}{2} - \frac{\lambda^2}{\tau} \left(\frac{dt}{\tau}\right)^{\left(\frac{2}{D\_r}\right)^{-1}} \nabla \cdot \mathbf{V}\_F \tag{22}$$

Equation (20) represents the specific momentum conservation law, while equation (21) represents the states density conservation law. By means of the fractal velocity, *VF,* the specific fractal potential *Q* is a measure of non-differentiability of the biological entities trajectories, *i.e.* of their chaoticity. The equations (20)-(22) define the fractal hydrodynamics model (FHM). In such a context, the biological system can be considered a fractal fluid.