Numerical Methods in Cancer ModelsNumerical Methods in Cancer Models Doron Levy Department of...

Motivation

Properties

numerical methods

Stability in the delays space

Numerical Methods in Cancer Models

Doron Levy

Department of Mathematics

and

Center for Scientific Computation and Mathematical Modeling (CSCAMM)

University of Maryland, College Park

Doron Levy Montreal, May 2013

Motivation

Properties

numerical methods


Plan

1 What is cancer?2 Delayed di↵erential equations3 Agent-based models4 PDEs


Motivation

Properties

numerical methods


Delayed Di↵erential Equations in Cancer ModelsAnalysis & Numerical Methods

Doron Levy

Department of Mathematics

and

Center for Scientific Computation and Mathematical Modeling (CSCAMM)

University of Maryland, College Park


Motivation

Properties

numerical methods


Outline

1 MotivationCancer ImmunologyStem Cell Transplantation

2 PropertiesZero CrossingsTime Scales

3 Numerical Methods

4 Stability in the delays space


Motivation

Properties

numerical methods


The immune response to leukemia

Stem cell transplantation

Outline



3 Numerical Methods



Motivation

Properties

numerical methods




What is leukemia?

Normal cells: stem cells turn intomature cells

Leukemia: A malignanttransformation of a stem cell or aprogenitor cell

Myeloid or Lymphocytic

Acute or Chronic


Motivation

Properties

numerical methods




CML

3 phasesChronic: uncontrolled proliferationAcceleratedAcute: Uncontrolled proliferations. Cellsdo not mature

Philadelphia chromosomeTranslocation (9;22)Oncogenic BCR-ABL gene fusionThe ABL gene expresses a tyrosinekinase. Growth mechanismsEasy to diagnoseDrug targeting this genetic defect (atyrosine kinase inhibitor)


Motivation

Properties

numerical methods




Treating leukemia

Chemotherapy

Bone Marrow or Stem Cell transplantChemo + radiotherapy +transplantation

Imatinib (Gleevec)Molecular targeted therapy -suppresses the corrupted controlsystem$32K-$98K/year


Motivation

Properties

numerical methods




Problems with existing therapies

Remission vs. Cure: Can CML be cured?Yes! but only with a bone marrow (or stem cell) transplantRequires a (matching) donorA risky procedure (+ unpredictable side e↵ects)

Imatinib? Does not cure the disease: stopping it causes a relapse

New Medical Data: There is an anti-leukemia immune response (Leelab)

The strength and dynamics of the specific anti-leukemia immuneresponse can be measured

Number of cellsActivity (count signaling molecules)


Motivation

Properties

numerical methods




The observed immune response

A di↵erent immune response for each patient. However:

At the beginning of the treatment: no immune response

Peak: around 6-12 months (after starting the drug treatment)

Later: waning immune response

Question:

What is the relation between the dynamics of the cancer, the drug, andthe immune response?


Motivation

Properties

numerical methods




A mathematical model

Leukemicstem cell (y0)

Progenitorcell (y1)

Differentiatedcell (y2)

Terminalcell (y3)

ay(ay’ ) b y(by’ ) c y(cy’ )

divider y

d0 + qCp(C,T) d1 + qCp(C,T) d2 + qCp(C,T) d3 + qCp(C,T)

sT

dT

n!

T cells (T)p0e-cnCkC

x2n

Ingredients:Leukemia cells: stem cells, , fully functional cellsMutations, Drug (Imatinib), Anti leukemia immune response

Kim, Lee, Levy: PLoS Computational Biology, ’08

Michor et al. (Nature 05). Cronkite and Vincent (69), Rubinow (69), Rubinow & Lebowitz (75), Fokas,

Keller, and Clarkson (91), Mackey et al (99,...), Neiman (00), Moore & Li (04), Michor et al (05), Komarova & Woodarz (05).


Motivation

Properties

numerical methods




Michor’s model + immune response

Cells without mutations:

y0

= [ry (1� u)� d0

]y0

� qcp(C ,T )y0

,

y1

= ayy0 � d1

y1

� qcp(C ,T )y1

,

y2

= byy1 � d2

y2

� qcp(C ,T )y2

,

y3

= cyy2 � d3

y3

� qcp(C ,T )y3

.

Anti-cancer T cells:

T = st � dtT � p(C ,T )C + 2nqTp(Cn⌧ ,Tn⌧ )Cn⌧ ,

p(C ,T ) = p0

e�cnCkT , C =X

(yi + zi ), Cn⌧ = C (t � n⌧).


