US20080275684A1

US20080275684A1 - Treatment protocol generation for diseases related to angiogenesis

Info

Publication number: US20080275684A1
Application number: US12/132,300
Authority: US
Inventors: Zvia Agur; Levon Arakelyan; Vladimir Vainstein
Original assignee: OPTIMATAL Ltd
Current assignee: Timpel SA; OPTIMATAL Ltd
Priority date: 2001-10-25
Filing date: 2008-06-03
Publication date: 2008-11-06
Also published as: WO2003036410A2; AU2002363044A1; US20030088237A1; DE60218229T2; WO2003036410A3; DK1485818T3; EP1485818A4; DE60218229D1; ATE354132T1; EP1485818A2; US7418374B2; EP1485818B1

Abstract

A computer-implemented method for determining an optimal treatment protocol for a disease related to angiogenesis, comprising creating an angiogenesis model including pro-angiogenesis and anti-angiogenesis factors. Effective vessel density (EVD) is incorporated as a factor regulating switching on and switching off of at least one component in the angiogenesis model. Effects of vasculature maturation and mature vessels destabilization are incorporated. Pro-angiogenesis and anti-angiogenesis factors, which can influence changes in state of a tissue are selected. Effects of drugs in the pro-angiogenesis and anti-angiogenesis factors are incorporated. A plurality of treatment protocols in a protocol space is generated. A best treatment protocol based on a pre-determined criteria.

Description

RELATED APPLICATIONS

This Application is a continuation of co-pending U.S. application Ser. No. 10/207,772, filed on Jul. 31, 2002, which in turn claims benefit of co-pending U.S. Provisional Patent Application Ser. No. 60/330,592 filed Oct. 25, 2001, the contents of both of which Applications are incorporated herein by reference.

FIELD

The present disclosure generally teaches techniques related to diseases and processes involving Angiogenesis. More particularly it teaches techniques for generating treatment protocols for diseases where angiogenesis is a factor. The techniques are also applicable to normal processes involving Angiogenesis even if no disease is present.

BACKGROUND

- 1. References

The following papers provide useful background information, for which they are incorporated herein by reference in their entirety, and are selectively referred to in the remainder of this disclosure by their accompanying reference numbers in brackets (i.e., <3> for the third numbered paper by Yangopoulos et al):
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<30> Davis S, Aldrich T H, Jones P F, Acheson A, Compton D L, Jain V, Ryan T E, Bruno J, Radziejewski C, Maisonpierre P C, Yancopoulos G D (1996) Isolation of Angiopoietin-1, a Ligand for the TIE2 Receptor, by Secretion-Trap Expression Cloning. Cell 87: 1161-1169. December 1996
<31> Suri C, Jones P E, Patan S, Bartunkova S, Maisonpierre P C, Davis S, Sato T S, Yancopoulos G D (1996) Requisite Role of Angiopoietin-1, a Ligand for the TIE2 Receptor, during Embryonic Angiogenesis. Cell 87: 1171-1180. December 1996
<32> Maisonpierre P C, Suri C, Jones P F, Bartunkova S, Wiegand S J, Radziejewski C, Compton D, McClain J, Aldrich T H, Papadopoulos N, Daly T J, Davis S, Sato T N, Yancopoulos G D (1997) Angiopoietin-2, a Natural Antagonist for Tie2 That Disrupts in vivo Angiogenesis. Science 277. 5322: 55-60. 4 Jul 1997
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<34> Eggert A, Ikegaki N, Kwiatkowski J, Zhao H, Brodeur G M, Himelstein B P (2000) High-Level Expression of Angiogenic Factors Is Associated with Advanced Tumor Stage in Human Neuroblastomas. Clinical Cancer Research 6: 1900-1908
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<45> Oh H. et al. (1999) Hypoxia and Vascular Endothelial Growth Factor selectively up-regulate angiopoietin-2 in bovine microvascular endothelial cells. J Biol Chem 274:15732-9.
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- 2. Introduction

