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Title: Resonance Phenomenon in Population Dynamics: From Mathematical Theory to Clinical Implementation in


1
Resonance Phenomenon in Population Dynamics
From Mathematical Theory to Clinical
Implementation in Oncology Zvia Agur
2
Agenda
  • The Need Why we are losing the war on cancer?
  • Changing the paradigm in medicine
  • Concept development
  • 1.     Problem-focused simple maths. models
  • 2.    Universal theories applied at multiple
    organization levels
  • 3.    Realistic treatment optimization concepts
  • 4. Experiments to verify maths-based theories
    (proof of concept)
  • 5.    Assist physicians by predictive models
  • i. On the whole organisms level
  • Precise modeling of drug-target physiological
    pathological processes
  • ii. Heuristic optimization methods
  • Validation

3
Need to predict effects of intra-cellular
interactions on the whole organism
March 22, 2004
4
1. Changing the paradigm in medicine
  • Concept development
  •   
  • Problem-focused simple mathematical models

5
Problem-oriented Simple Models Dynamics of
populations with complex life-cycle under random
catastrophes
  • 1. Intrinsic rate of increase large enough
  • 2. Biphasic life-cycle
  • resistant juveniles - longevity t
  • susceptible adults - breed cotinuously.
  • 3. No specific distribution of inter-disturbance
  • intervals, w. Constant disturbance duration
    d.
  • Now dynamics can be represented as a
    1-dimentional process on the time axis
  • This model was solved by elementary mathematics
    (Agur, 85)

6
Results Proved analytically
That the phenomenon of resonance in population
survival
exists for any population, undergoing a periodic
loss process, effective only during a portion of
the life-cycle When the period of the imposed
loss process coincides with the inherent
reproductive periodicity of the population there
is a preferential enhancement of the population
growth. In general, one can use forcing which
has a different periodicity than the internal
population periodicity to control population
growth. (Agur, Lect Notes Biomath 1982
Jour Theor Biol 1985, Theor Pop Biol, 1985)  
7
2. Universal theories applied at multiple
organization levels
Classical science has universal laws, those in
biology should be applicable at all levels of
biological organization
Application in population dynamics - marine
intertidal
Application in population dynamics - vaccination
policies
Application in population dynamics vs. immune
system
Application on cellular pop dynamics -
chemotherapy
8
2. Universal theories applied at multiple
organization levels
Population dynamics in perturbed environments
w
d
Marine intertidal
t/(dw)
This result may be especially interesting for
increasing selectivity of anticancer drugs as the
non-monotonicity property suggests that it may be
easy to select drug protocols that are very toxic
to the neoplasia (minimum persistence) and little
toxic to the susceptible host tissues (maximum
persistence).
Agur Deneubourg Theor. Pop. Biol, 1985
9
Define a simple optimization problem
3. Practical treatment optimization concepts The
Z-Method"
Aim Increase drug selectivityby
  • making tumors elimination time as short as
    possible compared to the elimination time of
    bone-marrow cells.
  • i.e.,
  • minimize
  • Define treatment coefficient

(Agur, Arnon, Schechter, Math. Biosci, 88)
10
3. Practical treatment optimization concepts The
Z-Method"

Simulating a drug targeting cancer origin
cells
(Agur, Arnon, Schechter, Math. Biosci, 88)
11
3. Practical treatment optimization concepts The
Z-Method"
The idea of the Z-method is to maximize the
destruction of tumor cells by using a treatment
period which avoids resonance, and to minimize
destruction of normal cells by a treatment period
which matches resonance (dosing at integral and
fractional multiples of the average age of cell
division).
Agur et al., Math Biosc, 1988 Agur, Proc. NY
Acad Sci., 1988
12
4. Experiments to verify theories
13
Agenda
  • The Need Why we are losing the war on cancer?
  • Changing the paradigm in medicine
  • Concept development
  • 1.     Problem-focused simple models
  • 2.    Universal theories applied at multiple
    organization levels
  • 3.    Realistic treatment optimization concepts
  • 4. Experiments to verify theories
  • 5.    Assist physicians by predictive theories
  • i. On the whole organisms level
  • Precise modeling of drug-target physiological
    pathological processes
  • ii. Heuristic optimization methods
  • Validation

