Dosage Optimization for a Virotherapy Model

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Dosage Optimization for a Virotherapy Model

• Matt Biesecker
• South Dakota St. Univ

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Discussion Outline

• Modeling Tumor Growth with ODEs
• Mathematical model relating the dynamics of tumor
  growth and virions
• Model Simulations
• Optimization of Therapy Protocols
• Future Research

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Tumor Dynamics 101

• Tumor Cells will divide periodically provided
  there are nutrients (Exponential Phase)
• As the tumor mass gets larger, competition for
  nutrients slows tumor growth (Lag Phase).
• Growth rates can be maintained if the tumor
  develops its own blood supply (angiogenesis)

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Modeling Tumor with Ordinary Differential Equatons
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r controls the growth rate during the
exponential growth phase
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epsilon affects the rate of transition from
exponential to the slower lag phase
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Modeling of Tumor Dynamics with Virotherapy

• Wu, Byrne, Kirn, and Wein. Bull of Math. Biol.
  63 (2001), 731
• Wein, Wu, Kirn. Cancer Res. 63 (2003) 1317
• Dingli, Carr, Josic, Russel, Bajzer Jour.
  Theo. Biol. 2008

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Principles of Virotherapy

• In the 1950s it was discovered that tumor
  growth ceased in some patients with a strain of
  the measles virus.
• Efforts are underway to produce viruses which
  preferentially infect tumor cells
• The virus alters the host cell in a such a way
  the tumor cell ceases to divide.
• Bonus Infected Cells can produce additional
  virus, thus amplifying the original dose.

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Unlike the movie I Am Legend, there is no known
risk of zombification as a result of virotherapy!
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Cartoon Model (from Bajzer, et al, JTB, 2008)
r, K, ?
?v
?yv
v
y
?x
?yx
? x
y(t) uninfected tumor cells x(t) infected
tumor cells/syncytia v(t) free virus particles
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Mathematical Model
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Model Assumptions

• y(t), x(t), v(t) measured in millions of cells
  
• Eradication Criteria The tumor is assumed to
  eradicated if at any time we have
• x(t) y(t)
• Eradication is (eventually) guaranteed if all
  tumor cells are infected That is,
• y(t)

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Model Parameters

• Generalized Gompertz Growth.
• r0.207, K2200, e1.65
• Various Fits for in vivo data for mice (See
  Bajzer, et al)
• a - viral replication rate
• d - death rate for infected cells
• ? infection rate parameter
• ? - rate of removal of free virions
• ? - rate constant for syncytia formation

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Parameter Fits For Mouse Data
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4 Possibilities for the Dynamics

• Therapy Failure
• Eradication of the Tumor
• Partial Reduction of Tumor Mass
• Period Fluctuations in Tumor Mass
• Important Caveat
• Treatment efficacy depends on model parameters
  and dosage volume.

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Therapy Failure

• Characterized by infected cells/scinthia and free
  virions vanishing before all tumor cells are
  infected.
• Possible Causes Slow Viral Replication Rapid
  Death of Infected Cells. Rapid clearance of free
  virions.

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Upon further review, we see that the free virions
v(t) and infected cell lines x(t) vanish at about
t300.
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Eradication of The Tumor
Immediate Eradication with extremely high viral
dosage.
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Later Eradication
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Partial Success
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Sustained Oscillations
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Questions Recently Investigated

• What is the minimum virus dosage necessary to
  eradicate the tumor?
• Can multiple dosages be more effective than a
  single dose? YES!!

Not in vivo, but with models simulations
e.g. in silico
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Modeling the Efficacy of a Single Dosage

• Treating smaller tumors is less effective than
  treating larger tumors.
• This seems to be independent of the parameter
  values!
• For parameter fits corresponding to currently
  available viruses, the doses required for
  eradication are impossible to achieve in a
  clinical setting.

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Methodology Brute Force Calculation

• Solve the ODE system for (almost) every possible
  combination of initial conditions.
• Let v(0), y(0) vary over a range
• from 10-6 to 104
• Plot all points (v(0),y(0)) where the tumor is
  eradicated within 1000 days.