Motivation

Properties

numerical methods




Accounting for the immune response

0 10 20 30 40 500

0.01

0.02

0.03

0.04

0.05

0.06

T cells

Leukemia

No immuneresponse

Time (months)

Cel

l Con

cent

ratio

n (k

/µL)


Motivation

Properties

numerical methods




Stopping imatinib (simulation)

0 5 10 15 200

5

10

15

20

25

30

35

40

Leukemia

1000 x T cellconcentration

Time (months)

Cel

l Con

cent

ratio

n (k

/µL)

Imatinib removed at month 12


Motivation

Properties

numerical methods




Cancer vaccines: a mathematical design

0 2 4 6 8 10 12 14 160

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Leukemia cells

Anti-leukemia T cells

Time (months)

Cel

l Con

cent

ratio

n (k

/µL)

Inactivated leukemiacells (vaccines)

0 5 10 15-20

-15

-10

-5

0 5 10 15-20

-15

-10

-5

0

0 5 10 15-20

-15

-10

-5

0

0 5 10 15-15

-10

-5

0

5

log 10

[con

cent

ratio

n]

SC PC

Time (months)

log 10

[con

cent

ratio

n]

DC

Time (months)

TC

Inactivated leukemia cells

V = �dVV � qcp(C ,T )V + sv (t)

Anti-cancer T cells

T = st � dtT � p(C ,T )(C + V ) + 2np(Cn⌧Tn⌧ )(qTCn⌧ + Vn⌧ )


Motivation

Properties

numerical methods




Model populations

Host CellsCancerAnti-donor T cellsGeneral blood cells

Donor cellsAnti-cancer T cells(cancer-specific)Anti-host T cellsGeneral blood cells


Motivation

Properties

numerical methods




Everything takes time


Motivation

Properties

numerical methods




Anti-Cancer T Cells

Survive

Perish

Ignore

ReactTC

Death

!!

kTDTC

TD /T

C InteractionT C/C

Inte

ract

ion

n"#Proliferate

Reload

p2TC/C

p1TC/C

p1TD/TC

p2TD/TC

x2n

kCTC

dTC

q1TC/C

q2TC/C

q3TC/C

Die or Become anergic

$


Motivation

Properties

numerical methods




Anti-Host T Cells

!

"

dTH

kCTH

p2TH/C

TH /T

D Interaction

T H/H

Inte

ract

ion

p2TH/H

p1TH/H

kHTHn# $

x2n

x2n

q1TH/H q2

TH/H

"

$

p1TH/C

q1TH/C

q2TH/C

n#Proliferate

Reload

"

!n#

$Reload

Proliferate

x2n

kTDTH

T H/C

Inte

ract

ion

p2TH/TD

p1TH/TD

q2TH/TD

q1TH/TD

Death!

Killed by TD

TH

q3TH/C

Ignore

React

Ignore

ReactDie orbecome anergic

Survive

Perish

Ignore

Reactp1TD/TH

p2TD/TH

q3TH/TD =0

No flow


Motivation

Properties

numerical methods




Anti-Donor T CellsT

D /TH Interaction

kTHTD

q2TD/TH

q1TD/TH

Killed by TH

!

"n#

$Reload

Proliferate

x2n

p2TD/TH

p1TD/TH

Survive

Perish

Ignore

Reactp1TH/TD

p2TH/TD

q3TD/TH =0

kTCTD

p2TD/TC

T D/D

Inte

ract

ion

p2TD/D

p1TD/D

kDTD

q1TD/D q2

TD/D

p1TD/TC

q1TD/TC

q2TD/TC

T D/T

C In

tera

ctio

n

No flow

q3TD/TC = 0

"

!

n# $

x2n

x2n

!

$

n#Proliferate

Reload

"

Ignore

React

Ignore

React

dTD

TD

Death

No flow


Motivation

Properties

numerical methods




General Donor and Host Blood Cells

Death

dD

TD /D

Interaction

Stem Cells

D

Death

dH

kHTH TH /H

Interaction

p1TH/H

Stem Cells

H

SD SH

kDTDp1TD/D

!

Perish

!

Perish


Motivation

Properties

numerical methods




Cancer Cells

PerishPerishC

Death

kTCCkTHCp1C/TH p1

C/TC

C/T

C InteractionC

/TH

Inte

ract

ion

rC

Death rate is includedin net logistic growth term

logistic growth

!!