14
5. Assist physicians by predictive theories
From theory to the clinic
Patient-tailored treatments
Precise quantitative predictions
Detailed models of pathology and physiology
Complexity of the described processes must be
significantly increased
Precise modeling of drug-target physiological
pathological processes
15
Methodology of constructing a biomathematical
model
Solid Tumor Model as an example of a Virtual
Disease
First, we create a verbal model of the biological
processes
Process Outline
This example shows the basic components in the
growth of a solid tumor, including Angiogenesis
16
Methodology of constructing a biomathematical
model
Solid Tumor Model as an example of a Virtual
Disease
Then, for each process component we develop an
algorithm describing its operation.
Process Outline
This is an example of the algorithm describing
the function of the Maturation Box
17
Methodology of constructing a biomathematical
model
Solid Tumor Model as an example of a Virtual
Disease
Process Outline
Next, we express each biological algorithm in
mathematical equations.
Finally, the equations are implemented in a
computer software.
18
Methodology of constructing a biomathematical
model
Solid Tumor Model as an example of a Virtual
Disease
The computerized model is now ready for
verification and use.
Process Outline
Biological Algorithm
Mathematical Model
To verify the Solid Tumor model we used
experimental tumor growth curves of ovarian
carcinoma (0.905 mm3) sub-cutaneously implanted
in nude mice.
We take tumor GA as an example
19
Methodology of constructing a biomathematical
model
Solid Tumor Model as an example of a Virtual
Disease
Specific parameters of tumor GA are identified
using the parameter identification module for
accurately simulating growth of tumor GA.
Process Outline
Biological Algorithm
Mathematical Model
Simulation - black line Experiment purple line
Simulated living cells blue line Simulated
necrotic cells green line Y-axis Tumor size
(cell count), X-axis Time (days)
20
Validation of the Solid Tumor Model
Demonstrating the accuracy of the simulations
Accurate prediction of crucial biological
dynamics Effective vessel density (EVD) as an
example
The vascularization profile of tumor GA was
predicted and compared with the actual EVD,
experimentally measured by MRI.
VCPs Predictions
MRI readings
SI
tIme
21
Examples of Physiology Models Hematopoiesis
Thrombopoiesis
22
Validation of physiology Models Demonstrating
the accuracy of the simulations
Granulopoiesis Accurate simulation of human
neutrophil profiles under DoxorubicineG-CSF
The simulation accurately retrieves neutrophil
counts during chemotherapeutic treatment
supported by G-CSF. Regimen Three 14-days
cycles 75mg/m2 doxorubicin followed by infusion
of G-CSF for 11 days. Experimental results taken
from M.H. Bronchud et al Br J Cancer, 1989
23
Thrombopoietin (TPO) development as a thrombo-
cytopenia alleviating drug was arrested due to
its immunogenicity
Thrombopoiesis Model
The biological process is broken-down to its key
elements, outlining the schematic structure of
the verbal model. Next, the dynamics of the
process are formulated in mathematical equations
(e.g., the dependence of MKB cells maturation
time on TPO concentration) for more details, see
Skomorovsky et al. BJH, 2003.
24
Thrombopoietin (TPO) development as a thrombo-
cytopenia alleviating drug was arrested due to
its immunogenicity
Accurate simulation of platelet profiles under
TPO in human subjects
0.3 µg/Kg TPO
1.2 µg/Kg TPO
Plts Counts
Days
2.4 µg/Kg TPO
Experimental data Vadhan-Raj, 1997, Ann.
Intern. Med.
25
Model-generated optimal TPO schedule for plts
donors was first validated in a murine model
Predictions of murine platelet profiles under VCP
recommended optimal TPO regimen were validated in
vivo.
Prediction Same efficacy can be achieved, with
no hypersensitivity, using 45 of the standard
drug dose. Purple single dose of 17.5
mg/kg Blue 4 daily doses of 2 mg/kg each
Experimental Validation TPO was administered to
mice according to the model predicted schedules
(left graph) achieving precisely the predicted
results (Skomorovsky et al. BJH, 2003).
26
Model-suggested safe TPO regimen was validated in
monkeys
Our non-toxic highly effective schedule for TPO
is now routinely used in platelets donors
(Rhesus)
Outcome of Standard Treatment
Outcome of Optimatas Treatment
Maximizing efficacy and minimizing immunogenicity
validated in Rhesus, thus relieving No-Go from
drug development (Skomorovsky et al. BJH, 2003).
No Immunogenicity
27
Example Prospective validation of Models
prediction accuracy in 3 cancer patients treated
in Soroka Hospital
28
and suggests better treatment regimens
Cancer model accurately simulates a patients
lung metastasis progression under approved
schedule
CurrentTreatment
100 mg/m2 of Taxotere every 3 weeks, IV infusion
Response
Efficacy
Tumor size
27.5 mm
15 mm
7.5 mm
Toxicity
Neutrophil count
Grade III
The suggested schedule (right) is expected to
reduce tumor burden by at least 50 and alleviate
Neutropenia
29
Resonance phenomenon in population survival
from theory to the clinic (Agur et al.,
1983-2005)
  • Models of population dynamics pinpoint the
    Resonance Phenomenon by which population growth
    is maximized when disturbance periodicity is an
    integer or fractional multiple of the population
    characteristic periodicity.
  • Analysis, suggests that drug therapy can be
    optimized by schedules, employing the Resonance
    Effect in conjunction with known differences in
    cell-cycle distributions of host and cancer cells
    (Z-Method).
  • The Z-Method, which uses the Resonance
    phenomenon, was verified experimentally and new
    optimization methods were developed.
  • Quantitative predictions about the optimal
    administration of the chemotherapy supportive
    drug, TPO, were validated in preclinical trials
    the maths-based TPO schedule is now in clinical
    use.
  • Tox. models were validated clinically and it was
    suggested that the developed models and their
    validation procedures provide solid grounds for
    treatment personalization in oncology.