For a better understanding of this disclosure, all the terms and parameters used in this disclosure are listed in the Table shown in FIG. 5.
Angiogenesis, or neovascularization, is a process of new blood vessels formation by budding from the existing ones. Neovascularization provides tissue with vital nutrients and growth factors and enables clearance of toxic waste products of cellular metabolism. Angiogenesis has been conventionally recognized as a biological mechanism of dual clinical effect. On the one hand, it allows survival of normal tissues when they become ischemic. That is, it enables functional development of normal tissues, for example, wound healing and embryogenesis. On the other hand, angiogenesis enables tumor tissue growth.
Intensive research of angiogenesis during the last 15 years has led to better understanding of this complex process <1-7, 57-60>. However, cause and effect relationships in the process of angiogenesis are yet to be clarified. Moreover, the massive research in the field of angiogenic therapy still suffers from the lack of tools for predicting the potential effects of PRO-and ANTI-angiogenic factors.
The two major determinants of new vasculature formation are thought to be the genetic features of the tissue and the availability of oxygen and nutrients <5,6>. The dependence of vessel formation on nutrients or oxygen deprivation was shown to be mediated by vascular endothelial growth factor (VEGF), which is a potent inducer of endothelial cell proliferation and migration <8-15>. VEGF is preferentially expressed by tissue cells in the nutrient-deprived areas <7, 16-20, 61-64>. In contrast, basic nutrient-independent VEGF production by the tissue is determined by genetic factors <20>. Consequently, VEGF-induced angiogensis depends on both the aforementioned vasculature growth determinants, namely genetics and nutrient/oxygen availability. VEGF-induced angiogenesis leads to increase in nutrient supply to the tissue. Accordingly, nutrient- and oxygen-dependent VEGF expression is down-regulated. When VEGF level becomes low enough, the newly formed blood vessels regress <21-26>, consequently leading to nutrients and oxygen deprivation again. This negative feedback can produce successive cycles of growth and regression of blood vessels. This phenomenon was demonstrated in the mouse xenograft tumor model <12>.
Blood vessels can be rendered insensitive to fluctuations of VEGF concentration by the process of maturation (coverage of capillaries by periendothelial cells 25,26). This process involves pericytes (smooth muscle-like cells) which form an outside layer covering the endothelial cells of the newly formed vessel . The major pericyte-stimulating factor is a platelet-derived growth factor (PDGF) <27-29>. Interactions between endothelial cells and pericytes, which apparently lead to maturation, are governed by the angiopoietin system. This system includes two soluble factors—Angiopoietins 1 and 2 (Ang1 and Ang2, respectively), and their receptor, Tie2, which is specifically expressed on endothelial cells <30-34>. Ang1 is Tie2 agonist that promotes maturation, while Ang2 is its natural antagonist <31,32>. Regulation of Ang1 and Ang2 expression is not completely understood. According to recent publications it can be influenced by tumor cell- as well as endothelial cell-specific factors <35,42-49>. These factors depend on the tissue and the host type <42-44>, and can be, accordingly, taken into account in the presented model. High Ang 1/Ang 2 ratio and pericytes' presence induces maturation of newly formed vessels. Alternatively, low Ang 1/Ang 2 ratio induces destabilization of mature blood vessels, while newly formed vessels remain immature and susceptible to VEGF fluctuations <25>.
Therefore, it would be advantageous to have techniques for generating and selecting treatment protocols for diseases where angiogenesis is a factor. Further, it is also advantageous to adapt the techniques to study the progression of processes involving angiogenesis.

- 3. Clinical Significance

The clinical significance of angiogenesis as an “ultimate” target for cancer therapy was first recognized in 1971 by J. Folkman <36>, and got wide acceptance in early nineties after the discovery of the first specific antiangiogenic substances <37,38>. Apparent advantages of this approach include its universality for different solid tumors, lack of prominent side effects and lack of resistance development during repetitive treatment cycles.
Angiogenesis is implicated in the pathogenesis of a variety of disorders: proliferative retinopathies, age-related macular degeneration, tumors, rheumatoid arthritis, psoriasis <1; 51-56, 66, 67> and coronary heart diseases <50>. The use of exogenous agents to selectively target neovasculature, or stimulate the growth of new blood vessels into ischaemic tissue is a potentially revolutionary therapy in a wide variety of clinical specialties, which opens new avenues for the diagnosis and therapy of diseases where angiogenesis is a factor; such as, cancer, blinding ocular disorders, rheumatoid arthritis and others.<65, 66>.

- 4. Problems in Practicing Anti/Pro-Angiogenic Therapy

In order to establish optimal pro-angiogenic or anti-angiogenic treatment protocols (either as a monotherapy or in combination with chemotherapy or radiotherapy) the dynamics of angiogenesis must be better understood. Recent studies <25,26> have shown that newly formed vasculature is very dynamic—blood vessels undergo constant remodeling that involves maturation in response to local levels of angiogenic and maturation factors. Mature and immature vessels may differentially respond to certain PRO and ANTI-angiogenic drugs during tissue growth, myocardial ischemia, macular degeneration and other diseases, leading to success or failure of the treatment <25>.
Mathematical models and computer simulation of angiogenesis and PRO- and ANTI-angiogenic therapy can be constructed, in order to predict the most promising treatment protocols thus eliminating the need for lengthy and expensive clinical trials.

- 5. Previous Mathematical Models

The construction of a mathematical model for angiogenesis includes I) in-depth understanding of the biology of angiogenesis, II) the selection of appropriate patient populations for clinical trials, choice of therapeutic end points and means of their assessment, choice of therapeutic strategy (gene versus protein delivery), route of administration, and the side effect profile.<68>
Several mathematical dynamic models have been proposed, each one of them constructed to illuminate specific aspects of angiogenesis <39-41>. Some of these models examine vascular tree formation in vitro, irrespectively of tumor dynamics, and consequently are not suitable for tumor growth modelling <39>. Others assume that the growing vascular tree is a subject to some optimization with regard to the target tissue perfusion <41>. This optimization, while possibly holding true for normal tissue development, can hardly account for tumor growth, since it is known that tumor vasculature is highly disorganized.
Logistic-type model, proposes by Hahnfeldt et al. <40>, analyzes the general vascular dynamics (“carrying capacity of current vascular tree”) with regard to production of pro- and antiangiogenic factors by the tumor. Analysis of experimental data of Lewis lung carcinoma growth in mice allowed the authors to estimate the model parameters and to examine the effects of antiangiogenic factors angiostatin and endostatin. The main problematic assumption of this model is the constant production rates of these factors, as we know that VEGF, for example, production is tightly regulated by tissue hypoxia.
Model by Tong S and Yuan F <69> focused on two-dimensional angiogenesis in the cornea. The model considered diffusion of angiogenic factors, uptake of these factors by endothelial cells, and randomness in the rate of sprout formation and the direction of sprout growth.
None of the aforementioned models takes into account vasculature maturation and mature vessels destabilization, which are very fundamental constituents of angiogenesis dynamics. Moreover, these models, due to their relative abstraction, cannot account for drug-induced, or other, molecular changes in angiogenic dynamics. Note, that, since PRO and ANTI-angiogenic drugs interfere with the dynamics described above at the molecular level, the model which can serve as a tool for predicting drug effect on this process must take into account all the molecular complexity of angiogenesis, including the dynamics of neovasculature maturation and mature vessels destabilisation.
Therefore, it is desirable to provide a techniques, including computer systems, that overcomes some of the disadvantages noted above.