Motivation

Properties

numerical methods




Equations...

dTH

dt= �dTH

TH � kCTH � kTDTH � kHTH

+ pTH/C2

kC (t � �)TH(t � �) + pTD/TH2

pTH/TD2

kTD(t � �)TH(t � �)

+ pTH/H2

kH(t � �)TH(t � �)

+ 2npTH/C1

qTH/C1

kC (t � ⇢� n⌧)TH(t � ⇢� n⌧)

+ 2npTH/H1

qTH/H1

kH(t � ⇢� n⌧)TH(t � ⇢� n⌧)

+ 2npTD/TH2

pTH/TD1

qTH/TD1

kTD(t � ⇢� n⌧)TH(t � ⇢� n⌧)

+ pTH/C1

qTH/C2

kC (t � ⇢� �)TH(t � ⇢� �)

+ pTH/H1

qTH/H2

kH(t � ⇢� �)TH(t � ⇢� �)

+ pTD/TH2

pTH/TD1

qTH/TD2

kTD(t � ⇢� �)TH(t � ⇢� �).


Motivation

Properties

numerical methods




Time Delays

Time for reactive T cell-antigen interaction = 5min

Time for unreactive interactions = 1min

Time for cell division = 0.5-1.5 day

T cell recovery time after killing another cell = 1 day


Motivation

Properties

numerical methods




Relapse

0 20 40 60 80 1000

0.5

1

1.5

2

Time in Days

Cel

l Con

cent

ratio

n in

103

cells

/µL

0 200 400 600 8000

0.2

0.4

0.6

0.8

1

1.2x 10

!6

Time in Days

Cel

l Con

cent

ratio

n in

103

cells

/µLGeneral host cells H

Anti!host T cells TH

Cancer cells C

never goes to 0

eventually overwhelms TH


Motivation

Properties

numerical methods




Remission

0 20 40 600

0.2

0.4

0.6

0.8

1

1.2x 10

!6

Time in Days

Cel

l Con

cent

ratio

n in

103

cells

/µL

0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

4

Time in Days

Cel

l Con

cent

ratio

n in

103

cells

/µL

General host cells H

Anti!host cells TH

Cancer cells C

C = 0 at time 25.7276


Motivation

Properties

numerical methods




Oscillations

0 500 1000 15000

0.2

0.4

0.6

0.8

1

B: Unstable Oscillation

Time in Days

Cell C

once

ntra

tion

in 1

03 c

ells

/µL

0 500 1000 15000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4A: Stable oscillation

Time in Days

Cell C

once

ntra

tion

in 1

03 c

ells

/µL

Anti!host T cellsCancer cellsGeneral host cells


Motivation

Properties

numerical methods




Extinction instead of stability

8 8.05 8.1 8.15!25

!20

!15

!10

!5

0

5

10

15

20

25Without state constraint

Time in Days

Cel

l Con

cent

ratio

n in

103

cells

/µL

0 2 4 6 80

5

10

15

20

25With state constraint

Time in Days

Cel

l Con

cent

ratio

n in

103

cells

/µL

The value of C crosses 0at time 8.0192.

Cancer cells C



Cancer cells C

The value of C crosses 0at time 8.0192, and doesnot recover.


Motivation

Properties

numerical methods




Initial anti-host cells vs. initial host cells

1 1.5 2 2.5 3 3.5 4 4.5 5

2

4

6

8

10

12

14

16

Initi

al a

nti!

host

T c

ell c

once

ntra

tion

T H,0

(cel

ls/µ

L)

Initial host cell concentration H0 (103 cells/µL)

Results up to time 2000SuccessfulUnsuccessfulUnresolved

Higher initial host blood cellconcentrations improve thechances of a successful cure

Greater initial anti-host T cellconcentrations slightly favor thechances of cure


Motivation

Properties

numerical methods




Number of cell divisions vs. cancer growth rate

0.03 0.035 0.04 0.045 0.055

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

6

Cancer growth rate rC (1/day)

Ave

rage

# o

f T c

ell d

ivis

ions

n

Results up to time 1000SuccessfulUnsuccessfulUnresolved

A higher average number of Tcell divisions favor completeremission

Higher cancer growth rate makecomplete remission slightlymore likely


Motivation

Properties

numerical methods


Zero Crossing

Time Scales

Outline



3 Numerical Methods



Motivation

Properties

numerical methods


Zero Crossing

Time Scales

Zero crossing: Example

A simple DDE:

dx

dt= �rx(t � 1), x(t) = 1, t < 0.

Solve: t 2 [0, 1).dx

dt= �rx(t � 1) = �r

Thenx = �rt + c = 1� rt.