30
Selected Literature
  • Agur Z., Hassin R., Levy S. Optimizing
    chemotherapy scheduling using local search
    heuristics, Operations Research, in press.
  • Y. Kheifetz, Y. Kogan Z. Agur. Long-range
    predictability in models of cell populations
    subjected to phase-specific drugs growth-rate
    approximation using properties of positive
    compact operators. Mathematical Models Methods
    in the Applied Sciences. In Press.
  • Vainstein V., Ginosar Y., Shoham M., Ranmar D.,
    Ianovski A., Agur Z. The complex effect of
    granulocyte on human granulopoiesis analyzed by a
    new physiologically-based mathematical model.
    Jour Theor Biol. 234(3), 2005 (pp.311-27).
  • Ribba B., Marron K., Alarcon T., Maini P, Agur Z.
    A mathematical model of doxorubicin treatment
    efficacy for non-Hodgkins lymphoma
    Investigation of the current protocol through
    theoretical modelling results Bull. Math. Biol.,
    67, 2005 (pp. 79-99).
  • Arakelyan L, Merbl Y., Agur Z. Vessel maturation
    effects on tumour growth validation of a
    computer model in implanted human ovarian
    carcinoma spheroids  Eur. Jour. Cancer, 41, 2005
    (pp.159-167).
  • Kheifetz Y., Kogan Y., Agur Z. Matrix and
    compact operator description of resonance and
    antiresonance in cell populations subjected to
    phase specific drugs. Jour. Medical Informatics
    Technologies Vol. 8, MM-11 MM-29, 2004.
  • Skomorovski K., Harpak H., Ianovski A., Vardi M.,
    Visser TP., Hartong S., Van Vliet H., Wagemaker
    G., Agur Z. New TPO treatment schedules of
    increased safety and efficacy pre clinical
    validation of a thrombopoiesis simulation model.
    Br. Jour. Haematol, 123 (4), 2003 (pp. 683-691).
  • Arakelyan L, Vainstain V., Agur Z. A computer
    algorithm describing angiogenesis and vessel
    maturation and its use for studying the effects
    of anti-angiogenic and anti-maturation therapy on
    vascular tumor growth, Angiogenesis, 5 (3), 2002
    (pp. 203-14).
  • Hart D., Shochat E., Agur Z. The growth law of
    primary breast cancer tumors as inferred from
    mammography screening trials. British
    Jour.Cancer, 78 (3) 1998 (pp. 382-387).
  • Ubezio P., Tagliabue G., Schechter B., Agur Z.
    Increasing 1-?-D-Arabinofuranosylcytosine
    efficacy by scheduled dosing intervals based on
    direct measurement of bone marrow cell kinetics,
    Cancer Res 54, 1994 (pp. 6446 - 6451).
  • Agur Z., Dvir Y. Use of knowledge on ?n series
    for predicting optimal chemotherapy treatment.
    Random Computational Dynamics 2(34), 1994
    (pp. 279-286).
  • Harnevo L., Agur Z. Drug resistance as a dynamic
    process in a model for multi-step gene
    amplification under various levels of selection
    stringency. Cancer Chemo. Pharmacol., 30,1992
    (pp. 469 - 476).
  • Cojocaru L., Agur Z. Theoretical analysis of
    interval drug dosing for cell-cycle-phase-specific
    drugs. Math. Biosci., 109, 1992 (pp. 85 - 97).
  • Agur Z., Arnon R., Schechter B. Effect of the
    dosing interval on myelotoxicity and survival in
    mice treated by cytarabine. Eur. Jour. Cancer,
    28A(6/7),1992 (pp. 1085 - 1090).
  • Agur Z., Arnon R., Schechter B. Reduction of
    cytotoxicity to normal tissues by new regimens of
    phase-specific drugs. Math. Biosci. 92, 1988
    (pp.1-15).

31
QA
agur_at_imbm.org
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