SUMMARY

To realize the advantages discussed above, the disclosed teachings provide a computer-implemented method for determining an optimal treatment protocol for a disease related to angiogenesis, comprising creating an angiogenesis model including pro-angiogenesis and anti-angiogenesis factors. Effective vessel density (EVD) is incorporated as a factor regulating switching on and switching off of at least one component in the angiogenesis model. Effects of vasculature maturation and mature vessels destabilization are incorporated. Pro-angiogenesis and anti-angiogenesis factors, which can influence changes in state of a tissue are selected. Effects of drugs in the pro-angiogenesis and anti-angiogenesis factors are incorporated. A plurality of treatment protocols in a protocol space is generated. A best treatment protocol based on a pre-determined criteria.
In another specific enhancement, the model comprises a tissue volume model, an immature vessel model and a mature vessel model.
In another specific enhancement, steps to regulate dynamics which influences EVD are incorporated.
In another specific enhancement, the model simultaneously accounts for tissue cell proliferation, tissue cell death, endothelial cell proliferation, endothelial cell death, immature vessel formation and immature vessel regression, immature vessel maturation and mature vessel destabilization.
In another specific enhancement, the model incorporates temporal parameters that characterize response rate of at least one element associated with angiogenesis.
More specifically, EVD is calculated by combining immature vessel density and mature vessel density.
In another specific enhancement, parameters incorporated into the model comprises tissue volume, number of free endothelial cells, number of free pericytes, volume of mature vessels, volume of immature vessels and concentration of regulator factors.
More specifically, the regulatory factors comprise vascular endothelial growth factor (VEGF), platelet derived growth factor (PDGF), angiopoietin 1 (Ang1) and angiopoietin 2 (Ang2).
More specifically, EVD is a function of a duration of insufficient perfusion and vice versa .
More specifically, the model incorporates threshold levels of regulatory factors and parameter ratios.
Even more specifically, the threshold level is at least one of: a) VEGF concentration below which no endothelial cells proliferation takes place (A); b) minimum number of receptors for VEGF above which endothelial cells proliferation takes place (B); c) VEGF concentration below which endothelial cells, both in the free state as well as when incorporated into immature blood vessels, are subject to apoptosis VEGF_thr; d) the minimal number of free pericytes which stimulates the onset of maturation of immature vessels (C); e) Ang 1/Ang 2 ratio below which mature vessels are destabilized, and above which maturation of immature vessels is enabled (K); f) EVD value that influences the rate of cell proliferation and death; and g) EVDS_ssvalue for which the system is in steady state. Even more specifically, the tissue volume model calculates the tissue volume by a process comprising: comparing EVD against an −EVD_ss. If EVD is equal to EVD_ssthen use a programmed tissue cell proliferation and a programmed tissue cell death (apoptosis) to compute tissue volume. If EVD >EVD_ssthen use increased tissue proliferation and decreased tissue cell death to compute tissue volume. If EVD<EVD_ssthen use decreased tissue proliferation and increased tissue cell death to compute tissue volume.
Even more specifically, Ang1 and Ang2 induction are incorporated into appropriate steps above following the computation of tissue volume.
More specifically, the immature vessel model calculates the immature vessels by a process comprising comparing EVD against an EVD_ss. If EVD is equal to EVD_ssthen set VEGF to a VEGF_ssand PDGF to a PDGF_ss. If EVD>EVD_ssthen use decreased VEGF and decreased PDGF. If EVD<EVD_ssthen useg increased VEGF and increased PDGF. Compare VEGF against A. Factor endothelial cell proliferation if VEGF>A. Compare VEGF against a VEGF threshold. Factor free endothelial cell deaths if VEGF<VEGF threshold. Compare VEGF receptor number against B. If VEGF receptor number is less than B then consider no angiogenisis prior to computing immature vessel regression. If VEGF receptor number is not less than B then compute growth of immature vessels. If VEGF<A then consider no angiogenesis and compute immature vessel regression. Compute mature vessels based on growth immature vessels, immature vessel regression and mature vessel destabilization.
Even more specifically, immature vessels computation considers no maturation if Ang2/Ang1>K or if number of free pericytes<C.
Even more specifically, mature vessel destabilization considers ang1/Tie2 interaction blocking.
Even more specifically, no destabilization occurs if Ang2/Ang1 is not greater than K
More specifically, mature vessel model is computed using a procedure comprising computing immature vessels. Determining if Ang1/Ang2<K. Determine if number of free pericytes<C. Considering immature vessel maturation if both the above steps are false. Factoring no destablization if number of free pericytes is not less than C.
More specifically, effects of a drug affecting EC proliferation are factored in computing immature vessels.
More specifically, effects of a drug affecting VEGF receptors are factored in computing immature vessels.
More specifically, effects of a drug affecting pericyte proliferation are factored in computing immature vessel computation.
More specifically, effects of a drug affecting VEGF are factored in computing immature vessels.
More specifically, effects of a drug affecting PEGF are factored in computing immature vessel computation.
More specifically, effects of a drug affecting Ang1 are factored in computing immature vessels.
More specifically, effects of a drug affecting Ang2 are factored in computing immature vessel computation.
In another specific enhancement, model takes into account duration of a tissue cell proliferation, tissue cell death, endothelial cell proliferation, endothelial cell death, pericytes proliferation, immature vessel regression, immature vessel maturation and mature vessel destabilization.
In another specific enhancement, model takes into account the duration of VEGF induction, PDGF induction, Ang1 and Ang2 induction by tissue cells and Ang1 and Ang2 induction by endothelial cells.
Another aspect of the disclosed teachings is an optimal treatment protocol for a disease related to angiogenesis, comprising an angiogenesis model including pro-angiogenesis and anti-angiogenesis factors; a treatment protocol space generator that generates a protocol space of possible treatments for the disease; a treatment selector that selects an optimal protocol, wherein effective vessel density (EVD) is a factor regulating switching on and switching off of at least one component in the angiogenesis model; wherein the model incorporates effects of vasculature maturation and mature vessels destabilization; and wherein the system is adapted to effect selection of pro-angiogenesis and anti-angiogenesis factors which can influence changes in state of a tissue and incorporating effects of drugs in the pro-angiogenesis and anti-angiogenesis factors.
A computer program product including computer readable media that comprises instructions to implement the above techniques on a computer are also part of the disclosed teachings.