If r > 0 then x(t) = 0 for T 2 [0, 1)


Motivation

Properties

numerical methods


Zero Crossing

Time Scales

Zero crossing: example

A simple DDE:

dx

dt= �rx(t � 1), x(t) = 1, t < 0.

Proceed: t 2 [1, 2)

dx

dt= �rx(t � 1) = �r + r2t � r2

Then

x = 1� rt +r2

2(t � 1)2.


Motivation

Properties

numerical methods


Zero Crossing

Time Scales

Zero crossing: example

A simple DDE:

dx

dt= �rx(t � 1), x(t) = 1, t < 0.

The general solution: t 2 [n, n + 1)

x(t) =n+1X

k=0

(�r)k(t � k + 1)k

k!

Question:

For what r does that exist a T 2 [n, n + 1) such that

n+1X

k=0

(�r)k(T � k + 1)k

k!= 0?


Motivation

Properties

numerical methods


Zero Crossing

Time Scales

Time Scales

A toy problem

xn+1

= (1� yn)xn

yn+1

= �x2n + k + yn

The map iterated twice

xn+2

= (1 + x2n � k � yn)(1� yn)xn

yn+2

= �((1� yn)2 + 1)x2n + 2k + yn


Motivation

Properties

numerical methods


Zero Crossing

Time Scales

This corresponds to...


Motivation

Properties

numerical methods


Outline



3 Numerical Methods



Motivation

Properties

numerical methods


Approach #1

A mesh in time that is based on the delay.

Numerical methods for ODEs that use the mesh points only(multistep methods)

Example:⇢

y 0(t) = f (t, y(t), y(t � ⌧(t))) , t0

t tf ,y(t) = �(t), t t

0

.

A set of meshpoints:

� = {t0

, t1

. . . , tN = tf },

such that tn � ⌧(tn) 2 �.Forward Euler: yn+1

= yn + hn+1

f (tn, yn, yq), q < n.Same idea with Adams-like methods, Heun, etc.


Motivation

Properties

numerical methods


Approach #2: Feldstein

Free the mesh selection from the delayUse extranodal points for the approximation of the delayed termy(t � ⌧(t)).Example: (t

0

↵(t) t)⇢

y 0(t) = f (t, y(t), y(↵(t))) , t0

t tf ,y(t

0

) = y0

,

Assume h = (tf � t0

)/m. For any tn = t0

+ nh, define

q(n) = floor

✓↵(tn)� t

0

h

◆.

A numerical method:⇢yn+1

= yn + hf (tn, yn, zn),y0

= y(t0

),

where zn = yq(n), i.e., a piecewise-constant approximation of y(↵(t)).Alternatively, a piecewise-linear approximation:

zn = (1� r(n))yq(n) + r(n)yq(n)+1

.


Motivation

Properties

numerical methods


Approach #3: Bellman’s method of steps

Assume a constant delay. In the first interval [t0

, t0

+ ⌧ ] the DDE has theform: ⇢

y 0(t) = f (y(t),�(t � ⌧)),y(t

0

) = �(t0

).

In the second interval [t0

+ ⌧, t0

+ 2⌧ ], define y1

= y(t � ⌧) andy2

(t) = y(t). Then:

8>><

>>:

y 01

(t) = f (t � ⌧, y1

(t),�(t � 2⌧)),y 02

(t) = f (t, y2

(t), y1

(t)),y1

(t0

+ ⌧) = �(t0

),y2

(t0

+ ⌧) = y(t0

+ ⌧)

And so on...For every timestep, a larger system. However, this system can be solvedusing standard methods for ODEs.


Motivation

Properties

numerical methods


Approach #4: Methods based on continuous extensions

Continuous Extension: very low cost method to get an accurateapproximation of the solution at every point in the interval

⌘(tn + ✓hn+1

) = �n,1(✓)yn + . . .+ �n,in+1

(✓)yn�in

+hn+1

(yn, . . . , yn�in ; ✓, g⌘,�0n), 0 ✓ 1.

whereg⌘(f , y) = f (t, y , ⌘(t � ⌧(t, y))).

This is how every Matlab routine provides solutions at the sampledpoints


Motivation

Properties

numerical methods


Approach #4: Methods based on continuous extensions

Consider the DDE⇢

y 0(t) = f (t, y(t), y(t � ⌧(t, y(t)))) , t0

t tf ,y(t) = �(t), t t

0

.

