Review Etc.

Modified Tumor-Immune Model

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dNdt

= a1N2 / 3(1−bN) − c1NE

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dEdt

= s+ a2Ng+ N

E − c2NE − dE

Dimensional Analysis

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a1 Effective growth rate for tumor cells (density)1/3/time

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1/b Carrying capacity for tumor cells density

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c1 Immune-mediated death rate 1/(density*time)

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s Tissue localization rate density/time

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a2 Maximum immune cell stimulation rate 1/time

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g Half-saturation constant for stimulation density

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c2 Tumor-mediated death rate 1/(density*time)

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δ Natural death rate of immune cells 1/time

Parameter ValuesParameter Value 180 b 2 x 10-9

1.1 x 10-7

1.3 x 104

0.12452.0 x 107

3.5 x 10-10

0.0412

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a1

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c1

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s

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a2

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g

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c2

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δ

The old a1 needs to be multiplied by density1/3 to get the new a1. The max density is about 1 x 109, so new a1 ~ 1000 old a1.

Nonidemnsionalization

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n = N1/b( )

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τ =a1b1/ 3t

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e = Ea1b

1/ 3 /c1( )

• Time is scaled relative to the tumor’s effective growth rate• The immune population is scaled relative to the number

of tumor cells inactivated by each immune cell per unit of rescaled time

• The tumor population is scaled relative to its maximum size, ie the carrying capacity

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σ =sc1

a12b2 / 3

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α =a2

a1b1/ 3

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γ=gb

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μ =c2

a1b4 / 3

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δ =da1b

1/ 3

Nondimensional Equations

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dndt

= n2 / 3(1− n) − ne

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dedt

=σ + αnγ + n

e −μne −δe

The Healthy Steady State

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n1 = 0

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e1 = σδ

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a11 = ∂f∂n

0,σδ

⎛ ⎝ ⎜

⎞ ⎠ ⎟= limn→0

1n1/ 3 − σ

δ ⎛ ⎝ ⎜

⎞ ⎠ ⎟> 0

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a12 = ∂f∂y

0,σδ

⎛ ⎝ ⎜

⎞ ⎠ ⎟= 0

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a21 = ∂g∂n

0,σδ

⎛ ⎝ ⎜

⎞ ⎠ ⎟= ασγ

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a22 = ∂g∂y

0,σδ

⎛ ⎝ ⎜

⎞ ⎠ ⎟= −δ

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β =limn→0

1n1/ 3 − σ

δ ⎛ ⎝ ⎜

⎞ ⎠ ⎟−δ > 0

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γ=−δ 1− σδ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Stability

The healthy state appears to be unstable always!! This is different from the model we analyzed in class. To verify this we need to look at trajectories in the phase plane.

The Disease States

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dndt

= n2 / 3(1− n) − ne

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dedt

=σ + αnγ + n

e −μne −δe

• Nullclines– N-nulliclines

– E-nullcline

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n = 0

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e = 1− nn1/ 3

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e =γ + n( )σ

δ + μn( ) γ + n( ) −αn

Graph of Nullclines for Small σ

The parameter values used are μ = 0.7651, δ = 0.1842, γ = 0.0404, α = 0.5567, σ = 0.0572.

Graph of Nullclines for Large σ

The parameter values used are μ = 0.7651, δ = 0.1842, γ = 0.0404, α = 0.5567, σ = 0.2859.

Summary of Steady States• It is clear from the graph and our in class plots of the

effect of moving the e-nullcline that, for this model, there will always be at least two steady states. – This is because the elimination state always exits and the

e-nucline will always have a portion in the first quadrant that will insect with the n-nullcline.• This is different from what we found in class, where it was possible

to increase σ enough to eliminate all non-healthy states– Decreasing the tissue localization rate of the effector cells

results in the addition of two new disease states, representing higher tumor burdens.

Phase Portrait for Small σ

Four steady states are present. The healthy state is unstable as is the steady state representing medium tumor burden. The dormancy and immune escape states are both stable. This is similar to what we found with the model analyzed in class.

Phase Portrait for Large σ

Two steady states are present. The elimination state is again unstable and the steady state representing dormancy is stable.

Conclusions• Using the Von Bertalanffy growth function we found

that it is possible to obtain four steady states and to capture all three phases of the cancer immunoediting hypothesis.

• The major difference between the model with Von Bertanlanffy and the one with Logistic growth is that when using Von Bertalanffy growth, the healthy tissue state is always unstable making it impossible to achieve elimination of the tumor. – Therefore dormancy is the best possible outcome in this

case.

Review

• The final exam will be comprehensive.• It will cover material in the text book from

Chapters 1 – 7.• It will also cover material not in the text

book that was provided to you in the lecture notes.

• Please see Exam 1 review for what to except from chapter 1- 6.2– All of that information still applies

Review – Discrete Models

• Exam 1– Covered some mathematical aspects of

discrete equations (problem 1)• Exam 2

– Will likely cover modeling aspects of discrete equations

• Model development, analysis, and interpretation

Review – Continuous Models

• Exam 1– Covered some model interpretation for

single-species continuous equations (problem 2)

• Exam 2– This is also fair game for Exam 2

• I may ask you a biological question and you need to decide what mathematics to do to answer that question

Review – Continuous Models

• Exam 1– Covered the analysis of a system of two

equations (problem 3)• Exam 2

– This is also fair game for Exam 2• Nondimensionalization• Produce a phase portrait• Produce a bifurcation diagram

Review – Continuous Models

• Exam 1– Covered model development

• Exam 2– This is also fair game for Exam 2

• The the description of a biological problem, you should be able to derive a system of equations to model it

Review – Specifics

• Chapter 6: 6.1 – 6.4 and 6.6– Interacting Species

• Predator-Prey (Lab 6)• Competition (Chapter 6 Assigned Reading)

– SIR Models• Be able to recognize, develop, analyze

and interpret the results from these types of models

Review – Specifics

• Predator-Prey – Population A grows normally in the

absence of population B. Population B has a significantly reduced growth rate in the absence of population A. The presence of population B significantly reduces the growth rate of population A. The presence of population A increases the growth rate of population B.

Review – Specifics

• Competition– Population A grows normally in the

absence of population B. Population B grows normally in the absence of population A. The presence of population A reduces the growth rate of population B and the presence of population B reduces the growth rate of population A.

Review – Specifics

• SIR Models– Model development– Model reduction– Conditions for an epidemic– Properties of an epidemic

• Severity• Doubling time• How many get infected, ...

Review – Specifics

• Chapter 7: Receptor-Ligand Binding 7.1 – 7.4– Be able to develop, analyze and interpret the

results from these types of models– Draw and interpret reaction diagrams– Derive mathematical equations

• Law of Mass Action– Reduce model equations

• Conservation• QSSA

Review – Specifics

• Infectious Disease Models– Analyze and interpret models

• Tumor Growth– Derive, Analyze and Interpret Models