BRIEF DESCRIPTION OF THE DRAWINGS

The above objectives and advantages of the disclosed teachings will become more apparent by describing in detail preferred embodiment thereof with reference to the attached drawings in which:

FIG. 1 (a)-(c) depicts a flowchart that shows an implementation of the overall techniques.

FIG. 2 (a)-(c) depicts the flowchart of FIG. 1 with possible effects due to Pro-Angiogeneses and Anti-Angiogeneses drugs.

FIG. 3 (a)-(f) shows graphs that illustrate the effects of Anti-VEGF Drugs.

FIG. 4 shows graphs that illustrate the effects of a combination of Anti-VEGF and Anti-Ang1 Drugs.

FIG. 5 (a)-(c) shows a table with description of terms used in the Equations included in the specification.

DETAILED DESCRIPTION

IV.A. Overview of Exemplary Implementations
The disclosed techniques are embodied in exemplary computer systems and exemplary flowcharts that are implemented by computers. The implementations discussed herein are merely illustrative in nature and are by no means intended to be limiting. Also it should be understood that any type of computers can be used to implement the systems and techniques. An aspect of the disclosed teachings is a computer program product including computer-readable media comprising instructions. The instructions are capable of enabling a computer to implement the systems and techniques described herein. It should be noted that the computer-readable media could be any media from which a computer can receive instructions, including but not limited to hard disks, RAMs, ROMs, CDs, magnetic tape, internet downloads, carrier wave with signals, etc. Also instructions can be in any form including source code, object code, executable code, and in any language including higher level, assembly and machine languages. The computer system is not limited to any type of computer. It could be implemented in a stand-alone machine or implemented in a distributed fashion, including over the internet.
The technique shown in the flowchart take into account for the dynamic interactions between tissue volume, angiogenesis (growth and regression of immature blood vessels), and vascular maturation and destabilization. The technique shown in the flowchart is combined with a quantitative mathematical model that is described in detail herein. A combination of the technique shown in the flowchart and the mathematical computations described would allow a skilled artisan to practice the disclosed technique; including for example, to quantify the dynamics of tissue vascularization and the effect of drug on this process at any given moment.
The technique describes the interactions between molecular regulatory factors, cell types and multi-cellular structures (such as vessels) which together influence the tissue dynamics.
The technique takes into account, the temporal parameters which characterize the response rates of each one of the elements included in the angiogenesis process.
The technique includes a series of simulation steps. The parameter values that are output from each simulation step are taken as initial conditions for the next simulation step. These parameter values are compared with the threshold levels. Their current values are calculated according to the arrows shown in the flowchart of FIG. 1. At least six major processes are taken into account simultaneously, namely tissue cells proliferation and death, endothelial cells proliferation and death, immature vessels formation and regression, immature vessels maturation and mature vessels destabilization, and possibly others.
The techniques depicted in FIG. 1 includes three interconnected modules: tissue cell proliferation, angiogenesis (immature vessels growth) and maturation (formation and destabilization of mature vessels). Further, each module operates on three scales: molecular, cellular and macroscopic (namely, vessel densities and tissue volume).
The tissue module includes tissue cell proliferation sub-module and cell death sub-module. Further, each is subdivided into i) time-invariant, cell type-specific, genetically determined sub-block, and ii) time-variant, nutrient-dependent sub-block. Nutrient-dependent cell proliferation and nutrient-dependent cell death rates are directly or inversely proportional, respectively, to the effective vascular density (EVD), which is a perfused part of vascular tree <40>.
Two additional quantities are calculated in the tissue module, namely VEGF and PDGF production. They are inversely related to EVD so that increasing nutrient depletion results in increasing secretion of pro-angiogenic factors <7-9>. The tissue growth module interacts with the angiogenesis and the maturation modules via the relevant regulatory proteins.
In the angiogenesis module, volume of immature vessels is calculated. Immature vessel volume increases proportionally to VEGF concentration, if VEGF is above a given threshold level. The volume regresses if VEGF is below a given, possibly different, threshold level. The latter threshold is generally referred to as “survival level” <21-24>.