Using continuous extensions, solving the DDE amounts to solvingthe ODE:⇢

w 0n+1

(t) = f (t,wn+1

(t), x(t � ⌧(t,wn+1

(t)))) , tn t tn+1

wn+1

(tn) = yn,

where

x(s) =

8<

:

�(s), s t0

,⌘(s), t

0

s tn,wn+1

(s), tn s tn+1

,

and ⌘ is the continuous extension interpolant.


Motivation

Properties

numerical methods


Additional Reading

Bellen and Zennaro, Numerical Methods for Delay Di↵erentialEquations, Oxford

Shampine and Thompson, Numerical Solution of Delay Di↵erentialEquations


Motivation

Properties

numerical methods


Outline



3 Numerical Methods



Motivation

Properties

numerical methods



d2x

dt2+

dx

dt+

dx(t � ⌧1

)

dt+

dx(t � ⌧2

)

dt+ 8x = 0


Motivation

Properties

numerical methods


Stability crossing curves

Ref:

Gu, Niculescu, Chen, On stability crossing curves for general systems withtwo delays, J. Mathematical Analysis & Applications, 311 (2005), pp.231–253.

Stability crossing curves: The set of delays for which thecharacteristic equation has at least one imaginary zero (or pair ofimaginary zeros).

Associated with change of stability


Motivation

Properties

numerical methods


The two delays case

A DDE with two constant delays

x(t)+c1

x(t�⌧1

)+c2x(t�⌧2

)+c3

x 0(t)+c4

x 0(t�⌧1

)+c5x 0(t�⌧2

) = 0.

The characteristic equation:

h(s) = h0

(s) + h1

(s)e�⌧1

s + h2

(s)e�⌧2

s .

Let ak(s) = hk(s)/h0(s). Then

a(s, ⌧1

, ⌧2

) = 1 + a1

(s)e�⌧1

s + a2

(s)e�⌧2

s = 0.

Stability: a question of the number of the roots of the characteristicequation with a real part on the right hand side of the plane.


Motivation

Properties

numerical methods


The two delays case

For an imaginary s = i! to satisfy a(s, ⌧1

, ⌧2

) = 0, the vectorcorresponding to the three terms must form a triangle:

a(s, ⌧1

, ⌧2

) = 1 + a1

(s)e�⌧1

s + a2

(s)e�⌧2

s = 0.For an imaginary s = iZ to satisfy

the vector corresponding to the three terms must form a triangle.

Hence, their magnitudes must satisfy the triangle inequalities:

Hence, their magnitudes must satisfy the triangle inequalities:

|a1

(i!)|+ |a2

(i!)| � 1,

�1 |a1

(i!)|� |a2

(i!)| 1.


Motivation

Properties

numerical methods


The two delays case

The triangle inequalities determine which i! may be zeros of a(s)

The set of all such ! are the crossing set ⌦.

Any given ! defines a collection of pairs (⌧1

, ⌧2

).

⌧1

=\a

1

(i!) + (2u � 1)⇡ ± ✓1

!� 0, u = u±

0

, u±0

+ 1, . . .

⌧2

=\a

2

(i!) + (2v � 1)⇡ ⌥ ✓2

!� 0, u = v±

0

, v±0

+ 1, . . .

where from the law of cosine:

✓1,2 = cos�1

✓1 + |a

1,2(i!)|2 � |a2,1(i!)|2

2|a1,2(i!)|

◆,

and u±0

, v±0

are the smallest possible integers such that thecorresponding ⌧

1,2 are nonnegative.


Motivation

Properties

numerical methods


The two delays case

The crossing set ⌦ always consists of a finite number of intervals offinite length

• The triangle inequalities determine which iZ may be zeros of a(s).

• We call the set of all such Z the crossing set :.

• The crossing set : always consists of a finite number of intervals

of finite length.

(symmetrical, because imaginary roots come in conjugate pairs)Any interval of !’s defines a collection of curves in R2

The general case is a union of the following sets:

• Any given Z defines a collection of pairs (W1, W2)

• Any interval of Z’s defines a collection of curves in R2.

• The curves glue together in different arrangements, depending on which triangle inequalities correspond to the endpoints of the intervals.


Motivation

Properties

numerical methods


Example

DDE:

dx

dt= �2x(t � ⌧

1

) + x(t � ⌧2

).Regions are marked with the numberof zeros in right half plane.


Numerical Methods in Cancer ModelsNumerical Methods in Cancer Models Doron Levy Department of...

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Transcript of Numerical Methods in Cancer ModelsNumerical Methods in Cancer Models Doron Levy Department of...