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A plot of the time at which at which every cell
is infected. The white region consists of
points y(0),v(0) where the tumor is not
eradicated
a0.053 ?0.00096, d0.015, ?0.001, ?0.5
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A plot of the time at which at which every cell
is infected. The white region consists of
points y(0),v(0) where the tumor is not
eradicated.
y(0)
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Strange Results Are Possible !!

• If the viral replication rate is large enough,
  eradication is possible with extremely small
  dosages.

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?0.00059,d0.021,?0.14, ?0.3, a1.3Color
Bar Time when uninfected cell line y(t) is
eradicated
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Change a to 0.1, and the hole disappears. Color
Bar Time when uninfected cells y(t) vanish.
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Modelling Multiple Dosages
Dosages Volumes
Delivery Times
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Evaluating Dosage Effectiveness

• For a given dosage scheme, we solve the ODE over
  the time horizon 0,T and compute two
  quantities

Goal of Treatments Obtain mM below a desirable threshold.
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Designing an Objective Function
The constants a and b are weights corresponding
to the competing priorities of eradication vs.
Keeping the tumor volume below 1200 cubic mm.
I use a10, b0.001 in the following results

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Constraints on Treatments
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Optimization Plan

• 1. Choose Vtot (total virus constraint)
• 2. Find treatment (?,t) plan that minimizes the
  objective function J.
• If J(?,t)0, then both our goals of eradication
  and containment.
• If so, we go back to step 1 and reduce Vtot and
  then repeat step 2. We quit stop reducing Vtot
  when it appears we can no longer achieve J(?,t)
  0

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Optimization Details

• Once the constraint V_tot is set, we use the
  Sequential Linear Programming (SLP) method. In
  particular, we perform the following iterative
  procedure.
• 1. Choose initial treatment plan

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Optimization Details, contd

• 2. Then we compute solution of the ODE
• and determine the values of
• min y(t) and maxx(t) y(t)
• Over the given time horizon 0,T.
• Then we get the value of the objective function
  J(?(0),t(0))

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Optimization Details, contd

• 3. Estimate the gradient vectors

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Optimization Details, contd

• 4. Form a local minimization problem for the
  linearized objective function.

where
Our original constraints imply the constraints on
dt and d? form a 2n-dimensional simplex!
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Optimization Details, contd

• Minimize the localized objective function and
  obtain the search directions dt and d?
• The we perform line searches to determine
  non-negative constants
• ß1 1 and ß2 1
• such that the new treatment protocol

reduces the value of J objective function. Then
go back to step 1 and do it all over again!
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Results 2 Doses Can be Better Than 1 !
Optimal Dosage of v0.13, and v1.08 delivered
around t0.5 and t40, respectively.
Parameters a0.053 ?0.00096, d0.015,
?0.001, ?0.5
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• Total volume for 2 doses v1.21.
• Minimal single dose to eradicate When
  y100 , we need v 3.58
• When y1200, we need v1.42

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Comparison of Various Treatments
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If 2 is better than 1, then 4 HAS to be better
than 2 ?
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Not Really!

• Opimization Results for 4 doses (Average of
  100 simulations)

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• Dosage Timing is Crucial !

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Good News Conjecture Giving
a second dose after the tumor mass peaks is more
likely to result in total elimination of the
uninfected cells.
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Bad News

• For some model parameter sets, it is possible
  that ill-timed secondary treatments can do more
  harm than good!

Second Treatment Given when t3
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Future Efforts

• For the simple ODE model, we observed strange
  phenomena
• -Large tumors are easier to treat
• - Variable doses (properly timed) dosages can
  be substantially more effective than periodic
  doses of equ al volume.
• Do the same phenomena arise in other models?
• 4 variable ODE model developed by Bajzer,
  Dingli.
• PDE models (w/ B. Jungman)
• Celluar Automata models (w/ T. Wilson)