In the maturation module, volume of mature vessels is calculated according to pericyte concentration <41-43>and according to the Ang1/Ang2 ratio <44>. Pericytes proliferate proportionally to PDGF concentration <25-26>. Ang1 and Ang2 are continuously secreted by tissue cells and immature vessels, respectively <27, 28, 32-34, 41-43, 45>. Additionally, Ang1 and Ang2 can be secreted by tissue cells, if the latter are nutrient-depleted <45>. It is assumed that maturation of immature vessels occurs if pericytes concentration and Ang1/Ang2 ratio are above their respective threshold levels, while under these thresholds immature vessels do not undergo maturation, while mature vessels undergo destabilization and become immature <29-33>.
It is clear that the parameters used in the technique can include tissue volume (determined as a function of tissue cell number); number of free endothelial cells and pericytes; volume of immature and mature vessels; and concentrations of the regulatory factors such as VEGF, PDGF, Ang1 and Ang2.
Moreover, several relative parameters (ratios) are calculated, such as Ang2/Ang1, immature vessel density and mature vessel density (denoting vessels volume divided by tissue volume). The latter two densities are combined into effective vessel density, EVD. EVD is a critical model variable, which at any moment determines tissue cells proliferation and death, as well as the production of factors, such as VEGF and PDGF. Resistance of tissue cells to antiangiogenic drugs may emerge from tissue adaptation to hypoxia.
In order to account for the possible adaptation of tissue cells to insufficient nutrition and to hypoxia it is assumed that EVD is a function of the duration of insufficient perfusion, (denoted below by EVDn).
The technique takes into account the threshold levels of regulatory factors and parameter ratios, such as:
VEGF concentration below which no endothelial cells proliferation takes place (denoted below by A);
The minimum number of receptors for VEGF above which endothelial cells proliferation takes place(denoted below by B);
VEGF concentration below which endothelial cells (both in the free state as well as when incorporated into immature blood vessels) are subject to apoptosis (this is denoted below by VEGF_thr;
The minimal number of free pericytes which stimulates the onset of maturation of immature vessels(denoted below by C);
The Ang 1/Ang 2 ratio below which mature vessels are destabilized, and above which maturation of immature vessels is enabled(denoted below by K).
The EVD value influences the rate of cell proliferation. The EVD value for which the system is in steady state (tissue cell proliferation rate being equal to tissue cell death rate) is denoted below by EVD_ss. At EVD>EVD_sstissue cells proliferation prevails, so that tissue volume increases. At EVD<EVD_sstissue cell death prevails, and the tissue shrinks. The EVD_ssis determined by genetic properties of a given tissue and a given host. VEGF, PDGF, Ang 1, Ang2 secretion level at the steady state of the system will be denoted VEGF_ssPDGF_ss, Ang1_ssand Ang2_ss.
The inputs to the represented system includes the tissue volume, blood vessels density, and the inherent parameters characterizing this tissue type at initiation of the process. The output at any given moment are parameters like tissue volume, mature and immature vessels sizes, EVD.
IV.B. Detailed Description of the Exemplary Implementation
The flowchart shown in FIG. 1 is discussed in detail herein with reference to specific mathematical equations describing the principal interactions affecting vascular tissue growth. The technique describes the interrelationships between tissue growth, the formation of new vessels (angiogenesis) and the maturation of the newly formed vessels. The interactions occur across three organization levels: molecular, cellular, and organic level. The arrows in the flowchart indicate the specific module interaction. The rectangular boxes indicate the point at which a specific sub-process calculation occurs. The parameter Tx in a box denotes the characteristic reaction time of the action calculated in the box. The diamonds indicate the conditions, which determine the direction of processes.
EVD_ssis the value for which the system is in steady state. VEGF_ssis the VEGF secretion level at the steady state of the system. VEGF_Thris the VEGF concentration below which endothelial cells, both in the free state as well as when incorporated into immature blood vessels, are subject to apoptosis. PDGF_ssis the PDGF secretion level at the steady state of the system.
In this mathematical model EVD_nin a certain moment n is represented as the sum of a density of immature (EVD_n ^im) and density of mature vessels ( EVD_n ^mat) at the moment “n”.
EVD _n =EVD _n ^im +EVD _n ^mat; (1)
In FIGS. 1 and 2 the effective vessel density as discussed above is calculated in block 1.1. The mature and immature vessel densities, in turn, are calculated in blocks 1.2 and 1.3 using the following equations:
$\begin{matrix} {EVD}_{n}^{im} = \frac{α^{im} * {Vves}_{n}^{im}}{{Vtis}_{n}}; {EVD}_{n}^{mat} = \frac{α^{mat} * {Vves}_{n}^{mat}}{{Vtis}_{n}}; & (2) \end{matrix}$
The EVD_n ^imand EVD_n ^matis the relation of volume of vessels feeding the tissue, to a number of living tissue cells. The amount of immature vessels at a moment “n” depends on an amount of both immature and mature vessels at the previous moment “n−1”.
All above described processes have an effect on the changes of the amount of vessels. The volume of immature vessels (block 1.4) at the moment “n” is a function of the volumes of immature and mature vessels at the moment “n−1”. This function has 5 terms, corresponding to the five exponential terms below. They are computed in blocks 1.5, 1.6, 1.7, 1.8 and 1.9.
$\begin{matrix} V_{n}^{im} = V_{n - 1}^{im} * e^{(\begin{matrix} A_{n - 1}^{im \Rightarrow new} \frac{T_{0}}{T_{3}} - \\ A_{n - 1}^{im \Rightarrow reg} \frac{T_{0}}{T_{4}} - A_{n - 1}^{im \Rightarrow mat} \frac{T_{0}}{T_{6}} \end{matrix}) * ϕ} + V_{n - 1}^{m} * e^{(\begin{matrix} A_{n - 1}^{mat \Rightarrow new} \frac{T_{0}}{T_{5}} + \\ A_{n - 1}^{mat \Rightarrow im} \frac{T_{0}}{T_{7}} \end{matrix}) * ϕ} - 1); & (3) \end{matrix}$
The generation of immature vessels by immature vessels (A^im
^new) is accounted for by block 1.5. The generation of immature vessels by mature vessels (A^mat
^new)is accounted for by block 1.6. The destabilization of mature vessels (A^mat
^im) is accounted for by block 1.7. The maturation of immature vessels (A^im
^mat) is accounted for by block 1.8. The degeneration of immature vessels (A^im
^reg) by is accounted for by block 1.9. The volume of mature vessels (block 1.10) at the moment “n” is also a function of the volumes of immature and mature vessels at the moment “n−1”. This function has 2 terms corresponding to the two exponential terms as shown below. They are calculated in blocks 1.7 and 1.9 respectively.
$\begin{matrix} V_{n}^{m} = V_{n - 1}^{im} * (e^{A_{n - 1}^{im \Rightarrow mat} \frac{T_{0}}{T_{6}} * ϕ} - 1) + V_{n - 1}^{m} (2 - e^{A_{n - 1}^{mat \Rightarrow im} \frac{T_{0}}{T_{7}} * ϕ}); & (4) \end{matrix}$
The maturation of immature vessels (A^im
^new) is accounted for by block 1.7 and the destabilization of mature vessels (A^mat
^im) is accounted for by block 1.9.
Every sub process described in functions (3) and (4) has its characteristic time, denoted by T₁to T₇. Resolution is denoted by T₀(the period between “n” and “n−1”). Factor φ represents a factor of the conformity.
The terms in eqns. (3) and (4) are functions of the following concentrations: the generation of immature vessels is a function of the concentration of VEGF with the coefficient λ_im ^ec, λ_mat ^ec, μ_ecand ρ_ec ^imand Eqns. (5) and (6);
$\begin{matrix} {\begin{matrix} A_{n - 1}^{im \Rightarrow new} = ρ_{ec}^{im} * (λ_{im}^{ec} * \frac{{VEGF}_{n - 1}}{{VEGF}_{thr}} - μ_{ec} / \frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}) \\ A_{n - 1}^{im \Rightarrow new} = 0 IF (λ_{im}^{ec} * \frac{{VEGF}_{n - 1}}{{VEGF}_{thr}} - μ_{ec} / \frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}) \leq 0 \\ A_{n - 1}^{im \Rightarrow new} = ρ_{ec}^{im} IF (λ_{im}^{ec} * \frac{{VEGF}_{n - 1}}{{VEGF}_{thr}} - μ_{ec} / \frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}) \geq 1 \end{matrix} & (5) \end{matrix}$
The degeneration of immature vessels is also a function of the concentration of VEGF, level VEGF_thr, with the coefficient μ_im.
$\begin{matrix} {\begin{matrix} A_{n - 1}^{mat \Rightarrow new} = ρ_{ec}^{im} * (λ_{mat}^{ec} * \frac{{VEGF}_{n - 1}}{{VEGF}_{thr}} - μ_{ec} / \frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}) \\ A_{n - 1}^{mat \Rightarrow new} = 0 IF (λ_{mat}^{ec} * \frac{{VEGF}_{n - 1}}{{VEGF}_{thr}} - μ_{ec} / \frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}) \leq 0 \\ A_{n - 1}^{mat \Rightarrow new} = ρ_{ec}^{im} IF (λ_{mat}^{ec} * \frac{{VEGF}_{n - 1}}{{VEGF}_{thr}} - μ_{ec} / \frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}) \geq 1 \end{matrix} & (6) \end{matrix}$
The destabilization of mature vessels (block 1.9) is a function of the ratio between Ang1 and Ang2 with the coefficient μ_mat ^im, Eqn. (8).
$\begin{matrix} A_{n - 1}^{im \Rightarrow reg} = μ_{im} / \frac{{VEGF}_{n - 1}}{{VEGF}_{thr}}; & (7) \\ A_{n - 1}^{mat \Rightarrow im} = μ_{mat}^{im} / \frac{Ang 1_{n - 1}}{Ang 2_{n - 1}}; & (8) \end{matrix}$
The maturation of immature vessels (block 1.7) is a more complicated function, Eq. (9).
$\begin{matrix} A_{n - 1}^{im \Rightarrow mat} = λ_{im}^{mat} * \frac{Ang 1_{n - 1}}{Ang 2_{n - 1}} * (\frac{{Nper}_{n - 1}}{ρ_{mat}^{per}}) / V_{n - 1}^{im}; & (9) \end{matrix}$
Maturation in a given moment is a function of a ratio of Ang1/Ang2 at the same moment, with the coefficient λ_im ^mat. Maturation is also a function of the volume of immature vessels and of the number of free pericytes. The term (N_per/ρ_mat ^per)/V^imgives the fraction of immature vessels can potentially mature (If (N_per/ρ_mat ^per)/V^im>=1 then all immature vessels can mature).
The egn. 10 shows the functional dependence VEGF from EVD with characteristic time T₈and T_n−1 ^VEGF.
$\begin{matrix} {VEGF}_{n - 1} = V_{n - 1}^{tis} * λ_{EVD}^{VEGF} ({EVD}_{b} - {EVD}_{n - 2}) * \frac{T_{0}}{T_{8}} + {VEGF}_{n - 2} * e^{(\begin{matrix} - 0.693171805 * \\ \frac{T_{0}}{T_{h - 1}^{VEGF}} \end{matrix})} & (10) \end{matrix}$
IF VEGF_n−1>VEGF_maxTHEN VEGF_n−1=VEGF_max
IF VEGF_n−1<VEGF_enTHEN VEGF_n−1=VEGF_en
The initial level VEGF_sscharacterizes the amount of VEGF secreted when effective tissue vessel density is EVD_ss.
In a similar way we obtain the dependence of PDGF (11).
$\begin{matrix} {PDGF}_{n - 1} = V_{n - 1}^{tis} * λ_{EVD}^{PDGF} ({EVD}_{b} - {EVD}_{n - 2}) * \frac{T_{0}}{T_{14}} + {PDGF}_{n - 1} * e^{(\begin{matrix} - 0.693171805 * \\ \frac{T_{0}}{T_{h - 1}^{PDEGF}} \end{matrix})} & (11) \end{matrix}$
IF VEGF_n−1>VEGF_maxTHEN VEGF_n−1=VEGF_max
IF VEGF_n−1<VEGF_enTHEN VEGF_n−1=VEGF_en
The characteristic time T₁₄and T_n−1 ^PDGF. The above equations 10 and 11 are involved in blocks marked 1.11 & 1.12.
$\begin{matrix} Ang 2_{n - 1} = (\begin{matrix} (Ang 2_{en}^{ec} + Ang 2_{ss}^{ec}) * {EC}_{n - 2} + \\ Ang 2_{ed}^{ec} * ({EDV}_{ss} - {EVD}_{n - 2}) * {EC}_{n - 2} \end{matrix}) * \frac{T_{0}}{T_{10}} ++ (\begin{matrix} (Ang 2_{en}^{tc} + Ang 2_{ss}^{tc}) * N_{n - 2}^{tum} + \\ Ang 2_{ed}^{tc} * ({EVD}_{b} - {EVD}_{n - 2}) * N_{n - 2}^{tum} \end{matrix}) * \frac{T_{0}}{T_{15}} + Ang 2_{n - 2} * e^{(\begin{matrix} - 0.693171805 * \\ \frac{T_{0}}{T_{h - 1}^{Ang 2}} \end{matrix})} & (12) \end{matrix}$
In Eqn. (12, 14) Ang2 and Ang1 also depends on the numbers of endothelial cells in immature vessels, which is determined by the Eqs. (13), and the numbers of tissue cells Ang2_en ^ec, Ang2_b ^ec, Ang2_ed ^ec, Ang2_en ^tc, Ang2_b ^tc, Ang2_ed ^tc, T_n−1 ^Ang2, ρ_V _im ^ec, Ang1 _en ^ec, Ang1_b ^ec, Ang1_ed ^ec, Ang1_en ^tc, Ang1_b ^tc, Ang1_ed ^tc, T_n−1 ^Ang1. The Ang1 induction and Ang2 induction are factors in bocks 1.4, 1.14 and 1.15.
EC _n−1 =
K ₅
ρ_V _im ^ec *V _n−1 ^im;
The characteristic reaction time for Ang2 generation is T₁₀and T₁₅, (12), for Ang1 generation it is T₉and T₁₁(14).
$\begin{matrix} Ang 1_{n - 1} = (\begin{matrix} (Ang 1_{en}^{tc} + Ang 1_{ss}^{tc}) * N_{n - 2}^{tum} + \\ Ang 1_{ed}^{tc} * ({EVD}_{ss} - {EVD}_{n - 2}) * N_{n - 2}^{tum} \end{matrix}) * \frac{T_{0}}{T_{9}} ++ (\begin{matrix} (Ang 1_{en}^{ec} + Ang 1_{ss}^{ec}) * {EC}_{n - 2} + \\ Ang 1_{ed}^{ec} * ({EVD}_{ss} - {EVD}_{n - 2}) * {EC}_{n - 2} \end{matrix}) * \frac{T_{0}}{T_{16}} + Ang 1_{n - 2} * e^{(\begin{matrix} - 0.693171805 * \\ \frac{T_{0}}{T_{h - 1}^{Ang 1}} \end{matrix})} & (14) \end{matrix}$
The addition of free pericytes (block 1.17) at any given moment depends on the level of PDGF at the previous moment, the replication of free pericytes, and on the number of free pericytes released from mature vessels (15). Accordingly, these two processes have the coefficients λ_bou ^perand λ_fr ^per. It is also necessary to take into account the characteristic reaction time of these processes T₁₂and T₁₃.
$\begin{matrix} {Nper}_{n - 1} = {PDGF}_{n - 2} * (λ_{bou}^{per} * V_{n - 2}^{mat} * \frac{T_{0}}{T_{12}} + ({Nper}_{n - 2} - ρ_{mat}^{per} * (V_{n - 1}^{mat} - V_{n - 2}^{mat}) * λ_{fr}^{per} * \frac{T_{0}}{T_{13}}) & (15) \end{matrix}$
The number of tissue cells (block 1.18) in the moment, n, depends on their number in the previous moment multiplied by a factor describing the process of cell proliferation and death, r_n−1.
V _n ^tis =V _n−1 ^tis *e ^r ⁿ⁻¹ ^*φ (16)
r_n−1depends on the mitotic index M(I) (mitotic time being T(1), apoptotic index A (I) (apoptotic time being T (2)), rate of tissue cell growth λ and the rate of the death of tumor cells, μ. The two terms in the equation below are involved in blocks 1.191 and 1.201 respectively. Clearly, they are also factors in blocks 1.192 and 1.202 as well as 1.193 and 1.203.
$r_{n - 1} = (M_{I} + λ_{n - 1} - ɛ_{1}) * \frac{T_{0}}{T_{1}} - (A_{I} + μ_{n - 1} + ɛ_{2}) * \frac{T_{0}}{T_{2}}$
The proliferation rate λ and the death rate μ are assumed to be standard sigmoids. Hence we obtain λ and μ as follows:
$\begin{matrix} μ_{n - 1} = 1 - A_{I} - \frac{(1 + ɛ_{2}) * {EVD}_{n - 1}^{\frac{1 + ɛ_{2}}{2 * A_{I} - 1 + ɛ_{2}}}}{\begin{matrix} {EVD}_{n - 1}^{\frac{1 + ɛ_{2}}{2 * A_{I} - 1 + ɛ_{1}}} + \\ (\frac{A_{I} + ɛ_{2}}{1 - A_{I}} * {EVD}_{ss}^{}) \end{matrix}} & (18) \\ λ_{n - 1} = \frac{(1 + ɛ_{1}) * {EVD}_{n - 1}^{\frac{1 + ɛ_{1}}{1 - 2 * M_{I} + ɛ_{1}}}}{\begin{matrix} {EVD}_{n - 1}^{\frac{1 + ɛ_{1}}{1 - 2 * M_{I} + ɛ_{1}}} + \\ (\frac{1 - M_{I} + ɛ_{1}}{M_{I}} * {EVD}_{ss}^{}) \end{matrix}} - M_{I} & (19) \end{matrix}$
IV.C. Tissue Control by Pro and Anti Angiogenic Drugs.
Possible drug effects on the pro and anti angiogenesis process indicated in FIG. 2. Note that the blocks in FIG. 2 are identical to those in FIG. 1, except for the additional drug effects shown. The drug effects on the overall process are analyzed by setting the selected drug schedule (number of doses, the dose and the dosing interval). For example the analysis of anti-VEGF drug activity shows that a drug which inhibits VEGF has an optimal efficacy when given by certain treatment protocol. Increasing the administered dose above the optimum can bring about the undesired effect of tissue proliferation as shown in FIG. 3. In addition, the technique enables one to predict the effects of various drug combination, for example as shown in FIG. 4.
Other modifications and variations to the invention will be apparent to those skilled in the art from the foregoing disclosure and teachings. Thus, while only certain embodiments of the invention have been specifically described herein, it will be apparent that numerous modifications may be made thereto without departing from the spirit and scope of the invention.

Claims

1. A computer-implemented method for determining an optimal treatment protocol for a disease related to angiogenesis, comprising:

creating an angiogenesis model including pro-angiogenesis and anti-angiogenesis factors;

incorporating effects of vasculature maturation and mature vessels destabilization;

selecting pro-angiogenesis and anti-angiogenesis factors, which can influence changes in state of a tissue;

incorporating effects of drugs in the pro-angiogenesis and anti-angiogenesis factors; generating a plurality of treatment protocols in a protocol space; and

selecting a best treatment protocol based on a pre-determined criteria.

2. A system for determining an optimal treatment protocol for a disease related to angiogenesis, comprising:

an angiogenesis model including pro-angiogenesis and anti-angiogenesis factors;

a treatment protocol space generator that generates a protocol space of possible treatments for the disease;

a treatment selector that selects an optimal protocol,

wherein the model incorporates effects of vasculature maturation and mature vessels destabilization;

wherein the system is adapted to effect selection of pro-angiogenesis and anti-angiogenesis factors which can influence changes in state of a tissue and incorporating effects of drugs in the pro-angiogenesis and anti-angiogenesis factors.

3. A computer program product, including computer-readable media comprising instructions to implement procedures for determining an optimal treatment protocol for a disease related to angiogenesis, said procedure comprising:

incorporating effects of drugs in the pro-angiogenesis and anti-angiogenesis factors;

generating a plurality of treatment protocols space; and

selecting a best treatment protocol based on a pre-determined criteria.